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The Universe is Only Spacetime

Particles, Fields and Forces

Derived from the Simplest

Starting Assumption

John A. Macken

Santa Rosa, California

[email protected]

Original Draft – February 2010

Revision 8.0 – July 2015

©2010, 2012

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Dedication: I dedicate this book to my wife, Enid, who has encouraged me and tolerated my idiosyncrasies over the years.

Acknowledgments: I wish to thank my friend, Dr. Nikolai Yatsenko, for volunteering to spend many hours teaching me aspects of quantum mechanics and general relativity. During these lessons I began the first steps of formulating a conceptually understandable model of the universe. Also Dr. Chris Ray has provided helpful comments on the text and my son, Jim Macken, provided many of the computer generated figures Finally, I would like to thank Francis William Browne for editing the book.

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Table of Contents

1 Confined Light Has Inertia 1-1 •LightinaReflectingBox•ConfinedBlackBodyRadiation•deBroglieWaves•8Particle‐likePropertiesofaConfinedPhoton

2 Definitions and Concepts from General Relativity 2-1•SchwarzschildSolution•DefinitionofGravitationalGammaГ•SchwarzschildCoordinateSystem•CoordinateSpeedofLight•ShapiroExperiment•GravityIncreasesVolume•ConnectionBetweentheRateofTimeandVolume

3 Gravitational Transformations of the Units of Physics 3-1WhyAre the Laws of Physics Unchanged When the Rate of Time Changes? • NormalizedCoordinate System • Length and Time Transformations • Transformations Required toPreservetheLawsofPhysics•InsightsFromtheTransformations

4 Assumptions 4-1 •ThisBook’sBasicAssumption•QMModelofSpacetime•GRModelofSpacetime•DipoleWaves inSpacetime •PlanckLength/TimeLimitationOnDipoleWaves • ImpedanceofSpacetime•5Wave‐AmplitudeEquations •EnergyDensityofSpacetimefromGR•BulkModulusofSpacetime•TheSingleFundamentalForce•SingleFundamentalField

5 Spacetime Particle Model 5-1 • Superfluid Properties of Vacuum Energy • SpacetimeVortex •ParticleDesignCriteria •Inertia from Confined Energy • The Rotar Model of a Particle • Quantum Radius andQuantumVolumeofaRotar•RotatingGravField•AnalysisofLobes•StrainAmplitudeHβ•SolitonCondition

6 Analysis of the Particle Model 6-1 •ParticleSizeAnalysis•RotarEnergyTest•Rotar’sAngularMomentumTest•QuantumAmplitude Equalities • Circulating Power •Rotar’s TheoreticalMaximum Force StrongForce •ElectromagneticForceatDistanceRq•Gravity•Rotar’sGravitationalForceatDistanceRq•WeaknessofGravity•Force–AmplitudeRelationship•The4ForcesAreIntimatelyConnected•Rotar’sInertia•HiggsFieldNotNeeded

7 Virtual Particles, Vacuum Energy and Unity 7-1 •Virtual Particle Pairs • Vacuum Energy • Harmonic Oscillators in Spacetime • EnergyDensityofVacuumEnergy•EnergyDensityEqualsPressure•BulkModulusofSpacetime•VacuumEnergyIsaSuperfluid •StabilityofParticlesMadeofWaves•VacuumEnergyStabilizationofaRotar•MinimumRequiredEnergyDensity•MaximumAttractingForce•AsymptoticFreedom •RotationofMolecules •UnityHypothesis •Entanglement‐UnityConnection

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8 Analysis of Gravitational Attraction 8-1 •NonlinearEffectsofWavesinSpacetime•NewtonianGravitationalEquationDerivation•Connection between Gravitational Force and Electromagnetic Force •Equivalence ofAcceleration and Gravity Examined • Comparison of a Rotar’s Rotating Grav Field andGravitationalAcceleration•EnergyDensityintheRotatingGravField•GravitationalEnergyStorage

9 Electromagnetic Fields and Spacetime Units 9-1 •Spacetime Interpretation of Charge • Charge Conversion Constant • Impedance ofSpacetimeConversion•WhatIsanElectricField?•ElectricFieldConversion•ProposedExperiments•MagneticFieldanalysis•SpacetimeUnitsConversionTable

10 Rotar’s External Volume 10-1 •GravitationalandElectromagneticStrainamplitudes•ElectricFieldCancelation•ModeloftheExternalVolumeofaRotar•Wavelets•deBroglieWaves•ΨFunction•RelativisticContraction•ComptonScattering•DoubleSlitExperiment

11 Photons 11-1 •HowBigIsaPhoton?•Photon‐Definition•WavesinVacuumEnergy•WaveModelofanElectron‐Positron Annihilation • Entanglement • Single Photon Model • Photon’sMomentum Uncertainty Angle • Compton Scattering Revisited •Limits on Absorption•Entanglement•PhotonEmissionfromaSingleAtom•Recoil•Huygens‐Fresnel‐KirchhoffPrinciple

12 Bound Electrons, Quarks and Neutrinos 12-1 •ElectronsBoundinAtoms•PhysicalInterpretationoftheΨFunction•IntrinsicEnergyofQuarks•EnergyofBoundQuarks•CalculationofaProton’sRadius•RemovalofaQuarkfromaHadron •Gluons •ModelingaNeutrinoWithRestMass •ModelingaNeutrinoWithoutRestMass

13 Cosmology I – Spacetime Transformation Model 13-1 •ComovingCoordinates•TheΛ‐CDMModel•PlanckSpacetime•MaximumEnergyDensity•CreationofNewProperVolume•BackgroundGravitationalGammaoftheUniverseГu•ImmatureGravity•ImplicationofanIncreasingГuintheUniverse•EnergyDensityofPlanckSpacetime •Proposed Alternative Model of the Beginning of the Universe • CosmicExpansionfromГu•StartingtheUniversefromPlanckSpacetime•RadiationDominatedEpoch•LostEnergyBecomesVacuumEnergy•EstimatesoftheCurrentValueofГu•DarkMatterSpeculation

14 Cosmology II – The Big Picture 14-1 •AlternativeToTheBigBangModel•ShrinkingMeterSticks•NoEventHorizon•ConstantEnergyDensityWhenVacuumEnergyIncluded•RedshiftAnalysis•EstimatingtheDensityof Vacuum Energy • Units of Physics in the Spacetime TransformationModel • 10120Calculation•DoesDarkEnergyExist?•CoolingoftheUniverse•BlackHoles•Time’sArrow•AreAllFramesofReferenceEquivalent?•TheFateoftheUniverse

15 Definitions, Symbols and Key Equations 15-1

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IntroductionIn most introductory classes on quantum mechanics, the physics professor starts out byexplaining to thestudents that theyaregoing to learnaboutcertainpropertiesofsubatomicparticlesthataresimplynotconceptuallyunderstandable.Forexample,subatomicparticlescandiscontinuouslyjumpfromonepointtoanotherwithoutpassingthroughthesurroundingspace.Fundamental particles exhibit “spin”which is an “intrinsic” fromof angularmomentum thatcannotbeexplainedbyclassicalconceptsofrotation.An isolatedmoleculecanonlyrotateatspecificfrequencies.Twoentangledphotonscancommunicatefasterthanthespeedoflightoverlargedistances–etc.Evenspecialandgeneralrelativityhavetheirshareofincomprehensiblemysteries.Howisitpossibleforobserversmovingatdifferentvelocitiestoallobservetheexactsamespeedoflight?Howdoesmattercauseadistantvolumeofspacetimetocurve?

Students are told that they should accept the fact thatmodernphysics hasmanypropertieswhich are simply not able to comprehend by the human intellect. Classical physics gaveconceptuallyunderstandexplanations,butnowthestudentsmustlearntobecomecomfortablethatmodernphysicscannotbeunderstoodinthesameway.Quantummechanicsisthemostsuccessful scientific theory ever developed, so “shut up and calculate”. A quote popularlyattributedtophysicistRichardFeynmanis:“Ifyouthinkyouunderstandquantummechanics,thenyoudon’t”.

Tobefair,astrictapplicationoftheprinciplesofquantummechanicsdoesnotstrivetogiveaphysicallyunderstandableexplanationofthesephenomena.Instead,theobjectiveofquantummechanics is to describe rules of each quantum mechanical operation and mathematicalequationswhichdescribe theseoperations. Forexample,anelectron isdescribedasapointparticlebecausethismathematicalsimplificationisadequatetoobtainusefulequations.AsPaulDiracsaid,theaimis“notsomuchtogetamodelofanelectronastogetasimpleschemeofequationswhichcanbeusedtocalculatealltheresultsthatcanbeobtainedfromexperiment”1.

Today,physicistshave learned to suppress their innatedesire for conceptualunderstanding.The explanation given for our inability to conceptually understand quantum mechanicalphenomenaultimately implies that thehumanbrain evolved tounderstand themacroscopicworld. Therefore,wesimplyshouldnotexpect toconceptuallyunderstand thepropertiesofsubatomic particles or photons. These concepts are just too far removed from our huntergathererroots.Overtimewereluctantlylearntoacceptabstractquantummechanicalconceptsandregardthedesireforconceptualunderstandingasaremnantofclassicalphysics.However,theimportanceofamodelthatgivesconceptualunderstandingshouldnotbeminimized.Iftherereallyisanunderlyingsimplicitytoallofphysics,thenunderstandingthismostbasicmodelnotonlysimplifiestheteachingofphysics,butalsomakesiteasiertoextendthismodelandmaketestablepredictionswhichadvancescience. Suchamodelwouldbeapowerfulnewtoolthatwouldgreatlyacceleratetherateofnewdiscoveries.

1 P. A. M. Dirac, “Classical Theory of Radiating Electrons,” Proc. Royal Soc., vol. 168, (1939)

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There is clear evidence that the current starting assumptions for quantum mechanicalcalculationsareeitherincompleteorcontainatleastoneerror.Whencalculationsfallapartandyield an impossible answer such as infinity, these equations are screaming that a rigorousextensionofthestartingassumptionsgivesnonsense. Renormalizationmightseemtofixtheproblem,butthisismerelyartificiallyadjustedtheanswersothatitisnolongerlogicallyderivedfrom the starting assumptions. Instead the unreasonable answer should be taken as anindicationthatthemodelbeinganalyzedeithercontainsatleastoneerroneousassumptionorismissinganessentialassumption.Everytimeanincorrectassumptionisutilizedorthemissingassumptionisnotused,themathematicalanalysismustyieldanincorrectanswer garbagein,garbageout .Ourinabilitytoconceptuallyunderstandpartsofquantummechanicsisafurtherindicationthatweareusinganinadequatemodel.Theapproachtakeninthisbookistoexplorethe possibility that there really is an underlying simplicity beneath the counter intuitivecomplexityofquantummechanics.Thisisequivalenttosayingthatwearelookingforthefabled“Theory of Everything”. A logical starting point for this search should be to start with thesimplestpossiblestartingassumptionwhichis:Theuniverseisonlyspacetime.Anotherwayofsaying this is that spacetime is the single energetic field responsible for everything in theuniverse. Obviously, thismodelofspacetimegoes farbeyondthegeneralrelativitymodelofspacetime.

Thissimplestpossiblestartingassumptionimpliesthatthereisonlyonefundamentalforce,onlyonefundamentalfieldandonlyonefundamentalbuildingblockofallparticles. Furthermore,even though forces, fields and particles appear to be very different, the starting assumptionimpliesthatonadeeperleveltheyarealljustdifferentaspectsof4dimensionalspacetime.Ifthisstartingassumptioniswrong,itshouldquicklybecomeobviousbecausethepropertiesof4dimensionalspacetimearevery limiting. Unlike thecurrentstateofphysicswhich is free topostulateanynumberofdimensions,multipleuniverses,vibratingstrings,messengerparticles,etc.,thisstartingassumptionisveryrestrictive.Allparticles,fieldsandforcesmustbederivedfromonlythepropertiesof4dimensionalspacetime.

BeforethetimeofGalileo,theworldwasviewedasbeingfullofmysteriousoccurrenceswhichwere either not questioned or attributed to the supernatural. With the dawn of the age ofanalytical science, it was realized that many of the mysteries were knowable and could beexplainedwithlawsofphysics.Themoon,planetsandstarswerenotheldabovetheearthbycrystalspheres.Thesamegravitythatcausedarocktofallhereonearthalsocausedthemoontohaveapredictableorbitaroundtheearth.Whatwaspreviouslyconsideredanunknowablemysterywasexplained throughdeductive reasoning,mathematicsand theapplicationof thescientificmethod.

WehavecomealongwaysincethetimeofGalileo,buteventodaywehavemanymysteriesthatareconsideredbeyondhumanunderstanding.Forexample,wehaveamultitudeofmysteriouseffectsthatweattributetodifferenttypesof“fields”.Wecanwriteequationsthatdescribetheeffects of these “fields”, but we really do not attempt to understand the fields in terms of

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somethingmorebasic.IseeananalogytothetimebeforeGalileoandourpresentdayattitudetowardsfieldsbeingphysicallyunknowable.Forexample,howandwhydoesmattercauseacurved spacetime field? What is the physical difference between the field produced by apositivelychargedparticleandanegativelychargedparticle?Doelectricandmagneticfieldsproduceadistortionofspacetime?

Iftheuniverseisonlyspacetime,thenallother“fields”shouldbeknowableintermsofeffectsonthemostbasicfield:spacetime. Forexample,anewconstantofnaturewillbeproposedthatquantifiesthedistortionofspacetimeproducedbyanelectricfield. Alsotheconceptthattheuniverseisonlyspacetimehasprofoundimplicationsforscience.Thisbookmakesthecasethatallparticles,fieldsandforcesaremadeofthesinglebuildingblockof4dimensionalspacetime.However,thisisnotthequiet,smoothlycurvingspacetimeenvisionedbyEinstein.Hismodelofspacetimeonlydescribesspacetimeonthemacroscopicscale.Alltheaction energy isonthequantummechanicalscalewherespacetimeisfullofactivity.Thisquantummechanicalmodelofspacetimeisanalyzedandfoundtopossessthepropertiesthatallowittobethesinglebuildingblockofeverythingintheuniverse.

Gravityisverywellunderstoodononelevel,butitalsohasmanymysteries.Whyistheforceofgravityvastlyweaker thantheother3 fundamental forces? Whydoesgravityhaveonlyonepolarity alwaysattracts ?Isgravityevenatrueforceormerelytheresultofthegeometryofspacetime?Cangravitybeunifiedwiththeotherforces?Ifweassumethattheuniverseisonlyspacetime, then we bring a new perspective to explaining the mysteries of gravity. Bothfundamental particles and the forceof gravity canbederived from this starting assumption.Thereisnoneedtomakeananalogytoaccelerationtoexplaintheforceofgravity.Furthermore,this approach makes a prediction about a previously unknown relationship between thegravitationalforceandtheelectromagneticforce.Thispredictioniseasilyprovencorrect.Thisbookshowshowthesetwoforcesplusthestrongforceareallcloselyrelated.

Thestandardmodelofparticlephysicshaspassedmanytests.However,manymysteriesstillremain. TheconventionalmodelssaythatfundamentalparticlesareeitherpointparticlesorPlancklengthvibratingstringsthatarevirtuallypointparticles.Neitherofthesemodelsexplainsnumerous quantum mechanical properties of fundamental particles. For example, how dofundamentalparticlesdiscontinuouslymovefrompointtopointwithoutpassingthroughtheintermediatespace?Howdotheypossessangularmomentumof½ħwhentheyarevirtuallypoints?Howmuchofanelectron’senergyisstoredinitselectric/magneticfield?Howdoesitexhibitbothwaveandparticleproperties?Isafundamentalparticlemadeofsomeevenmorefundamentalbuildingblock?Thisbookshowsthatwhenfundamentalparticlesareassumedtobe made of spacetime the quantum mechanical model of spacetime , then these counterintuitive properties can be explained. They become conceptually understandable andmathematically quantifiable. Furthermore, this model gives predictions about gravity andelectricfields.

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Tomostscientiststhisstartingassumption theuniverseisonlyspacetime willinitiallyseemimpossible. How canmatter, light, galaxies and the forces of nature be obtained fromwhatappears to be the empty vacuum of spacetime? Well, the quantum mechanical model ofspacetime proposed here is far from being a featureless void. Besides having well knownpropertiessuchaspeedoflightandagravitationalconstant,italsohasimpedanceandabulkmodulus. Most important, the quantum mechanical version of spacetime is full of activity.Vacuum fluctuations zero point energy imply that vacuum has a vast energy density. Yetexperimentsandanalysisseemtoindicateanemptyvoid. Themeasurableenergydensityofspacefromcosmologyisabout10120timeslessthantheenergydensityimpliedbythequantummechanicalmodelofavacuum.Thecommonassumptionisthatsomethingmustcancelouttheimplied tremendous energy density of vacuum. The alternative proposed here is that thevacuumreallydoeshavethistremendousenergydensity,butthereisakeydifferencebetweentheenergythatwecandetectandtheenergythatwecannotdetect.Itisnotnecessarytoassumeenergycancelation,itisonlynecessarytounderstandthedifferencebetweenthetwotypesofenergy fermions/bosonsversusvacuumenergy .

This book attempts to show that the biggest mysteries of quantum mechanics becomeconceptuallyunderstandablewhenweadoptthemodelthatbuildsparticlesandforcesfromthequantummechanicalpropertiesofspacetime.Limitationsofthehumanintellecthavenothingtodowithour inabilitytoconceptuallyunderstandthemysteriesofquantummechanicsandgeneralrelativity.Wehavebeenusingthewrongmodels!Thehumanintellectcanunderstandanythinginnatureprovidedthatweareusingthecorrectmodel.

Themodelof theuniversedescribedhere is shown tobe compatiblewith existing laws andequationsofphysics.Theconceptsarecheckedtoconfirmplausibilitybynumerouscalculations.Most of these are algebraic calculations so that the book is accessible to awide audience ofreaders that are scientifically knowledgeable but not necessarily specialists. The fewcalculationsthatgobeyondthisbasiclevelarehandledinawaythatthereadercangrasptheresultwithoutnecessarilyfollowingthemathematics.

Thecontentofthisbookwasnotfirstpresentedintechnicalpapersbecausethesubjectisjusttoolarge.Itisnecessarytolayoutaseriesofintroductoryideas,andthenweavetheseconceptsintoasinglecoherenttheory.Alargepartoftheappealofthisapproachishowasinglestartingassumptioncananswersomanydiversequestionsinphysics.

Note:Thefirstthreechapterslayimportantgroundworkthatpreparesthereadertounderstandthe proposed model. These chapters have a strong emphasis on physical interpretation anddefinitions. To establish a commonbaseof physical interpretation, it is necessary to sometimespresent explanations about quantum mechanics or general relativity that experts will findelementary. The objective of introducing these elementary concepts is to provide a sharedexplanation for the physical interpretations that are being used in the remainder of the book.Developmentofthewavebasedmodeloftheuniversestartsinearnestinchapter#4.

Referencescitedinthebookareshownasfootnotesatthebottomofthepagethatcitesthereference.

The Universe Is Only Spacetime ©2012 [email protected] 1-1

Chapter1

ConfinedLightHasInertia

AttheendofthischapterthereisAppendixAthatgivesamorerigorousmathematicalanalysisoftheconceptsfirstpresentedusingonlyalgebraicequations.Itisthereforepossibletoreadthischapterontwolevels.Light in a Reflecting Box:Theconceptspresentedinthisbookstartedwithasingleinsight.Irealized that if it was possible to confine light in a hypothetical 100% reflecting box, theconfinedlightwouldexhibitmanyofthepropertiesofafundamentalparticle.Inparticular,aconfinedphotonwouldpossessthesameinertia restmass andsameweightasaparticlewithequalinternalenergy E mc2 .Iftheboxismoving,aconfinedphotonalsoexhibitsthesamekinetic energy, same de Broglie waves, same relativistic length contraction and same timedilationasanequalenergyparticle.Itisanaxiomofphysicsthataphotonisamasslessparticle.Masslessparticlesdonothavearest frame of reference. They are moving at the speed of light in any frame of reference.However, if light is confined in a box, it is forced to have a specific frameof reference. This“confined light” then exhibits properties normally associatedwith a restmass of equivalentenergy m E/c2 in the frameof referenceof thebox.Thiswill firstbeanalyzedusing thefollowingspecialrelativityequation:

m2 – where:pismomentumandEisenergy equation1

Notetothereader:Thefirsttimesymbolsareused,theywillbeidentifiedinthetext.Allthesymbolsused in thebookand the importantequationsarealsoavailable inChapter15. It isrecommended that you takeamomentand lookat chapter15. If youare reading thisbookonline, it is recommended that you print out Chapter 15 10 pages as an essential quickreference.Ifa“particle”hasenergyofE pc,thensubstitutingthisintotheaboveequationgives:

m p cc

–pc

0

Inotherwords,whenE pc,thena“particle”hasnorestmass.Now,momentumisavector,soaveryinterestingthinghappenswhenweapplyequation#1toconfinedlight.Forexample,


a single photon confined between two reflectors is a wave traveling both directionssimultaneously. The total momentum of this photon is zero because the two oppositemomentum vectors nullify each other. Substituting p 0 into the equation #1 yields:m E/c2 .Inotherwords,confinedlightsatisfiesthedefinitionofrestmass.Anotherexampleofaphotongainingrestmass isaphotonpropagatingthroughglass. If theglasshasanindexofrefractionof1.5,thenthephotonpropagatesatonly2/3thespeedoflightinavacuumandthemomentumofthephotonisreduced.Thephotondoesnotchangeenergywhenitenterstheglass,butsomeofitsmomentumisimpartedtotheglassuponentranceandthismomentumisreturnedto thephotonuponexit. While thephoton ispropagating in theglass,itcanbethoughtofaspossessingsomerestmassbecauseE pc.Inotherwords,glassthathas lightpropagating through ithasmoretotalmass more inertia thanthesameglasswithout any light propagation. It is also possible to analyze this more deeply and get intoforwardscatterandphase shifts introducedby theatomsof theglass. Thisanalysis impliesthatthelighthasundergonepartialconfinementasitpropagatesthroughtheglassatlessthanthespeedoflightinavacuum.Thispartialconfinementaddsasmallamountof“restmass”totheglass.Thefollowingexamplegivesadeeperphysicalinsightintohowitispossibleforconfinedlightto exhibit mass. Suppose that a laser cavity has a 1 meter separation between two highlyreflectivemirrors.Thisisa2m 6.67ns roundtripforlightreflectingwithinthiscavity.Lightexertsphotonpressureonabsorbingorreflectingsurfaces.TheforceexertedonanabsorbingsurfaceisF P/cortwicethisforceisexertedonareflectingsurfaceF 2P/cwhereF forceandP power. Ifthelaserinthisexamplehad1.5 108wattscirculatingbetweenthetwomirrors reflectingsurfaces ,theenergyconfinedbetweenthetwomirrorswouldbeequalto1Joule and a force exerted by the light on eachmirrorwould be oneNewton in an inertialframeofreference.Supposethatthelaserisacceleratedinadirectionparalleltotheopticalaxisofthelaser.Intheacceleratingframeofreference,therewouldbeaslightdifferenceinthefrequencyofthelightstriking the two mirrors because of the mirror acceleration that occurs during the timerequiredforthelighttotravelthedistancebetweenthetwomirrors.Thefrontmirrorintheaccelerationdirectionwouldbereflecting lightthathasDopplershiftedtoa lowerfrequencycomparedtothelightthatisstrikingtherearmirror.Ifwereturntotheexampleof1.5 108wattsoflightcirculatingbetweentwomirrorsseparatedbyonemeter,thentheforceexertedagainstthefrontmirrorwouldbeslightlylessthanoneNewtonandtheforceexertedagainsttherearmirrorwouldbeslightlymorethanoneNewtonbecauseofthedifferenceinDopplershifts.Thisforcedifferencecanbeinterpretedastheforceexertedbytheinertiaof1Jouleofconfinedlight.Theinertiaof1Jouleofconfinedlightexactlyequalstheinertiaofamasswith1Joule of internal energy 1.11 10‐17 kg . For comparison, thismass is equal to about 6.6billionhydrogenatoms.


While general relativity treats energy in any form the same, particle physics does not. TheStandardModelofparticlephysicssuggeststhat leptonsandquarksrequirethehypotheticalHiggs field to create the inertia of these particles. Therefore, in this example 6.6 billionhydrogenatomsrequireaHiggsfieldforinertiabutanequalenergyofconfinedlightexhibitsequal inertiawithout theneedofaHiggs field. In fact, the inertiaexhibitedbytheconfinedlightisultimatelytraceabletotheconstancyofthespeedoflight.If we place the laser with 1 Joule of confined light stationary in a gravitational field, theconfinedlightwillexertanetforceonthetwomirrorsequivalenttotheweightexpectedfrom6.6billionhydrogenatoms. Ifthelaserisorientedwithitsopticalaxisvertical,thenthenetforce difference comes fromwhat is commonly called the gravitational red/blue shift. Thisnameisamisinterpretationthatwillbediscussedlater.Thepointisthatmoreforceisexertedonthelowermirrorthantheuppermirrorbecauseofthegravitationalgradientbetweenthesetwomirrors. If the laser is orientedhorizontally, therewill be gravitational bendingof thelight.Themirrorcurvaturenormallyincorporatedintolasermirrorseasilyaccommodatesthisslightmisalignment.However,thebendingoflightintroducesadownwardvectorcomponentintotheforcebeingexertedagainstbothmirrors.Thisvectorcomponentistheweightofthelight.Itistruethatgeneralrelativityteachesthatenergyinanyformexhibitsthesamegravity.Therefore the gravitational similarity is expected. However, this does not automaticallytranslateintogivinginertiatoquarksandleptons.Theconceptthatconfinedlighthasweightandinertiahasbeenexplaineddifferentlyinthearticle“LightIsHeavy”1Confinedlightalsoexhibitskineticenergywhenitisconfinedinamovingframeofreference.Supposethatthelaserwith1Jouleofconfinedlighttravelsataconstantvelocityrelativetoa“stationary” observer. Also suppose that theobserver sees themotion as travelingwith theopticaxisofthelaserparalleltothedirectionofmotion.Thestationaryobserverwillperceivethat light propagation in the direction of motion is shifted up in frequency and lightpropagatingintheoppositedirectionisshifteddowninfrequency.Combiningtheseperceivedchanges in frequency result in a net increase in the total energy of the confined light.AppendixA This energy increase is equal to the kinetic energywhichwould be expectedfromtherelativemotionofamassofequal internalenergy. AppendixAalsoshowsthattheenergyincrease kineticenergy iscorrectevenforrelativisticvelocities.Thekineticenergyoftheconfinedlightisultimatelytraceabletotheconstancyofthespeedoflight.Confined Black Body Radiation:Thusfar,theexampleofconfinedlighthasusedalaserwithhighly reflectingmirrors. An alternative example could use an ordinary cardboard box at atemperatureof300°Kwithan internalvolumeof1m³. Theblackbody radiationwithin thisboxwould have infrared light being emitted and reabsorbed by the internalwalls. For thestated conditions, the blackbody radiation within the box would be about 6.1 10‐6 J of

1 http://www.tardyon.de/mirror/hooft/hooft.htm


radiationinflightwithinthevolumeoftheboxatanyinstant.Thisenergyisequivalenttotheannihilation energyof about40,000hydrogenatoms. Even though theblackbody radiationexampleisslightlyhardertosee, theresult issimilartohavingreflectingwalls.Theconfinedblack body radiation exhibits inertia, weight and relative motion exhibits kinetic energy.Carrying this blackbody radiation example further, let’s consider the sun with a coretemperature of about 15,000,000°K. At this temperature, the radiation is primarily at x‐raywavelengths. This confinedx‐ray radiationhas inertiaequivalent toabouta gramper cubicmeter.Atahighertemperaturewhereastarcanburncarbon,theinertiaoftheconfinedx‐raysisequivalenttotheinertiaofanequalvolumeofwater density 1000kg/m3 .Onceagain,noHiggsfieldisrequiredforconfinedradiationtoexhibitinertia.Theexamplesusedabovehadbidirectional lighttravelinginalaserormultidirectional lighttravelingwithinablackbodycavity. Itwouldalsobepossible toconfine lightbyhavingthelighttravelinginasingledirectionaroundaclosedloop.Forexample,lightcouldbeconfinedby traveling around a loop made of a traveling wave tube or fiber optics. Also, it is notnecessarytolimitthediscussiontolight. Gravitationalwavesarealsomasslessbecausetheypropagateenergyatthespeedoflight.Therearenoknownreflectorsforgravitationalwaves,butitishypotheticallypossibletoimagineconfinedgravitationalwaves.Iftherewereconfinedgravitational waves, they would also exhibit the rest mass property of inertia and exhibitkineticenergywhentheconfiningvolumeexhibitsrelativemotion.Aphotonisoftendescribedasa“masslessparticle”.Wenowseethataqualificationshouldbeaddedbecauseonlyafreelypropagatingphotonismassless.Aconfinedphotonpossessesrestmass possesses inertia . Both photons and gravitational waves are examples of energypropagatingatthespeedoflight.Inchapter4anotherformofenergypropagatingatthespeedof light will be introduced waves in spacetime . This also exhibits inertia when confined.Fromtheseconsiderations,thefollowingstatementcanbemade:Energy propagating at the speed of light exhibits rest mass (inertia) when it is confined to a specific frame of reference.Constraint on Higgs Mechanism:Imaginewhatitwouldbelikeifconfinedlight orconfinedgravitationalwaves exhibitedadifferentamountofinertiathanaparticleofequalenergy.Forexample,supposeanelectronandpositronareconfinedinthe100%reflectingbox.Aftersometimetheelectronandpositroninteracttoformtwogammarayphotonsofequalenergy.Wouldthe total inertia of the box be any different when it contained the electron and positroncomparedtocontainingconfinedphotonsofequalenergy?Iftherewasanydifference, evenatrelativistic velocity , then thiswould be a violation of the conservation ofmomentum. Theequal energy confined photonsmust have exactly the same inertia as the confined particlesunderallconditions.


The standard model of particle physics explains the inertia of a fundamental particle afermion results froman interactionwith thehypotheticalHiggs field. Thisexplanationsaysthat a muon interacts more strongly than an electron, therefore a muon has more inertia.However, the inertia imparted by the Higgs field does not have a precise requirement forexactlyhowmuchinertiaamuonoranelectronshouldpossess.Nowwelearnthattheinertiaofanelectronwith511KeVof internalenergymustexactlymatch the inertiaof511KeVofconfinedphotons.Similarly,amuonwithinternalenergyof106MeVmustexactlymatchtheinertiaof106MeVofconfinedphotons. Matchingtheinertiaofafundamentalparticletotheinertiaofanequalamountofenergypropagatingatthespeedoflightbutconfinedtoaspecificvolumeaddsanadditionalconstrainttoanyparticlemodel.Theparticlemodelproposedlaterinthisbookperfectlymatchestherequiredinertiaconstraint.TheHiggsmechanismdoesnotcurrentlysatisfythisrequirement.de Broglie Waves:Thesimilaritybetweenconfinedlightandparticlesdoesnotendwiththeconfinedlightpossessingrestmass,weightandkineticenergywhenthereisrelativemotion.NextwewillexaminethesimilaritybetweenthewavecharacteristicsofconfinedlightandthedeBrogliewavepatterns of fundamental particles. For example, particleswithmassm andvelocity v that pass through a double slit produce an interference pattern which can beinterpretedashavingadeBrogliewavelengthλdgivenbytheequation:λd h/pwhereλd deBrogliewavelength;h Planck’sconstant;p momentum

λd h/γmovwhereγ ⁄mo particle’srestmass

νd E/hwhereνd deBrogliefrequencyE totalenergy


ThedeBrogliewaveshaveaphasevelocitywd c2/vandagroupvelocityud v.Thephasevelocitywdisfasterthanthespeedoflightandthegroupvelocity, ,equalsthevelocityoftheparticle,v.ThereisastrikingsimilaritybetweenthedeBrogliewavecharacteristicsofamovingparticleand thewavecharacteristicsofconfined light inamoving laser. Figure1‐1showsamovinglaserwithmirrorsAandBreflectingthelightwavesofalaserbeam.Figure1‐1isacompositebecausethelightwavedepictselectricfieldstrengthintheYaxiswhilethemirrorsareshownincrosssection.Ifthelaserisstationary,thestandingwavesbetweenthemirrorswouldhavemaximumelectricfieldamplitudethatisuniformatanyinstant.However,thelaserinFigure1‐1 ismoving in the direction of the arrow shown at velocity v. From the perspective of a“stationary”observer,lightwavespropagatinginthedirectionofvelocityvareDopplershiftedup in frequency, and light waves moving in the opposite direction are shifted down infrequency. When these electric field amplitudes are added, this produces the modulationenvelope on theDoppler shifted bidirectional light in the laser as perceived by a stationaryobserver. Thismodulationenvelopepropagates in thedirectionof the translationdirectionbut the modulation envelope has a velocity which is faster than the speed of lightwm c2/v calculated in appendix A . This is just an interference pattern and it canpropagate faster than the speed of light without violating the special relativity prohibitionagainst superluminal travel. Nomessage can be sent faster than the speed of light on thisinterferencepattern.Themodulationenvelopehasawavelength where:

λm λm modulationenvelopewavelength;λγ wavelengthofconfinedlight

Asseeninfigure1‐1,onecompletemodulationenvelopewavelengthencompassestwonullsortwolobes.Itwillbeshownlaterthatthereisa180degreephaseshiftateachnull,sotoreturntotheoriginalphaserequirestworeversals twolobesperwavelength .Thesimilarity todeBrogliewavescanbeseen ifweequate theenergyofasinglephotonofwavelength λγ to the energy of a particle of equivalent mass m. This will assume thenon‐relativisticapproximation.AppendixAwilladdressthemoregeneralrelativisticcase.

E mc2 equatingphotonenergytomassenergythereforem

λd λ deBrogliewavelength;

λd λmdeBrogliewavelengthλ modulationenvelopewavelength


Themodulation envelope not only has the correctwavelength, it also has the correct phasevelocity wd wm c2/v . The “standing” optical waves also have a group velocity of v.Thereforethesewavesmovewiththevelocityofthemirrorsandappeartobestandingrelativetothemirrors.

de Broglie Waves in Radial Propagation: It iseasytoseehowtheopticalequivalentofdeBrogliewavescan form in theexampleabovewithpropagationalong theaxisof translation.However, it isnotasobviouswhatwouldhappenifwetranslatedthelaserinadirectionnotalignedwiththelaseraxis.Wewilltakethistothelimitandexaminewhathappenswhenthewaves propagate radially into a 360° plane that is parallel to the translation direction. Tounderstandwhathappens,wewillfirstlookatfigure1‐2thatshowstheDopplershiftedwavepattern produced by waves propagating away from a point source in a moving frame ofreference.Thesourceismovingfromlefttorightasindicatedbythearrow.Wavesmovinginthedirectionofrelativemotion totheright areseenasshiftedtoashorterwavelengthandwavesmovingoppositetothedirectionoftravelareshiftedtoalongerwavelength.Figure1‐3issimilartofigure1‐2exceptthatonlywavespropagatingtowardsthesourceareshown.


Figure1‐4showswhathappenswhenweaddtogether theoutwardand inwardpropagatingwaves shown in figures 1‐2 and 1‐3. Also a cross‐sectioned cylindrical reflector has beenadded to figure 1‐4. This reflector can be thought of as the reason that there are wavespropagatingtowardsthecenter.Thecentrallobeoffigure1‐4canbethoughtofasalinefocusthatrunsdowntheaxisofthecylindricalreflector.The vertical dark bands in figure 1‐4 correspond to the null regions in the modulationenvelope. These null regions can be seen in figure 1‐1 as the periodic regions ofminimumamplitude.Thereisa180°phaseshiftatthenulls.Thiscanbeseenbyfollowingaparticularfringethroughthedarknullregion.Ifthewaveisrepresentedbyayellowcolorononesideofthe null, this same wave is a blue color on the other side of the null. This color changeindicates that a 180° phase shift occurs at the null. In figure 1‐1, the reason that thewavelengthofthemodulationenvelopeλmisdefinedasincludingtwolobesisbecauseofthisphasereversalthathappensateverynull.Thereforeittakestwolobestoreturntotheoriginalphaseandformonecompletewavelength.


Themainpurposeof this figure is to illustrate thatdeBrogliewaveswithaplanewavefrontappear even in light that is propagating radially. This is amodulation envelope that is theequivalentofaplanewavemovinginthesamedirectionastherelativemotion,butmovingataspeed faster than the speed of light. Figure 1‐4 represents an instant in a rapidly changingwavepattern.Therehasalsobeenanartisticlicensetakeninthisfiguretohelpillustratethepoint.Normallywewouldexpecttheelectricfieldstrengthtobeverylargealongthefocallineatthecenterofthe cylindrical reflector and decrease radially. However, accurately showing this radialamplitudevariationwouldhidethewavepatternthatisthepurposeofthisfigure.Thereforetheradialamplitudedependencehasbeeneliminatedtopermittheotherwavepatternstobeshown. Another artistic license is the elimination of the Guoy effect at the line focus. Thecentrallobeofacylindricalfocusshouldbeenlargedby¼wavelengthtoaccommodatethe90°phaseshiftproducedwhenelectromagneticradiationpassesthrougha linefocus. Ultimatelywewillbetransferringtheconceptsillustratedheretoadifferentmodelthatdoesnotrequirethisslightenlargement.

Figure1‐5isa3dimensionalrepresentationofthewavepatternpresentinfigure1‐4.Infigure1‐5theZaxisisusedtorepresenttheelectricfield.Thecylindricalreflectorhasbeenremovedfrom the illustration to permit the waves to be seen. Also as before, the radial amplitudedependencehasbeeneliminatedtopermitthesubtlemodulationenvelopetobeseen.Iffigure1‐5wasset inmotion, theconcentriccircularwavepatternwouldmoveasaunit. However,


superimposed on this is themoving envelope of waves that are moving through this wavestructure wavesonwaves .Thismovingenvelopeofwavesismovingfasterthanthespeedoflightinthesamedirectionasoverallmotion wm c2/v .The surprising part of figure 1‐4 and 1‐5 is that we obtain a linear modulation envelopeimposed on the radial propagating waves. It does not make any difference what thepropagationangleis,theequivalentofdeBrogliewavesareproducedforallangles.Theonlyrequirement is that the wave has bidirectional propagation. If later we are successful inestablishing a model of fundamental particles that exhibits bidirectional wave motion, thatmodelwillalsoexhibitdeBrogliewaves.Nextwearegoingtotalkaboutrelativisticlengthcontraction.Forillustration,wewillreturntofigure 1‐1. This figure shows the wave frozen in time and designates the distance thatapproximately corresponds to the laserwavelength λ. Actually, this distance only preciselyequals the laser wavelengthwhen there is no relativemotion. In the example illustrated infigure1‐1,thereisrelativemotion.ThewaveillustratedistheresultofaddingtogetherawavethathasbeenDopplershiftedupinfrequencytoawavethathasbeenDopplershifteddowninfrequency.Thecombinationproducesapeak topeakdistance that isequal to therelativisticcontractionofthelaserwavelength.Thisisreasonablewhenyouconsiderthatthereareafixednumberofstandingwavesbetweenthetwomirrors.Ifthedistancebetweenthetwomirrorsundergoesarelativisticcontraction,the standing waves must also exhibit the same contraction to retain the fixed number ofstandingwaves.However,itispossibletoreversethisreasoning.Ratherthansayingthatthestandingwavesmustcontracttofitbetweentherelativisticcontractedmirrorseparation,itispossible to say that we might be getting a fundamental insight into the mechanics of hownatureaccomplishesrelativisticcontractionofphysicalobjects.Ifallfundamentalparticlesandforces of nature can ultimately be reduced to bidirectional waves in spacetime, then thesebidirectionalwaves,will automatically exhibit relativistic contraction and themechanism ofrelativisticcontractionofeventhenucleusofanatomwouldbeconceptuallyunderstandable.Similarly, the mechanism of relativistic time dilation would also become conceptuallyunderstandable.Ifweweretotimetheoscillationfrequencyofindividualwavesinthelaseroffigure1‐1,wewould find that theoscillation frequency that resultswhenweadd these twoDoppler shifted waves together slows exactly as we would expect for the relativistic timedilation of amoving object. Again this is traceable to the constant speed of light producingdifferentDoppler shifts on the components of the bidirectional light. The sumof these twofrequenciesexhibitsanetslowingthatcorrespondstorelativistictimedilation.


Therefore the analysis in this chapter and appendix A shows that a confined photon in amovingframeofreferencehasthe following8similaritiestoa fundamentalparticlewiththesameenergyandsameframeofreference:

1 Theconfinedphotonhasthesameinertia restmass : ħ ⁄ 2 Theconfinedphotonhasthesamekineticenergy:ke ½mv2 ½ ħ ⁄ v23 Theconfinedphotonhasthesameweightastheparticle.4 Theconfinedphoton bidirectionalpropagation hasthesamemomentum.5 Theconfinedphoton’senvelopewavelengthλsisthesameastheparticle’sdeBroglie

wavelength:λs λd6 Theconfinedphoton’smodulationenvelopephasevelocityisthesameastheparticle’s

deBrogliephasevelocity:vs w ⁄ 7 Thephoton’sgroupvelocityisthesameastheparticle’sgroupvelocity:vd u8 Theconfinedphotonhasthesamerelativisticlengthcontraction: ⁄

It is hard to avoid the thought that perhaps a particle is actually a wave with componentsexhibitingbidirectionalpropagationat thespeedof lightbut somehowconfined toa specificvolume. This confinement produces standing waves that are simultaneously moving bothtowardsandawayfromacentralregion.Dowehaveanytrulyfundamentalparticles?IfIdefinedafundamentalparticlebytheancientGreek standardof indivisibility and incorruptibility, then there are none. An electron and apositroncanbeturnedintotwophotons andviceversa .Anisolatedneutron 2downquarksand1upquark willdecayintoaproton 2upquarksand1downquark plusanelectronandanantineutrino.Infact,all12“fundamental”fermionsofthestandardmodelcanbeconvertedinto other fermions and into photons. The simplest explanation for this easy conversionbetween“fundamental”particlesisthatthereisawavestructuretothesefermions.Thetrulyfundamental building block of all fermions is the underlyingwave in spacetime that allowsthese easy transformations. It is on the level of this truly fundamental building block thatthere is a similarity between confined light and particles. This thought process will becontinuedinchapter4.Note:Chapters2and3laygroundworkthatpreparesthereadertounderstandtheproposedmodel.Developmentofthespacetimebasedmodelofparticlesandforcesstartsinearnestinchapter4.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Appendix A - 1

Appendix AExamination of the Similarities

Between Confined Light and a ParticleChris Ray

§ Confined Light

This appendix will investigate a photon confined in per-fectly reflecting resonator. It will be shown that such a con-fined photon exhibits many particle-like properties includingrest mass, relativistic contraction and a moving wave patternthat is similar to de Broglie waves.

We will begin by examining a standing wave in a res-onator as viewed from a frame of reference in which the res-onator is moving.

§ First View: Counter Propagating Waves

A 1-D standing wave can be modeled as a superpositionof right and left moving plane waves.

ψ = ei(k0x−ω0t) + ei(−k0x−ω0t)

where k0 = ω0/c.In the frame of reference where the resonator is at rest

there are standing waves in the resonator set up by thecounter propagating waves.

ψ = ei(k0x−ω0t) + ei(−k0x−ω0t)

= (eik0x + e−ikox)e−iωot

= 2 cos(k0x)e−iωot

In the frame of reference where the resonator is movingto the right with velocity v, we have counter propagatingwaves with different frequencies, because the waves have beendoppler shifted: ωR = γ(1+β)ω0 and ωL = γ(1−β)ω0, whereβ = v/c and γ = 1/

√1− β2. The wave then is given by

ψ = ei(kRx−ωRt) + ei(kLx−ωLt)

where kL = −ωL/c and kR = ωR/c.Define ω+ and ω− as follows

ω+ ≡ 12 (ωR + ωL)

= 12 (γ(1 + β)ω0 + γ(1− β)ω0)

= γω0

ω− ≡ 12 (ωR − ωL)

= 12 (γ(1 + β)ω0 − γ(1− β)ω0)

= γβω0

Now define k+ and k− in a similar way.

k+ ≡ 12 (kR + kL) =

12c

(ωR − ωL) =ω−c

= γβω0

c

k− ≡ 12 (kR − kL) =

12c

(ωR + ωL) =ω+

c= γ

ω0

c

Note thatkL = k+ − k−kR = k+ + k−

ANDωL = ω+ − ω−ωR = ω+ + ω−

Now we can write the wave asψ = ei(kLx−ωLt) + ei(kRx−ωRt)

= ei((k+−k−)x−(ω+−ω−)t) + ei((k++k−)x−(ω++ω−)t)

=[e−i(k− x−ω− t) + ei(k− x−ω− t)

]ei(k+x−ω+t)

= 2 cos(k− x− ω− t)ei(k+x−ω+t)

The imaginary part of this is graphed below (in red) forβ = 0.085 at t = 0.

This is a product of two traveling waves. We can com-pute wavelengths and velocities of these two parts.

v+ =ω+

k+=

ω+

ω−/c=c

β=c2

v

v− =ω−k−

=ω−ω+/c

= βc = v

λ+ =2πk+

=2πcγβω0

=λ0

γβ

λ− =2πk−

=2πcγω0

=λ0

γ

where λ0 = 2πcω0

is the wavelength in the rest frame of theresonator.

First let us consider the “-” part of the wave. First wenote that this part of the wave moves with the velocity ofthe resonator. Second we see that the wavelength has shrunkby a factor of γ relative to the wavelength in the rest frame.Thus the same number of wavelengths will fit in the similarlylength-contracted resonator. Thus the cos() standing wavepattern has shrunk to fit the moving resonator and moveswith the resonator.

Now consider the “+” part of the wave. This part ofthe wave moves with a velocity c2/v. Which is the same asthe phase velocity of a de Broglie plane wave for a massiveparticle: ψd = eipx/h̄e−iEt/h̄.

−→ vphase =E/h̄

p/h̄=E

p=γmc2

γmv=c2

v

We can also see that the wavelength for the de Broglie planewave: λd = 2πh̄

p = 2πh̄Ev/c2 = 2πh̄c

Eβ = 2πh̄cγE0β

is also the same, ifwe assume that there is a single photon in the resonator andthus that the energy in the rest frame is E0 = h̄ω0. Since

λ+ =2πk+

=2πcγβω0

=2πh̄cγβE0

Thus we see that the “+” part of the resonator wave has thewavelength and phase velocity of a de Broglie plane wave of

Appendix A - 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Appendix A - 2

a massive particle with a rest energy equal to the energy ofthe photon in the resonator.

§ General From

In the rest frame of a general standing wave the ampli-tude of the wave is given by

ψ′(x′, y′, z′, t′) = f(x′, y′, z′)e−iω0t′

Where it is understood that the physical wave is the real partof the complex wave ψ. Note that the amplitude function f()is a real valued function.

We will assume that the energy of this wave isE0 = h̄ω0.

We will also consider this energy in the rest frame of thewave, divided by c2, to be the mass of the system:

m ≡ E0

c2=h̄ω0

c2.

Since the wave is a standing wave the total momentum iszero: p0 = 0.

In a General Frame

We want to know what the standing wave will look likein a frame in which the rest frame of the standing wave ismoving in the positive x direction, with a velocity v. We canfind this, if we assuming that the amplitude of the wave ata given space time point is the same in each frame, so that

ψ(x, y, z, t) = ψ′(x′, y′, z′, t′)with x′ and t′ related to x and t via the following Lorentztransformation, while y′ = y and z′ = z.[

ct′

x′

]=[

γ −γβ−γβ γ

] [ctx

]=[γ(ct− βx)γ(x− βct)

]So that

ψ(x, y, z, t) = ψ′(x′, y′, z′, t′)= ψ′ ( γ(x−βct), y, z, γ(ct−βx)/c )

= f(γ(x− βct), y, z)e−iω0γ(ct−βx)/c

= f(γ(x− vt), y, z)eiω0γβx/ce−iω0γt

= f(γ(x− vt), y, z)ei(kx−ωt)

In the last line we used the following definition.ω ≡ γω0

k ≡ γβω0/c

As was the case with the counter propagating plane waves,the wave function in the general frame is manifestly in theform of a plane wave (ei(kx−ωt)), with wavelength and veloc-ity of a de Broglie wave, modulated with an standing wavepattern (f) that moves in the positive x direction with ve-locity v. In addition the characteristic length of the standingwave pattern has been length contracted in the x direction bythe factor γ compared with the length in the rest frame. Forexamples suppose that there are two features in the standingwave, one at the position x′ = a and the other at the positionx′ = b. The distance between these features is L = b−a. Thefeatures could for example be two null points in the standingwave. These two features will also exist in the general frame,

though they are moving. Let xa and xb the location of thesefeatures, then we know that

γ(xa − vt) = a

γ(xb − vt) = b

Solving these two equations for xb − xa we find that

L′ = xb − xa =b− aγ

=L

γ

So the distance between the features has been contracted bya factor γ.

We also see, as in the case of counter propagating waves,that one part of the wave moves with the velocity v and theother moves with the velocity

ω

k=

γω0

γβω0/c=c

β=c2

v

Energy and Momentum

We can also find the energy and momentum in the newframe, using the relativistic transformation of energy andmomentum. [

Epc

]=[γ γβγβ γ

] [E′

p′c

]=[γ γβγβ γ

] [E0

p0c

]=[γ γβγβ γ

] [h̄ω0

0

]=[γh̄ω0

γβh̄ω0

]=[h̄ωh̄kc

]We see that the frequency and wavenumber of the plane wavepart of the wave are proportional to the energy and momen-tum of the wave.

E = h̄ω

andp = h̄k

in accordance with de Broglie waves.Using the definition m = h̄ω0/c

2, we consider the massof the light in our resonator to be equal to the energy in therest frame divided by c2. Thus we can rewrite the above as.

E = γmc2

andp = γmv

Appendix A - 2


Chapter2

DefinitionsandConceptsfromGeneralRelativity

Theprimarypurposeofthisbookistoshowhowitispossibleforthefundamentalparticlesandtheforcesofnaturetobeconceptuallyexplainedusingonly4dimensionalspacetime.Thisisnothing less than a new model of the universe. Presentation of this model would be animpossiblylargetaskforasinglepersonifeveryaspectofthemodelneedstobeanalyzedwiththedetailthatmightbeexpectedfromatechnicalpapercoveringaspecificaspectofamaturesubject. Therefore, the concepts will be introduced using simple equations that involveapproximations and ignore dimensionless constants such as 2π or½. These simplificationspermitthekeyconceptsofthisverylargesubjecttobeexplained.Later,otherscananalyzeandexpanduponthislargesubjectinmoredetail.Wewillstartby lookingatthegravitationaleffectsonspacetimeinthe limitingcaseofweakgravity.Muchoftheanalysistofollowinsubsequentchapterswilldealwiththegravityexhibitedbyasinglefundamentalparticlesuchasanelectronoraquark.Workingwithsingleparticlesorthe interaction between two fundamental particles allows the proposed structure of suchparticlestobeconnectedtotheforcesthatareexhibitedbytheseparticles.Atthesametime,theextremelyweakfieldspermitsimplificationsintheanalysis.In the following discussion, a distinction will be made between the words “length” and“distance”.Normally,thesewordsaresimilar,butwewillmakethefollowingdistinction.Lengthisaspatialmeasurementstandard.Thisisnotjustastandardizedsizesuchasameterorinch,butitalsocanincludeaqualificationsuchasproperlengthorcoordinatelength.Aunitoflengthcanbedefinedeitherbya rulerorby thespeedof lightanda time interval. Theconceptofdistanceasusedhere isbest illustratedby thephrase “thedistancebetween twopoints”. Adistancecanbequantifiedasaspecificnumberoflengthunits.This chapter starts off with a discussion of the Schwarzschild solution to the Einstein fieldequationandphysicalexamplesoftheeffectofgravityonspacetime.Thiswillseemelementaryto many scientists, but new terminology and physical interpretations are introduced.Understandingthisterminologyandperspectiveisarequirementforsubsequentchapters.Schwarzschild Solution of the Field Equation:Einstein’sfieldequationhasanexactsolutionfor the simplified case of a static, nonrotating and uncharged spherically symmetric massdistribution inanemptyuniverse.Thismassdistributionwith totalmassm is locatedat theoriginofasphericalcoordinatesystem.Thestandard nonisotropic Schwarzschildsolutioninthiscasetakestheform:


dS2 c2dτ2 1/Г2 c2dt2–Г2dR2–R2dΩ2

Г≡dt/dτ explainedbelow

dS cdτ The invariantquantitydS is the lengthof theworld linebetweentwoevents in4dimensionalspacetime.R circumferentialradius circumference/2π explainedbelow Ω asolidangleinasphericalcoordinatesystem dΩ2 dθ2 sin2θdΦ2 c thespeedoflightconstantofnaturet coordinatetime timeinfinitelyfarfromthemass‐effectivelyzerogravity τ propertime–timeintervalonalocalclockingravityCircumferential radius:Gravitywarpsthespacearoundmass,sothatthespacearoundthetestmasshasanon‐Euclidiangeometry.Thecircumferenceofacirclearoundthemassdoesnotequal2πtimestheradialdistancetothecenterofmass.Toaccommodatethiswarpedspace,theSchwarzschildequationusesaspecialdefinitionofdistancetospecifythecoordinatedistanceinthe radial direction. Names like “R‐coordinate” and “reduced circumference” are sometimesusedtodescribethisradialcoordinatethatcannotbemeasuredwithameterstickorapulseoflight.Thenamethatwillbeusedhereis“circumferentialradius”anddesignatedwiththesymbolR.Thisisadistancethatiscalculatedbymeasuringthecircumferenceofacirclethatsurroundsamass,andthendividingthiscircumferenceby2π.Ifwemeasuretheradiususingahypotheticalmeterstickortapemeasure,thenthe“proper”radialdistancewillbedesignatedbyr.WecanseefromtheSchwarzschildmetricthatifwesetdt 0,dS cdτ dranddΩ 0then:dr ГdRGravitational Gamma Γ:ThemetrichasbeenwrittenintermsofthequantityГwhichthisbookwillrefertoasthe“gravitationalgammaГ”.ThebasicdefinitionofГis:

Г≡dt/dτ

whereГ gravitationalgammaandφ –Gm/RgravitationalpotentialГ dt/dτinthestaticcasewhendR 0anddΩ 0.gooisametriccoefficientcommonlyusedingeneralrelativityThesymbolofuppercasegammaГwaschosenbecausethisequationcanalsobewrittenasfollows:


Г whereVe≡2

escapevelocity

ThesimilaritybetweenГandγofspecialrelativityisobvioussince:γ

Thisanalogybetweentheescapevelocityingeneralrelativityandtherelativevelocityinspecialrelativityextendsfurtherintheweakgravityapproximation.Thetimedilationduetogravityapproximatelyequalsthetimedilationduetorelativemotionwhentherelativemotionisequaltothegravitationalescapevelocity weakgravity .TheSchwarzschilduniversewithonlyonemasshaseffectively “zerogravity”atany locationinfinitelyfarfromthemass.AtsuchalocationГ 1.TheoppositeextremeofthemaximumpossiblevalueofГistheeventhorizonofablackholewhereГ ∞.Iftheearthwasinanemptyuniverse,thenthesurfaceoftheearthwouldhaveagravitationalgammaof:Г 1 7 10‐10.ItisimportanttorememberthatГisalwayslargerthan1whengravityispresent.AstationaryclockinfinitelyfarawayfromthemassinSchwarzschild’suniverseisdesignatedthe “coordinate clock” with a rate of time designated dt. In the same stationary frame ofreference,thelocalrateoftimeingravity nearthemass isdesignateddτ.Therelationshipis:Г dt/dτThereisausefulapproximationofΓthatisvalidforweakgravity.

Γ 1 weakgravityapproximationofГ

Gravitational Magnitude β: ThegravitationalgammaΓhasarangeofpossiblevaluesthatextends from 1 to infinity. There is another related concept where the strength of thegravitationaleffectonspacetimerangesfrom0to1where0isalocationinzerogravityand1istheeventhorizonofablackhole.Thisdimensionlessnumberwillbecalledthe“gravitationalmagnitudeβ”andisdefinedas:

β≡1– 1 1– 1–Гβ gravitationalmagnitude

Inweakgravitythefollowingapproximationisaccurate:

β andГ 1 βweakgravityapproximations


Forexample,theearth’sgravity,intheabsenceofanyothergravityisβ 7 10‐10.Thesun’ssurfacehasβ 2 10‐6andthesurfaceofahypotheticalneutronstarwithescapevelocityequaltohalfthespeedoflightwouldhaveβ 0.13.In common usage, the strength of a gravitational field is normally associated with theacceleration of gravity. However, the acceleration of gravity depends on the gradient ofgravitationalpotential.Incontrast,thegravitationalmagnitudeβandthegravitationalgammaГ are measurements of the effect of gravity on time and distance without regard for thegravitational gradient. For example, there is an elevation in Neptune’s atmosphere whereNeptunehasapproximatelythesamegravitationalaccelerationastheearth.However,Neptunehasroughly16timestheearth’smassandroughly4timestheearth’sradius.ThismeansthatNeptune’sgravitationalmagnitudeβisroughly4timeslargerthanearth’satlocationswherethegravitationalaccelerationisaboutthesame.The gravitational magnitude approximation β Gm/c2r will be used frequently with weakgravity.Forexample,thisapproximationisaccuratetobetterthanonepartin1036forexamplesthatwillbepresentedlaterinvolvingthegravityofasinglefundamentalparticleatanimportantradial distance. Therefore, this approximation will be considered exact when dealing withfundamentalparticles.NotethatthisapproximationincludesthesubstitutionofproperradialdistancerforthecircumferentialradiusR. r R .Usingtheapproximationβ Gm/c2rwealsoobtainthefollowingequalitiesforβinweakgravity:

β φ –Gm/RandRs Gm/c2 Schwarzschildradius* seenotebelow

β therateoftimeapproximationforweakgravity

All of these approximationswill be considered exactwhendealingwith the extremelyweakgravityofsinglefundamentalparticlesinsubsequentchapters.

Note: The Schwarzschild radius of a nonrotating black hole is rs 2Gm/c2 and theSchwarzschildradiusofmaximallyrotatingblackhole,suchasaphotonblackhole, isRs Gm/c2.Thisbookwilloftenignorenumericalconstantsnear1,butthereisanotherreasonforusingRs Gm/c2forparticles.TheparticlemodelproposedlaterinthisbookhasenergyrotatingatthespeedoflightwhichwouldhavegravitationaleffectsscalewithRsratherthanrs.

Zero Gravity: Schwarzschild assumedanemptyuniversewithonly a singlemass. Suchauniverseapproacheszerogravityasthedistancefromthemassapproachesinfinity.However,isthere anywhere in our observable universe that can truly be designated as a zero gravitylocation?Therearevastvolumeswithvirtuallynogravitationalaccelerationcomparedtothecosmicmicrowavebackground.However,thisisnotthesameassayingthatthesevolumeshaveagravitationalgammaofГ 1 ahypotheticalemptyuniverse .Everywhereintherealuniverse


thereisgravitationalinfluencefromallthemass/energyintheobservableuniverse.Later,inthechaptersoncosmology,anattemptismadetoestimatethebackground omni‐directional gravitationalgammaofourobservableuniversecompared toahypotheticalemptyuniverse.WhilethepresenceofauniformbackgroundГfortheuniversehasimplicationsforcosmology,wecanonlymeasuredifferencesinГ.ThereisampleevidencethatgeneralrelativityworkswellbysimplyignoringtheuniformbackgroundГoftheuniverse.EffectivelyweareassigningГ 1to the background gravitational gammaof the universe and proportionally scaling from thisassumption.Therefore,adistantlocationwhichwedesignateashavingβ 0orΓ 1willbereferredtoasa“zerogravitylocation”orsimply“zerogravity”.Theterm“zerogravity”incommonusageusuallyimpliestheabsenceofgravitationalaccelerationasmightbeexperiencedinfreefall.However,inthisbook“zerogravity”literallymeansthatweareusingtheSchwarzschildmodelofadistantlocationwhichhasbeenassignedcoordinatevaluesofβ 0andГ 1.Therateoftimeatthiscoordinate locationwillbedesignateddtandcalled“coordinaterateoftime”. Aclockatthislocationwillbedesignatedasthe“coordinateclock”.Gravitational Effect on the Rate of Time: The equation dt Γdτ is perhaps themostimportantandeasiesttointerpretresultoftheSchwarzschildequation.ItsaysthattherateoftimedependsonthegravitationalgammaГ.Thisequationhasbeenprovencorrectbynumerousexperiments.TodaytheatomicclocksinGPSsatellitesareroutinelycalibratedtoaccountforthedifferentrateoftimebetweenthelowergammaattheGPSsatelliteelevationandthehighergammaattheEarth’ssurface. Withoutaccountingforthisgravitationalrelativisticeffect,theGPSnetworkwouldaccumulate errors and cease to functionaccurately after aboutoneday.Thereisalsotimedilationcausedbytherelativemotionofthesatellite.Thisisamuchsmallercorrectionthanthegravitationaleffectandintheoppositedirection. Thedifference in therateof timewithrespect toradialdistance ingravitywillbecalled the“gravitationalrateoftimegradient”.Thegravitationalrateoftimegradientisnotatidaleffect.Anacceleratingframeofreferencehasnotidaleffects,yetitexhibitsarateoftimegradient.Ourobjectiveistoprovideanequationthatrelatestheaccelerationofgravity“g”totherateoftimegradientandutilizesproperlengthintheexpressionoftherateoftimegradient.Forexample,inside a closed room it is possible to measure the gravitational acceleration. If we cannotmeasure any tidal effects, there is no information about themass anddistanceof the objectproducingthegravity.Isitpossibletodeterminethelocalrateoftimegradient expressedusingproperlengthandpropertime fromjustthegravitationalacceleration?It has been shown1 that a uniform gravitational fieldwith proper acceleration g measuredlocally ,hasthefollowingrelationshipbetweenredshiftandgravitationalacceleration:

1 Edward A. Desloge, “The Gravitational Redshift in a Uniform Field” Am. J. Phys. 58 (9), 856-858 (1990)


υ/υo 1–g /c2where:υo frequencyasmeasuredatthesourcelocationwithrateoftimedτoυ frequencyasmeasuredatthedetectorlocationwithrateoftimedτ verticaldistanceusingproperlengthbetweenthesourceanddetector

g accelerationofgravityReference 1 showsthatthisequationisexactifthefollowingqualificationsareplacedontheabovedefinitions.Thesequalificationsare:1 thesourcelocation subscripto shouldbeatalowerelevationthanthedetectorlocation.2 theseparationdistance shouldbetheproperlengthasmeasuredbyatimeofflightmeasurement radarlength measuredfromthesourcelocation.Aslightlydifferentradarlengthwouldbeobtainedifthisdistancewasmeasuredfromthedetectorelevationormeasuredwitharuler.However,inthelimitofagradient infinitelysmall ,thisdiscrepancydisappears.Thereforewiththesequalifications:υ/υo 1–g /c2isexact.Itshouldbenotedthattheheightdifference isaproperdistanceradarlengthmeasuredfromthesource andnotcircumferentialradius.Aswillbeshowinthenextchapter,thegravitationalredshiftisreallycausedbyadifferenceintherateoftimeatdifferentelevations.Thereisnoaccumulationofwavelengths,soυo 1/dτoandυ 1/dτ.Aftermakingthesesubstitutions,thisequationbecomes:

g c2

This is also an exact equation if the above qualifications are observed. Here the ratiodτ‐dτo /dτd willbereferredtoasthegradientintherateoftime.Therearetwopointstobenoticed.First,thegradientintherateiftimeisabletobedeterminedfromtheaccelerationofgravitywithnoknowledgeaboutthemassordistanceofthebodyproducingthegravity.Forexample,agravitationalaccelerationofg 1m/s2 isproducedbyarateof timegradientof1.113 10‐17seconds/secondpermeter.Theearth’sgravitationalaccelerationof9.8m/s2neartheearth’ssurfaceiscausedbyarateoftimegradientofabout10‐16seconds/secondpermeterofelevationdifferenceintheearth’sgravity.Themostaccurateclockpresentlyavailable 2015 isan87Sropticallatticeclock2whichhasanaccuracyof2.1x10‐18.Aclockwiththisaccuracyhasaresolutioncomparableto2cmelevationchangeintheearth’sgravity.Thesecondimportantpointisthattherateoftimegradientisafunctionofproperlengthintheradial direction. Even though dt/dτ is a function of circumferential radius, the relationshipbetweenrateoftimegradientandgravitationalaccelerationisnotafunctionofcircumferentialradius.Thisfactwillbecomeimportantinthenextchapterwhenweexaminehownaturekeepsthe laws of physics constant when there is an elevation change. The connection between

2 http://www.nature.com/ncomms/2015/150421/ncomms7896/full/ncomms7896.html


gravitationalaccelerationandrateoftimegradientwillalsobeanimportantconsiderationwhenweexaminethecosmologicalmodeloftheuniverse chapters13&14 .Previously,wedefinedtheconceptof“gravitationalmagnitudeβ”as:β 1–dτ/dt.It isalsopossibletorelatetheaccelerationofgravitytothegradientinthegravitationalmagnitude.

g c2

Inertial Frame of Reference: Theconcept thatgravitycanbe simulatedbyanacceleratingframeof reference sometimes leads to theerroneous interpretation thatan inertial frameofreferenceeliminatesalleffectsofgravity.Beinginfreefalleliminatestheaccelerationofgravity,butthegravitationaleffectontherateoftimeandthespatialeffectsofthegravitationalfieldremain.AnotherwayofsayingthisisthattheeffectsofthegravitationalgammaΓonspacetimeare still present, even if amass is in an inertial frame of reference. A clock in free fall stillexperiencesthelocalgravitationaltimedilation.Arigorousanalysisfromgeneralrelativityconfirmsthispoint,buttwoexampleswillbegiventoalsoillustratetheconcept.Supposethattherewasahollowcavityatthecenteroftheearth.Aclockinthiscavitywouldexperiencenogravitationalaccelerationandwouldbeinaninertialframeofreference. Thegravitationalmagnitudeβinthiscavityisabout50%largerthanthegravitationalmagnitudeonthesurfaceoftheearth ~10.5 10‐10comparedto7 10‐10 .Forexample,ignoringairfriction,theescapevelocitystartingfromthiscavityishigherthanstartingfromthesurfaceoftheearth.Theclockinthecavityhasaslowerrateoftimethanaclockonthesurface.Theinertialframeofreferencedoesnoteliminatetheothergravitationaleffectsontherateoftimeandthegravitationaleffectonvolume.Asecondexampleisinterestingandillustratesaslightlydifferentpoint.TheAndromedagalaxyis2.5millionlightyears ~2.4 1022m awayfromEarthandhasanestimatedmassofabout2.4 1042kg includingdarkmatter .Thegravitationalaccelerationexertedbythisgalaxyatthedistance of the Earth is only about 2.8 10‐13m/s2. To put thisminute acceleration inperspective,a10,000kgspacecraftwouldaccelerateataboutthisratefromthe“thrust”ofthelight leavinga1watt flashlight. In spiteof theminutegravitationalacceleration, thedistantpresenceofAndromedaslowsdowntherateoftimeonthesurfaceoftheearthabout100timesmore than the Earth’s own gravity. This is possible because the gravitational magnitudeβ Gm/c2r forweakgravity decreasesatarateof1/rwhile thegravitationalaccelerationdecreaseswith1/r2.Attheearth’ssurface,Andromeda’sgravitationalmagnitudeisabout:β Gm/c2r G/c2 2.4 1042kg/2.4 1022m 7 10‐8Andromeda’sβatearthSincetheearth’sgravityproducesβ 7 10‐10atthesurface,Andromeda’seffectontherateoftimeattheearth’ssurfaceisabout100timesgreaterthantheeffectoftheearth’sgravity.Itdoes


notmatterwhetheraclockisinfreefallrelativetoAndromedaorwhethertheclockisstationaryrelativetoAndromedaandexperiencestheminutegravitationalacceleration.Inbothcasesthegravitationaleffectontimeandvolumeexist.Thisexamplealsohintsthatmass/energyinotherpartsoftheuniversecanhaveasubstantialcumulativeeffectonourlocalrateoftimeandourlocalvolume.Thisconceptwillbedevelopedlaterinthechaptersdealingwithcosmology.Schwarzschild Coordinate System:ThestandardSchwarzschildsolutionusescoordinatesthatsimplifygravitationalcalculations.ThissphericalcoordinatesystemusescircumferentialradiusRascoordinatelengthintheradialdirectionandusescircumferentialradiustimesanangleΩfor the tangentialdirection. While there isnodistinction inproper length for theradialandtangentialdirections,wewilltemporarilymakeadistinctionbydesignatingproperlengthintheradial direction as LR and designating proper length in the tangential direction as LT. Thisdistinction does not exist in reality since: cdτ dL dLT dLR. However, using thesedesignations,therelationshipbetweenproperlengthandSchwarzschild’scoordinatelengthis:dLR ΓdRradiallengthLRconversiontoSchwarzschildradialcoordinateRdLT RdΩtangentiallengthLTtoSchwarzschildtangentialcoordinatelengthThe equation dLR ΓdR is obtained by setting dt 0. Ifwe are using the proper distancebetweentwopointsasmeasuredbyaruler,orthecalculatedcircumferentialradius,thenthiszerotimeassumptionisjustified.NextwewillcalculatethecoordinatespeedoflightCfortheradialandtangentialdirectionsbystartingwiththestandardSchwarzschildmetric:dS2 1/Г2 c2dt2–Г2dR2–R2dΩ2forlightsetdS2 0,1/Г2 c2dt2 Г2dR2 R2dΩ2

c2 Г4 Г2

Ifweseparatethiscoordinatespeedof light into itsradialcomponent CR and its tangentialcomponent CT ,weobtain:R dR/dt c/Г2 R coordinatespeedoflightintheradialdirection dΩ 0 T RdΩ/dt c/Г T coordinatespeedoflightinthetangentialdirection dR 0

Thisapparentdifferenceinthecoordinatespeedoflightfortheradialandtangentialdirectionsisnotexpressingaphysicallymeasurabledifferenceintheproperspeedoflight.Thedifferencefollows fromthestandard nonisotropic form of theSchwarzschildmetric. Ifwechoose theisotropic formof theSchwarzschildmetric thedifferencewilldisappearand R T c/Γ2.


However,theisotropicformhasitsownsetofcomplexities,sowewillbeusingthestandardSchwarzschildmetric.

The Shapiro Experiment:Next,wearegoingtoswitchtoadiscussionaboutthegravitationaleffectonproperlength,propervolumeandthecoordinatespeedoflight.In1964,IrwinShapiroproposedanexperimenttomeasuretherelativisticdistortionofspacetimecausedbytheSun’sgravity.Thisnon‐NewtoniantimedelayisobtainedfromtheSchwarzschildsolutiontoEinstein’sfield equation. The Sun is a good approximation of an isolated mass addressed by theSchwarzschildsolution.Theimplicationisthatgravityaffectsspacetimesothatittakesmoretimeforlighttomaketheroundtripbetweentwopointsinspacewhenthemass gravity ispresentthanwhenthemass gravity isabsent.ShapiroandhiscolleaguesusedradartotracktheplanetVenusforabouttwoyearsasVenusandtheEarthorbitedtheSun.Duringthistime,VenuspassedbehindtheSunasseenfromtheEarth nearlysuperiorconjunction .TheorbitsofVenusandtheEarthareknownaccurately,so itwaspossibletomeasuretheadditionaltime


delayintheroundtriptimefromtheearthtoVenusandback.Theeffectofthesun’sgravityonthisroundtriptimecouldbecalculatedfrommultiplemeasurementsmadeoverthetwoyeartimeperiod.Figure2‐1showsShapiro’sgraphoftheexcesstimedelayoverthetwoyearperiod.Thepeakdelayatsuperiorconjunctionwas190μsonahalfhourroundtriptransittime.Variationsofthisexperimenthavebeenrepeatednumeroustimesinthenormalcourseofthespaceprogram.SpacecraftontheirwaytotheouterplanetsoftenstartwithanorbitalpaththatatsomepointresultsinnearlysuperiorconjunctionrelativetotheEarth. ThemostaccuratemeasurementtodatewaswiththeCassinispacecraft.Itwasequippedwithtranspondersattwodifferentradarfrequencies,thereforeitwaspossibletodetermineandremovetheeffectoftheSun’scoronaonthetimedelay.Theresultwasanagreementwiththetimedelaypredictedbygeneralrelativityaccurateto1partin50,000.Withthistypeofagreement,itwouldseemasifthereareno remainingmysteriesabout thiseffect and thephysical interpretation shouldbeobvious.Howexactlydolength,timeandthespeedoflightcombinetoproducetheobservedtimedelayintheShapiroeffect?Forsimplicity,wewouldliketolookatthetimedelayassociatedwitharadarbeamtravelingonlyintheradialdirection.Inthelimit,wecanimaginereflectingaradarbeamoffthesurfaceoftheSun.Thispathwouldbepurelyradial.Tomakeameasurementweneedtohavearoundtrip,butforsimplicityofdiscussion,wewilltalkaboutthetimedelayforaonewaytrip.ForlightdS 0,sothemetricequationgivesusthatforlightmovingintheradialdirection:

Гcdt ГdR→dt

ГdR

WecancomputethetimeΔtittakestomovebetweentwodifferentradii:r1andr2 wherer2 r1 .

cΔt

cΔt 2 Г

Гweakgravity:

Г

Г 0

Δt

SubstitutingtheSun’smass,radiusanddistancegivesΔt 50μs.Therefore,inadditiontoanon‐relativistictimedelay about8minutes ,theonewayrelativistictimedelaywouldbeaboutanadditional50μs.The190μsdelayobservedbyShapiroisroughly4timestheoneway50μsdelayfromtheearthtothesunbecauseoftheadditionallegtoVenusandthentheroundtripdoublingofthetime.


Normally,onEarthwewouldinterpreta50μsdelayinaradarbeamasindicatinganadditionaldistanceofabout15km.Howmuchdoesthesun’sgravitydistortspaceandincreasetheradialdistancebetweentheearthandthesun’ssurfacecomparedtothedistancethatwouldexistifwehad Euclidian flat space? The additional non Euclidian path length will be designated ΔL .Startingfrom:dLR ΓdR

ΔL

ΔL setr2 1.5 1011m,r1 7 108mandm 2 1030kg

ΔL 7.5kmnon‐EuclidianadditionalproperdistancebetweentheEarthandSunSupposethatitwaspossibletostretchatapemeasurefromtheearthtothesurfaceofthesun.Thedistancemeasuredbythetapemeasure properdistance wouldbeabout7.5kmgreaterthanadistanceobtainedfromanassumptionofflatspaceandaEuclidiangeometrycalculation.Theuseofatapemeasuremeansthatweareusingproperlengthasastandard.Gravity Increases Volume: Ifweuseproper length as our standardof length rather thancircumferentialradius,thenwemustadopttheperspectivethatgravityincreasesthevolumeoftheuniverse.However,aninterpretationbasedonpropervolumeisoftenignoredsincetheuseofcircumferentialradiusascoordinatelengtheliminatesthisvolumechangecausedbygravity.In theShapiroexperiment,wecalculatedthat therewasanon‐Euclidian increase indistancebetweentheearthandthesunof7.5km.Supposethatweimagineasphericalshellwitharadiusequaltotheaverageradiusoftheearth’sorbit.Thisisaradialdistanceequaltooneastronomicalunit AU 1.5 1011m .Thesun’smassis 2 1030kgandthesun’sSchwarzschildradiusisrs 2950meters.Whatisthechangeinvolume ΔV insidethissphericalshellifwecomparetheEuclidianvolumeoftheshell Vo andthenon‐Euclidianvolumeoftheshell V whentheSunisatthecenteroftheshell?V 4 Г Afterintegration,thedifferenceΔV V–Voisapproximately:ΔV 5π/3 rs r22–r12 setr2 1.5 1011m,r1 7 108mandrs 2950metersΔV 3.46 1026m3nonEuclidianvolumeincreaseToputthisnonEuclidianvolume increase inperspective, thesun’sgravityhas increasedthepropervolumewithinaradiusof1AUbyabout3.5 1026m3whichismorethan300,000timeslargerthanthevolumeoftheearth earth’svolumeis 1.08 1021m3 .Statedanotherway,the volume increase is about 20% smaller than the volume obtained by multiplying thenon‐Euclidianradiallengthincrease ~7,500m timesthesurfaceareaofthesphericalshell


witharadiusof1AU.ObviouslythisnonEuclidianvolumeincreasewouldbemuchlargerifwehad chosen a larger shell radius for example, the size of the observable universe . Theimplicationsofthiswillbeexploredinthechaptersoncosmology.Concentric Shells Thought Experiment:Theconceptthatgravityincreasesthevolumeoftheuniverseisimportantenoughthatanotherexamplewillbegiven.Supposethattherearetwoconcentric spherical shells around an origin point in space. The inside spherical shell is4.4 109mincircumference aboutthecircumferenceoftheSun .Theoutsideshellis2πmeterslargercircumference.Withnomassattheoriginandinfinitelythinshells,thismeansthatthereisexactlya1metergapbetweentheshells.Wecouldconfirmthe1meterspacingwithameterstickorapulseoflightandaclock.NextweintroducetheSun'smassattheorigin.ThisintroducesgravityintothevolumebetweenthetwoshellswithanaveragevalueofaboutГ 1 2 10‐6.Thecircumferenceofeachshellproperlength doesnotchangeafterweintroducegravity.However,thedistancebetweenthetwoshellswouldnowbeabout1 2 10‐6meters.Thisis2micronslargerthanthezerogravitydistance.This2micron increase inseparation increases thepropervolumebetweenthe twoshellsbyroughly1013m3.Inthisexampleoftwoconcentricshellsweacceptedtheproperlengthofthecircumferenceoftheshells tangentialproperlength .Thequestioniswhethertherewasalsoadecreaseofthistangentiallength relativetoa“flat”coordinatesystem whenthesun’smasswasintroducedattheorigin.Itispossibletoconsiderbothradialandtangentialdirectionsaffectedequally.Thisresultsingravityproducinganevenlargerincreaseinpropervolumethanpreviouslycalculated.Thiswillbediscussedfurtherinthechaptersoncosmology.Connection Between the Rate of Time and Volume: WearegoingtocomputetheeffectofthegravitationalgammaГonpropervolumeusingthestandardSchwarzschildmetric.TheuseofthismetricmeansthatthestandardSchwarzschildconditionsapply:astaticnonrotatinganduncharged spherically symmetric mass distribution in an empty universe. This calculationinvolvesterminologyfromgeneralrelativitythatisnotexplainedhere.Readersunfamiliarwithgeneral relativity should skip the shaded calculation section below and move on to theconclusion.


Ifweknowmetricequations,wecancomputethe3‐dimensionalvolumeandthe4‐dimensionalvolume includestime .Theeasiestwaytodoitistousethefollowingdiagonalmetric:dS2 –g00 dx0 2 g11 dx1 2 g22 dx2 2 g33 dx3 2Thenthe3‐dimensionalvolumedV 3 is:dV 3 g11 g22 g33 1/2dx1dx2dx3Andfor4‐dimensionalvolumedV 4 dV 4 –g00 g11 g22 g33 1/2dx0dx1dx2dx3IntheparticularcaseofthestandardSchwarzschildmetric:g00 –1/Г2;g11 Г2;g22 R2,g33 R2sin2θThedifferentialsof4dimensionalcoordinatesinthiscaseare:dx0 cdt; dx1 dR; dx2 dθ; dx3 dΦSo:dV 3 Г2 R2 R2sin2θ 1/2dR dθ dΦdV 3 ГR2sinθdRdθdΦnotethatvolume 3 scaleswithГV 4 – –1/Г2 Г2 R2 R2sin2θ 1/2 cdt dR dθ dΦdV 4 R2csinθdtdRdθdΦnotethatthisisindependentofГThe above calculation shows that proper volume 3 spatial dimensions scales with thegravitational gamma Г. This supports the previous examples involving the volume increaseinterpretationoftheShapiroexperimentandalsothevolumeincreasethatoccursinthethoughtexperimentwithtwoconcentricshells.Whenweincludethetimedimensionandcalculatetheeffectof thegravitygeneratedbyasinglemassonthesurroundingspacetime,weobtaintheanswerthatthe4dimensionalspacetimevolumeisindependentofgravitationalgammaГ.Theradialdimensionincreases Г dLR/dR andthetemporaldimensiondecreases Г dt/dτ .Theseoffseteachotherresultinginthe4dimensionalvolumeremainingconstant.


Thereisasimplerwayofexpressingthisconcept.SinceГ therefore:

dτdLR dtdRThis concept will be developed further when we develop particles out of 4 dimensionalspacetimeanditalsohasapplicationtocosmology.


Chapter3

GravitationalTransformationsoftheUnitsofPhysics

Covariance of the Laws of Physics: Fromtheeventhorizonofablackholetothemostisolatedvolumeintheuniverse,therearebigdifferencesintheratesoftimethroughouttheuniverse.Howdothelawsofphysicsremainthesamewhentherateoftimeisdifferentbetweenlocations?Whydoesarateoftimegradientalsonotaffectthelawsofphysics?Itisanoversimplificationto imagine thatchanging therateof time issimilar torunningamovie inslowmotionwhilekeepingthelawsofphysicsunchanged.Wearegoingtobelookingathownaturemaintainsthelawsofphysicswhentherateoftimechangeswithgravitationalgamma.Thisisnotjustanacademicquestion.Gravityproducesarateof time gradient and a gradient in the coordinate speed of light. Therefore, even in earth’sgravity,thesimpleactofliftinganobjecttoadifferentelevationmeansthattheobjectismovedtoa locationwhere there isadifferentrateof timeandadifferentcoordinatespeedof light.Acknowledging that there are changes in the rate of time leads to surprising new physicalinsights.Whentherateoftimeisdifferentbetweentwolocations,butthelawsofphysicsarethesame,theremustalsobeotherchangesintheunitsofphysicstooffsetthedifferenceintherateoftime.For example,momentum scalesproportional to 1/t, force scales proportional to1/t2, powerscalesproportionalto1/t3andthefinestructureconstantisindependentoftime 1/t0 .Thisistimeraisedtofourdifferentpowers,yetthelawsofphysicsareconstantevenwiththisdifferenceintimedependence. Whatadditionalchangesarerequiredtooffsetthechangeintherateoftimeandpreservethelawsofphysicsunchangedindifferentgravitationalpotentials?Ifthereisacoordinaterateoftimeinazerogravitylocationthatisdifferentfromtherateoftimeinalocationwithgravity,andifthecoordinatespeedoflightisdifferentinthetwolocations,shouldn’ttherealsobeadifferenceinatleastsomeoftheotherunitsofphysics?Forexample,isoneJouleofenergyoroneNewtonofforcealsodifferentinthezerogravitylocationcomparedto the gravity location? Tomake ameaningful comparison of the units of physics betweenlocationswithdifferentgravitationalpotentials,itwouldbenecessarytouseasinglerateoftime.Thispointiseasytosee.Themoredifficultquestionis:Howdowetreatlengthinthisexercise?Itisimpossibletodirectlycomparelengthbetweentwolocationswithadifferentgravitationalpotential. Also vectors are ambiguous when compared between locations with differentgravitationalpotential.Forexample,thedirectionofavectorcanbedifferentdependingonthepathchosentotransportavectorbetweentwolocationswithdifferentgravitationalpotentials.


Thereforeadoptingthelocallymeasuredproperlengthasastandardoflengthforthatlocationeliminatesambiguity,butisitthelengthstandardweareseeking?Tobeclear,thisexerciseisnotinterestedincalculatingthegeneralrelativisticeffectsonspaceandtime.Wewillobtainthisinformationfromstandardgeneralrelativitycalculations.Wewillpresumethatwealreadyknowthegravitationalgamma Г dt/dτ foreachlocationofinterest.InsteadweareinterestedinunderstandinghowthelawsofphysicsaccommodatethespatialandtemporaldifferencesassociatedwiththesedifferentvaluesofГ.Thelawsofphysicsalwaysscaleinawaythatkeepsthespeedoflightconstant c dL/dτ .Forexample,azerogravityobservermightperceivethatthatalocationingravityhasaslowrateoftime.However,thezerogravity observer also perceives that this location in gravity also has a proportionately slowcoordinatespeedoflight.Aspeedoflightexperimentperformedingravityalwaysresultsintheuniversalconstantcbecauseazerogravityobserverperceivesaslowcoordinatespeedoflightbeingtimedbyaslowclock.Thisresultsinnotonlyaconstantproperspeedoflight c butalsothezerogravityobservercanconsiderproperlengthasconstant independentofГ .Inotherwords,whenthezerogravityobserverapplieshis/herrateoftimeandadjustsforthedifferentcoordinatespeedoflight,thentheunitoflength L canbeconsideredconstant.Alltheforcesscalewithproperlength.Thisistruefornotonlytheelectromagneticforce,butevengravitationalaccelerationscaleswithproperlength.Inthelastchapterweshowedthatg c2ΔT/Δ dτ.InthisequationΔ isanincrementofproperlengthintheelevationdirection.GeneralrelativitytellsusthatthereisadifferencebetweencircumferentialradiusRandproperlengthL.Thereisalsooneperspectivewherethetangentialproperlengthdecreasesrelativetocoordinatetangentiallengthwhengravityisintroducedintoavolumeofspacetime.However,fermions, bosons and forces know nothing about the general relativistic effects involvingcircumferentialradiusorcoordinatetangentiallength.Theseparticlesandforcesallscalewithproper length. IfgravityproducesnonEuclidianspatialgeometry, theseparticlesand forcesmerelyacceptthepropervolumeataparticularlocationandscalewithproperlength.Thereforesince fundamental particles and forces scalewithproper length andproper volume, for thisexerciseweneedtoadoptacoordinatesystemthatrecognizesproperlengthasastandard.Normalized Coordinate System: Theconclusionofthisisthattheanalysiswewishtoperformonthecovarianceofthelawsofphysicsisbestaccomplishedbyadoptingacoordinatesystemthatusescoordinaterateoftimefromgeneralrelativityasthetimestandardandproperlengthasourlengthstandard.Thisisanunconventionalcoordinatesystemthatisahybridbetweenthe Schwarzschild coordinate system coordinate time and circumferential radius and thestandardcoordinatesthatusepropertimeandproperlength. Thiscoordinatesystemwillbeused to analyze the covariance of the laws of physics when two locations have differentgravitationalpotentialsanddifferentratesoftime. Inthisanalysiswealwaysassumebothaconstant distance between locations and static gravity. Further support for the use of this


coordinate system will be offered by actually performing this analysis using this hybridcoordinatesystemandseeingiftheresultsarereasonable.Thehybridcoordinatesystemthatusesproperlengthandcoordinaterateoftimewillbecalledthe“normalized”coordinatesystem.Thespeedoflightutilizingthiscoordinatesystemwillbecalled the “normalized” speedof light. Wewill also be referring to the “normalized” unit ofenergy, force, etc. All of these units use proper length and zero gravity rate of time. Thenormalized coordinates cannot be used for general relativity calculations to determinespacetimecurvature. Theequationsbecomesosimplifiedthat important information is lost.Instead,thenormalizedcoordinatesystemacceptsthevalueofГobtainedfromgeneralrelativityandutilizesthisinformationtoanalyzeotheraspectsofphysics.Byadoptingproperlengthasourcoordinatelengthandzerogravityrateoftimeweachieveacoordinatesystemthatworkswellwithquantummechanicsandgivesinsightsintotheforcesofnature.Length and Time Transformations: Thefollowinganalysiswillusedimensionalanalysisandthereforewewillbeusingthesymbolsofdimensionalanalysis.Theseare:M,L,T,Q,andΘtorepresentmass, length,time,chargeandtemperaturerespectively. Forexample, theunitsofenergyare:kgm2/s2.Theconversiontodimensionalanalysisterminologyis:kgm2/s2→ML2/T2.Also, the calculations to followwill bemaking transformations between the variousunits ofphysicswhenГchanges.Forexample,tounderstandhowthelawsofphysicsaremaintainedgoingfromahypotheticallocationinzerogravity Г 1 toalocationwithstronggravity Г 1 ,wewill beworkingwith discrete units such as a Joule or a Newton. Thismeans that thetransformation of our coordinates also requires the use of discrete units of length and timeratherthanthedifferentialform.Forexample,dt Гdτrelatestherateofcoordinatetime dt totherateofpropertimeingravitydτ .Inthiscasedt dτ.However,supposewecompareaunitoftime,suchasonesecondinalocationwithzerogravitytoonesecondinalocationwithgravity.Ifeachlocationsentoutalightpulselasting1second accordingtoalocalclock ,thenanyobserver independentofГ wouldagreethatthelightpulsefromthelocationingravitylastedlongerthanthelightpulsefromthezerogravitylocation.ThiswillberepresentedasTg To.Thesubscript“g”representsa location in gravity and “o” represents a location with zero gravity. Therefore, Tg Torepresents thataunitof time ingravity is larger longer thanaunitof time inzerogravity.Whenweconvertdt Гdτtoexpressarelationshipbetweenunitsoftimeitbecomes:To Tg/ГunitoftimetransformationfromzerogravitytogravityThereisnonewphysicsbeingexpressedhere.Thedifferenceiscomparabletocomparingarateof pulses expressed aspulse per second compared to the timebetweenpulses expressed assecondsperpulse.


Sinceproperlengthisadoptedasourstandardoflength,thismeansthatwearenotmakingadistinctionbetweenproperlengthinanylocationororientation.Thewaythatthisisexpressedis:Lo LgunitoflengthtransformationfromzerogravitytogravityNormalized Speed of Light: Whenthenormalizedcoordinatesystemusesproperlengthandthezerogravityrateoftime,thenthenormalizedspeedoflight designatedwithacapital“C” becomes:C dL/dt.Inotherwords,thenormalizedspeedoflightinthenormalizedcoordinatesystemisthechangeinproperlengthdividedbytherateofcoordinatetimedt.C≡dL/dt cdτ/dt c/ГC normalizedspeedoflightCo cCo normalizedspeedoflightinzerogravity whenГ 1 Cg c/ГCg normalizedspeedoflightinalocationwithgravity whenГ 1 Co ГCgrelationshipbetweenCoandCgIftherearetwolocations 1and2 thathavegravitationalgammasΓ1andΓ2respectively,thentheywillhavenormalizedspeedoflightofC1andC2.Therelationbetweenthesetwodifferentnormalizedspeedsoflightis:Γ1C1 Γ2C2 c.IntheequationCo ГCgwehaveeliminatedtheneedforΓ1becauseinzerogravityΓ 1andtheneedtomentionΓ1disappears.ItisinformativetogiveanexampleofCo ΓCg.ThegravitationalgammaΓatthesurfaceofthesunis:Γ 1.000002.IfwesetthenormalizedspeedoflightinzerogravitytoCo 1, thenthesurfaceofthesunhasCg 0.999998. Sinceproper lengthiscoordinatelengthinthenormalizedcoordinatesystem,thenon‐Euclidianpropertiesofspaceareinterpretedasgravitycreatingadditionalpropervolumeinthespacesurroundingamass.Thereforethenon‐Euclidianvolumesurroundingthesunismerelyacceptedbythenormalizedcoordinatesystem.IfabeamoflightpassesthroughthisnonEuclidianvolumeofspace,thenthedifferenceinopticalpathlengthacrossthewidthofthebeamistakenintoaccount. Thisdifferenceinpathlengthcontributestobendingofthelight.Comparison of Coordinate Speed of Light: WepreviouslydesignatedthecoordinatespeedoflightusingSchwarzschildcoordinatesas Rand T.ThecomparisonofthenormalizedspeedoflightingravityCgtotheSchwarzschildcoordinatespeedoflightis:LengthCoordinateSpeedofLightCoordinatesTransformationSpeedofLightConversionLo LgCg dL/dtc ΓCgNormalizedspeedoflightconversiondR 1/Γ dLR R dR/dtc Γ2 RSchwarzschildcoordinates radial RdΩ dLT T RdΩ/dtc Γ TSchwarzschildcoordinates tangential


Thenormalizedspeedoflightissimilartotheproperspeedoflightcinthesensethatbothareindependentoforientation radialor tangential . Theonlydifference is that thenormalizedspeedoflightusescoordinatetimewhichisanabsolutestandardfortherateoftimeandalsoisa faster rate of time compared to the proper rate of time in gravity stationary frame ofreference .Internally Self Consistent: Anotherimportantsimilaritybetweenthenormalizedspeedoflightandproperspeedoflightis:dL Cdt cdτThe above relationship indicates that the normalized coordinate system is internally selfconsistent.Bydefinition,thefollowingisalwaystrue:cdτ dL anyorientationorgravitationalГ .Soalsothefollowingisalwaystrue:Cdt dL anyorientationorgravitationalГ .Ingravity,thenormalizedspeedoflightisslow Cg c/Г .However,aunitoftimeingravityTgislongerthanthesameunitoftimeinzerogravity Tg ГTo .Thecombinationofthesetwofactorsoffseteachother,therebyproducingaconstantlength:CgTg c/Г ГTo Lo. ThecombinationofthesetwofactorsachievesthesamelengthinanyГororientation.ThereforeitispossibletosaythatLo Lgandhaveacoordinatesystemthatisinternallyconsistent.Energy Transformation: Weknowthetransformationofunitsoflength Lo Lg andtimeTo Tg/Г ,butweneed todetermine the transformation forunitsofmassbeforeall othertransformationscanbeeasilycalculated.Itisnotobviouswhattransformationmasswouldbeifweusedastandardunitofmassinzerogravity Mo toquantifythesameproperunitofmassingravity Mg .Massisnotsynonymouswithmatter.Forexample,oneelectroninzerogravitytransforms intooneelectron ingravity. However,do theyhave the same inertia? Mass isaquantificationofinertiawhichimpliesforceandacceleration.Bothoftheseinvolvetime,soitshouldbeexpectedthatmass inertia mayhavesomedependenceontherateoftime.Sincewecannot directly reason to themass transformation, it is necessary to determine some othertransformationbetweenzerogravityandgravitythatcanbedeterminedbyphysicalreasoning.Thenwewillusethattransformationtodeducethemasstransformationindirectly.Fortunately,thereare twoadditional transformations thatcanbedeterminedbyphysicalreasoning. Theconservationofmomentumimplies thataunitofmomentuminzerogravityequalsaunitofmomentumingravity.Thiswillbewrittenaspo pg.Thesecondtransformationthatcanbedeterminedfromphysicalreasoninginvolvesenergy.ItwillbeshownthatEo ГEg.Before actually deriving this, I just want to review the meaning of Eo ГEg. The term Eorepresentsaunitofenergy suchas1 Jouleor1eV ina locationwithzerogravity Г 1 .Similarly,Egrepresentsthesameunitofenergyinalocationwithgravity Г 1 .Furthermore,


weassumethatbothsourcesofenergycanbeconsideredessentiallystationaryrelativetoeachother.Thereforesincegammaisgreaterthanone Г 1 theequationEo ГEgsaysthat1Jouleinzerogravityrepresentsmoreenergythan1Jouleinalocationwithgravity Г 1 .TheratioofthesetwoenergiesisEo/Eg Г.The simplified energy transformation involves an approximation while the momentumtransformation does not. However, we will start with the energy transformation anywaybecauseitbetterillustratesthemeaningofterms.Tomakeacomparisonofenergyindifferentgravitationalpotentials,bothEoandEgmustbemeasuredusingthesamestandardofenergywhichimpliesusingthesamerateoftimeforbothmeasurements.Perhapsitisconvenienttoimagineusingthezerogravity coordinate rateoftimeforbothenergymeasurements,buttheonlyrequirementisthatthesamerateoftimebeused.Forexample,itwillbeshownthatanelectron in gravity has less energy than an electron in zero gravity when the energies arecomparedusingthesamerateoftime.Bothelectronshave511,000eVmeasuredlocally,buttheenergystandard 1eV changes.Theproportionalityconstantisthegravitationalgamma:

Thisconceptisbestexplainedwithathoughtexperiment.Supposethatthereisaplanetthatisinahighlyellipticalorbitaroundastar.Theplanet’skineticenergychangesfromaminimumkineticenergyattheorbitalapogeetoamaximumkineticenergyattheperigee.Doesthischangeinkineticenergyproduceanychangeinthegravityproducedbythecombinationoftheplanetand star as the planet orbits the star? Assume the measurement is made far from thestar/planet .Weknowfromgeneralrelativitythatthetotalgravityproducedbyaclosedsystemremainsconstantwhenthereisnotransferofenergyintooroutoftheclosedsystem. Sincegravityscaleswithenergy,theimplicationisthattheplanet’stotalenergyremainsconstantinallpartsofthehighlyellipticalorbit.Itisobviousthattheplanet’skineticenergychangesfromapogeetoperigee.Sincethegravityproducedbytheorbitingplanetisconstant,theimplicationisthatthegaininkineticenergyequalsthelossofinternalenergyastheplanetentersastrongergravitationalfield.Themoregeneralprincipleisthatabodyinfreefallmaintainsaconstanttotalenergy internalpluskinetic asitfalls.Thiscanbeexpressedas:Eo Eg Ekwhere:Eoistheinternalenergyofamassminzerogravity Eo moc2 measuredusingthezerogravitystandardofenergy.Egistheinternalenergyofamassingravitybutmeasuredusingthezerogravitystandardofenergy measuredusingthecoordinateclock .EkisthekineticenergyofmassminfreefallfrominfinitytodistancerinthegravityoflargermassM.AlsoEkismeasuredusingthezerogravitystandardofenergy.


Since the definition of Ek references zero gravity, then Ek GmoM/r Eo GM/c2r . In thefollowingcalculationwewillusetheapproximation:Eo GM/c2r Eg GM/c2r .Normallythisapproximationwould not be allowed sincewe are attempting to determine the relationshipbetweenEoandEg,howeverinthiscaseitisacceptablebecauseinweakgravity GM/c2r 1..Eo Eg EkEk Eg GM/c2r approximationfromaboveEo Eg 1 GM/c2r set:Г 1 GM/c2rEo ΓEgthisapproximationisexactinarigorousanalysisThe gravitational red/blue shift can cause some confusion in the discussion of standards ofenergyandwillbediscussedlater. Also, thisequationcaneasilybemisinterpretedifproperunitsofenergyareusedtomeasureEgratherthanalwaysusingnormalizedunitsofenergy.Mass Transformation from Energy: NextwewillsolveforthemasstransformationusingEo ГEg;Lo LgandTo Tg/Г.Energyhasunitsofkgm2/s2whichindimensionalanalysistermswillbeexpressedas:E→ML2/T2

Eo ГEgset:Eo→ andEg→

Г set:Lg LoandTg ГTo

ГГ

Mo Mg/ГunitsofmasstransformationobtainedfromtheenergytransformationAgain, both Mo and Mg represent the same units of stationary mass such as 1 kilogram.Furthermore, both units of mass are measured using a single rate of time. As suspectedpreviously,theconnectionbetweenmassandinertiameansthatthenormalizedunitofmassinertia hasadependenceontherateoftime Гdependence .The transformationMo Mg/Г looks strange because it says that the normalizedmass unitincreasesasgravityincreases asГincreases .Thiswillbeanalyzedlater,butitrelatestotheinertiameasuredbyazerogravityobserver.Mass Transformation from Momentum: Nextwewillcheckthemasstransformationbyalsoderivingthemasstransformationbyassumingtheconservationofmomentum po pg .Thisbecomes:


setLo LgandTo Tg/Γ

Mo Mg/ГNowthatwehaveestablishedthemasstransformation,wecancombinethiswiththelengthandtimetransformationstogeneratealloftheothertransformationsnaturerequirestomaintainthelawsofphysicswhentherateoftimechangesbecauseofachangeingravitationalpotential.Tosummarize,herearethekeytransformations:Lo LgunitoflengthtransformationTo Tg/ГunitoftimetransformationMo Mg/ГunitofmasstransformationAppendix B at the end of this chapter shows the details of derivation of the varioustransformations. Essentially this is justdimensionalanalysiswhere thedimensionsofmass,lengthandtimearetransformedfromzerogravitytogravity.Thefollowingtransformationofforceistypicaloftheothertransformations.

Force F:Fo→ Г →ГFg

Fo ГFgforcetransformation

Thetableonthefollowingpagegivesalloftheimportanttransformations.


GravitationalTransformationofUnitsandConstants:ThefollowingaretransformationsofunitsofphysicsfromzerogravityΓ 1toalocationingravityΓ 1.Therelationshipsareexpressedassuming a single rate of time andproper length. The gravitational gammaΓ is defined as:Г≡

2 1 Gm/c2R.ThesymbolsofdimensionalanalysisL,T,M,QandΘareused

torepresentlength,time,mass,chargeandtemperaturerespectively.NormalizedTransformationsLo LgunitoflengthtransformationTo Tg/ГunitoftimetransformationMo Mg/ГunitofmasstransformationQo Qgunitofchargeexpressedincoulombs–notstatcoulombsΘo ΘgunitoftemperaturetransformationCo ΓCgnormalizedspeedoflighttransformationdL ΓdRproperlengthandcircumferentialradiustransformationEo ΓEgenergyvo ΓvgvelocityFo ΓFgforcePo Γ2PgpowerGo Γ3GggravitationalconstantUo ΓUgenergydensityℙo Γℙgpressureωo Γωgfrequencyρo ρg/Гdensityko ΓkgBoltzmann’sconstantσo Γ2σgStefan‐BoltzmannConstanto Γ gelectricalcurrento Γ gvoltage

εoo εog/Гpermittivityofvacuumµoo µog/ГpermeabilityofvacuumUnitsandConstantsThatDoNotChangeinGravitypo pgmomentumisconservedo gangularmomentumisconservedħo ħgPlanck’sconstant angularmomentumisconserved αo αgfinestructureconstant dimensionlessconstantisconserved Ωo Ωgelectricalresistanceo gmagneticfluxdensity

Zoo ZogimpedanceoffreespaceZso ZsgimpedanceofspacetimeFundamentalEquationsEo Eg EkrelationshipofinternalenergyandgravitationalkineticenergyEkEo Eg–EporelationshipofinternalenergyandgravitationalpotentialenergyEpo


Conversion to Normalized Perspective: When we calculate energy, velocity, force, mass,power,voltage,etc.usingpropertime,becauseofthecovarianceofthelawsofphysics,weobtainananswerthatnumericallyequalsthevalueinzerogravity.Therefore,toconvertthispropervalue into normalized values, we merely substitute the proper value into the abovetransformationsbyreplacingthezerogravityterm Eo,Vo,Fo,etc. andsolveforthenormalizedvalue Eg, Vg, Fg, etc. . For example, an electron has energy of 8.187 10‐14 Joules. In thenormalizedperspective, this isreallytheenergyofanelectron inzerogravity. However, thecovarianceofthelawsofphysicsallowsustousethisenergyinlocationswithΓ 1.Toconverttonormalizedenergy,wemerelysubstitutethezerogravityenergy 8.187 10‐14Joules forEointheequationEg Eo/ΓandsolveforEg.SinceΓ 1foralocationingravity,thismeansthatEg Eo.Forexample,atthesurfaceofthesunГ 1 2 10‐6.Thereforeanelectronatthesurfaceofthesunhasonly0.999998theenergyofanelectroninzerogravity.

InsightsfromTransformationsEnergy Transformation and Calculation: Wenormally closely associatemassandenergy.However,thenormalizedtransformationstreatmassandenergydifferently.Whenanobjectismovedtoalocationwithalargergravitationalgamma andslowerrateoftime ,thenormalizedenergyoftheobjectdecreasesandthenormalizedmass inertia oftheobjectincreases.Themasstransformationisdiscussedbelow,butwhenwetransformE mc2intothenormalizedtimeperspective, thegravitationaleffecton thenormalizedspeedof light issquaredandthegravitationaleffectonmassisraisedonlytothefirstpower.Theresultis:Eo ГEg.Thisequationappliesonlytostationaryobjectsbecausethiswasanassumptionusedinthederivation.Theterm“stationary”meansnochangeasafunctionoftimeintheopticalpathlengthbetweentwoobjectsorpoints.TheequationEo ГEgsaysthataunitofenergyinzerogravityislargerthanthesameunitofenergyinalocationwithgravity.Thisappliestoallformsofenergysuchas:theannihilationenergyofmass,theenergystoredincapacitors,theenergyofchemicalreactions,thermalenergy,ortheenergyofatomictransitions.Inallcasesthenormalizedenergyofstationaryobjectsingravity is diminishedby the gravitational gamma factor. The loss of energywhen an objectmovestostrongergravityiseasytosee.Ameteorstrikingtheearthgeneratesheat.Thisheatisthelostinternalenergyofthemeteor.Theatomsofthemeteorthatremainintheearth’sgravityhavelessinternalenergythantheyhadinspace oncetheheatisremoved .Ifweelevateaonekilogrammassby1meterinearth’sgravity,wesaythatwehavegiventhemasspotentialenergyof9.8Joules.Whereisthisenergystored?Iwanttoseeandunderstandthismysteriousgravitationalpotential energy. Ifwe ignore the change in timeover theone


meterelevation,thenthesourceofgravitationalpotentialenergyisamystery.However,ifweacknowledgethattherateoftimeisdifferentwhenwechangeelevation,thenwearriveattheconclusionthattheenergyexpressedinnormalizedunitsofenergyisalsodifferentatthetwoelevations. This insight allowsus toobtainapartial insight into the storageofgravitationalpotential energy. The proposed particle model presented later will give a more completeexplanation. Below,wewillusethetransformationEo ГEgandcompareEgattwodifferentelevations 1and2whereelevation2ishigherthan1 .Wewilluseweakgravityapproximationsandthefollowingsymbols:Normalizedenergyingravityatelevation1and2:Eg1andEg2,Largemass planet andsmalltestmassMandmRadiusfromthecenterofthelargemasstoelevation1and2:r1andr2Gravitationalgammaandbetaforelevation1and2:Γ1,Γ2,β1,andβ2

Eg2 andEg1 normalizedenergytransformations

Eg2–Eg1 – Eo –

Г Eo β1–β2 set: 1–β

Eg2–Eg1 – mc2 – GMm

since – substitute: – –

Eg2–Eg1 r2–r1 msetr2–r1 Δ andacceleration g

Eg2–Eg1 gmΔ gravitationalpotentialenergyTherefore, thechange in thenormalizedenergybetween level1and2 Eg2–Eg1 equals thegravitationalpotentialenergyformassmingravitationalaccelerationgforaheightchangeofΔ r2–r1.Itisinterestingtodoanumericalexampleusingaonekilogrammassbeingelevatedbyonemeterintheearth’sgravity.M 5.976 1024kgmassoftheearthr1 6.378 106mradiusoftheearthΓ1 1 GM/c2r1 1 6.977 10‐10gravitationalgammaoftheearthatsealevelΓ2 Γ1 Г1–1 r2–r1 /r1 1 6.977 10‐10 1.091 10‐16Γ2–Γ1 1.091 10‐16differenceinΓwhenΔ r2–r1 1meterE2–E1 Γ2–Γ1 E1 1.091 10‐16 mc2mc2 8.99 1016Jfor1kgmassE2–E1 9.8Jdifferenceinnormalizedenergyofa1kgmasselevated1mTherefore,whenweusenormalizedunits coordinatetimeandproperlength ,wefindthattheinternalenergyofamasschangeswithelevationbyexactlytheamountofgravitationalpotentialenergy.Forexample,aonekilogrammasshas9.8Joulesmoreenergywhenitisatanelevation


1meterabovesealevelthanithaswhenitisatanelevationofsealevelwhenenergyisexpressedinnormalizedunitsofenergy.Thereisalsoachangeinthenormalizedspeedoflightandinthenormalizedmass.Thecombinationofthesefactorsresultsinachangeinthenormalizedenergyofamassthatiselevatedorloweredinagravitationalfield.TherelationshipbetweeninternalenergyinzerogravityEo,thenormalizedinternalenergyingravityEgandgravitationalpotentialenergyEφis:Eo Eg–EφTheminussigninfrontofEφistheresultofconsideringpotentialenergytobeanegativenumber.ThisequationleadstotheequationEo Eg Ekdiscussedabove.Abodyinfreefalldoesnotchangeitsnormalizedenergy.Previously,astarandplanetinanellipticalorbitwereusedinanexample.Sensingthegravityofthestar/planetcombinationbytheuseofadistantprobemassisequivalenttosensingthezerogravityenergyofthestar/planetcombination.Mass Transformation:ThenormalizedmasstransformationMo Mg/Гlookscounterintuitivebecause it indicates that normalizedmass increaseswhen gravitational gamma Γ increases.Thereisnoadditionalmatterbeingcreated,thereisjustachangeintheperceivedinertiawhenweconverttoasinglerateoftimeandquantifyunitsofinertiainlocationswithdifferentvaluesofГ.Theinertia mass exhibitedbyabodyisdefinedbytheforcegeneratedwhenabodyisaccelerated.Bothforceandaccelerationincorporateunitsoftimethereforemass inertia alsoexhibitsadependenceontherateof time. Acombinationof factorsresults inaunitofmassinertia ingravitybeinglargerthanthesameunitofmass inertia inzerogravityEventhoughnormalizedmassincreasesingravity,normalizedenergydecreasesasexplainedabove.Sinceitiseasiertoconceptuallyunderstandenergydecreasingingravity,perhapsitiseasier to imagineenergybeingmore fundamental thanmass inertia . It isonlyahistoricalaccidentthatweusemassasoneofthe5dimensionalunitsofphysics.Inparticlephysics,energyis consideredmore fundamental thanmass and units of eV orMeV are common. If energyreplacedmassasoneofthe5dimensionalunits,thentheenergytransformationwouldbethesinglefactorthatoffsetsthegravitationaleffectontime.Theweakequivalenceprinciplesaysthatthereisnodifferencebetweengravitationalmassandinertialmass.Thisistruebecausetheybothscaleproportionatelytothegravitationalgammawhenaconstantrateoftimeisused.Merelyelevatingamassintheearth’sgravitationalfieldchanges both the normalized gravitational mass and the normalized inertial mass. Thegravitationalfieldproducedbytheonekilogrammassscaleswithtotalenergy,soelevatingthismassincreasestheenergyoftheelevatedmass.Thisenergyincreaseexactlyoffsetsthedecreaseinenergyofwhatevermeanswasusedtoelevatethemass. Thetotalnormalizedenergyandtotalgravitationalfieldoftheearthisunchanged.


Gravitational Redshift: Thegravitationalred/blueshift isoftenmisinterpreted12. Aboveitwasshownthattheinternalenergyofanatomchangeswithelevationbyexactlythedifferenceinthegravitationalpotentialenergy.Notonlyisthereachangeintheinternalenergyofanatomwhenitchangeselevation,thereisalsoaproportionalchangeintheenergyoftheatom’senergylevelswhenthereisachangeinelevation.Therefore,fromtheperspectiveofsomeoneinzerogravity normalized time , a particular atomic transition in gravity is less energetic and thistransitionemitsacomparativelylowenergyandlowfrequencyphoton.Thislowenergyisnotdetectablelocallybecauseallenergycomparisons suchas1ev havebeensimilarlyshiftedtoalowerenergybythegravitationaleffectontimeandmass.Nowwewilladdressthegravitationalredshift.Theformulaforthered/blueshiftis:λo Гλgwhere:λg wavelengthingravityandλo wavelengthinzerogravitySupposeaphotoningravitystartsatalowerelevation level1 andendsatahigherelevationlevel2 .Ifthephoton’senergyismeasuredatlevel1and2withlocalinstruments,thenalossofenergyisobservedatlevel2.Thisredshiftappearstobeadecreaseinfrequency,adecreaseinenergyandanincreaseinwavelength.Ifitwaspossibletomeasurewavelength,frequencyandenergyfromasingleelevation singlerateoftime ,thenitwouldappearasiftherewasnochange in energy, no change in frequency, but the same increase in wavelength that wasobservedwithalocalmeasurement. Theenergyandfrequencydisagreementoccursbecausedifferentrateoftimeandenergyscalesarebeingusedatdifferentelevations.TheagreementinwavelengthoccursbecausethetransformationLo Lgsaysthatallobserversareusingthesamelengthstandardtomeasurewavelength.Ifwelookonlyatwavelength,thenthereissuchathingasgravitationalredshift.Thisisbecausefromallgravitationalpotentials, thesamechangeinwavelengthisobserved. However, ifwelookateitherenergyorfrequencyusingzerogravityrateoftime,thenthereisnosuchthingasgravitationalredshift.Theapparentchangeinfrequencyandenergyoccursbecausewemeasureenergyandfrequencyusing localstandardsof thebeginningandendingelevations. At thesedifferentelevations differentvaluesofΓ ,itisourlocalstandardsofenergyandtimethathavechanged,notthephoton’senergyandfrequency.ThereisaninterestingquestionthatIwouldliketopropose.Weknowthatthereappearstobea gravitational blue shift when a photon is generated at a high elevation height 2 and isanalyzed at a lower elevation height 1 . In otherwords, the photon seems to have gainedenergy.Thequestionis:Whatwouldhappenifwetrappedaphotoninareflectingboxatheight

1 L. B. Okun, K.G. Selivanov and V.L. Telegdi," On the Interpretation of the Redshift in a Static Gravitational Field", Am.J.Phys.68:115,2000; arXiv:physics/9907017. 2 L. B. Okun, "Photons and static gravity, Mod.Phys",Lett.A15:1941,2000; arXiv:hep-ph/0010120


2,thenloweredtheboxtoheight1?Wouldthephotoninaboxhavethesameenergyasafreelypropagatingphotonwhenitreachedheight1?Icontendthatthephoton lowered inaboxwouldappeartohave lessenergythanthe freelypropagatingphoton. Fromalocalperspective,thefreelypropagatingphotonappearstogainenergy blue shifted and thephoton inaboxwouldappear tohave thesameenergyas theenergy at height 1. From the perspective of a zero gravity observer, the freely propagatingphotonretaineditsoriginalenergywhenitpropagatedfromheights2to1andthephotoninabox lost energy. This lost energywas removed from the photon in the lowering process. Aspreviouslyexplained,aconfinedphotonexhibitsweight.Loweringaboxcontainingaconfinedphoton transfersenergy fromthephoton to theapparatusused to lower thebox. TherearenumerouswaystoanalyzethisproblemandIcontentthattheyallgivetheanswerdescribedhere.However,thisisalittleoffthesubject,soIwillnotelaboratefurther.Anotherquestionis:Howisitpossibleforthewavelengthtochangewithelevationifthereisnochangeinthenormalizedfrequency?Theansweristhatthenormalizedspeedoflightchangeswithelevation Co ГCg .Ifaphotonpropagatesfromalowerelevationtoahigherelevation,thereisnochangeinfrequency,butthenormalizedspeedoflightCgincreaseswithelevation.This increase in thenormalized speedof light increases thedistance traveledper cycle timeincreasesthewavelength . Thischange inwavelength isobvious fromany location,buttheconstantfrequencyisonlyobservablewhenallmeasurementsaremadefromasingleelevationasinglerateoftime .Finally,awordofcautionaboutnotusingtheredshiftformula λo Гλg intransformationsasasubstituteforlength.ItisnotcorrecttoequatewavelengthwiththeunitoflengthwhenthereisachangeinГ.Aphotongeneratedfromalocalatomhasawavelengththatisagoodstandardoflength.However,aphotongeneratedatanothergravitationalpotentialhasawavelengththatchangeswithΓ.Therefore,aphotonthatisnotgeneratedlocallycannotbeusedasastandardoflength.Electrical Charge Transformation: The transformation of electric charge needs specialexplanation.Thetransformationsof isonlycorrectifchargeisexpressedincoulombsratherthanstatcoulombs.Coulombsandstatcoulombsarefundamentallydifferent.TheMKSunitofcoulombis6.24 1018electronsbuttheCGSunitofstatcoulombisrelatedtoelectrostatic

force and has units of √ / . The result is that charge is conserved in gravitationaltransformations if charge is expressed as a number of electrons coulombs , but it is not

conservedifchargeisexpressedinstatcoulombswithunitsof√ / .IfanelectrostaticforceequationiswritteninCGSunits,thenεoismissingandthereisnonormalizationofpermittivity.Thefollowingtransformationmustbeusedforchargeexpressedinstatcoulombs:Qo √ГQg chargeinstatcoulombs


Testing:Theobjectiveofthesetransformationsistooffsetthegravitationaleffectontherateoftimeandkeepthelawsofphysicscovariantinanygravitationalpotential.Itisinterestingtomakesubstitutionsintovariousequationsofphysicsandseethatthetransformationsdoindeedkeepthesameequationsofphysicswhenthereisatransformationofgravitationalgamma.Thisissayingthatthesetransformationsexhibitinternalselfconsistency.However,itisalsopossibletoseetheimpliedphysicsbehindthesetransformationsuponcloseexamination.Shapiro Revisited: In the last chapterwe talked about the Shapiro experiment detecting arelativistic increase in the timerequired forradar totravelradially in thesun’sgravity. Thedelaywasequivalenttoabouta50μsdelayinthetimerequiredforlighttotravelonewayfromtheearthtothesun’ssurface.IfweuseSchwarzschildcoordinates,thenthe50μsdelayisdueentirelytotheslowingofthecoordinatespeedoflightwhichscalesaccordingtothelocalvalueofГ2alongtheradialpathbetweentheearthandthesun c Γ2 R Г2dR/dt .Howdoweinterpretthe50μsdelayusingthenormalizedcoordinates?Ifatapemeasurecouldbe used tomeasure the distance between the earth and the surface of the sun, the distancemeasuredbya tapemeasuredwouldbeabout7.5kmlonger thanthecircumferentialradiusdistancecalculatedbydividingthecircumferenceby2π.Thenormalizedcoordinatesystemusesproper length tape measure length as coordinate length. Therefore, the normalizedcoordinates gives the radar pulse credit for having traveled the additional 7.5 km ofnon‐Euclidiandistancebetweentheearthandthesun.Ittakesabout25μsforspeedoflighttraveltocover7.5km,sothenormalizedcoordinatesattributeshalfthe50μsdelaytothetimerequiredtotravelthenon‐Euclidiandistancegeneratedbythesun’sgravity.Theotherhalfisduetothenormalizedspeedoflightbeingslowedaccordingtoc ГCg dLR/dt.Integratingover the changing Г along the optical path gives the additional 25 μs due to this slowing.Therefore,the50μstotaldelayisthesame,buttheinterpretationisdifferent.The followingchapterswillprimarilyuse thestandarddefinition for thespeedof light. Thisstandarddefinitionwillbedesignatedbylowercase “c”.Occasionallywewillswitchintousingnormalizedspeedoflighttogiveanotherperspective.Inthiscasewewilluseanuppercase“C”.Attentionwillbecalledtothischange.


AppendixB

This appendix gives the details of the derivation of some of the additional transformationsenumerated in the table titled “Gravitational Transformations of Units and Constants”. Thisappendixcanbeskippedwithoutthelossofanyimportantinformationtothemainpointsofthisbookifthisbackupinformationisnotofinteresttothereader.

Velocityv:vo→ Г→Гvg

vo Гvgnormalizedvelocityvgdecreasesingravity justlikeC

GravitationalConstantG:Go→ Г Г

→Γ3Gg

Go Γ3Ggnormalizedgravitationalconstant seecommentbelow

EnergyDensityU:Uo→ Г

Г

→ГUg

Uo ГUgnormalizedenergydensityElectricalCharge:NextwecometotransformationsthathavedimensionsthatincludeelectricalchargeinCoulombs.Theseincludepermittivityεo,permeabilityμo,currentI,VoltageVandtheimpedanceoffreespaceZo.Forthisexercise,thesymbolεoowillrepresentεoinzerogravityandεogwillrepresentεoingravity.Similarlywewillusethesymbolsμoo,μog,Zoo,andZog.AunitofchargewillberepresentedbyQoandQg.It isnotpossible touse theabovesubstitutions todetermine the transformationof aunitofelectricalchargewhencomparingchangebetweenzerogravityQoandaunitofchargeingravityQgusingasinglerateoftime.Itistruethatthedimensionofcharge expressedinCoulombs doesnotcontaineithertimeormass,sothetwodimensionsknowntohaveΓdependencearemissing statCoulombswillbediscussedlater .SosuperficiallyitseemsasifthereshouldbenochangeinaunitofchargewhentherateoftimechangesduetoachangeinΓ.However,weneedtosupplementthiswithsomeadditionalphysicalreasoningusingthelawsofphysics.FromtheconservationofchargeandtheFaradaylawweknowthatchargeisconservedwhenthereisachangeinelevation.ThisindicatesthatQo Qg.ThereisadditionalsupportforthiscontentionbecausetheimpedanceoffreespaceZohasunitsthatscalewith1/Q2 dimensionalanalysissymbolQ .Iftherewasagravitationaldependenceoncharge,thentheimpedanceof


freespacewouldhaveagravitationaldependence.Therewouldbeaslightimpedancemismatchwhenlightchangeselevationingravity. Thisimpedancemismatchwouldcausescatteringofelectromagneticradiationfromgravitationalfields.Forallthesereasonswewillassumethat:Qo QgAunitofelectricalchargeingravity Coulombs equalsaunitofelectricalchargeinzerogravity.Wewillnowusethistransformationtogenerateadditionalelectricaltransformations.

Permittivity εo:εoo→Г

→εog/Г

εoo εog/Гnormalizedpermittivity

Impedance of Free Space Zo: Zoo→ Г

Г

→Zog

Zoo Zogimpedanceoffreespace

Voltage V:Vo→ Г

Г

→ГVg

Temperature Θ: Finally we come to the transformation of Boltzmann’s constant k andtemperature. Boltzmann’s constant is typically described as: k 1.38 10‐23 Joule/moleculeOKelvin.Thisnumber,measuredlocally,doesnotchangewhengravitationalgammaischanged.However, the standardofwhat constitutesaunit of energy Joule changesaccording to thepreviouslyderivedtransformation:Eg Eo/Γ.Therefore,toanobserverusingnormalizedtime,the energy per molecule per degree Kelvin decreases in gravity. Therefore, an acceptableinterpretation for the zero gravity observer is that temperature in gravity equals the sametemperatureinzerogravity,buttheBoltzmann“constant”dependsonΓ.Θo ΘgtemperatureisunaffectedbyanchangeingravityBoltzmann Constant: kB→ML2/T2ΘkB 1.3810‐23Joule/moleculeOKelvin→ML2/T2Θmolecule

kBo→ molecule Г

Г

molecule→ГkBg

kBo ГkBgnormalizedBoltzmann’s“constant”


Stefan-Boltzmann Constant: σo→ Г

Г

→σg/Γ2

σo Γ2σgnormalizedStefan‐Boltzmann“constant”The Stefan‐Boltzmann constant is the constant associatedwith the intensity of a blackbodyemission σεT4whereσ Stefan‐BoltzmannConstant,ε emissivity,TistemperatureandΘisthedimensionoftemperature.Theequation σεT4supportstheideathatΘg Θobecausetemperature is raised to the fourthpower. If therewasa temperaturedependenceonΓ,wewouldhaveaΓ4dependence.


Chapter4

Assumptions

“Therehasneverbeenalawofphysicsthatdidnotdemand

‘space’and‘time’foritsstatement.”JohnArchibaldWheeler

Starting Assumption: Physics today has a large body of experimental observations andmathematical equations that correspond to the experimental observations. Therefore, on asuperficiallevelitwouldappearthatwehaveagoodtheoreticalunderstandingofnatureuptoalimitthatwillbecalledthefrontierofknowledge.However,therearemanycounterintuitivephysical interpretations of themathematical equations and experimental observations. Thisbook proposes alternative physical interpretations that not only fit the equations andexperiments,butalsoofferimprovedconceptualunderstandingandnewinsights.Ifwe are looking for the fabled “theory of everything", it is best to start the questwith thesimplest possible starting assumption. Only if the simplest assumption is proven to beinadequate,shouldwereluctantlymoveontoamorecomplexassumption.WhenIexaminedthesimilaritiesbetweenlightconfinedinareflectingboxandparticles,Iwasstruckbyabigidea.Thisideais:BasicAssumption:The universe is only spacetime.This ideawill be taken as a basic assumption for the remainder of this book. If this simplestartingassumptioniscorrect,itshouldbepossibletoinventamodeloftheuniversethatusesonlythepropertiesof4dimensionalspacetime.Ultimately,allmatter,energy,forces,fieldsandlawsofphysicsshouldlogicallybeobtainablefromjust4dimensionalspacetime.Thisisalargeprojectthatencompassesallofphysics.Itgrewintothisbooklengthexplanationratherthanafewtechnicalpapers.Initially,thismightseemimpossiblebecausespacetimeappearstobejustaquietvacuumthatpossessesthreespatialdimensionsplustime.However,thequantummechanicalmodelofthevacuumhasavastenergydensityalsoknownasvacuumenergy,quantum fluctuations, zeropointenergy,etc.AsJohnArchibaldWheelersaid“Emptyspaceisnotempty…Thedensityoffieldfluctuationenergyinthevacuumarguesthatelementaryparticlesrepresentpercentage‐wisealmostcompletelynegligiblechangeinthelocallyviolentconditionsthatcharacterizethevacuum.”Thespacetimemodelthatiscapableofformingmatter,energy,forcesandfieldsisacompositeofthequantummechanicalvacuumonthemicroscopicscaleandgeneralrelativisticcharacteristicsofspacetimeonthemacroscopicscale. Besidesaspecificspeedof lightandagravitationalconstant,spacetimealsopossessesimpedance,bulkmodulus,energydensityetc.


Thecombinationofthesepropertiespermitsspacetimetobecomethebasicbuildingblockforallmatterandforces.Spacetime does have some real advantages as the basic building block of everything in theuniverse. Spacetimeisthestiffestofallpossiblemediumsthatsupportwavepropagation. Adisturbance in spacetime propagates at the speed of light. The characteristics of spacetimepermit it to support any frequency wave up to Planck frequency ~1043 Hz . This is atremendousadvantageifweareattemptingtofindamediumthatcanhypotheticallysupportthelargeenergydensityrequiredtobuildaproton,forexample.Somewavesinspacetimewillbeshowntobecapableofmodulatingthespatialandtemporalpropertiesofspacetime.Thiscanserveasthebasicbuildingblockofmatter,forcesandfields.Thesimplicityofthestartingassumptiondoeshaveoneadvantage.Itshouldberelativelyeasytoproveordisprove.Unlikestringtheory,thisstartingassumptionhardlyprovidesany“wiggleroom”. If theassumptioniswrong,theerrorshouldbequicklyevident. If theassumptioniscorrect,theextremelimitationsdefineanarrowpaththatshouldleadtobothconformationsandnewinsights.Iwillsummarizetheconclusionsofchapter#1.

Aconfinedphotoninamovingframeofreferencehasthefollowing8similaritiestoafundamentalparticlewiththesameenergyandsameframeofreference:1 thesameinertia,2 thesameweight,3 thesamekineticenergywhenmoving4 thesamedeBrogliewavelength 5 the samedeBroglie phase velocity, 6 the samedeBroglie group velocity, 7 the same relativistic length contraction, 8 the samerelativistic time dilation. It is hard to avoid the thought that perhaps a particle isactuallyawavewithcomponentsexhibitingbidirectionalpropagationatthespeedoflightbutsomehowconfinedtoaspecificvolume.Thisconfinementproducesstandingwavesthataresimultaneouslymovingbothtowardsandawayfromacentralregion.

The assumption that theuniverse is only spacetime causesus to explore thepossibility thatwavesinspacetime dynamicallycurvedspacetime arethebasicbuildingblocksofparticles,forces and fields. It also offers the opportunity to give a physical description of quantummechanicaloperationssuchasthecollapseofthewavefunctionormakingameasurementofaquantummechanicalstate.Theultimatetestiswhetherthisassumptionlogicallyleadstogravityandcompatibilitywithquantummechanics.The starting assumption that the universe is only spacetime requires that the reader change perspective about what is a cause and what is an effect. The standard physical interpretation of general relativity is that matter causes curved spacetime. The reverse perspective would be that curved spacetime causes matter. In this perspective spacetime is the single universal field responsible for everything in the universe including matter. Waves in spacetime can be


thought of as dynamically curved spacetime. Static curved spacetime and particles will both be shown to be the result of dynamically curved spacetime.Creative Challenge: Istartedwithtwopositionsthatarecurrentlynotconnected.Ontheonehand,thereisthecurrentunderstandingofthefundamentalparticles,forcesandphysicallaws.Ontheotherhand,thereisthebasicassumptionthattheuniverseisonlyspacetime.Anattemptto bring these two disconnected positions together requires a creative look at both theproperties of spacetime and the properties of particles, forces and physical laws. Theexperimentally verified physical facts and equations are assumed to be correct, but it is notnecessary to adopt the physical interpretations currently used to explain these facts andequations.Forexample,theequationsofgeneralrelativityaccuratelydescribegravityandtheuniverse. However, the accuracy of these equations does not guarantee the accuracy of thephysicalinterpretationscurrentlyassociatedwiththeseequations.Theideathatgravityisnotaforce,buttheresultofthegeometryofspacetimeisaphysicalinterpretationoftheequationsofgeneralrelativity.Ratherthanfocusingonexplaininggravityorunitingquantummechanicsand general relativity, I merely start with what I believe is the simplest possible startingassumption the universe is only spacetime and attempt to reconcile this assumptionwitheverythingknowntoexistintheuniverse.Planck Units:In1899MaxPlanckproposedasystemofunitsbasedonlyonconstantsofnature.HeusedthereducedPlanckconstantħ 1.055 10‐34kgm2/s, theNewtoniangravitationalconstantG 6.672 10‐11m3/s2kg,thespeedoflightc 2.998 108m/sandtheCoulombconstant1/4πεo 8.987 109m3kg/s2C2.Thiscombinationofconstantscanbeusedtomakeunitsoflength,time,mass,andcharge.Extrapolatingfurther,itispossibletomakeallotherunitssuchasunitsofforce,energy,momentum,power,electricfield,etc.ItwasimmediatelyrecognizedthatPlanckunitswerefundamentalbecausetheywerederivedfromconstantsofnature.However,itlaterbecamerecognizedthatPlanckunitsheldanevenmorespecialplaceinphysics.Planckunitswereactuallybasedonthepropertiesofspacetimeandtheyrepresentedthelimitingvalues maximumorminimum forasinglefermionorboson.Forexample,agroupofparticlescanhavemassinexcessofPlanckmass mp 2.176 10‐8kg .However,thetheoreticallimitforasinglefundamentalparticle asinglefermion isPlanckmass.IftherewasafermionwithPlanckmassitwouldformablackhole.TheinverseofPlancktimeisthemaximumpossiblefrequencyforaphoton.Plancklengthlp 1.616 10‐35mrepresentsthe theoretical limiting value device independent of any length measurement. Similarly,Plancktimetp 5.301 10‐44srepresentsthelimitingaccuracyofanytimemeasurement.Whenweareattemptingtobuildtheuniverseoutofonlyspacetime,Planckunitsbecomethenaturalunitsforthisanalysis.


SpacetimeModelsSpacetime: The Quantum Mechanical Model: Quantum mechanics does not specifically have a model of spacetime. However, quantum mechanics does describe the properties of the energetic vacuum which we will be defining as the properties of spacetime on the microscopic scale. For comparison, the general relativity description of spacetime will be characterized as the macroscopic properties of spacetime. Thequantummechanicalviewofthevacuum includingQEDandQCD isalocallyviolentmediumfilledwithvacuumfluctuations.Atthebasisoftheuncertainty principle there is energetic spacetime that is in a continuous state of flux. Thedistancebetweentwopointscanonlybespecifiedtoalimitedaccuracy Plancklength becauseof the effect of these fluctuations on spatial measurement. Similarly the energy at a pointundergoeswild fluctuations. This isusuallyconsideredas justificationforthe formationandannihilationofvirtualparticlepairs,butthisenergyfluctuationcanalsobeconsideredmerelyanenergeticdistortionofspacetime.Eventherateoftimeatadjacentpointsin“flat”spacetimenogravitationalacceleration canfluctuateslightlyproducingvariations differencesbetweenclocks thatcandifferbyPlancktime.Thesefluctuationsproducemeasurableresults.Forexample,theCasimireffectproducesaforceon two closely spacedmetal plateswhich have beenmeasured to an accuracy of 5% of thetheoreticalprediction.Inahydrogenatomthereisaninteractionbetweentheelectronandthevacuum fluctuations thatproducesa small shift in theenergyof the 2S1/2 energy level. This“Lambshift”hasbeenaccuratelypredictedandexperimentallymeasured.InQED,thevacuumfluctuationscanmixwiththeexcitedstatesofanatomresultingintheinitiationofspontaneousemissionofaphoton fromtheatom1 2.Also, thesevacuum fluctuationscanproducevacuumpolarizationwhichchangesCoulomb’slawnearanelectricallychargedparticle.

An electron has a magnetic moment which would be precisely equal to 2 except for theanomalousmagneticdipolemomentcausedbyvacuumfluctuations.Thiselectronsping‐factorgs has been predicted by QED calculations and experimentally verified to better than 10significant figures for the anomalous contribution. The result is: gs 2.00231930436. Thismeansthatthemagneticmomentofanelectronisthemostaccuratepredictioninallofphysics.Thisaccuracydependsontheaccuracyofthequantummechanicalmodelofthefluctuationsinspacetime.Alloftheseexamplesaremeanttoillustratethatquantummechanicsrequiresthatvacuumhasvacuumfluctuationsataverylargeenergydensity.

1 Hiroyuki Yokoyama & Ujihara K (1995). Spontaneous emission and laser oscillation in microcavities. Boca Raton: CRC Press. p. 6. ISBN 0-8493-3786-0. 2 Marian O Scully & M. Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. p. §1.5.2 pp. 22–23. ISBN 0-521-43595-1


Onewayofquantifyingthesefluctuationsistovisualizeavacuumasbeingfilledwithharmonicoscillatorsatatemperatureofabsolutezero.Fromfieldtheory,thelowestquantummechanicalenergyofeachoscillator3isE ½ħω ½ħc/λ wherec ωλandλispronouncedlambdabar .Thisisthefamouszeropointenergy.EachoscillatorcanbevisualizedasoccupyingavolumeofV kλ3wherekisanumericalfactornear1.Forexample,awavecanbeconfinedinareflectingcavitythatis½wavelengthonaside.Thewaveamplitudeiszeroatthewallsandmaximuminthecenterforthissizecavity.Sincewearestandardizingontheuseofωforfrequencyandλforwavelength,itispossibletosaythatthewavehasbeenconfinedtoavolumeofλ3ifweignorenumericalfactorsnear1.Usingthisvolumedesignation,thismeansthattheenergydensityUatfrequencyωis:Uω ħω/λ3 ħc/λ4 ħω4/c3.Whatisthetotalenergydensityofzeropointenergyatallfrequenciesuptoacutofffrequencyofω?Whenall frequenciesarepresentwiththecharacteristicsoftheharmonicoscillatorsofspacetime,Milonni previousreference hasshownthatthereareinteractionswhichresultonthetotalenergydensityisequaltoUt ħω/λ3 ħω4/c3.Thisisthesameasbeforeexceptthatnowthefrequencyrangesfromzerotoamaximumofω.Ifωisassumedtohavenolimit infinitefrequency thentheimpliedenergydensitywouldalsobeinfinite.Ifweassumethatzeropointenergy is associated with the properties of spacetime proven later then the maximumfrequencythatspacetimecansupportisPlanckfrequencyωpwhichistheinverseofPlancktimeωp 1/Tp c5/ħG 1/2 1.85 1043s‐1. AssumingPlanck frequencyωp, the impliedenergydensityofthequantummechanicalmodelofspacetimeisapproximatelyequaltoPlanckenergydensityUp ħωp4/c3 4 10113J/m3. anumericalconstantisbeingignored .Thisshockinglargenumberwillbeextensivelyanalyzedseveraldifferentplaceslaterinthisbook.Fornowwewillmerelyrecognizethatthisispartofthequantummechanicalmodelofspacetime.Spectral Energy Density:Thenormalwayoftreatingenergydensityataparticularfrequencyistodesignatethe“spectralenergydensity”whichisenergydensityperunitfrequencyinterval.Wewilldesignatethisspectralenergydensityas:U ω dω.EverypointinspacetimeistreatedlikeitisaquantizedharmonicoscillatorwithenergyE ½ħω.ThisconceptleadstoaspectralenergydensityU ω dωthatis:

U ω dω kħ

dω

“Thisspectrumwithitsω3dependenceofspectralenergydensityisuniqueinasmuchasmotionthroughthisspectraldistributiondoesnotproduceadetectableDopplershift. ItisaLorentzinvariantrandomfield.Allinertialobserversareequivalent.Anyparticularspectralcomponentundergoes a Doppler shift, but other components compensate so that all components takentogetherdonotexhibitaDopplershift.Thereforethisspectralenergydistributionsatisfiesthe

3 Milonni, P. W.: The quantum vacuum: an introduction to quantum electrodynamics. p. 49 Academic Press Inc., San Diego (1994)


requirementthatitshouldnotbepossibletodetectanydifferenceinthelawsofphysicsinanyframeofreference atleastuptothecutoffatPlanckfrequency .Itshouldalsobenotedthatneithercosmologicalexpansionnorgravityaltersthisspectrum”4.Theimplicationsofhavingafinitecutofffrequencyarediscussedaspartofthecosmologicalanalysisinchapter14.Quantum Foam:In1955,JohnWheelerproposedthatspacetimeishighlyturbulentatthescaleofPlancklength.HeproposedthatasthescaleoftimeandlengthapproachesPlancktimeandPlancklength,theenergyfluctuationsinspacetimeincrease.Thesefluctuationsonthesmallestscale possible cause spacetime to depart from its smooth macroscopic characteristic. JohnWheelersuggestedtheterm“quantumfoam”todescribespacetimeonthissmallestscale.Inthebook“Einstein'sVision”,JohnWheelerproposedthatelementaryparticleswereexcitedenergystates resonances of the vacuum energy fluctuations. He pointed out that the density of anucleuswas~1018kg/m3andthisdensityisnegligiblysmallcomparedtotheequivalentdensityof spacetime ~ 1097 kg/m3 or an energy density of ~ 10113 J/m3 . While John Wheeler’sdescriptionofspacetimehasthesamebasiccomponentsasthespacetimemodelproposedinthisbook,hisconceptofhowspacetimeformsparticlesandforcesisdifferent.Tosummarize,thequantummechanicalmodelofvacuum,spacetimeisaseaofenergeticactivitythat can be visualized several different ways. The uncertainty principle has distance,momentum, time and energy undergoing fluctuations. Field theory has a sea of harmonicoscillators,eachwithzeropointenergyofE ½ħω.Particlephysicshasvirtualparticlepairsandvirtualphotonscomingintoexistenceandgoingoutofexistence.QCDhasvirtualparticleswith both color charge and electrical charge producing vacuum polarization. The quantummechanicalmodelofvacuumrequiresaminimumvacuumenergydensityofatleast1050J/m3for many QCD calculations and some variations require the full Planck energy density of~10113J/m3.Spacetime: The General Relativity Model: We will be viewing the general relativitydescriptionofspacetimeasdescribingonlythemacroscopicpropertiesofspacetime.Generalrelativity visualizes spacetime as a smooth, well behaved medium consisting of 3 spatialdimensionsplustime.Spacetimecanbecurvedbyenergyinanyformbutitisnotsubjecttotherandomfluctuationsofthequantummechanicalmodel.GeneralrelativityisaclassicaltheorythatdoesnotrecognizePlanck’sconstant,PlancklengthorPlancktime.Thedistancebetweentwocloselyspacedpointsisnotconsideredtofluctuatebutthereisalimitastotheprecisionofthemeasurement.Thisprecisionlimitissetbythepossibilityofformingablackholeiftoomuchenergyisrequiredtomakethemeasurement.However,eventhislimitisamixtureofquantummechanicsandgeneralrelativity.Ingeneralrelativitytherearenoquantizedoperations.

4 Puthoff, H.E. Phys. Rev. A Volume 40, p.4857, 1989 Errata in Phys. Rev A volume 44, p. 3385, 1991 See also New Scientist, volume 124, p.36, Dec. 2, 1989


Generalrelativity GR teachesthatenergyinanyformgeneratesgravity.Accordingtogeneralrelativity theuniversewould collapse into ablackhole if the energydensity of theuniverseexceedsthe“critical”energydensity.Accordingtocosmologicalobservation,spacetimeis“flat”on the largescaleexceedingabout½billion lightyears. Theobservedenergydensityof theuniverse about10‐9J/m3 appearstobewithinthemarginoferror within1% ofequalingthecriticalenergydensityoftheuniverse.Therefore,accordingtogeneralrelativitythequantummechanicalmodelofvacuummustbewrongbecausethequantummechanicalmodelrequiresenergydensityvastlyexceeding the criticaldensityofabout10‐9 J/m3. According togeneralrelativity,anenergydensityof10113J/m3isridiculous.ThisenergydensitywouldformablackholeevenforaspherethatisPlancklengthinradius.General relativitydoeshave its shareofpredictionsnot sharedbyquantummechanics. Forexample,therateoftimedependsongravityinGRwhilequantummechanicsconsiderstherateoftimetobeconstant.Also,GRpredictsthatpropervolumealsoisaffectedbygravity.Quantummechanicsdoesnotrecognizeagravitationaleffectonvolume.Reconciling the QM and GR Models: Thediscrepancy between the quantummechanicalenergydensityofvacuum ~10113J/m3 andthecosmologicallyobservedenergydensityoftheuniverse ~10‐9J/m3 isthelargestnumericaldiscrepancyinallofphysics.Thedifferenceisafactorofabout10122butthisisusuallyroundedoffto“merely”afactorof10120.Thestandardinterpretation is that theremustbe someother effect that cancelsoutwhat appears tobearidiculouslylargequantummechanicalenergydensity.However,thereisgoodevidencethatthevacuum fluctuations exist. They are required formany currentquantummechanical effects.Theycannotsimplybecanceledbyanothereffectthatsomehoweliminatesalltheeffectsofthesefluctuations.Furthermore,cancelingout10113J/m3wouldrequireanequallylargeeffectintheoppositedirection.Noeffectthatcancels10113J/m3hasbeenproposed.Oncloseexaminationwereallydonotneedatruecancelationofenergy.Wemerelyneedoneormoremechanismsthatallowthequantummechanicalvacuumenergytoexistbutnotinteractwith us or our observable universe except through the quantum mechanical interactionsmentioned.ItwillbeproposedlaterthatthequantummechanicalmodelofspacetimeiscorrectregardingtheenergydensityofspacetimeatthequantumscaleofPlancklengthandPlancktime.Also, thegeneralrelativitymodel iscorrectregardingtheenergydensityofspacetimeonthemacroscopicscalethatdoesnotrecognizefluctuationsatthescaleofPlancklengthandPlancktime. Since theGRpredictionsarevirtuallyuniversallyaccepted,wewill concentrateon theenergy density predictions of quantum mechanics which are generally presumed to beeliminatedbysomeunknownoffsettingpropertyofspacetime.Thisseemsobvioussincewedonotmacroscopicallyinteractwiththistremendousenergydensitynorhasitcausedtheuniversetocollapseasimpliedbygeneralrelativity.


It is proposed here that spacetime is a composite of the quantum mechanical model and the general relativity model. The quantum mechanical model of spacetime has quantum fluctuations and tremendous energy density. It is describing undetectable waves in spacetime that lack quantized angular momentum and are as homogeneous as quantum mechanics allows. Usually we ignore this by renormalization because it is a uniform background energy density of the vacuum. The general relativity model only recognizes the small portion of the energy in the universe that possesses quantized angular momentum (fermions and bosons). This portion is capable of forming energy concentrations such as massive bodies which distort the macroscopic homogeneity of the quantum mechanical model to form curved spacetime.Therestofthisbookisdevotedtoexplainingvariousaspectsoftheabovestatement.Thefactorof10120discrepancybetweenthetwomodelsisthedifferencebetweentheminutefractionofenergy particles,photons,etc. thatpossessesquantizedangularmomentumandthevastlylargervacuumenergydensitythatdoesnotpossessquantizedangularmomentum.Thisvacuumenergywilllaterbeshowntobeahomogeneoussuperfluidandhaveotherpropertiesthatpreventgravitationalcollapse.Thesestatementsareonlymadetoalertthereaderthattheobviousobjectionswillbeaddressedlater.Anotherobjection addressed later is the contention that the largeenergydensity of vacuumfluctuations is impossible because the volume of the universe is expanding yet the vacuumenergydensityisperceivedtoremainconstant.Thisseemstoimplythatavastamountofnewenergyisbeingaddedtotheuniverseeachsecondtoaccompanythenewvolumebeingcreated.Thisanswerrequirestwochapters 13&14 foracompleteexplanation,butakeypointinthisexplanationisthatspacetimeisundergoingatransformationthatstartedattheBigBangandcontinues today. The expansion of the proper volume of the universe is one result of thistransformation.Anotherresultisthatourstandardofaunitofenergyisshrinking.Adecreasingstandardwillmakeaconstantamountofenergyonanabsolutescaletoappeartogrowonourshrinkingscale.Newenergyisnotbeingaddedtotheuniversebutthepropertiesofthevacuumfluctuationsarechanging.Thiswillbeexplainedinmoredetailinchapters13and14.Vacuum Fluctuations and Vacuum Energy: Intheremainderofthisbooktheterms“vacuumenergy”and“vacuumfluctuations”willbeusedinterchangeably.Inbothcasestheyrefertothequantummechanicalmodelofspacetimewithenergydensityofabout10113J/m3.Forexample,“vacuumfluctuations”impliesquantummechanicalfluctuationsofthevacuum fluctuationsofspacetime whichwillbedescribedinthenextsectionofthisbook.Suchfluctuationsalsocanbedescribedas“vacuumenergy”.Thereasonforusingtwodifferenttermsisbecausesometimesitisdesirabletoemphasizetheenergycharacteristicsandsometimesitisdesirabletoemphasizethefluctuationcharacteristic.


Inthisbooktheterms“vacuumenergy”and“vacuumfluctuations”willneverimplydarkenergyor thecosmologicalconstant. Theseareconcepts fromcosmology that implyavastly lowerenergydensityandadifferentexplanation.Darkenergywillbediscussedinchapters13and14.A model of the universe will be presented that is based on spacetime undergoing atransformationthatproducestheobservedincreaseinpropervolumeandotherobservationswithouttheneedofdarkenergy.

DipoleWavesinSpacetimeThusfarwehavetalkedaboutvacuumfluctuationsandinferredthatthesecanbeconsideredwavesinspacetime.Nowitistimetobemorespecificaboutthepropertiesofthesewavesinspacetime.Sincethestartingassumptionofthisbookisthattheuniverseisonlyspacetime,thegoal is tosee if it ispossible toprove thatallparticles, fieldsand forcesare formedoutof4dimensionalspacetime.Acriticalstepisto“invent”amodelofwavesinspacetimethatcouldpossibly be the universal building block of all particles and forces. Once such a model ispostulated,itmustbetestedtoseeifitactuallycorrespondstoreality.Iffundamentalparticlesareultimatelyconfinedwavesinspacetime,itisnecessarytolookforanexplanationthatincorporateswavesinspacetimewithcharacteristicsthatcanbethebasicbuilding block for all matter and forces. Gravitational waves do not have the necessarypropertiestobebothvacuumfluctuationsandthebasicbuildingblockofallparticlesandforces.Weare looking forawave inspacetimethatchangesboththerateof time distorts the timedimension andchangesthedistancebetweenpointsinawaythatchangespropervolume.Weknowfromgeneralrelativitythatmassaffectsboththerateoftimeandpropervolume masscurvesspacetime .Therefore, if we are trying to build matter out of waves in spacetime, we must use waves in spacetime that possess the ability to affect both the rate of time and the distance between two points. Wemustusewavesthathavetheabilitytodynamicallycurvespacetime.Theonlywaveinspacetimethatcanaffecttherateoftimeandpropervolumeisahypotheticaldipolewaveinspacetime.The immediateproblem is thatdipolewaves in spacetimeare forbiddenon themacroscopicscaleaddressedbygeneralrelativity.Instandardtextsongeneralrelativitythesubjectofdipolewaveswarrants just a briefmention because they are considered impossible. For example,perhaps the most authoritative text on general relativity is the 1300 page tome titled“Gravitation”byCharlesMisner,KipThorneandJohnArchibaldWheeler.Onpage975ofthe24thprinting,dipolewavesinspacetimereceiveathreelinementiontotheeffectthattherecanbenomassdipoleradiationbecausethesecondtimederivativeofthemassdipoleiszero d p 0 .Thisconclusionultimatelyfollowsfromtheconservationofmomentum.Thegenerationofdipolewavesinspacetimewouldrequirethecenterofmassofaclosedsystemtoacceleratein violation of the conservation ofmomentum. Furthermore, if a dipolewave in spacetimesomehowexisted,thepassageofthiswavepastanelectricallyneutral,isolatedmasswouldcause


the center of mass to undergo an oscillating displacement which is also a violation of theconservationofmomentum.Clearly,dipolewavesinspacetimecannotexistonthemacroscopicscalegovernedbygeneralrelativity.However, if we are exploring the possibility of constructing the entire universe out of 4dimensionalspacetime,dipolewavesinspacetimehavealotofappealanddeserveacloserlook.Matteraffectsboththerateoftimeandpropervolume mattercurvesspacetime .Therefore,dipole waves in spacetime need to be considered as the spacetimewave building block formatter.Arethereanyconditionswheredipolewavesinspacetimewouldbepermitted?Theanswerisyesprovidedthatthedipolewavesconformtoaseverelimitation.Thefollowingisthesecondkeyassumptionofthisbook:SecondAssumption: Dipole waves in spacetime are permitted by the uncertainty principle provided that the displacement of spacetime caused by the dipole wave does not exceed Planck length or Planck time. This restriction will be called the “Planck length/time limitation”.The spacetime basedmodel proposes that dipolewaves in spacetime can exist on the scalegovernedbyquantummechanics.Thisistosaythattherearedisplacementsofspacetimethataresosmallthatthedisplacementsarebelowthequantummechanicaldetectablelimitsetbytheuncertaintyprinciple.Thisisanalogoustothereasoningthatpermitsvirtualparticlepairstotemporarilyexistprovidedthattheyarepermittedbytheuncertaintyprinciple.ItisproposedthatdipolewavesinspacetimecanexistindefinitelyprovidedthatthespatialdisplacementofspacetimedoesnotexceedPlancklengthandthetemporaldisplacementofspacetimedoesnotexceedPlancktime.Thesesmalldisplacementswouldbeundetectableevenwithaninfinitelylongobservationtime. Thesedipolewavesare thebackground“noise”of thevacuum. Theyactuallyarethecauseoftheuncertaintyprinciple.Asuperficialanalysisoftheminimumdetectablechangeinlengthmeasuredoveralongtimewould use the uncertainty principle equation: ΔxΔp ½ħ and substitute for Δp the largestpossible value of momentum uncertainty for a single quantized unit. This would be PlanckmomentumwhichisthemomentumofahypotheticalphotonwithPlanckenergy. Usingthismaximummomentumforasinglequantizedunit,theminimumvalueofΔxis:Δx Lp Plancklength .However,thisquestionoftheminimumlengthmeasurementhasbeengivenamorerigorousexaminationthatincludesbothquantummechanicsandgeneralrelativity.Theconclusionofall


thesearticles5,6,7,8,9,10 is that that there isa fundamental limit to lengthmeasurement deviceindependent ontheorderofPlancklength Lp 1.6 10‐35m .Asimilaranalysisoftime2,3hasconcluded that there is a fundamentalminimumdetectableunit of time differencebetweenclocks whichisontheorderofPlancktime Tp 5.4 10‐44s .LikeallPlanckunitdefinitions,thePlancklength/timelimitationignoresnumericalfactorsnear1.Therefore,amoreprecisestatementofthislimitationmightincludeanumericalfactornear1thatisbeingignoredhere.Since we cannot be sure if there is a numerical factor near 1 that is being ignored in anydiscussionofPlancklengthorPlancktime,thisoffersjustificationforignoringnumericalfactorsnear 1 in the plausibility calculations made in the remainder of the book. Presumably,subsequentanalysisbyotherscaninserttheignoredfactorsnear1iftheyareneeded.However,itwillbeshownthroughouttheremainderofthebookthattheplausibilitycalculationsgivethecorrectanswersbyignoringnumericalfactorsnear1.Onlyinthecosmologychapters 13and14 willweneedtouseawell‐knownnumericalfactornear1.Wewill accept thePlanck length/time limitationandproceedassuming thatdipolewaves inspacetime can have this displacement and yet have the wave properties fundamentallyundetectable.Thislimitationisconceptuallyunderstandableifitisviewedasasignaltonoiselimitation.IftheverynatureofspacetimeisthattherearequantumfluctuationsofPlancklengthandPlancktime,thenitisquitereasonablethatitisimpossibletomakephysicalmeasurementsbelowthisnoiselimit. Therefore,quantummechanicspermitsadipolewaveinspacetimetodisplace a particle’s center of mass by Planck length without violating the conservation ofmomentum. Similarly, a dipolewave candisplace time thedifferencebetween clocks at aspecific location by Planck time compared to the surrounding volumewithout violating anyconservation requirement. Even though thewave properties of spacetime dipole waves areundetectable,thisdoesnotimplythatspacetimedipolewavesareinconsequential. Itwillbeshown that any dipole wave that possesses quantized angular momentum any fermion ofboson willproducedetectableinteractionswithoutpermittinganexperimentalmeasurementofthePlancklengthandPlancktimedisplacementsofspacetimewhichconstitutesomeofitswaveproperties.Another characteristic of dipolewaves in spacetime is that they arenot strictly longitudinalwavesortransversewavesbecausetheyarereally4dimensionalwaves.Awavethatmodulatestherateoftimesuchthatperfectclockscandifferby Tpispresentinall3spacialdimensionsofaphysicalvolume.Thereforeithasbothatransverseandlongitudinalquality.Similarly,a

5 Padmanabhan, T.: Limitations on the operational definition of spacetime events and quantum gravity. Class. Quantum Grav. 4 L107 (1987) 6 Garay, L. J.: Quantum gravity and minimum length. Int. J. Mod. Phys. A 10, 145-166 (1995), arXiv:gr-qc/9403008 7 Baez, J. C., Olson, S. J.: Uncertainty in measurements of distance. Class. Quantum Grav. 19:14, L121-L125 (2002) 8 Calmet, X., Graesser, M,. Hsu, S. D.: Minimum length from quantum mechanics and general relativity. Phys. Rev. Lett. 93, 211101 (2004) [arXiv:hep-th/0405033v2] 9 Calmet, X.: Planck length and cosmology. Mod. Phys. Lett. A 22, 2027-2034 (2007), [arXiv:0704.1360v1] 10 Calmet, X.: On the precision of length measurement. Eur. Phys. J. C 54: 501-505, (2008) [arXiv:hep-th/0701073v1]


wavethatmodulatesthe3spatialdimensionsby Lpisproducingavolumechangewhichisbothtransverse and longitudinal. Again all 4 dimensions are being modulated simultaneouslybecausethereisalsoacoordinatedmodulationoftherateoftime.Whentherateoftimeslows,thevolumeincreasesandviceversa.Itisproposedthatvacuumfluctuations,quantumfoam,zeropointenergy,vacuumenergy,theuncertaintyprincipleetc.arealljustdifferentwaysofdescribingdipolewavesinspacetimewitha spatial displacement amplitude in the range of Planck length ΔL Lp and a temporaldisplacementamplitudeofPlancktime ΔT Tp .Thissaysthatthedimensionlessstrainwaveamplitude As forareducedwavelengthofλexpressedusingspatialpropertiesandtemporalpropertieswouldbe:As ΔL/L Lp/λstrainamplitudeexpressedusingPlancklengthandλAs ΔT/T TpωstrainamplitudeexpressedusingPlancktimeandωFigures5‐3and5‐4 in chapter5 aredrawn for adifferentpurpose,but they canbeused toillustratestrainamplitude.Figure5‐3hasadisplacementamplitudeof Lp.TheXaxisonthisgraph is designated in units of reducedwavelengths λ. The strain amplitude at any instantcorresponds to the slope of the graph. For example, themaximum slope maximum strainamplitude isLp/λwhichoccurswhenthewavecrossesthe“X”axis y 0 .Similarly,figure5‐4relatestothetemporalpropertiesofadipolewaveinspacetimewithamaximumandminimumtemporaldisplacementof Tp.Therefore,maximumtemporalstrainamplitudecorrespondstoTpω.ThePlancklength/timelimitationimpliesthefollowingcorollary:

CorollaryAssumption:The maximum strain amplitude (Amax) permitted for a dipole wave in spacetime is Lp/λ in the spatial domain and Tpω in the temporal domain. Amax = Lp/λ. = Tpω Static versus Dynamic Units:Wearegoingtointerruptthediscussionofdipolewavestoinsertanoteaboutunits.Manytimesinthisbook,Plancklengthwillberepresentingawaveamplitudedimension dynamicdimension ratherthanastaticdistanceasmightbeequatedtorulerlength.Iwillattempttoclearlyspecifywhenlengthrepresentsawaveamplitudeorastraininspacetimewithunitsoflengthbutnottobeconfusedwithstaticlength distance .Itisalsopossibletohaveawave amplitude associatedwith Planck time. For example if awave in spacetime causedprefectclockstospeedupandslowdownbyPlancktime,thenPlancktimewouldbeexpressingawaveamplitude adynamicunit ratherthantheconventionalideaofatimeintervalsuchas1second.What Are Dipole Waves In Spacetime? Wewill start theexaminationofdipolewaves inspacetimebymakingananalogytoelectromagneticdipoles.Forexample,acarbonmonoxidemoleculehasthecarbonatomnegativelychargedandtheoxygenatompositivelycharged.The


bond between these two atoms has similarities to a mechanical spring. When the carbonmonoxidemoleculeisgivenenergy,itwillvibrateandrapidlyrotatearoundatransverseaxis.This isa rotatingelectromagneticdipolewhichemitsaphoton intoacharacteristicemissionpatternassociatedwithdipoles.TheCOmoleculeoscillatingdipoleemissionisalwayscombinedwithachangeofoneunitofangularmomentumassociatedwithachangeinrotationalenergylevel.Theemissionofaphotonalwaysremovesħofangularmomentum.Next,supposethatwehadadiatomicmoleculewherebothoftheatomswerepositivelycharged.Oscillatingorrotatingthismoleculewouldalsoproduceelectromagneticradiation,butatamuchlowerefficiencybecausebothatomshavethesamecharge.Thisproducesadifferentemissionpattern known as a “quadrupole emission pattern”. Even though quadrupole emission isdifferent thandipoleemission, thephotonsproducedbyquadrupoleemissionhavethesamefundamental properties as photons produced by dipole emission. Both are electromagneticradiationwithtransversewaveproperties.Thisiswheretheelectromagneticanalogytodipolewaves inspacetimebreaksdownbecausewaves inspacetimehavea fundamentaldifferencebetweendipolewavescomparedtoquadrupolewaves gravitationalwaves .Masshasonlyonepolarity.Therefore,itisimpossibletogeneratedipolewavesinspacetimebyoscillating or rotating two connected masses. The lowest order wave obtainable byunsymmetrical acceleration of mass such as an oscillating ellipsoid or a rotating rod isquadrupolegravitationalwaves.Gravitationalwavesaretransversewaveswhichpropagateatthe speedof light in themediumof spacetime. For example, a gravitationalwavepassing aspherical volume would convert the spherical volume into an oscillating ellipsoid. Onedimensiontransversetothepropagationdirectionelongateswhiletheorthogonal transversedimension contracts. These offset each other so that the oscillating ellipsoid has the samevolumeasthesphericalvolume.Also,agravitationalwavedoesnotchangetherateoftimeasitpassesatestvolume.Toobtaindipolewavesinspacetimebyacceleratingmass, itwouldbenecessarytohavethecenter ofmass of a closed system accelerate in violation of the conservation ofmomentum.Anotherimpossiblealternativewouldrequireonemassofordinarymatterandanothermassthatexhibitednegativeenergy.This“anti‐gravitymatter”wouldproduceaninversecurvatureofspacetimecomparedtoordinarymatter.Asdistancetotheanti‐gravitymatterdecreased,therate of time would increase and proper volume would decrease the opposite of ordinarymatter .Amasswithanti‐gravitywouldbetheequivalentofanegativegravitationalchargeanditwouldalsohavetobehypotheticalnegativeenergy.Ifaparticleofordinarymatterwasattachedtoahypotheticalantigravityparticleandthecombinationrotated,thewaveemittedwouldcausetherateoftimeatanypointintheequatorialplanetospeedupandslowdown.Alsopropervolumewouldexpandandcontractattheselocations.Thehypotheticalwaveinspacetimethatwouldbe


generatedwouldbeadipolewaveinspacetime.Theeffectonspacetimewouldbetheequivalentofamodulatedgravitationalfield.Electricallyneutralparticles,nomatterhowmassive,wouldundergoanoscillatingaccelerationtowardsandawayfromthehypotheticalrotatingobject.Thisdipole wave in spacetime is the simplest type of hypothetical wave, but it would also be aviolationoftheconservationofmomentumif itoccursonthemacroscopicscale largerthanPlancklength .Thereisnowayofgeneratingadipolewaveinspacetimewithordinarymatter.Envisioning a Dipole Wave in Flat Spacetime: Chapter 5will give some figures to helpquantifyadipoleinspacetimeincludingfigures.Thereisnowayofgeneratingtotallynewdipolewaves newenergy inspacetime.Planckamplitudedipolewavesinspacetimeandquantizedangularmomentumareallpartof thepropertiesof spacetimeand these fluctuationsexistedsincetheBigBang.Imagineanemptyvoidwithnovacuumfluctuationsatanyscale.Thisvoidwouldhavenophysicalproperties–nospeedoflight,noimpedance,nogravitationconstant,nopermittivity εo ,nofieldofanykind.Thisquietvoidgoestoofar,soinsteadwewillimagineperfectlyflatspacetimewithaperfectlyuniformrateoftimeandperfectlyuniformEuclidiangeometryevenatthesmallestscale.Next,wewillimaginethisvolumefilledwithmultiplesmallamplitudedipolewavesinspacetime.Thetemporalpartofadipolewavecausestherateoftimetospeedupandslowdownslightly.If therewereperfectpointclocksdistributedthroughout thisvolume,someclocks located involumesof“fasttime”wouldberunningslightlyfasterthanclocks locatedinthe“slowtime”volumes of spacetime. The dipolewaves are oscillations, therefore a clock that is runningslightly fast one instant would run slightly slow at a later time. There is also chaoticrearrangements of the dipole waves, so the distribution also changes. If it was possible tomomentarily freeze thedipolewaves inspacetime,wewould find that thevolume isnot flatspacetimeeither.Thelocationsthathaveslowrateoftime,haveaslightlylargervolumethanexpectedfromEuclidiangeometry.ThelocationsthathavefastrateoftimehaveslightlylesspropervolumethanexpectedfromEuclidiangeometry.Thisconnectionbetweentherateoftimeandvolumealsooccursonthemacroscopicscale.Aswasdiscussedinchapter2,avolumethatexhibitsafastrateoftimealsoexhibitsanon‐Euclidiandecreaseinpropervolumecomparedtothesurroundingspace.Similarly,avolumethatexhibitsaslowrateoftimealsoexhibitsnon‐Euclidianenlargedpropervolume.Likeallwaves,dipolewavesinspacetimealwayshaveequalamountsofwavemaximumsandwaveminimums.Forexample,afastrateoftimecanbeconsideredawavemaximumandaslowrateoftimecanbeconsideredawaveminimum. Thepoint is thateventhoughthere isarandomqualitytothedipolewaves,thereisalwaysapairingofmaximumandminimumspacetimedistortionsoveradistanceofonewavelength.Ifwewereflippingcoinstodeterminethedistributionoffastandslowvolumesofspacetime, itwouldbepossible to finda few locations thatbychancehadagroupingofalmostall fastorslowtimedistortionsandthiswouldbedetectable.However, ifthereneedstobeapairingoffastandslowtimecomponentsonthescaleofperhapstwotimes


Plancklength,thenthisdictatesadifferentrangeofpossibilities.Allthatispossibleisaslightvariation in the orientation of fast and slow locations. This is a plausible mechanism thatenforces the Planck length/time limitation over a time period long enough to make ameasurementatthespeedoflightacrossatestvolumeofspacetime.Dipolewaves in spacetime represent the simplest type of wave distortion of spacetime. Incontrast,avolumefilledwithchaoticgravitationalwaves quadrupolewaves wouldhavenooscillationoftherateoftimeandnooscillationinpropervolume.Therewouldbeaneffectonthedistancebetweenpoints,butanincreaseindistanceinonedimensionisoffsetbyadecreaseindistanceinanorthogonaldimensionsothatthereisnonetchangeinvolume.Thewaythatgravitationalwavesaffectspacetimemeansthattheycanproduceameasurableoscillationinthedistancebetweentwopoints.Inotherwords,gravitationalwavescanproducechangesinthedistancebetween twopoints that ismuch greater thanPlanck length. This is a fundamentaldifferencebetweendipolewavesinspacetimeandgravitationalwaves.Thewavepropertiesofgravitationalwavesaredetectable, thewavepropertiesofdipolewaves inspacetimearenotdetectableasdiscretewaves.However,itwillbeshownthattherearemanyobservableeffectsofthesePlanckamplitudewavesinspacetime. “Spacetime Field”: What is a field? John Gribbin11 describes a field as a physical quantity that has a value for each point in space and time. John Archibald Wheeler says a field “occupies space‐Itcontainsenergy.Itspresenceeliminatesatruevacuum.”12RichardFeynmansaid,"Afieldhassuchfamiliarpropertiesasenergycontentandmomentum,justasparticlescanhave".13AlbertEinsteinequated“field”to“physicalspace”.Thestandardmodelisafieldtheory. Fundamentalparticlesaredescribedas“excitations”oftheirassociated“fields”. Therefore,eventhestandardmodelhasthevacuumfilledwithfieldenergy.Sinceparticlesareexcitationsoffields,andfieldsexertforces,itmightbesaidthatallparticlesandforcesarederivedfromfields.Inotherwords,thestandardmodelimpliesthat“theuniverseisonlyfields.”Itisashortjumpfromthisto“theuniverseisonlyspacetime”.Thestandardmodelcurrentlyhas17namedfundamentalparticlesbutthisnumbercouldbeincreased if the “color” difference is counted as separate particles. This implies theunappealingprospectthatthevacuumcontainsatleast17overlappingfields.Thestandardmodel describes these particles, but it does not give a physical description of the fieldsthemselves. Lacking this base, physics currently contains manymysteries which can bedescribedmathematicallybutarenotunderstoodconceptually.

11 John Gribbin (1998). Q is for Quantum: Particle Physics from A to Z. London: Weidenfeld & Nicolson. p. 138. ISBN 0-297-81752-3 12 John Archibald Wheeler (1998). Geons, Black Holes, and Quantum Foam: A Life in Physics. London: Norton. p. 163. 13 Richard P. Feynman (1963). Feynman's Lectures on Physics, Volume 1. Caltech. pp. 2–4


Theproposalpresentedinthisbookisthatthereisonlyonetrulyfundamentalfieldwhichwillbenamedthe“spacetimefield”.Thisistheseaofdipolewavesinspacetimepreviouslydescribed Planckfrequency,displacementamplitudeof Lp,TpandPlanckenergydensity .Besides wave properties, the spacetime field will be shown to possess impedance, bulkmodulus, pressure and superfluid characteristics which will be described. All thefundamentalparticleswillbeshowntobeafewcombinationsoffrequency,amplitudeandspinwhichachievearesonancewithinthespacetimefield.Thereisonlyoneuniversalfieldthespacetimefield whichhasmultipleresonances stabilityconditions correspondingtothe fundamental particles. Rather than 17 separate fields which lack conceptuallyunderstandablephysicsandunification,asinglefieldwillbedescribedinenoughdetailtoeventuallypermitcomputermodelingofthequantummechanicaloperationsofphysics.Predominant Frequency of Dipole Waves in Spacetime: Zeropointenergy isdescribedasharmonicoscillators,eachwithenergyofE ½ħω.TheusualdescriptionisthateachharmonicoscillatoroccupiesavolumeofV kλ3 k c/ω 3wherekisanumericalfactornear1. Thisleads to theconceptof thespectralenergydensitydiscussedpreviously in thischapter. TheusualassumptionisthatallfrequenciesarecontinuouslypresentuptoamaximumfrequencyequaltoPlanckfrequency.The“spectralenergydensity”ofthesewavesscalewithω3,thereforethehigherfrequenciesdominatetheenergydensity.I previously used thiswave frequency and density description in earlier drafts of this book.However,Ialwaysknewthattherewasanotheralternativemodelofthefrequencydistributionwhichisslightlydifferent.Icannotchoosebetweenthesetwoalternatives,soIwillmentionthealternativealso.InthisalternativethespacetimefieldisdipolewavesatPlanckfrequencywhenviewed from the rest frameof referenceof the cosmicmicrowavebackground CMB .Otherfrequencies canexist in spacetimeas spatial and temporalmodulationsof this seaofPlanckfrequency dipole waves in spacetime. Therefore, the presence of these other frequenciesbroadensthespectrumaroundPlanckfrequency.Inthisalternativemodel,theBigBangonlyinitiallygeneratedPlanckfrequency ∿1043s‐1 dipolewaves.Thesemonochromaticwavesthenbecamethefoundationofthespacetimefield.Allthelowerfrequencydipolewavesthatwillbeshown to be required to make particles and forces are wave distortions on this energeticbackgroundofPlanckfrequencywaves.BothalternativeshavelowfrequencywavesontopofpredominantlyPlanckfrequencywaves.It should be mentioned that the spacetime field is a completely different concept than thegranularity or pixelationproposedby “loopquantumgravity”. In loopquantumgravity, thepredictedgranularityofspacetimehasdimensionsontheorderofPlancklength.ThereisalsoaminimumunitoftimeequaltooneunitofPlancktime. Therefore, the loopquantumgravitymodel has spacetime broken into static pixels of volume and time. These pixels are not


oscillatingdipolewaves inspacetimeand thereforedonotrepresent the tremendousenergydensityofspacetimerequiredtoexplainzeropointenergy.Planck Scale:ThementionofPlancklengthandPlancktimedoesnotnecessarilyimply“Planckscale”.Theterm“Planckscale”hascometoimplytheconditionsthatwouldexistifparticlesorphotons had Planck energy Ep 1.22 1019 GeV or 1.96 109 Joule . For example, ahypothetical particle with Planck energy Planck mass would have the force of gravity becomparable to the strong force or the electromagnetic force. Such a particle would have aComptonfrequencyequaltoPlanckangularfrequencywhichistheinverseofPlancktime.ThenaturalunitoflengthofsuchahypotheticalparticleisPlancklength.AllofthesepropertiesarehypotheticalbecauseaPlanckmassfundamentalparticledoesnotexist.Itwouldbethesmallestpossibleblackhole.Fermionswillbeshowntobeawaveinthespacetimefieldthatpossessesquantizedangularmomentumwhichisconfinedtoaspecificvolume.Thisisarotatingdistortionofthespacetimefieldatfrequenciesintherangeof1020Hzto1026Hz.TherotatingdistortionofthespacetimefieldhasadisplacementamplitudeofPlancklengthandPlancktimebutthefrequencyismanyordersofmagnitudelessthanPlanckfrequency.ThereforetheenergyismuchlessthanPlanckenergyanddoesnotfitthedefinitionof“Planckscale”.Properties of Dipole Waves in Spacetime: Itisimportanttoalsounderstandthatdipolewavesinthespacetimefieldtravelatthespeedoflightbuttheydonotfreelypropagatelikephotonsorgravitationalwaves. Sincedipolewaves affect the rateof timeand theproper volume, theyinteractwitheachother.Herearesomeotherproposedpropertiesofdipolewavesinspacetimethatarepresentedhereinsummaryformandexplainedlater.

1 Everypartofadipolewave in thespacetime fieldbecomesthesourceofanewwavecalledawavelet .

2 Thesewaveletspropagateinalldirections.3 Theadditionofwaveletstendstoconstructivelyinterferepredominatelyintheforward

andbackwardpropagationdirectionsofthepreviouslyexistingwavefronts.4 These wavelets explore an infinite number of possible trajectories to achieve an

amplitudesumatanypoint intensityisamplitudesquared .5 This is proposed to be the physical explanation that is being modeled by Richard

Feynman’spathintegralformulation.Thesepropertieswillbeexplainedlater.


ForceIftheuniverseisonlyspacetime,andifenergyisawaveinspacetime dynamicspacetime ,thenforcemustalsobetheresultofadynamicdistortionofspacetime.Thefollowingassumptioncanbemade:

ThirdAssumption:There is only one fundamental force: Fr = Pr/c. This is a repulsive force

that occurs when waves in the spacetime field, traveling at the speed of light, are deflected.Thissinglefundamentalforceexertedbythedeflectionofwavesinspacetimepropagatingatthespeedof lightwill be called the “relativistic forceFr.”Also,Pr represents “relativisticpower”whichispowerpropagatingatthespeedoflight.Forexample,lightandgravitationalwavesarebothpowerpropagatingatthespeedoflight.Dipolewavesinspacetimewillalsobeshowntobeanotherformofenergypropagatingatthespeedoflight.Therelativisticforceistheonlyforcedeliveredbydipolewavesinspacetime.Thisenergyalwayspropagatesatthespeedoflight,evenwhenitseemstobeconfinedtoalimitedvolume. Thelimitedvolumeistheresultofspeedoflightpropagationinaclosedloopandinteractingwiththesurroundingvacuumenergy explainedlater .Iproposethattherelativisticforceistheonlytruly fundamental force in the universe. All other forces of nature are just differentmanifestationsof this truly fundamental force.Therelativistic force isderived fromtheonlyfundamentalformofenergyintheuniverse,dipolewavesinthespacetimefield.ItisacommonassumptionamongphysiciststhattheforcesofnaturewereallunitedatthehighenergyconditionsthatexistedshortlyaftertheBigBang.ItistruethatahypotheticalparticlewithPlanckmasswouldhaveagravitationalforceroughlycomparabletotheelectromagneticforceorevencomparabletothestrongforceatshortdistances.Accordingtothecommonlyheldview, the forces of nature separated when the universe expanded and the energy densitydecreased.Theimplicationisthattodaytheforcesofnaturearefundamentallydifferent.Thefundamentalforcesareusuallythoughttobetransferredby“messengerparticles”suchasvirtualphotons, virtual gluons and gravitons. Furthermore, gravity is not included in the standardmodel.Apopularphysicalinterpretationofgeneralrelativityconsidersgravitytobeageometriceffectandnotatrueforce.Therefore, the above assumption is a radical departure from conventional thought. It isproposed thatall forces, thestrong force, theelectromagnetic force, theweak forceand thegravitationalforce ,aretheresultofthedeflectionofwavesinthespacetimefieldtravelingatthespeedoflight. Thecasewillbemadethateventodaythefourforcesofnature includinggravity arestillcloselyrelatedbecausetheyareallderivedfromtherelativisticforce.


TheforceF Pr/ciswellknownastheforceassociatedwithphotonpressurewherePristhepowerof abeamof light. Forexample, theemissionorabsorptionof3 108wattsof lightproducesaforceof1Newton.Theword“deflection”isusedtocoveranychangeinpropagation.Forexample,eventheabsorptionofaphotonbyanelectroninanatomischaracterizedhereasaninteractionbetweenwavesinthespacetimefieldthatinvolvesa“deflection”.Inlaterchaptersit will be shown that the electromagnetic force, the strong nuclear force and even thegravitationalforcearealltheresultofdipolewavesinthespacetimefieldinteractingandbeingdeflected.Energy Density Equals Pressure: Photon pressure is always repulsive. In fact, pressure in any form is always repulsive. EnergydensityUisfundamentallyequivalenttopressureℙwhenwearedealingwithenergypropagatingatthespeedoflight.EventhoughtheunitslookdifferentJ/m3versusN/m2 ,indimensionalanalysisnotationthedimensionsofbothenergydensityandpressurearethesame:M/LT2 mass/lengthtime2 .Forexample,blackbodyradiationinsideauniformtemperatureclosedcontainerhasradiationpressurebeingexertedonthewallsandhasaradiationenergydensityfillingthecontainer. Therelationshipbetweenenergydensityandpressureforblackbodyradiation electromagneticradiation isU 3ℙ. Thefactorof3 inacontainer filledwith blackbody radiation is traceable to 3 spatial dimensions. A laser withcollimatedelectromagneticradiationreflectingbetween2mirrorswouldeliminatethefactorof3andhaveU ℙwhereℙisthepressureexertedonthe2mirrors.Weareignoringnumericalfactorsnear1intheseconceptualequationsthereforewewillequateU ℙ.Therelationshipbetween energy density andpressure is important in cosmology because radiation pressureinsideastarpreventsthestarfromundergoingagravitationalcollapse.Forexample,thecenterof thesun isata temperatureofroughly15milliondegreesKelvin. At this temperature, thephotonenergydensityisabout3 1013J/m3andthephotonpressureisabout1013N/m2.Thisinternalpressurestabilizesthesun’soutputandmakeslifeonearthpossible.Therelationshipbetweenpressureandenergydensityinagasorliquidismorecomplex.Thesimplestexampleoftheenergystorageofthepressurecomponentinafluidcanbeillustratedby the followingexample. Imagine twoheliumatomscolliding inavacuum. Thiscollision isviewedfromtheframeofreferencewheretheatomsareinitiallypropagatingatequalspeedinopposite directions. The kinetic energy of each atom can be associated with a typicaltemperatureusing theBoltzmannconstant. When theatomscollide, the speedmomentarilydrops to zero in this frame of reference and the absolute temperature of each atom alsomomentarily drops to zero. The kinetic energy temperature is temporarily converted tointernalenergyineachatomresultinginadistortionoftheelectroncloudofeachheliumatom.Inahighpressuregastherearemanysuchcollisionspersecond.Theenergyassociatedwiththepressureofthegascanbetracedtothefactthatatomsinthegasspendpartofthetimewithadistortedelectroncloudthathashigherenergythananisolatedatomwithnodistortion.Ahighpressuremonatomicgashas3energycontributionstoitstotalenergydensity;1 theinternal


energy E mc2 oftheindividualatoms,2 thekineticenergyofthetemperatureand3 thepressurecomponentresultinginadistortedelectroncloud.Nowwewilllookatthemuchsimplercaseofconfinedlight.EachphotonhasenergyofE ħω,andthetotalenergydensityoftheconfinedlightisU 3ℙforchaotic3dimensionalpropagationsuch as confined black body radiation. It is proposed that the equivalence between energydensityandpressureappliesinallcasesbecausetheunitsarethesame M/LT2 andthephysicalinterpretationoftheseunitsisthesame.Thereareafewcasesinphysicswheretwodissimilardefinitionscanhave the sameunitswhenexpressedas length, timeandmass. Forexample,torqueandenergybothhaveunitsofML2/T2.However,itisclearthattorqueisforceappliedthrougharadiallengthrwithoutmotion nowork .EnergyisforceappliedthroughadistanceML/T2 L .Theunitsofenergyhavetobeinterpretedasrequiringmotionthroughadistancework .Thereisneveranexamplewhereaunitoftorqueisequaltoaunitofenergybecausethephysicalinterpretationoftheunitsisfundamentallydifferent.Inthecaseofenergydensityandpressure,thephysicalinterpretationisthesame.Itisproposedthatatthemostfundamentallevel,energydensityALWAYSimpliespressure.Itisproposedthateventheenergydensityofaprotonorelectronimpliespressure. Itisnotpossible to casually ignore theenergydensityof aprotonandassume that since there isnoobviouscontainerrestrainingtheimpliedpressurethattheequivalencebetweenenergydensityand pressure has somehow been broken. The standard model assumes that fundamentalparticleshavenointernalstructureandnovolume pointparticles .Evenstringtheoryhasonedimensional stringswithnovolume.Therefore,bothof these require infinite energydensitywhichimpliesinfinitepressure.Thelawsofphysics“breakdown”–endofstory!Thealternativeoffered by the spacetime based model of particles and forces is that everything isunderstandable.Thelawsofphysicsneverbreakdown.Distortionsofspacetimecanappeartobepointparticlesiftheexpectationisaclassicalobjectwithahardsurface.However,wewillshowhowitispossibleforfundamentalparticlestobeareasonablefinitevolumedistortionofspacetimewhichappearstobeapointparticleinexperiments.Attractive Forces: Thepreviousassumptionissurprisingbecauseitclaimsthatthereisonlyonefundamentalforceandbecauseitclaimsthatthissinglefundamentalforceisonlyrepulsive.The obvious question is: How can attractive forces such as gravity, the strong force or theelectromagneticforcebetheresultofasingleforcethatisonlyrepulsive?Thedetailedanswertothisquestionrequiresadditionalinformationcoveredinsubsequentchapters.However,itispossibletogiveabriefintroductoryexplanationhere.ItwaspreviouslyexplainedthatvacuumfluctuationshaveenergydensityequaltoPlanckenergydensity. From the equivalence of energy density and pressure, it follows that vacuumfluctuationsarecapableofexertingamaximumpressureequaltoPlanckpressure 10113N/m2.Lateritwillbeshownthattheproposedspacetimebasedmodeloffundamentalparticleshasa


specific energy density and this requires that vacuum energy/pressure exert an offsettingpressuretoachievestability.Forexample,aprotonhasaknownradiusofabout10‐15m.Thisvolumecombinedwiththeproton’senergyimpliesthataprotonhasenergydensityofabout1034 J/m3. Thisenergydensity impliesthat thewaves formingaprotongeneratean internalpressure of about 1034 N/m2. Stated another way, an isolated proton is stabilized by thespacetime field exerting a repulsive force on all sides of the proton. An electric field is adistortionofthespacetimefieldthatwillbediscussedinchapter9.Iftheprotoncomesnearanelectron,theprotonexperienceswhatweconsidertobeaforceofelectrostaticattraction.Itwillbeshownthatthisisactuallyanunbalancedrepulsiveforce.Thevacuumpressurerequiredtostabilizeaprotonisunbalancedbythedistortionofthespacetimefieldcausedbytheelectron’selectricfield.Thisresultsinwhatappearstobeaforceofattraction.Inthismodeltherearenoexchangeparticlesthatsomehowachieveattraction.Allactionatadistanceisultimatelytraceabletoalocalizedimbalanceinvacuumpressure.Therearealsonoattractiveforces.Thereisonlyanunbalancedrepulsiveforce unbalancedpressure exertedonfundamentalparticlesby thedipolewaves thatare thevacuumfluctuationsof thespacetimefield.Thisintroductoryexplanationlacksmanyessentialdetailsthatwillbeprovidedlater.

ImpedanceofSpacetimeThefirststepinunravelingthe10120discrepancybetweenthequantummechanicalmodelandthe general relativity model is to see if there is anything in general relativity that actuallysupports the idea of a large vacuum energy density. It is proposed that an analysis ofgravitationalwaves indeedgivessupport to thequantummechanicalmodelof thespacetimefield.GravitationalwavesareaformofenergythatpropagatesatthespeedoflightastransversewavesINspacetime.Theyproduceadynamicdistortionof2ofthe4dimensionsofthespacetimefield.Thedistortionconvertsthesphericalvolumeintoanoscillatingellipsoid.Onetransverseaxisoftheellipsoidelongateswhiletheorthogonaltransverseaxiscontracts.Thisoscillationoftheellipsoidproducesnonetchangeinvolumefromtheoriginalsphericalvolumeandthereisnochangeintherateoftime.Sincegravitationalwaves quadrupolewaves donotmodulatevolume or the rate of time, gravitationalwaves can have a displacement of spacetimemuchgreaterthanthePlancklength/timelimitationthatappliestodipolewavesinspacetime.Gravitational waves transfer energy and angularmomentum. In 1993 the Nobel Prize wasawardedtoRussellHulseandJosephTaylorfortheproofthatabinaryneutronstarsystemwasslowingdownitsrotationbecauseitwasemittingabout1025wattsofgravitationalwaves.Theamountofslowingwaswithin0.2%oftheamountpredictedbygeneralrelativity.Theemissionofgravitationalwavesproducesaretardingforceontherotatingbinarystars,thusproducingan


observableslowingoftherotation lossofenergyandangularmomentum .Ifitwaspossibleto reverse the direction of these gravitational waves, the gravitational waves would returnenergyandangularmomentumtothebinaryneutronstarsystem.Thereasonthatgravitationalwavesareintroducedintoadiscussionabouttheenergydensityofspacetimeisthatgravitationalwavesarepropagatinginthespacetimefield.Theyareanalogoustosoundwavespropagatinginanacousticmedium.Thesamewaythattheequationsforsoundpropagation give information about the acoustic medium, so also the gravitational waveequationscangiveinformationaboutthepropertiesofthespacetimefield.Zs – The Impedance of Spacetime:Inacoustics,allmaterialsofferoppositiontoacousticflowwhenanoscillatingacousticpressureisapplied.Forexample,tungstenhasthehighestacousticimpedance which is about 2.5x106 times greater than the acoustic impedance of air.Electromagneticradiationalsoexperiencesacharacteristicimpedanceasitpropagatesthroughspace.Theelectricfield andmagneticfield arerelatedbythe“impedanceoffreespaceZo”.Therelationshipis:

Zo≡ / = 376.7Ωimpedanceoffreespace

Gravitational waves also experience impedance as they propagate through spacetime. IidentifiedtheimpedanceexperiencedbygravitationalwaveswhenIfirststartedworkingonthisproject.IwassurprisedthatIinitiallycouldnotfindanyotherreferencetothis.Afterabout5years, Idiscoveredthattheimpedanceofspacetimehadbeenpreviously identifiedbyBlair14fromananalysisofgravitationalwaveequationsandreportedinthe1991bookTheDetectionofGravitationalWaves.However,eveninthatbooktheimpedanceofspacetimeisonlycasuallymentionedandisnotusedinanycalculations.Sincethen,theimpedanceofspacetimeappearstobeignoredbythescientificcommunity.Aswillbeseen,theimpedanceofspacetimeisthekeytoquantifyingthepropertiesofthespacetimefield.MostofthecalculationsintheremainderofthisbookdependonthisimpedancewhichisidentifiedbyBlairas:

Zs c3/G 4.038 1035kg/sZs impedanceofspacetime

The reasoning that ledme to independentlydiscover the impedanceof spacetimestartedbycomparinggravitationalwavestoacousticwaves.Allpropagatingwavesinvolvethemovementofenergy.Inotherwords,propagatingwavesofanykindareaformofpower.Thereisageneralequation thatapplies towavesofanykind. Themostcommon formof thisequationrelatesintensity“ ”,thewaveamplitudeA,thewaveangularfrequencyω,theimpedanceofthemediumZandadimensionlessconstantk.Theintensity canbeexpressedinunitsofw/m2.

14 Blair, D. G. (ed.): The Detection of Gravitational Waves. p 45. Cambridge University Press, Cambridge New York Port Chester (1991)


kA2ω2ZWe will first illustrate the use of this general equation using acoustic waves. The acousticimpedanceis:Za ρcawhereρisdensityandcaisthespeedofsoundinthemedium acousticspeed .Acousticimpedancehasunitsofkg/m2susingSI dimensionalanalysisunitsofM/L2T .Theamplitudeofanacousticwaveisdefinedbythedisplacementofparticlesoscillatinginanacousticwave.Theamplitudeterminacousticequationshasunitsoflengthsuchasmeters.When the equation k A2ω2Z is used for gravitational waves, the amplitude term is adimensionlessratiowhichinitssimplestformcanbeexpressedasstrainamplitudeA ΔL/L.Thisratioisexpressingastraininspacetimewhichcanalsobethoughtofasthemaximumslopeof a graph that plots displacement versus wavelength. When the amplitude term isdimensionlessstrainamplitude,thenforcompatibilitytheimpedanceofspacetimeZsmusthavedimensionsofmass/time M/T .Eventhough kA2ω2Zisauniversalwave‐amplitudeequation,itcanonlybeusedifamplitudeAandimpedanceZareexpressedinunitscompatiblewithintensity watts/m2 inthisequation.Forexample,electromagneticradiationisusuallyexpressedwithamplitudeinunitsofelectricfieldstrengthandthe impedanceof freespaceZo inunitsofohms. Thiswayofstatingwaveamplitudeandimpedancedoesnothavethecorrectunitsrequiredforcompatibilitywiththeaboveintensityequation.Asdiscussedinchapter9,thereareotherwaysofexpressingthesetermsthatmakeelectromagneticradiationcompatiblewiththisuniversalequation.The intensity of gravitationalwaves can be complex because of nonlinearities and radiationpatterns. However,thisintensitycanbeexpressedsimplyifweassumeplanewavesandtheweak gravity limit.15 Using these assumptions, the gravitational wave intensity is oftenexpressedas:

υ2A2where: intensityofagravitationalplanewaveandυ frequency

However,thiscanberearrangedtoyieldthefollowingequation: kA2ω2 c3/G k adimensionlessconstant;ω angularfrequencyAs ΔL/L strainamplitudewhereLismeasurementlengthandΔListhechangeinlength

15 D. G. Blair, The Detection of Gravitational Waves, Cambridge University Press, 1991, p. 34


Itisobviouscomparingthisequationtothegeneralequation kA2ω2Zthatthetwoequationshavethesameformandthattheimpedancetermmustbe:Z Zs c3/G5 Wave-Amplitude Equations:Nowthatwearearmedwiththeimpedanceofspacetime,theequation for intensity can be converted into equations that express energy density U ,energy E andpower P .Ifwearerestrictedtowavespropagatingatthespeedoflight,thenwecanalsoconvert the intensityequation intoanexpressionof the force F exertedbythepropagating wave. This conversion incorporates the equation F P/c where P is powerpropagatingatthespeedoflight.Thesewillbecalledthe“5wave‐amplitudeequations”.Theseequationsalsousethesymbolsof: area m2 ,V volume m3 andk dimensionlessconstantnear1

kA2ω2Z intensity w/m2 U kA2ω2Z/cU energydensity J/m3 U /c andU ℙ pressureE kA2ω2ZV/cE energy J E V/c P kA2ω2Z P power J/s P F kA2ω2Z /cF force N F /c These5equationswillbeusednumeroustimesintheremainderofthebook.Itisproposedthatallenergy,forceandmatterisderivedfromwavesinthespacetimefieldandthese5equationswillbeusedtosupportthiscontention.TheamplitudetermAneedsfurtherexplanation.Weare presuming waves propagating at the speed of light and we are temporarily excludingelectromagneticwavesuntilchapter9.Thisleavesgravitationalwavesanddipolewavesinthespacetimefield.Weneedtostandardizehowwedesignatetheamplitudeofthesewaves.Forgravitationalwaveexperimentswherethewavelengthismuchlongerthanthemeasurementpathlength λ L ,itisacceptabletodesignatethestrainamplitudeasAs ΔL/L.However,whenwearedealingwithanarbitrarywavelengthwhichmightbesmall,itisnecessarytospecifystrain as the maximum slope of a graph that plots displacement versus wavelength. Thismaximumslopeoccurswhenthedisplacementiszeroandthestrainismaximum seefigures5‐3and5‐4inchapter5 .IfwedesignatethemaximumdisplacementasΔL,andthewavelengthasλ,thenthemaximumstrain maximumslope isAs ΔL/λ.whereλ λ/2π.Thisexamplepresumesthatweareworkingwithadisplacementof length. Gravitationalwavesproducealength modulation with offsetting effects in orthogonal dimensions such that there is nomodulationofvolumeandnomodulationoftherateoftime.Therefore,gravitationalwavesarenot subject to thePlanck length/time limitation that applies todipolewaves. As previouslyexplained, dipole waves have amaximum spatial displacement amplitude of ΔL Lp and amaximumtemporalamplitudeofΔT Tp.Therefore,themaximumstrainamplitude Amax ofadipolewaveis:

Amax Lp/λ ω/ωp ħ ⁄


Impedance of Spacetime from the Quantum Mechanical Model:Nowthatweareequippedwiththe5wave‐amplitudeequations,thedipolewavehypothesisandAmax Lp/λ,itispossibletoanalyzezeropointenergyfromanewperspective.Ifzeropointenergyisreallydipolewavefluctuations inthemediumofspacetime, then itshouldbepossibletodoacalculationwhichsupports this idea. For review, the quantum mechanical model of the spacetime field hasspacetimefilledwithzeropointenergy quantumoscillators withenergyofE ½ħω.Ifweareignoring numerical factors near 1, therefore we can consider each quantum oscillator asoccupyingavolumeV λ3.Thismeansthattheenergydensityofthequantummechanicalmodelofzeropointenergy isU ħω/λ3 ħω4/c3.Nowwearereadytocalculatetheimpedanceofspacetimeobtainedfromacombinationof1 zeropointenergywithenergydensityU ħω4/c3;2 dipolewavesinspacetimewithmaximumamplitudeofAmax Lp/λ,and3 thepreviouslyobtainedequationforenergydensityU A2ω2Z/c.Rearrangingtermswehave:Z Uc/A2ω2

Set:U ħω4/c3andA Amax ħ ⁄

Zħ

ħ ZsSuccess!

Link between QM and GR Models of Spacetime:Thisisafantasticoutcome!Wetooktheenergydensityofzeropointenergyandcombinedthatwiththestrainamplitudeofadipolewavein the spacetime field and an equation from acoustics. Whenwe solved for impedanceweobtainedc3/G.Thisisthesameimpedanceofspacetimethatgravitationalwavesexperienceastheypropagate through spacetime. Tome, this implies that the characteristics of spacetimeobtainedfromgeneralrelativityagreewiththequantummechanicalmodelofthespacetimefieldfilledwithzeropointenergyandexhibitingenergydensityof10113J/m3.Howcanthisbe?Thegeneralrelativitymodelincorporatescosmologicalobservationandsetstheenergydensityoftheuniverseatabout10‐9J/m3.Actuallythisisanerroneouscomparison.Thequantummechanicalmodelofthespacetimefieldisgivingthehomogeneousinternalenergydensityofspacetimeitself.Whengravitationalwavespropagatethroughthespacetimefield,theyareinteractingwiththisinternalstructureofthespacetime field and thegravitationalwavesexperience impedanceofZs c3/G. Theenergydensityof10‐9J/m3obtainedbycosmologicalobservationisnotseeingtheinternalstructureofspacetime with its tremendous energy density of dipole waves. Instead, the cosmologicalobservationsarejustlookingattheenergydensityofthefermions,bosonsand“darkenergy”discussed later . This isnot thesame thingas the internal structureof thespacetime field.Gravitationalwaves can propagate through the spacetime field that contains no fermions orbosonsandstillexperienceZs c3/G.Assumingthatthetotalenergydensityoftheuniverseis


10‐9J/m3islikelookingonlyatthefoamonthesurfaceoftheoceanandignoringallthewaterthatmakesuptheocean.The first part of reconciling the difference between the general relativity and quantummechanicalmodelsof spacetime is toview thequantummechanicalmodelasdescribing theinternalstructure themicroscopicstructure of thespacetimefield. Meanwhile, thegeneralrelativitymodelisdescribingthemacroscopiccharacteristicsofspacetimeandtheinteractionswithmatter.If the spacetime field canpropagatewaves such as gravitationalwaves ordipolewaves , itimpliesthatthespacetimefieldmusthaveelasticity.Thiselasticityrequirestheabilitytostoreandreturnenergyasthewavepropagates.Themediumitselfmusthaveenergydensity.Thequantummechanicalmodelofspaceisfilledwithaseaofenergeticfluctuations dipolewaves .If these are visualized as energetic waves in the spacetime field, then a new wave can bevisualizedascompressingandexpandingthesepreexistingwaves.Ifthisnewwavecausesthepreexistingwavestoslightlychangetheirfrequencyanddimensions wavelength astheyarebeingcompressedandexpanded,thenthispictureprovidesthenecessaryelasticityandenergystoragetothespacetimefield.Thismightsoundlikeacircularargumentsinceeachwavecontributestotheelasticityrequiredbyallotherwaves.Whataboutthe“first”wave?Thissubjectwillbediscussedfurtherinthetwocosmologychapters13and14.However,itwillbeproposedthattherewasnofirstwave.Thespacetimefieldcameintoexistencealreadyfilledwiththesevacuumfluctuations.Energeticwavesaresimplyafundamentalpropertyofthespacetimefieldthatgivethevacuumpropertiessuchasεo,µo,c,G,Zs,etc..Infact,thespacetimefielddoesnothavewaves;thespacetimefieldIStheseaofvacuumfluctuations waves describedbythequantummechanicalmodel.Spacetimeneverwasthequietandsmoothmediumassumedbygeneralrelativity.Thereforethereneverwasatimewhenafirstwavewasintroducedintoaquietspacetime.ThiswavestructurewithitsPlancklength/timelimitationcanbeignoredonthemacroscopicscalebutspacetimehasaquantummechanicalbasis.Thetaskisnottofindamechanismthatcausescancelationofthistremendousenergydensity.Thisenergydensityisreallypresentinthespacetimefieldandisnecessarytogivethespacetimefieldthepropertiesdescribedbygeneralrelativity.Insteadthefocusneedstoturntofindingthereason that thishighenergydensity isnotmoreobviousandwhy itdoesnot itselfgenerategravity.Istheresomethingabouttheenergyinvacuumfluctuationsthatmakesitdifferentthantheenergyinmatterandphotons?Thisquestionwillbeansweredlater.Energy Density of Spacetime Calculated from General Relativity: Previouslyweshowedthatit was possible to deduce the impedance of spacetime Zs c3/G from quantummechanicalconsiderations,zeropointenergyandanequationfromacoustics.However,nowwewillshow


thatitispossibletocalculatetheenergydensityofthespacetimefieldusingjustequationsfromgeneral relativity and acoustics. Since general relativity and quantum mechanics are oftenconsideredtobeincompatible,itmightseemunlikelythatwewouldturntogeneralrelativitytoanalyzethequantummechanicalenergydensityofspacetime.Thereasonforsuspectingthatthismightbeafruitfulapproachisthatgravitationalwavesarelikesoundwavespropagatinginthemediumofspacetime.Itiswellknownthatanalyzingtheacousticpropertiesofamaterialcanreveal someof itsphysicalpropertiesof themedium including itsdensity. Gravitationalwaves are like sheer acoustic waves propagating in the medium of the spacetime field.Therefore,wewillmakeanalogiestoacousticsandattempttocalculatetheenergydensityofthespacetimefield.Thefollowingequationfromacousticsrelatesthedensityofthemediumρtointensity ,particledisplacementΔx,acousticspeedofsoundca,andangularfrequencyω. kρω2caΔx2. Thespacetimefielddoesnothaverestmasslikefermions,butgravitationalwavesdopossessmomentum.Aspreviouslyexplained,ifwecouldconfinegravitationalwavesinahypothetical100%reflectingbox,thenthegravitationalwaveswouldexhibitrestmess.Theboxismerelyturningtravelingwavesintostandingwaves.Thewavesthemselvespossesscharacteristicsthatcan be associated with not only energy density but also mass density under specializedconditions.Ifwecancalculatetheenergydensityofthespacetimefieldusingequationsfromacoustics and gravitational waves, then this will be important not only for establishing thequantummechanicalpropertiesofspacetime,butalsoformakingaconnectionbetweengeneralrelativityandquantummechanics.Earlierinthischapter,anequationwasreferencedwhichconnectstheintensity ofgravitationalwaveswiththefrequencyυandthestrainamplitudeAofthegravitationalwaves.Thisequationassumestheweakfieldlimitwherenonlinearitiesareeliminatedandalsoassumesplanewaves.That equation is repeatedbelow. The amplitudeA of the gravitationalwave is given as thedimensionless strain amplitude maximum slope of A ΔL/λ where ΔL is the maximumdisplacementofspacetimeandthereducedwavelengthis:λ λ/2π c/ω.

υ2A2 kA2ω2 k

Wewillsettheintensityoftheaboveequationequaltotheintensityoftheacousticequation kρω2caΔx2andsolvefordensityρ.ToachievethiswewillsettheacousticdisplacementΔxequaltothegravitationalwavespatialdisplacementΔLandsetacousticspeedca c.

set , ,λ c/ω,solveforρandU


set:λ r radialdistance whichisarequiredforphysicalinterpretation

Where: istheinteractivedensityofspacetimeUiistheinteractiveenergydensityofspacetimeUp c7/ħG2 10113J/m3 Planckenergydensity

ωp ħ⁄ 1.85 1043s‐1 PlanckangularfrequencyThe terms “interactive density” and “interactive energy density” are necessary because thespacetimefielddoesnothavedensityandenergydensityintheconventionaluseoftheterms.Whenwethinkofthedensityofanacousticmediumsuchaswater,thishasthesamedensityeveniftheacousticfrequencyisequaltozero.Thespacetimefieldonlyexhibitsan“interactivedensity”whenthereisawaveinspacetimewithafinitefrequency.Ifthefrequencyis0,thenρi 0andUi 0.Iwanttobrieflypointoutthattheaboveequationsderivetheenergydensityofspacetimethatmustbethereinorderforgravitationalwavestopropagate.Thepresenceofthisenergydensityandthefrequencydependencewasobtainedfromagravitationalwaveequationandanacousticequationwithnoassumptionsfromquantummechanics.Proceedingwiththespacetimefieldinterpretationoftheseequations,agravitationalwaveisoscillatingapartoftheseaofdipolewavesthatformsthespacetimefield.Thesedipolewavesareslightlycompressedandexpandedbythegravitationalwave,sotheyrevealtheenergydensitythatisactuallyinteractingwiththegravitationalwave.ThedipolewavesinthespacetimefieldareprimarilyatPlanckfrequencyωp 2 1043s‐1.IftherewassuchathingasaPlanckfrequencygravitationalwavefillingaspecificvolume,thenthisPlanckfrequencygravitationalwavecouldefficientlyinteractwithalltheenergydensityinthatspecificvolumeofthespacetime.Noknownparticlescouldgeneratethisfrequency,butthisrepresents the theoretical limits of the properties of spacetime. For example, suppose weimaginetwohypotheticalPlanckmassparticlesformingarotatingbinarysystem.TheywouldbothbeblackholeswithradiusequaltoPlancklengthLp.Astheyrotatedaroundtheircommoncenterofmass,theywouldgenerategravitationalwaves.Iftheywereclosetomerging,thenthefrequencywouldbeclosetoPlanckfrequency.Toexplorethislimitingcondition,wewillassume

a gravitationalwavewith Planck angular frequency and substituteω ωp ħ⁄ intoUi c2ω2/G.ThisgivesPlanckenergydensityUp c7/ħG2 4.63 10113J/m3.Beforeproceeding,weshouldpauseamomentandrealizethatthissimplecalculationhasjustproven that general relativity requires that spacetimemust have Planck energy density forspacetimetobeabletopropagategravitationalwavesatPlanckfrequency. Generalrelativityalso specifies howwaves less thanPlanck frequency interactwith the energy density of the


spacetime field.Wenormally thinkofgeneral relativityasbeing incompatiblewithquantummechanics. However,generalrelativityactuallysupportsandhelpstoquantifytheproposedquantummechanicalmodelofthespacetimefield.InteractiveEnergyDensity fromWave‐AmplitudeEquation: It is possible to gain a differentperspectiveontheinteractiveenergydensityofspacetimebyfindingthesubstitutionintotheequationU kA2ω2Z/crequiredtoyieldUi c2ω2/G.

A 1Therefore,theinteractiveenergydensityisgeneratedwhenwesettheamplitudetermAequalto the largest possible value which is A 1. Planck energy density is obtained when wesubstituteboththelargestamplitudeA 1andthehighestpossiblefrequencyω ωp.AtanyfrequencyωlessthanPlanckfrequency,theinteractiveenergydensityUirepresentsthelargestpossible energy density at frequency ω assuming the medium has impedance equal to theimpedanceofspacetime:Zs c3/G.Togeneralizetheinteractiveenergydensitysothatitappliestomore than just gravitationalwaves,we have view the entire universe even particles asentirelywave‐based.Thiswillbeprovenintherestofthisbook.Thesignificancehereisthatwecanextrapolatefromtheinteractiveenergydensityencounteredbyagravitationalwaveoverdistanceλtotheinteractiveenergydensitythatexistsoverasphericalvolumeofspacetimewithradiusr.Tocalculatethis,wecansubstituteλ rsothatUi Fp/λ2becomesUi Fp/r2.AnalysisofWavesLessthanPlanckFrequency:AtfrequencieslowerthanPlanckfrequency,agravitationalwaveexperiencesamismatchwiththespacetimefieldthatprimarilyhaswavesatPlanckfrequency.Thereisonlyapartialcouplingtotheenergydensityofthespacetimefield.

Thescalingofthelowerfrequenciesisgivenbytheequation .Anumericalexamplewillbegivenwhichassumesagravitationalwavewithanangularfrequencyof1s‐1andreduced wavelength of 3 108 m. For this wave, the frequency mismatch factor is

2.9 10‐87. Therefore, according to Ui Up the interactive energydensityencounteredbythisfrequencyis:Ui 1.35 1027J/m3orρi 1.5 1010kg/m3.Ifagravitationalwavewithangularfrequencyof1s‐1isassumedtohaveintensity 1w/m2,thenusing the previously stated gravitationalwave equation, the oscillating spatial displacementproducedoveradistanceequaltothereducedwavelengthis:ΔL 4.7 10‐10m.Iwillnotgothroughtheentirenumericalexample,butaλ3volumehasan interactivemassof4 1035kg.Ignoringnumericalconstants,theenergydepositedbythegravitationalwaveinthisvolumeisE λ2/ω 9x1016J.Ifyoucalculatethedistancethatthisenergywillmovea4x1035kgmassin

time1/ω,itturnsouttoalsobe4.7x10‐10m ignoringnumericalconstantsnear1 .Therefore,thedisplacementofspacetimeΔxobtainedfromgeneralrelativitycorrespondstothedistance4.7x10‐10m thattheinteractivemass orinteractiveenergy canbemovedinatimeof1/ω.


Thedipolewavesinspacetimecontainedinthegravitationalwavevolumecannotbephysicallymovedbecausetheyarealreadypropagatingatthespeedoflight.Instead,thegravitationalwaveiscausingaslightchangeinfrequencywhichproducesashiftinenergyequivalenttoimpartingkinetic energy to amass equal to the interactivemass discussed. Nowwe can conceptuallyunderstand why gravitational waves are so hard to detect. They are interacting with thetremendouslylargeenergydensityofthespacetimefield.Evenwithalargefrequencymismatch,thegravitationalwavesarestillchangingthefrequencyofaverylargeenergyofdipolewavesinspacetime.Connection to Black Holes: Sofar, thediscussionhascenteredofgravitationalwaveswithangular frequency ω and reduced wavelength λ interacting with the energy density of thespacetimefield. However,forgeneralusetheenergydensitycharacteristicsofthespacetimefieldshouldreallybeexpressedusingthesubstitutionλ r,whereristheradiusofasphericalvolumeofthespacetimefieldratherthanλorωpertainingtogravitationalwaves.Forexample,lateritwillbeproposedthatgravityandelectricfieldsbotharetheresultofadistortionofthespacetimefield.EventhoughthespacetimefieldhasPlanckenergydensity,thisimpliesaPlancklength interaction volume. A larger radius volume interacts in such a way that there is areductioninthecouplingefficiencysimilartotheeffectdescribedforgravitationalwaveswhenλ Lp.ThereforetheequationsforUiandρicanberewrittenusingradiusr.Therefore,wehave:

and

TheseequationsshouldbecomparedtotheequationsforablackholewithSchwarzschildradiusRs≡Gm/c2 explainedinchapter2asdefinitionusedhereforSchwarzschildradius .Theblack

holeenergydensityisdesignatedUbhandthedensityofablackholeisρbh.

and

Therefore, it can be seen thatwe have the same equations. This is another case of generalrelativityconfirmingtheenergydensitycharacteristicsofthespacetimefield.Thepicturethatwillemergeisthatblackholesoccurwhentheenergywithinasphericalvolumeofradiusrfromfermionsandbosonsequalstheinteractiveenergyofdipolewaves whenUbh Ui Anotherinsightintoblackholescanbegainedbyimagingtworeflectinghemisphericalshellsconfining photons at energy density of about 3 J/m3. This photon energy density striking areflecting surface generates pressure of ℙ 2 newton/m2. To hold together the twohemisphericalshellswouldtaketwoopposingforcesof2newtontimesthecrosssectionalareaofthehemispheres.Nextwewillimagineincreasingthephotonenergydensitytothepointthat


itmeetstheenergydensityofablackholewitharadiusequaltotheradiusofthehemisphericalshells.Ignoringgravity,theforcerequiredtoholdtheblackholesizesphericalshellstogethercan be easily calculated. For energy propagating at the speed of light, as previouslydemonstrated,energydensityequalspressure U kℙ . TheequationUi Fp/r2 becomesℙ Fp/Rs2.Ignoringconstantsnear1,PlanckforcemustbesuppliedbythespacetimefieldoverareaRs2tocontaintheinternalpressureofanysizeblackhole.ThesmallestpossibleblackholeconsistingofphotonswouldbeasinglephotonwithPlanckenergyinavolumePlancklengthinradius.Aconfinedphotonof thisenergydensitywouldgeneratePlanckpressure Fp/Lp2 10113N/m2butsincetheareaisonlyLp2,thetotalforcerequiredtoholdthetwohemispherestogether isPlanck force 1044N.A supermassiveblackholesuchas foundat the centerofgalaxieshasmuchlargerradiusandthereforemuchlowerenergydensity.Howeverevenasupermassivebackholerequiresthesameamountofforce Planckforce toholdtheshellstogether.Normallyphysicistsmerelyacceptthatgravitycangenerate this forceandtheydonot try torationalize the physics that causes the various “laws” of physics. In the case of gravity, thespacetimefieldwillbeshowntoapplyarepulsiveforce pressure whichweinterpretastheforceofgravity. Themaximumforcewhich thespacetime fieldcangenerate isPlanck force,thereforeallblackholes,regardlessofsize,requirethisforcetoconfinetheinternalenergy.Why Does the Energy Density of the Spacetime Field Not Collapse into a Black Hole?Theenergy in thespacetime fielddoesnotcollapseandbecomeblackholesbecause this formofenergyistheessenceofspacetime vacuum itself.Thesewavesformthebackgroundenergetic“noise”oftheuniverse.Somequantummechanicalcalculationsrequire“renormalization”whichassumesthatonlydifferencesinenergycanbemeasured.Thereforethebackgroundenergeticfluctuationswhichonlymodulatedistanceby Lpandtherateoftimeby Tpcanusuallybeignored.However,whenweareworkingonthescalewhichcharacterizesvacuumenergy,thenthese small amplitudewavesmust be acknowledged andquantified. These small amplitudewavesarethebuildingblocksofeverythingintheuniverse.Theyareultimatelyresponsiblefortheuncertaintyprincipleandtheygivespacetimeitspropertiesofc,G,ħ,andεo.Thestandardmodelhas17namedparticleswithatotalof61particlevariations colorcharge,antimatter, etc. . Each of the fundamental particles is described as an “excitation” of itsassociatedfield.Therefore,accordingtothestandardmodelthereareatleast17overlappingfields,eachwithitsassociatedenergydensity.Forexample,theHiggsfieldhasbeenestimatedtohaveenergydensityof1046J/m3.Therefore,eventhestandardmodelhasenergetic“fields”whichdonot collapse into blackholes. The spacetime‐basedmodelmerely replaces the17 separatefieldswithunknownstructurewithone“spacetimefield”withquantifiablestructuralproperties.Gravitationalwaveequationshavebeenshowntoimplytheexistenceofthisvacuumenergy density. Zero point energy has long characterized the vacuum as being filled with“harmonic oscillators” with energy of E ½ ħω and energy density of U kħω/λ3. Thespacetime‐basedmodel of the universe characterizes the vacuumenergy as dipolewaves in


spacetimewhichlackangularmomentum.Thishomogeneousenergydensityisresponsibleforthepropertiesofthequantummechanicalvacuum.Curvatureofthespacetimefieldoccurswhenenergypossessingquantizedangularmomentumfermionsandbosons isaddedtothishomogeneousenergydensity.Blackholeswithradiusrareformedwhentheenergydensityoffermionsandbosons quantizedangularmomentum equalstheinteractiveenergydensityofasphericalvolumewithradiusr whenUi Ubh .Inthis case r Rs and we need to measure the radius by the circumferential radius methodpreviouslyexplained. Statedanotherway, theenergydensityof the spacetime fielddoesnotcauseblackholes,itformsthehomogeneousvacuumwithnocurvature.Introducingfermionsandbosonsintothishomogeneousfielddistortsthisuniformbackgroundenergydensity.Whenthisdistortingenergyequalstheinteractiveenergydensityofthespacetimefield,thenthisisthelimitingcondition.This“contamination”distortsthespacetimefieldtotheextentthatitformsablackhole.Interactive Bulk Modulus:InpreviousdraftsofthisbookIalsoincludedaderivationofthe“interactivebulkmodulusofspacetime”designatedwiththesymbolKs.Thebulkmodulusofa

fluidisK≡ ℙ

⁄whichisthechangeinvolume/totalvolumeinresponsetoachangeinpressure

Δℙ.ThiswasatediouscalculationwhichendsupwiththesameanswerastheinteractiveenergydensityUi.Hereisthefinalconclusionofthebulkmoduluscalculation.

Ks Ks interactivebulkmodulusofspacetime

Actuallyinacousticsitisknownthatthebulkmodulusisthesameastheenergydensityforidealgasses ignoringconstantsnear1 .Thiscanbeunderstoodbecauseaspreviouslyshown,energydensityisthesameaspressureforenergypropagatingatthespeedoflightorforidealgasses.Atthepointthat ⁄ 1,thiscorrespondstoΔℙ ΔUwhichinthiscontextisequivalenttotheinteractiveenergydensityofspacetime.Therefore,thisderivationisnotrepeatedhere.Insights into the Speed of Light: Thekeyconceptofspacialrelativityisthatthespeedoflightisconstantinallframesofreference.Inthe19thcenturyphysicistspostulatedthattherewasafluid‐likemediumthatpropagatedlightwavescalledthe“luminiferousaether”. ThisconceptwaslargelyabandonedaftertheMichelsonMorleyexperimentshowedthatthespeedoflightwas constant in all frames of reference and Einstein’s special relativity generated equationswhichdidnotrequiretheaether.However,Einsteinhimselfcontinuedtorefertotheaetheror“physical space” until his death.16 The description of the aether was merely changed tocorrespondtothephysicalpropertiesofspace.Isthereanyreasontobelievethatthespacetimefieldpossessesthepropertythatwouldallowwaves propagating in the spacetime field to propagate at the speed of light in all frames of 16 L. Kostro, Einstein and the Ether, (2,000) Apeiron, Montreal, Canada


reference? Gravitational waves propagate in the medium of the spacetime field and theypropagateatthespeedoflight.Inanyframeofreference,anobserverwouldseeagravitationalwavepropagatingatthespeedoflight.IfitwaspossibletodoaMichelsonMorleyexperimentusinggravitationalwaves,nomotioncouldbedetectedrelativetothemediumofthespacetimefield.Ifspacetimeisvisualizedasamediumconsistingofinteractingdipolewavespropagatingat the speed of light, then it is reasonable that itwould be impossible to detect anymotionrelativetothismedium.ItcanbeshownthatwavesofanykindthatpropagateinthemediumwithimpedanceZs c3/GandstrainamplitudeAs Lp/λwillpropagateatc,thespeedoflight.Vacuum Energy and the Einstein Field Equation:ThereisalsoasimilaritytotheEinsteinfieldequationwhichcanbeconsideredastatementthatenergydensityequalspressure.Ignoringthecosmologicalconstant,theEinsteinfieldequationcanbewrittenas:

Tμυ 8

Gμυset Fpand k

Tμυ kFpGμυThe left side of this equation has Tμυ which is the stress energy tensor with units ofenergy/length3whichisenergydensity.TherightsideofthisequationhasPlanckforceandGμυwhichistheEinsteintensorthatexpressescurvaturewithunitsof:1/length2.Therefore,theright side of this equation is force/area pressure. Therefore from the dimensions a validinterpretation of this equation is that the field equation is an expression of:energydensity kpressure.TheproportionalityfactorisequaltoPlanckforce c4/G timesanumericalfactornear1.Inthelimitofmaximumcurvature,Einstein’sfieldequationsaysthatPlanckforceisthemaximumpossibleforceintheuniverse17. Thoughts on the Impedance of Spacetime: It shouldbeemphasized that the impedanceofspacetimeisoneofthefewtrulyfundamentalpropertiesofspacetime.Forexample,lateritwillbeshownthatitispossibletomakeasystemofunitsthatuseonlythepropertiesofspacetime.OneofthethreepropertiesofspacetimeusedforthissystemofunitsisZs, theimpedanceofspacetime.InPlanckunits underlined ,theimpedanceofspacetimeisequalto1 Zs 1 .Also,theimpedanceofspacetimehasthefollowingconnectiontootherPlanckterms:Zs mp/Tp mpωp Fp/c pp/Lpwhere: mp Planck mass, Tp Planck time; ωp Planck frequency; Fp Planck force;pp Planckmomentum,TheimpedanceofspacetimeisintimatelyconnectedtoallthePlanckterms.ThesePlancktermsrepresentthelimitingvaluesofmass,force,length,momentum,etc.Theimpedanceofspacetime

17 Gibbons, G. W.: The Maximum Tension Principle in General Relativity Found. Phys. 32, 1891 (2002) http://arxiv.org/pdf/hep-th/0210109v1.pdf


isthemaximumpossiblevalueofimpedance.Inorderforgravitationalwavestopropagateatthe speed of light in the medium of the spacetime field, the required impedance is Zs. Theimpedance of spacetime also connects the mass of a black hole to its radius. For exampleZs 4.038 1035 kg/s, and a mass of 4.038 1035 kg has a defined Schwarzschild radiusRs≡Gm/c2equaltothedistancelighttravelsin1second about3x108meters .Ifamediumhasimpedance,theimplicationisthatthemediumhaselasticityandenergydensity.Inthecaseofwavesinspacetime,theimpedanceofspacetimeissolargeZs 4.038 1035kg/sthatevenasmallstrainamplitudeproducesaverylargeintensityforagivenfrequency.Thisenormous impedance of spacetime can only be achieved if spacetime has the large energydensity about 10113 J/m3 implied by zero point energy. The quantummechanical basis ofspacetimeisreal.Inchapter3wesawhowachangeinthegravitationalgammaΓaffectedtheunitsofphysics.ItshouldbenotedthattheimpedanceofspacetimeZsisoneofthefewtermsthatisunaffectedbya change in Γ. There is an analysis not presented here that concludes that Zs must beindependentofΓinorderforallthelawsofphysicstobecovariantwhenthereisachangeingravitationalpotential.Exchange Particles: Thestandardmodelusesexchangeparticlestotransferforce.Forexample,theelectromagneticforceissupposedlytheresultoftheexchangeofvirtualphotonsbetweenchargedpointparticles.Thesevirtualphotonstravelatthespeedoflight,sotheelectrostaticforceiscommonlyexplainedasresultingfromtheemissionorabsorptionofenergytravelingatthespeedoflight.Therefore,thepower P ofvirtualphotonsrequiredtogenerateagivenforceisalso:F Pr/c.Similarly,gravitonsarebelievedbymanyscientiststobetheexchangeparticlethatconveysthegravitationalforce.Whilethespacetimebasedmodeloftheuniversedoesnotrequireexchangeparticles,thepointisthatgravitonssupposedlyalsotravelatthespeedoflightandtheforcetheygeneratewouldalsobeF Pr/c. Forexample,apersonweighing70kgissupposedlybeingpulledtowardstheearthbyabout200billionwattsofgravitonsandthisisbeingresistedbyabout200billionwattsofvirtualphotonsstrikingthebottomofaperson’sshoestokeepthepersonfromsinking intotheEarth. It isproposedherethatgravitonsandvirtualphotonsarereplacedwithanequallylargepowerofinteractingwavesinspacetime.Gluonshavebeenignoredsofar,buttheyarealsoviewedashavinganexplanationassociatedwithwaves in spacetime discussed later .Theweak forcehasalreadybeenunitedwith theelectromagneticforcetoformtheelectroweakforce.Therefore,inthediscussiontofollow,Iwillconcentrateondeterminingtherelationshipbetweenthestrongforce,theelectromagneticforceandthegravitationalforce.However,abriefexaminationoftheweakforcewillbemadelater.NotethatIhaveusedtheterms“thestrongforce”and“thegravitationalforce”.Bothofthesetermsarecurrentlyoutoffavoramongphysicists.Atonetime“thestrongforce”wascommonly


usedtodescribetheforcethatboundprotonsandneutronstogetherinthenucleusofanatom.Sincethediscoveryofquarks,itwasnecessarytonametheforcethatbindsquarkstogetherandtheterm“thestronginteraction”isnowcommonlyused.Withthischangeinterminology,theforcethatbindsprotonsandneutronsisnow“theresidualstronginteraction”.Ineedasimplenameforthestrongestofallforces.Ichoosetoresurrecttheterm“thestrongforce”andredefinethisastheresultantforcefromaninteractionbetweenquarks wavemodel andvacuumenergywavemodel thatbindsquarkstogether.Newton considered gravity to be a force, but a popular physical interpretation of generalrelativity considers gravity to be the result of the geometry of spacetime. The equations ofgeneral relativity are commonly interpreted as describing curved spacetime. However, theconceptofcurvedspacetimedoesnotleadtoaconceptuallyunderstandableexplanationofhowaforceisgeneratedwhenamassisheldstationaryinagravitationalfield.Itwillbeshownthatgravity isareal force that iscloselyrelatedtoboththeelectromagnetic forceandthestrongforce.Therefore,theterm“thegravitationalforce”willbeusedevenpriortoofferingthisproof.

FieldsOnecommonconceptualmodeloftheuniverseamongphysicistsisamodelwhereparticlesareemphasized.Discreteparticlesareviewedaspointsofenergyaccompaniedbyawavefunction.TheCopenhageninterpretationofthiswavefunctionisawaveofprobabilitythatinfluencesthelocationofthepointparticlewhenthereisaninteraction measurement .Inotherwords,thisschoolofthoughtconsiderstheparticleportionthedominantconceptualportionofthewave‐particleduality.However, there is anopposingviewheldbymanyprominentquantum fieldtheorists.Theconceptisthattheuniverseismadeofmultiplefieldsratherthanprimarilybeingparticlebased.Particlesareviewedasquantizedexcitations quantizedwaves existingwithinthesefields.Thishasbeendocumentedinapaper18byArtHobsontitled“Therearenoparticles,there are only fields”. This paper gives numerous references supporting this position. Forexample,asFrankWilczekexplains19,“Inquantumfieldtheory,theprimaryelementsofrealityarenotindividualparticles,butunderlyingfields.Forexample,allelectronsarebutexcitationsofanunderlyingfield,….theelectronfield,whichfillsallspaceandtime.” Inthisconceptualmodel,eachfundamentalparticlehasitsownfield.Eventhestandardmodelisreallyaquantumfieldtheory.All17particlesofthestandardmodelareassociatedwith17differentoverlappingfields and this does not even include the gravitational field or other possible fields if morefundamental particles are discovered. This multitude of superimposed fields creates anunappealingcomplexpicture.

18 A. Hobson “There are no particles, there are only fields” Am. J. Phys. 81 (3), March 2013 19 F. Wilczek, “Mass Without Mass I: Most of Matter,” Phys. Today 52(11), 11-13 (1999)


Albert Einstein said the following in his 1923 Nobel lecture, “The intellect seeking after anintegratedtheorycannotrestcontentwiththeassumptionthatthereexiststwodistinctfields,totallyindependentofeachotherbytheirnature”.Hewasreferringtotheelectricfieldandthegravitational field appearing tobedistinctbuthebelieved that theymustbe closely related.Einsteinworked forabout30yearsattempting tounite these two fields. Today,Einstein’s2fieldshavegrownto17distinctfields–andcounting.Apparentlytheproliferationoffieldshasmadetheconceptofunitingtheseintoasinglefieldisadistantmemory.However,ifthebasicassumption of this book is that the universe is only spacetime, then the following corollaryfollowsfromthisstartingassumption:CorollaryAssumption: There is only one truly fundamental field. This single field is the quantum mechanical model of the spacetime field with its dipole wave vacuum fluctuations. All fundamental particles and all forces are manifestations of this single spacetime field. Another way of phrasing the starting assumption the universe is only spacetime is thefollowing: The universe is made entirely of a single field and its quantized components. Thissingle field is known to us as the spacetime field, but more specifically it is the quantummechanical spacetimewave descriptionofthespacetimefield.ThespacetimefieldmodelhaschaoticspatialfluctuationswithdisplacementamplitudeequaltoPlancklengthandtemporalfluctuations differencebetweenclocks withdisplacementamplitudeequaltoPlancktime.Thechaoticseaofdipolewavesinthespacetimefieldthatarethebasicconstituentofspacetimewillbeshowntohaveknowablepropertiessuchas:waveamplitude,impedance,frequencyrange,bulk modulus, energy density, propagation characteristics, angular momentum, etc. Laterchapterswillexplainhowthissinglefieldcanformallparticles,allforcesandwhatappeartobeotherfieldssuchasgravitationalfieldsandelectricfields.Summary – Properties of Spacetime: In upcoming chapters we are going to attempt toconstruct the universe particles, forces and photons using only the quantum mechanicalpropertiesof spacetime. Besides the standardproperties of spacetimedescribedby generalrelativity,itisusefultosummarizetheadditionalpropertiesofspacetimethatwehaveaddedtoourtoolbag.Theseadditionsare:

1 WehaveconcludedthatspacetimehasaquantifiableimpedanceofZs mpωp c3/GandinteractiveenergydensityofUi Fp/λ2 c2ω2/G.

2 Thequantummechanicalmodelofspacetimehasaseaofhighfrequency,smallamplitudevacuum fluctuations at Planck energy density ~ 10113 J/m3. This model is adoptedbecauseeventheimpedanceofspacetimeobtainedfromgeneralrelativitysupportsthismodel.

3 Dipole waves are allowed to exist in spacetime but they are subject to the Plancklength/timelimitationpreviouslydiscussed.Vacuumfluctuationsandzeropointenergyare actually dipole waves in spacetime. The uncertainty principle and probability


characteristics are the result of the random fluctuations of these dipole waves inspacetime.

4 Wearearmedwiththe5wave‐amplitudeequationsobtainedbycombiningageneralwaveequation kA2ω2Z withtherelativisticforceequation Fr = Pr/c).

5 Wepresume that theonly truly fundamental force is therelativistic force Fr Pr/c which is the repulsive force exerted when energy traveling at the speed of light isdeflected.Thedipolewavesinspacetimearealwaysmovingatthespeedoflightevenwhentheyareconfinedtoalimitedvolume.

6 Fromtheequivalenceofenergydensityandpressure, it followsthatthe largeenergydensity of vacuum fluctuations dipole waves is exerting an equally large vacuumpressure.

7 Whatappearstobemultipleseparatefieldsaredifferentresonanceswithinasinglefield.


“Elementaryparticlesrepresentapercentage‐wisealmostcompletelynegligiblechangeinthelocallyviolentconditionsthatcharacterizethevacuum…Inotherwords,elementaryparticlesdonotformareallybasicstartingpointfordescribingnature.”JohnArchibaldWheeler


Chapter5

SpacetimeParticleModel

“Thinkofaparticleasbuiltoutofthegeometryofspace;thinkofaparticleasageometrodynamicexcitation.”JohnArchibaldWheeler

Early Wave-Particle Model: In1926,ErwinSchrodingeroriginallyproposedthepossibilitythatparticlescouldbemadeentirelyoutofwaves.However,inanexchangeofletters,HenrikLorentzcriticizedtheidea.Lorentzwrote,

“Awavepacketcanneverstaytogetherandremainconfinedtoasmallvolumeinthelongrun.Evenwithoutdispersion,anywavepacketwouldspreadmoreandmoreinthetransversedirection,whiledispersionpullsitapartinthedirectionofpropagation.Becauseofthisunavoidableblurring,awavepacketdoesnotseemtometobeverysuitableforrepresentingthingstowhichwewanttoascribearatherpermanentindividualexistence.”

Schrodinger’sideaofawave‐particlewasagroupofdifferentfrequencywavesthat,whenaddedtogether, formed a Gaussian shaped oscillating wave confined to a small volume Fouriertransformation .SchrodingereventuallyagreedwithLorentzthatthewavesthatformedsucha“particle”woulddisperse.However,thisinitialfailureshouldnotbeinterpretedascondemningallpossiblewaveexplanationsforparticles.Forexample,opticalsolitonsarecompactpulsesoflaserlight waves thatpropagateinnonlinearopticalmaterialswithoutspreading.Theyexhibitparticle‐likepropertiesandwillbediscussedlater.In this chapter a model of a fundamental particle will be proposed made entirely out of aquantizeddipolewave in the spacetime field. Even though thismodel givesa structureandphysicalsizetoanisolatedfundamentalparticle,thereaderisaskedtoreservejudgmentaboutthismodeluntilitcanbefullyexplained.Thespacetimebasedmodelofafundamentalparticlehaswaveswithphysicalsize,butthismodelwillbeshowntobeconsistentwithexperimentsthat indicate no detectable size in collision experiments. Also the spacetime based modelexplainshowanelectroncanformacloud‐likedistributionundertheboundaryconditionsofaboundelectroninanatom.

Brief Summary of the Cosmological Model: Wewill startwith a brief description of thecosmologicalmodelproposedtobecompatiblewiththestartingassumption.Thecosmologicalmodeliscoveredindetailinchapters13and14.Iftheuniverseisonlyspacetimetoday,itmusthavealwaysbeenonlyspacetime.Notonlyareallparticles,fieldsandforcesderivedfromthe


propertiesof4dimensionalspacetime,butallthecosmologicalpropertiessuchastheexpansionoftheuniversearealsoderivedfromthechangingpropertiesof4dimensionalspacetime.Thiswillbediscussedlater.ThehighestenergydensitythatspacetimecansupportisPlanckenergydensity ~10113J/m3 .Onepointofpossibleconfusionisthat~10113J/m3isboththeenergydensityatthestartoftheBigBangandthecurrentenergydensityofthespacetimefield.Whatisthedifference?Todaytheenergydensityoftheuniverseobtainedfromgeneralrelativityandcosmological observations is about 10‐9 J/m3. However, this is proposed to only be the“observable”energydensityof theuniversepossessedbyall the fermionsandbosons in theuniverse.Wecanonlydirectlyinteractwithfermionsandbosonswhicharewavesthatpossessquantizedangularmomentum.However,thevastlylargerenergydensity ~10113J/m3 isthe“unobservable”vacuumenergyofthespacetimefield.ThisisthePlanckamplitudewavesinthespacetimefieldwhichlackangularmomentum.Thisisthestructureofthespacetimefielditself.Thisstructuregivesthespacetimefieldsconstantssuchasc,ħ,εo,G,Zs,etc.

Fromourcurrentperspective,spaceappearstobeanemptyvoid.However,asJohnArchibaldWheeler1said,“Emptyspaceisnotempty…Thedensityoffieldfluctuationenergyinthevacuumarguesthatelementaryparticlesrepresentpercentage‐wisealmostcompletelynegligiblechangeinthelocallyviolentconditionsthatcharacterizethevacuum.”Itisthisenergeticvacuumthatwearecallingthe“spacetimefield”.Vacuumenergycanbesaidtohaveatemperatureofabsolutezerobecauseitlacksanyquantizedunitsandtemperatureisdefinedasenergyperquantizedunit.ThesameenergydensityatthestartoftheBigBangwasintheformof100%quantizedspin units. The energy per quantized unit was equal to Planck energy and therefore thetemperature at the start of the Big Bang was approximately equal to Planck temperature∿1032°K .

Even though10113 J/m3 isan incredibly largenumber, it isnota singularitywhichwouldbeinfiniteenergydensity.ForthespacetimefieldtoreachPlanckenergydensity Up spacetimemust have dipolewaves at the highest possible frequency ωp Planck frequency and thelargestpossibleamplitude A 1 .

U A2ω2Z/cset:A 1,ω ωp /ħ 1.9 1042s‐1Z Zs c3/G

U ħ

4.6 10113J/m3 Planckenergydensity

Unlike virtual particle pairs which have a very short lifetime, Planck amplitude waves inspacetime can last indefinitely because they are undetectable even if there is continuousobservationtime.APlancklengthdisplacementofspaceoraPlancktimedisplacementoftimeadvance or retard clocks is fundamentally undetectable. Only if a Planck amplitudewavepossessesquantizedangularmomentumofħor½ħdoesitbecomedetectable observable by

1 Misner, C. W., Thorne, K. S. and Wheeler, J A.: Gravitation. (W. H. Freeman and Company, New York 1973) P. 975.


us. Therefore today’s spacetime has vacuum energy density of 10113 J/m3 but none of thisvacuumenergypossessesquantizedangularmomentum.ThespacetimeatthebeginningoftheBig Bang also had energy density of 10113 J/m3 but then 100% of that energy density wasobservable. Photons propagating in spacetime today are undergoing a redshift because ofuniversal expansion. The photon’s lost energy possesses no angular momentum. All of thephoton’s angular momentum remains in the form of approximately the same number ofredshiftedphotons.In chapters 13 and 14 the casewill bemade thatwhatwe perceive as an expansion of theuniverse is actually a more complex transformation of spacetime. Even fermions such aselectrons and quarks are also losing energy on a proposed absolute scale. Ultimately, thistransformation of spacetime is responsible for converting spacetime from being 100%observable at a temperature of about 1032 °K Planck temperature times a constant at thebeginningoftheBigBang 100%quantizedangularmomentum totodaywherealmostalltheenergyof thespacetime field is in the formofwaves that lackangularmomentum. ItwillbeshownthatthisexplanationgivesthecorrectdifferenceintemperaturebetweenthestartoftheBigBangandtoday’sCMBtemperatureof2.725°K.

Ifwejumpforwardintime,todayonlyabout1partin10122ofthetotalenergyintheuniverseincludingvacuumenergy possessesquantizedangularmomentumofħor½ħ.Furthermore,thisfractioniscontinuouslydecreasingbecauseofthecosmicredshiftandanothercharacteristicdescribedlater.Allthefundamentalparticlesandforcesthatwecandetectarethe1partin10122thatpossessesquantizedangularmomentum.Thevastlylargerenergyintheuniverseistheseaofvacuumfluctuations dipolewavesinspacetime thatdoesnotpossessangularmomentum.TheonlyhintwehavethatthisvastenergydensityexistsisthequantummechanicaleffectssuchastheLambshift,Casimireffect,vacuumpolarization,theuncertaintyprinciple,etc.However,itwillbeshownthatthisseaofvacuumfluctuationsisessentialfortheexistenceoffundamentalparticlesandforces.Vacuum Energy Has Superfluid Properties: Ifwe are going to be developing amodel offundamental particles incorporating waves in spacetime, it is important to understand theproperties of the medium supporting the wave. In the last chapter we enumerated manypropertiesofthespacetimefield.Thepointwasmadethatthepropertiesofvacuumfluctuationsareanintegralpartofthepropertiesofthespacetimefield.However,onepropertyofvacuumenergy was intentionally saved for this chapter because it is particularly important in theexplanationoffundamentalparticlesformedoutofwavesinspacetime.It is proposed that vacuum energy has the property that it does not possess angular momentum. Any angular momentum present in the midst of the sea of vacuum energy is isolated into units that possess quantized angular momentum. These quantized angular momentum units have different properties than vacuum energy.


ThisconceptiseasiesttoexplainbymakingananalogytosuperfluidliquidheliumoraBose‐Einstein condensate. When the helium isotope 4He is cooled to about 2° K, it changes itspropertiesandpartlybecomesasuperfluid.Coolingtheliquidfurtherincreasesthepercentageoftheheliumatomsthatareinthesuperfluidstate.Coolingsomeotheratomstoatemperaturevery close to absolute zero changes their properties and a large fraction of these atoms canoccupy the lowest quantum state and exhibit superfluid properties. This is a Bose‐Einsteincondensate.SincesuperfluidliquidheliumisaspecialcaseofaBose‐Einsteincondensate,theywillbediscussedtogether.Whenagroupofatomsoccupyasinglequantumstate,thegroupmustexhibitquantizedspinona macroscopic scale. The quarks and electrons that form atoms individually are fermions.However,aBose‐Einsteincondensateorsuperfluid4Heexhibitsquantizedspinonamacroscopicscale.Thegroupoffundamentalparticlescanpossesseitherzerospinoranintegermultipleofspin units related to ħ. Ifwe have a group of atoms in a Bose‐Einstein condensate andweintroduceangularmomentum “stir” thecondensate , thenwecan form“quantizedvortices”that possess quantized angular momentum within the larger volume of Bose‐Einsteincondensatethatdoesnotpossessmacroscopicangularmomentum.Thereforeaquantumvortexisagroupofatomsthathasadifferentangularmomentumquantumstate differentspin thanthe larger group of surrounding atoms that forms the superfluid 4He or Bose‐Einsteincondensate.Thiseffectwasfirstdiscoveredwithsuperfluidliquidhelium2.DramaticpicturesarealsoavailableofmultiplequantumvorticesinaBose‐Einsteincondensate3,4.ItisproposedthatthequantumfluctuationsofspacetimeareaLorenzinvariant“fluid”thatisthemostidealsuperfluidpossible.Unlikethefundamentalparticles fermions thatformaBose‐Einsteincondensate,thevacuumfluctuationsdonotpossessanyquantizedangularmomentum.However,vacuumfluctuationsaresimilartoasuperfluidorBose‐Einsteincondensatebecauseitisolatesangularmomentumintoquantizedunits.Thesequantizedangularmomentumunitsthatexistwithinvacuumenergyareproposedtobethefermionsandbosonsofouruniverse.Thesamewaythatthequantizedvorticescannotexistwithoutthesurroundingsuperfluid,soalsothefermionsandbosonscannotexistwithoutbeingsurroundedbyaseaofvacuumenergy.ThetotalangularmomentumpresentinthequantumfluctuationsofspacetimeatthestartoftheBigBangprobablyaddeduptozero.However,eventhoughcounterrotatingangularmomentumcancancel,stillthereshouldbeoffsettingeffectsthatshouldstatisticallypreservethequantizedangularmomentumunitsfromtheBigBangtotoday calculatedinchapter13 .Dipolewavesinspacetimethatpossessangularmomentumwouldnotbethesameasthedipolewavesthat

2 E.J. Yarmchuk, M.J. Gordon and R.E. Packard, Observation of stationary vortex arrays in rotating superfluid helium, Phys. Rev. Lett. 43, 214-217 (1979) 3 K. W. Madison, F. Chevy, W. Wohlleben and J. Dalibard, Vortex lattices in a stirred Bose-Einstein condensate cond-mat/0004037[e-print arXiv]. 4Ionut Danaila, Three-dimensional vortex structure of a fast rotating Bose-Einstein condensate with harmonic-plus-quartic confinement, http://arxiv.org/pdf/cond-mat/0503122.pdf


formvacuumfluctuations. Aunitthatpossessesquantizedangularmomentumlosesitsidealsuperfluid properties because it can interact with another unit of energy that possessesquantizedangularmomentum forexample,exchangeangularmomentum .Spacetime Eddy in a Sea of Vacuum Energy: Itisproposedthatwhatweconsidertobethefundamentalparticlestoday quarksandleptons arethespacetimeequivalentofthevorticesthat carry quantized angular momentum in a superfluid. However, the quantized angularmomentumentitiesarebettervisualizedaschaoticeddiesthatexistsinthevacuumfluctuationsofthespacetimefield.Wecanonlydirectlyinteractwiththedipolewavesinthespacetimefieldthatpossessquantizedangularmomentum.Weareunawareofthevastamountofsuperfluidvacuumenergythatsurroundsusbecauseitonlyindirectlyhasanyinfluenceonus.In the spacetime based model of the universe, fundamental particles are dipole waves in spacetime that possess quantized angular momentum. They are living in a sea of superfluid vacuum fluctuations that cannot possess angular momentum. Fundamental particles cannot exist without the support provided by this sea of superfluid vacuum fluctuations.Spin and Particle Size: Before describing how fundamental particles can be formed out of the properties of spacetime, I would like to point out several problems with the vague “point particle” description of particles currently used in quantum mechanics. Molecules have experimentally observable size, so we will start there. There is experimental evidence which indicates that molecules possess physical angular momentum. For example, molecules suspended in a vacuum physically rotate at specific frequencies. Each isolated molecule has a fundamental rotational frequency and allowed rotational harmonics of that frequency. For example, a carbon monoxide molecule has a fundamental rotation frequency of 115 GHz. This molecule can only rotate at this frequency or at higher frequencies which are integer multiples of the fundamental frequency. This fundamental rotational frequency corresponds to the carbon monoxide molecule possessing ½ħ of angular momentum. The carbon monoxide molecule has physical dimensions which can be measured. The rotation can be visualized as a classical physical rotation. The quantized rotational frequencies are not classical, but even these are conceptually understandable if the molecule is visualized as existing within the superfluid spacetime field which enforces quantized angular momentum. The reason for going into this detail about the rotation of molecules is that the physical rotation of a molecule is going to be contrasted to fundamental particles which are said to possess an “intrinsic” form of angular momentum known as “spin”. The concept is that the fundamental particles somehow possess a quality with dimensions of angular momentum, but without any part of the fundamental particle undergoing a classical rotation. The problem is that the description of a fundamental particle as a point particle or a Planck length string does not allow for any physical rotation which achieves ½ ħ of angular momentum. For example, an electron has energy of 0.511 MeV. If a photon had this energy, it would have to propagate at the speed of light around a circle


larger about 10-13 m in order to have ½ ħ of orbital angular momentum. Since relativistic collision experiments seem to imply that an electron is no larger than 10-18 m, this 10-13 m physical size is rejected. The angular momentum of an electron is the first of three areas that imply a size mystery. The second conflict is the implied minimum size of an electron calculated from the electric field of an electron. An electron possesses an electric field with measurable energy density. The classical radius of an electron is calculated by finding the radius where the total energy contained in the electron’s electric field equals the electron’s energy (0.511 MeV). This “classical electron radius” equals 2.8×10-15 m. This is the implied minimum size of an electron based on our knowledge of electric fields. Since this is much larger than the previously mentioned 10-18 m, this is another size mystery. If an electron is actually approximately Planck length in radius as in some models, then the energy of the electron’ electric field should be about 1020 times larger than 0.511 MeV. Rather than admit that there is a possible problem with the point particle model, the calculated maximum size of an electron is dubbed the “classical” electron radius. The word “classical” is the kiss of death to any quantum mechanical explanation. A third problem occurs in quantum electrodynamics calculations. When the size of an electron is required for a calculation, then the analysis gives a ridiculous answer of infinity if the electron is assumed to be a point particle. The process of renormalization is required to eliminate the infinity. However, this is an artificial adjustment of the answer so that it is no longer a faithful extension of the starting assumptions. An analysis of this problem shows that in order to obtain a reasonable answer (not infinity), the radius of the electron must be assumed to be larger than the classical electron radius. This also implies a size mystery. These three examples are given here because they illustrate some of the problems with the point particle model used in quantum mechanics. The explanation usually given to students is that the point particle model is correct, but quantum mechanics is a mathematical subject. Physical explanations are simply beyond human understanding. How is it possible to conceptually understand a point particle possessing “intrinsic” angular momentum with nothing actually rotating? How is it possible for an electron to be much smaller than its classical radius? Will we ever be able to understand an electric field in terms of something more fundamental? Will we ever be able to understand how a fundamental particle causes curved spacetime? All of these questions will be answered in this book. The first step is to examine the proposed spacetime particle model which solves all of the size mysteries and many other related mysteries. Rotar: Thesimplest formofquantizedangularmomentumthatcanexist inaseaofdipolewaves in spacetime would be a rotating dipole wave that forms a closed loop that is onewavelengthincircumference.Adipolewaveinspacetimeisalwayspropagatingatthespeedoflight,evenifitformsaclosedloop.ThisagreeswiththehighlysuccessfulDiracequationwhichrequiresthatafundamentalparticlesuchasanelectronisalwayspropagatingatthespeedof


light.Diracexpressedthisrequirementmathematicallyas csoanelectroncanhaveanaveragevelocityofzerowhilehavingawavemovinginaconfinedvolumeataspeedofc.ThisrotatingdipolewaveinspacetimeisstillsubjecttothePlancklength/timelimitation,soitcanbethoughtofasbeingatthelimitofcausality.Itsrotationischaoticratherthanbeinginasingleplane.Ithasadefinableangularmomentum,butallrotationdirectionsarepermittedwithdifferentprobabilitiesofobservation. Theexactoppositeoftheexpectationdirectionhaszeroprobability .Therefore,theproposedmodelofanisolatedfundamentalparticleisadipolewaveinthespacetimefieldthatformsarotatingclosedloopthatisonewavelengthincircumference.Thisobviouslyisadrasticdeparturefromthestandarddefinitionoftheword“particle”. Thestandardmodelofa fundamentalparticle isamass thathasnodiscerniblephysicalsize,butsomehowexhibitswaveproperties,angularmomentumandinertia.Itisanaxiomofquantummechanicsthattheψfunctionhasnophysicalinterpretation.Thisvaguenesswillbereplacedwithatangiblephysicalmodelthatexplainsmanyofthepropertiesexhibitedbyfundamentalparticles.Anewnameisrequiredtodistinguishbetweenthestandardconceptofafundamentalparticleandtheproposedmodelofarotatingdipolewaveinspacetime.Thenewnameforthespacetimemodelofafermionwillbe:“rotar”. It iswithgreatreluctancethatanewwordiscoined,butthisisnecessaryforclarityandbrevityintheremainderofthisbook.Thenamerotarrefersspecificallytothespacetimedipolemodelofafundamentalparticlethatexhibitsrestmass afermion .Thismodel,anditsvariations,isdescribedindetaillater.Bosonswithoutrestmass,suchasaphoton,haveadifferentmodelandarenotcoveredbytheterm“rotar”.Therewillbeaspacetimemodelofaphoton,butthatwillbeintroducedlater.Theword“particle”willbeusedwheneverthecommondefinitionofaparticleisappropriateorwhenitisnot necessary to specifically refer to the spacetime particlemodel. For example, the name“particleaccelerator”doesnotneedtobechanged.Eventheterm“fundamentalparticle”willoccasionallybeused in theremainderof thisbookwhen theemphasis isondistinguishingaquarkorleptonfromcompositeobjectssuchashadronsormolecules.Trial and Error:TheangularmomentumpresenttodayintheformoffermionsandbosonswaspresentattheBigBangintheformofphotonswiththehighestenergypossiblewhichisPlanckenergy.Thisstartingconditionwillbediscussedmoreinthechaptesoncosmology.Eventhoughphotonsmakeupasmallpercentageoftheenergyintheuniversetoday,thephotonsofthe cosmic microwave background still possess the vast majority of the quantized angularmomentumintheuniverse.Theearlyuniversewasradiationdominated,butenergeticphotonscancombinetoformmatter/antimatterpairs.However,thereisaslightpreferenceformatteraboutonepartinabillion .Astheearlyuniverseexpanded transformed ,thechaoticdipolewavesinspacetimeexploredeveryallowedcombinationoffrequencyandamplitudeinaneffortto find themostsuitable formtohold theunwantedangularmomentum. By trialanderror,relatively long lived spacetime resonances were found that both held quantized angular


momentumandalsowerecompatiblewiththepropertiesofthevacuumenergydipolewavesinspacetime. These spacetime resonances are proposed to be the fundamental particlesfundamentalrotars .Itwillbeshownlaterthattheknownfundamentalrotarsareresonancesthathavefrequenciesbetweenabout1020and1025Hz.In theproposedearlystagesof theBigBang, themostenergeticspacetimeparticles rotars formed first. Forexample, tauons tau leptons formedbeforemuons. Thesewerepartiallystable resonances. They survived for perhaps 1011 cycles, but not indefinitely. They thendecayed into other energetic rotars and photons. The radiation dominated universe wasundergoinga large redshift. Energywasbeing removed fromphotonsand transformed intovacuumenergy. Thisloweredthetemperatureoftheobservableportionoftheuniversethatpossessesangularmomentum photons,neutrinosandrotars .Eventually,otherfundamentalspacetimeresonances rotars wereformedatlowerfrequencies lowerenergy . Eventually,trulystableresonancesformedandthesewereelectronsandtheupanddownquarksthatfoundstabilitybyformingprotonsandneutrons.Wave-Particle Duality: Before launching intoamoredetaileddescriptionandanalysisof arotar, it is interesting to initially stand back and look at the philosophical difference inperspectiverequiredtoimagineaparticlemadeentirelyofdipolewavesinspacetime.Atfirst,theideaofaparticlemadeoutofwavesinspacetimeseemstobeintuitivelyunappealing.Theessenceofaparticleissomethingthatactsasaunit.Waves,ontheotherhand,areimaginedtobeinfinitelydivisible. Whenaparticleundergoesacollision, itrespondsasasingleunit inacollision.Thispropertyseemsincompatiblewitharotarmadeentirelyofawave.Thesuperfluidpropertiesofthespacetimefieldmakesangularmomentumintoquantizedunitsof ½ ħ or ħ. The same way that a superfluid Bose Einstein condensate isolates angularmomentumintodiscretevorticies,eachwithħofangularmomentum,soalsospacetimeisolatesangular momentum into isolated units possessing quantized angular momentum.. It is notpossibletointeractwithjust1%ofaquantizedunitofangularmomentum.Itisallornothing.Therefore, dipole waves in spacetime possessing ½ ħ of angular momentum appear to beparticlesbecausetheyrespondtoapertibationasaunit.LaterinthisbookIwillpostulateanewpropertyofnaturecalled“unity”.Thispropertyiscloselyrelatedtoentanglement.Itpermitsadipolewavepossessingquantizedangularmomentumtocommunicateinternallyfasterthanthespeedoflightandrespondtoaperturbationasasingleunit. This property would impart a “particle like” property to a quantized dipole wave inspacetime. However, the quantized wave would not exhibit classical particle properties.“Finding”theparticlewouldbecomeaprobabilisticeventbecausewearereallydealingwithinteractingwithawavecarryingquantizedangularmomentumthatisdistributedoverafinitevolume.AquantizedwaveinthespacetimefieldcanexhibitbothangularmomentumandgiveaphysicalinterpretationtothepathintegralofQED.Thereisactuallyagreatdealofappealto


fundamentalparticlesbeingmadeofquantizedwavesinspacetime,providedthatthemodelisplausible.Anotherobjection to a rotarmodelmade entirely ofwaves is that our experiencewith lightseems to imply that waves do not interact with each other. Light does have a very weakgravitational interaction, but overall lightwaves exhibit almost no interaction. Thewaves inspacetimethatareproposedtobethebuildingblocksofallmatterandforcesmustbeabletointeractwitheachother.Any wave that exhibits nonlinearity will interact to some degree with a similar wave. Forexample,soundwaveshaveaslightnonlinearity.Thereisatemperaturedifferencebetweenthecompressionandrarefactionpartsofasoundwave.Thisslighttemperaturedifferenceproducesaslightperiodicdifferenceinthespeedofsound.Thisslightnonlinearityinasinglesoundwavemeansthattwosuperimposedsoundwavesdointeract. However,atcommonlyencounteredsoundintensities,theinteractionbetweentwosoundwavesisverysmall.Gravitational waves also have a slight interaction because general relativity shows thatgravitationalwavesarenonlinear.Oneoftheappealsofdipolewavesinspacetimeisthattheyexhibittherequiredabilitytointeractwitheachother.Infact,dipolewavesinteractsostronglythat they would cause a violation of the conservation of momentum without the quantummechanicalPlancklength/timelimitationpreviouslydiscussed.Theinteractionsbetweendipolewavesinspacetimewillbeshowntoberesponsibleforalltheforcesincludinggravity.Thespacetimefieldisthestiffestpossiblemedium.TheincrediblylargeimpedanceofspacetimeZs c3/G 4 1035kg/s permitsawavewithsmalldisplacementtohavetheveryhighenergydensityforagivenfrequency. Whenwehavefrequenciesinexcessof1020Hz,thenwavesinspacetimearecapableofachievingtheenergydensityoffundamentalparticles.WhenwepermitthefrequencytoreachPlanckfrequency,wecanachievetheenergydensityrequiredatthestartof the Big Bang Planck energy density . Also, a universe made only of spacetime andperturbationsofspacetimehasanappealingsimplicity.Particle Design Criteria:Everythingthathaspreviouslybeensaidinthisbookhassetthestageforthetaskofattemptingtodesignandanalyzeaplausiblemodelofafundamentalrotar. Indesigningarotarfromdipolewavesinspacetime,thereisonefactorthatwillbetemporarilyignored.Thisistheexperimentalevidencethatseemstoindicatethatfundamentalparticlesarepointswithnophysicalsize.Therotarmodelwillbeshowntoexhibitthisproperty,butthiswillbeanalyzedlater.Itshouldbeexpectedthatthefirstmodeloffundamentalparticleswillbeoverlysimplified.Forexample,thisfirstgenerationrotarmodelwillmakenodistinctionbetweenleptonsandquarks.Subsequentgenerationsof the rotarmodel shouldmakesuchadistinctionandexhibitother


refinements.Thehopeisthatthefirstgenerationrotarmodelwillpassenoughplausibilityteststhatotherswillbeencouragedtoimproveonthismodel.Thereare6considerationsthatwillbebroughttogetherinanattempttodesignafundamentalparticle.Theseare:

1 The universe is only spacetime. The energetic spacetime field contains waves inspacetime. Thewaves canbedipolewaves with thePlanck length/time limitation ,quadrupolewaves forexample,gravitationalwaves orhigherorderwaves.Ofthese,onlydipolewavesinspacetimemodulatetherateoftimeandmodulatevolume.Thesearethecharacteristicsrequiredtobethebuildingblocksofrotars.

2 The rotar model should exhibit inertia. As shown in chapter 1, this requires energytraveling at the speed of light, but confined to a limited volume in a way that themomentumvectorsgenerallycancel.

3 Therotarmodelshouldexhibitangularmomentum.Thiswillbeinterpretedasimplyingacirculation rotation ofthedipolewavesinspacetime.Furthermore,thereshouldbealogicalreasonwhyfundamentalrotars fermions withdifferentenergyallpossessthesameangularmomentum.

4 TherotarmodelshouldexhibitdeBrogliewaveswhenmovingrelativetoanobserver.Thisimpliesbidirectionalwavemotion,atleastinthe“externalvolume”.ThefrequencyoftheconfineddipolewavesinspacetimecanbecalculatedbyanalogytothedeBrogliewavesgeneratedbyconfinedlightdescribedinchapter1.

5 The rotarmodel should exhibit action at a distance without resorting tomysteriousexchangeparticles. Bothgravityandanelectric fieldshould logically follow fromtherotardesign.Toaccomplishthis,partoftherotar’sdipolewaveinspacetimemustextendintowhatweregardasemptyspacesurroundingtherotar.

6 Therotarmodelshould logicallyexplainhow it ispossible forparticles toexploreallpossiblepathsbetweentwoeventsinspacetime pathintegralofQED .

NotetoReader:Therestofthischapterpresentsthespacetime‐basedmodelofafundamentalparticle.Inthese11pagestheemphasiswillbeondescribingtheparticlemodelandtherewillbenoattempttojustifythismodel.Thisspacetime‐basedparticlemodelwillincludeunfamiliarconceptsthatmaybedifficulttoinitiallyvisualize.Chapters6,8and10aredevotedtotestingthisparticlemodel.Forexample,chapter6willsubjecttheparticlemodeltotestsofitsangularmomentum,inertia,energyandthegenerationofforces includinggravity .Thesetestswillalsohelptoexplainthemodelfurther.


ParticleModelFourthStartingAssumption:A fundamental particle is a dipole wave in spacetime that forms a rotating spacetime dipole, one wavelength in circumference. Inertia is a natural property of this particle design.Arotatingdipoleinthespacetimefieldcanbementallythoughtofasadipolewaveinspacetimethathasbeenformedintoaclosedloop,onewavelengthincircumference.Recallthatadipolewaveinspacetimeoscillatesboththerateoftimeandpropervolume.Forexample,oneportionofthewave,whichwewillnamethespatialmaximum,expandspropervolumeandslowstherateoftimerelativetolocalflatspacetime.Theoppositeportionofthewave,whichwewillnamethespatialminimumhasareductionofpropervolumeandanincreasedrateoftimerelativetolocalflatspacetime.Ifadipolewaveinspacetimepossessesquantizedangularmomentumof½ħ,itformsaclosedloopthatisonewavelengthincircumference.To visualize this, a single cyclewave has been given angularmomentum so that the spatialmaximumandminimuminaplanewavehavenowbecomethetwooppositepolaritylobesoftherotatingdipolewave.Thewaveisstilltravelingatthespeedoflight;itisjusttravelingatthespeedoflightaroundaclosedloop. Suchawaveisconfinedenergytravelingatthespeedoflight.Whilethereisangularmomentum,thenettranslationalmomentumofthisquantizedwaveis zero p 0 because of opposing vectors . Therefore, just like confined light or confinedgravitationalwaves,adipolewaverotatingatthespeedoflightsatisfiestheconditionrequiredforittoexhibitrestmassandinertia.Thisrotatingdipolemustbepicturedasan isolatedrotatingdipolewaveexisting inaseaofvacuumenergy/pressure that consists of other non‐rotating dipolewaves in spacetime. Therotarmodel implies internal pressure explained later .The vacuumenergy/pressure of thesurrounding spacetime field is capableof exerting a far greaterpressure than is required toconfine the energydensity of a rotar. For now the important point is that a rotar rotatingspacetime dipole can achieve stability by interacting with the surrounding sea of vacuumenergy thesurroundingspacetimefield .Therotatingdisturbanceisonlyaquantizedunitofangularmomentumthatcaneffortlesslymovethroughthesuperfluidspacetimefield.Illustrations of a Rotar:Figures5‐1and5‐2aretwodifferentwaysofdepictingtherotatingdipoleportionoftherotarmodel.ThespacetimedipoledepictedinFigure5‐1showstwodiffuselobesrepresentingstrainedvolumesofspacetimethatarerotatingintheseaofvacuumenergy.Theselobesaredesignated“dipolelobeA”and“dipolelobeB”.Eachlobeexhibitsbothaslightspatialandatemporaldistortionofthespacetimefield.Forexample,lobeAcanbeconsideredthe lobe that exhibits a proper volume slightly larger than the Euclidian norm the spatialmaximumlobe andarateoftimethatisslightlyslowerthanthelocalnorm.LobeBhastheoppositecharacteristics smallerpropervolumeandfasterrateoftime .Theselobesarealways


movingatthespeedoflight,soitisonlypossibletoinfertheireffectontimeorspacebywaveamplitudes.Also, the rotarmodelextendsbeyond thevolumeshown,but thatportion isnotillustratedhere.


Rotar Radius and Rotar Volume : This cross sectional view inFigure5‐1 showsa circledesignated“imaginaryboundaryoftherotatingdipolewave”.ThiscirclewhichisoneComptonwavelengthincircumference.Thiscirclewillsometimesbecalledthe“Comptoncircle”andthevolume within this circle will be called the “quantum volume”. The radius is equal to thefundamentalparticle’sComptonwavelengthdividedby2πwhichwillbedesignatedthereducedComptonwavelength λc . This circle and radius λc should not be considered as hard edgedphysicalentities.Instead,theyshouldbeconsideredasconvenientmathematicalreferencesforarotar.Thisissimilartothewaythatthecenterofmassisaconvenientmathematicalconceptformechanicalanalysis.Figure 5‐1 depicts spatial nd temporal variuations in the properties of the spacetime field.Missing from this figure is all the higher frequency dipolewaves in spacetime that are alsopresentinspacetime.Forexample,themodelofthespacetimefieldisdipolewaveswhichlackangular momentum primarily at Planck frequency with wavelength predominantly equal toPlancklength.Iffigure5‐1representsanelectron,thenadipolewavewithwavelengthofPlancklengthwouldbeabout1022timessmallerthantheradiusdepictedinfigure5‐1.Therefore,thevariationsshowninfigure5‐1canbethoughtofasmadefromavastnumberofsmallerwaveswhichhaveslightdifferencesindensityresultinginthecharacteristicsshowninthisfigure.Sincethedipolelobesarequantummechanicalentities,theycannotbeaccuratelydescribedbystaticpictures.Whilefigure5‐1depictsarotationinasingleplane,theintendedrepresentationisasemi‐chaoticrotationthathasanexpectationdirectionofrotation,butalsootherplanesofrotation occur with a probabilistic distribution. Also, this is merely quantized angularmomentumintheseaofpredominantlyhighfrequencydipolewavesthatmakeupthespacetimefield.Thischaoticenvironmentisatthelimitofcausality.Ithasanexpectationrotationalaxis,butthischaoticenvironmentcreatesthequantummechanicalspincharacteristicsofaparticle.ThecircledepictedinFigure5‐1shouldbeconsideredthecrosssectionofanimaginaryspherebecausethechaoticrotationalcharacteristicsallowallrotationalaxisexcepttheoppositeoftheexpectationaxis..Thevolumeofthisimaginaryspherewillbedesignatedthe“rotarvolumeVr”.Whilethisvolumeshouldbe 4/3 πλc3,oftenwearedroppingnumericalfactorsnear1inthisplausibilitystudy,sotherotarvolumewillbeconsideredVr λc3.Rotating Rate of Time Gradient: The presence of these lobes also implies that there is agradient in therateof timeandagradient inpropervolumebetween these lobes andevenoutsidetheselobes .IflobeAhasarateoftimethatisslowerthanthelocalnormandlobeBhasa rate of time that is faster than the local norm, then this implies that the rotarmodel alsocontainsavolumeofspacewithagradientintherateoftimethatisrotatingwiththelobes.Any


gradientintherateoftimeproducesacceleration.Inchapter2weshowedthattheaccelerationofgravitywasdirectlyrelatedtothegradientintherateoftime:

g c2 –

For example, a 1 m/s2 acceleration is produced by a rate of time gradient of: 1.11 10‐17seconds/secondpermeter.Therateoftimegradientintherotarmodelthereforeproducesavolumeofspacetimethatexhibitsaccelerationsimilartogravitybuttherearealsoimportantdifferencesexplainedbelow.Figure5‐2isintendedtoillustratetherotatingrateoftimegradientpresentintherotarmodel.In figure 5‐2 the lobes A and B have been replaced with a dashed outline showing theirapproximatelocation.Insteadofillustratingthelobes,figure5‐2showstherateoftimegradientthatexistsbetweenthelobes. Onlytherateoftimegradientinsidetherotarvolumeisshown .Thearrowsshowthedirection vector oftherateoftimegradientandthelengthofthearrowsisacruderepresentationoftheamountofrateoftimegradient.Thedirectionoftherateoftimegradientrotateswiththelobes,soFigure5‐2shouldbeconsideredasdepictingamomentintime.Rotating “Grav” Field: Anewnameisrequiredtodescribethisrotatingaccelerationfieldcausedbytherotatingrateoftimegradientillustratedinfigure5‐2.Thename“rotatinggrav field”willbeusedtodescribethisrotatingrateoftimefield.Itwillbeshownlaterthatthisisafirst order effect capable of exerting a force comparable to the maximum force of a rotar.Howeverthisforcevectorisrapidlyrotatingthereforewearenotawareofitseffect.Thegravityproducedbyarotarisavastlyweakerforce. However,thegravityvectorisnotrotatingandthereforeitisadditive.This“rotatinggravfield”fillingthecenteroftherotarvolumewillbeshowntohaveanenergydensitycomparabletotheenergydensityoftherotatingdipolewavewhich is concentrated closer to the circumference of the rotar volume. Therefore, the twodifferent types of energetic spacetime approximately fill the entire rotar volume with anapproximatelyuniformtotalenergydensity.Lobe A as described above produces an effect in spacetime that is similar to the effect onspacetimeproducedbyordinarymass verysnall slowing in the rateof timeandverysmallincreaseinvolume .LobeB,ontheotherhand,producesaneffectthatissimilartotheeffectofahypotheticalanti‐gravitymass.Itproducesaverysmallincreaseintherateoftimerelativetothelocalnormandaverysmalldecreaseinvolume.InlobeB,theverysmallincreaseintherateoftimeneverreachestherateoftimethatwouldoccurinahypotheticalemptyuniverse.Thiswill be discussed later, but it will be proposed that the entire universe has a backgroundgravitationalgammaГthatresultsintheentireuniversehavingarateoftimethatisslowerthanahypotheticalemptyuniverse.ItisthereforepossibleforlobeBtohavearateoftimethatis


faster than the surrounding spacetime field without having a rate of time faster than ahypotheticalemptyuniverse.Compton Frequency: We will return to figures later, but first we want to calculate therotational frequency of the rotating dipole. If we presume that a rotar is a confined wavetravelingatthespeedoflight,itisnecessarytoassignafrequencytothiswave.Isitpossibletoobtainanimpliedfrequencyfromaparticle’sdeBrogliewavecharacteristics?Inchapter#1weshowedthatconfinedlightexhibitsmanypropertiesofaparticle.TheseincludetheappearanceoftheopticalequivalentofdeBrogliewaveswhentheconfinedlightismovingrelativetoanobserver.IfwewereonlyabletodetecttheopticaldeBrogliewavespresentinamovinglaser,itwouldbepossibletocalculatethefrequencyofthelightinthemovinglaser.Similarly,wecanattempt to calculate a rotar’s frequency from its de Brogliewaves.We know a particle’s deBrogliewavelength λd h/mv andthedeBrogliewave’sphasevelocity wd c2/v . Fromtheseweobtainthefollowingangularfrequencyω.

υ frequency

2 ħ

ω ωc ħ ħ

Comptonangularfrequency

This calculation says that a rotar’s angular frequency is equal to a rotar’s Compton angularfrequencyωc.Wewillpresumethatthisisarotar’sfundamentalfrequencyofrotation.WhilethedeBrogliewavelengthandphasevelocitydependonrelativevelocity, thevelocity termscancel in the above equation yielding a fundamental frequency Compton frequency that isindependentofrelativemotion.Thereasoninginthiscalculationcanbeconceptuallyunderstoodbyanalogytotheexampleinchapter1ofthebidirectionalwavesinthemovinglaser.Arotar’sComptonwavelengthwillbedesignatedλc.Theconnectionbetweenarotar’sComptonwavelengthanddeBrogliewavelengthλdisverysimple.λc λd v/c where isthespecialrelativitygamma: 1– v/c 2 ‐1/2λc λd Ultrarelativisticapproximationwhenv/c 1Thesimplicityoftheseequationsshowtheintimaterelationshipbetweenarotar’sdeBrogliewavelengthandComptonwavelength. Foranotherexample,imagineageneric“particle”thatmightbeacompositeparticlesuchasanatomormolecule.This“particle”isatrestinourframeof reference. Suppose that this particle emits a photon of wavelength λγ. This photon hasmomentump h/λγ. Therefore the emission of this photon imparts the samemagnitude ofmomentum to the emitting particle but in the opposite vector direction recoil . Now, the


particleismovingrelativetoourframeofreference.WhatisthedeBrogliewavelengthoftherecoilingparticleinourframeofreference?λd h/psetp h/λγλd λγTherefore,weobtaintheveryinterestingresultthatthedeBrogliewavelengthoftherecoilingparticleequalsthewavelengthoftheemittedphoton.InAppendixAofchapter1itwasproventhataconfinedphotonwithaspecificenergyexhibitsthesameinertiaasafundamentalparticlewiththesameenergy.AnotherwayofsayingthisisthataparticlewithdeBrogliewavelengthλd exhibits the same magnitude of momentum as a photon with the same wavelength.Furthermore,inchapter1wesawthesimilaritybetweendeBrogliewaveswithwavelengthλdand the propagating interference patterns with modulation wavelength λm. Impartingmomentump h/λγtoeitherafundamentalparticlewithComptonwavelengthλcoraconfinedphotonwiththesamewavelengthwillproducetheresult:λd λm λγ.Therefore,itisproposedthatthisoffersadditionalsupporttothecontentionthatfundamentalparticlesarecomposedofaconfinedwaveinspacetimewithawavelengthequaltotheparticle’sComptonwavelengthλc.Intheremainderofthisbookwewilloftenuseanelectroninnumerousexamples.AnelectronhasthefollowingComptonfrequency,ComptonangularfrequencyandComptonwavelength:Electron’sComptonfrequencyνc 1.24 1020HzElectron’sComptonangularfrequencyωc 2πνc 7.76 1020s‐1Electron’sComptonwavelengthλc 2.43 10‐12mElectron’sreducedComptonwavelengthλc 3.86 10‐13m λc c/ωc Radius of a Rotar:Onceweknowtherotar’sfrequencyofrotation,wecancalculatetherotar’sradiusassuming speedof lightmotion.Thecircle inFigure5‐1 is an imaginarycirclewithacircumferenceoneComptonwavelength.TheradiusofthecircleoneComptonwavelengthincircumferenceisequaltotherotar’sreducedComptonwavelengthλc.λc c/ωc λc/2π ħ/mc ħc/EiWhere:,λc reducedComptonwavelength rotar’sradius;λc Comptonwavelength,Ei rotar’sinternalenergyInquantummechanics,thisdistanceλcisthelogicaldivisionwhereaparticle’squantumeffectsbecomedominant. Forexample,afundamentalparticleofmassmcanmovediscontinuouslyoveradistanceλc.Aparticlecangooutofexistence,orcomeintoexistence,foratimeequaltoλc/c.Essentially,thedistanceλcisarotar’snaturalunitoflengthand1/ωcisarotar’snaturalunitoftime.Inchapters6and8itwillbeshownthatthegravitationalandelectrostaticforceexerted by a fundamental particle become much easier to understand when the distancebetweenparticlesisexpressedanthenumberofreducedComptonwavelengthsratherthanthenumberofmeters.


Analysis of the Lobes:Supposethatitwaspossibletofreezethemotionoftherotatingdipoleandexaminethedifferencebetweenthetwolobes.Theslowtimelobe lobeA canbethoughtof as having a proper volume that exceeds the anticipated Euclidian volume as previouslyexplained.Thefasttimelobe lobeB canbethoughtofashavinglesspropervolumethantheanticipatedEuclidianvolume thespatialminimumlobe .Thisconnectionbetweenvolumeandtherateoftimeiswellestablishedfortheeffectsofgravity.However,gravityisastaticeffectonspacetime.Thiseffectonspaceproducedbyarotar’sdipolewaveinspacetimeresultsinthedistancebetweentwopointsontheComptoncirclechangingslightlyasthedipolerotates.


Similarly,ifitwaspossibletofreezetherotationwewouldfindadifferentrateoftimebetweenthetwolobes.Sincethelobesarealwaysmovingatthespeedoflight,theeffectisthattherateoftimefluctuatesatapointontheComptoncircle radius λc andthedistancebetweentwopointsontheComptoncirclealsofluctuates.Allquantizeddipolewaveshavemaximumspatialdisplacementamplitudeequal to Plancklength Lp as the quantumdipole rotates. To illustrate this concept, imagine two pointslocatedontheComptoncircleofFigure5‐1separatedbyacircumferentialdistanceequaltoλcseparatedbyoneradian .AsthelobesrotatetheymodulatevolumeandresultintheseparationdistancebetweenthesetwopointsincreasinganddecreasingbyPlancklength Lp .Figure5‐3isagraphofthespatialeffectproducedbytherotatingspacetimedipole.Infigure5‐3the“Y”axisisthespatialdisplacementproducedbytherotatingdipolebetweenthesetwopointsLp .The“X”axisislengthinunitsofλc.Itshouldbenotedthatthe“Y”axisisaboutafactorof

1023 smaller scale than the “X” axis ifwe presume that λc is an electron’s reducedComptonwavelength λc 3.86 10‐13mandLp 1.6 10‐35m .Figure5‐4isagraphofthetemporaleffectoftherotatingspacetimedipole.ItwaspreviouslystatedthatinFigure5‐1wecanconsiderlobe“A”asexhibitingarateoftimeslowerthanthelocalnormandlobe“B”asexhibitingarateoftimefasterthanthelocalnorm.Toillustratethisconceptfurther,wewill imagineathoughtexperimentwhereweplaceahypotheticalperfectclockatapointonthecircumferencepreviouslydesignatedthe“Comptoncircle”inFigure5‐1.Previouslyweimaginedfreezingtherotationofthedipole.Nowinfigure5‐4weimaginehavingthedipolerotateandwearemonitoringthetimeatonlyonepointontheimaginaryComptoncircleandcomparingthistoa“coordinateclock”atanotherlocationinflatspacetime.Theclockmonitoringapointontheedgeofthedipolewillbecalledthe“dipoleclock”.AslobesAandBrotatepastthedipoleclocklocation,thedipoleclockwouldspeedupandslowdownrelativetothecoordinateclockthatisunaffectedbytherotatingdipole.Bothclocksarestartedatthesamemoment.Inflatspacetime,wewouldexpectbothclockstoperfectlytrackeachother.Figure5‐4plotsthetemporaldisplacementofspacetimeproducedbytherotatingdipolewave Yaxis versustimeasexpressedinunitsof1/ω Xaxis .Forexample,anelectronhasangularfrequencyofωc 7.76 1020s‐1.Therefore,foranelectron1/ωc 1.29 10‐21s.Ascanbeseeninfigure5‐4thedipoleclockspeedsupandslowsdownrelativetothecoordinateclock. The maximum time difference between the two clocks is plus or minus PlancktimeTp ~5 10‐44 s .Thismaximumtimedifference isaquantummechanical limit foradisplacementofspacetimethatisundetectable.Thisoscillationoftherateoftimeiswhathasbeencalled“dynamicPlancktimeTp”.Figure5‐4onlyshowswhathappensduringashorttimeperiod ~10‐20s afterstartingthedipoleandcoordinateclocks. Thetimedifferenceoveralongertimewillbediscussedinalaterchapterandshowntoresultinanetreductionintherateoftime.


Strain Amplitude – Aβ:Thestrainamplitudeofthewavedepictedinfigures5‐3and5‐4isjustthemaximumslopeofthesewaves.Thedashedlineinfigure5‐3representsthemaximumslopewhichoccurswhen thesinewavecrosseszero.Thismaximumslopecanbeadimensionlessnumberif the“X”and“Y”axishavethesameunitswhichcancelwhenexpressingslope. Forexample,infigure5‐3boththe“X”and“Y”axishaveunitsoflength.Themaximumdisplacementis one unit of Planck length Lp 1.6 10‐35m . The “X” axis is length units expressed asmultiplesofλc.Foranelectronλc 3.86 10‐13m.ThemaximumslopeoccursatY 0.Themaximumslopeinfigure5‐3isLp/λc.Thisdimensionlessmaximumslopewillbedesignatedthe“rotar’sstrainamplitude”anddesignatedwiththesymbolAβ.Therefore,onewayofexpressingarotar’sstrainamplitudeiswiththeratiooflengths:Aβ Lp/λc strainamplitudeexpressedwithlengthratioFigure5‐4issimilartofigure5‐3exceptthat5‐4ischaracterizingtheeffectontherateoftime.TheYaxisofthisfiguredepictsthedifferencebetweenthedipoleclockandthecoordinateclockshortlyafterwestartbothclocks.Thisdifferencebetweenclockscanreach Plancktime Tp .The “X” axis of figure 5‐4 is in units of time expressed as 1/ωc which for an electron is1/ωc 1.29 10‐21s.ThestrainamplitudeAβofthedipolewavecanalsobeexpressedusingtimerelatedsymbols:

Aβ ωc/ωp TpωcstrainamplitudeexpressedusingfrequencyandtimeThereforethedipolewavesstrainamplitudecanbeexpressedeitherasastrainofspace Lp/λc or as a strain in the rate of time ωc/ωp Tpωc . For an electron λc 3.86 10 ‐13m andωc 7.76 1020s‐1.Therefore,anelectron’sdimensionlessstrainamplitudeis:Aβ 4.18 10‐23.Thiswillbediscussedinmoredetailinthenextchapter. OtherrotarshavedifferentstrainamplitudesbecausetheyhavedifferentComptonangularfrequenciesanddifferentvaluesofλc.Notethatthesinewavesinfigures5‐3and5‐4areshiftedbyπradians 180° .Thisisbecausethelobewithmaximumpropervolumecorrespondstothelobewiththeminimumrateoftimeandviceversa.Conceptual Examples of Wave Amplitude: Therotarmodelisbasedontheseaofvacuumfluctuationsthatformspacetimebeingdynamicallystrained.Itisimportanttohaveamentalpictureof the incrediblysmalldisplacementsof timeandspacerequired for thismodel. Forexample,foranelectronAβ 4.18 10‐23whichistheratioofLp/λcorTpωc.Thisspatialstrainofthespacetimefieldcausestheorbitsofthetwolobestoexhibitdifferencesincircumferenceand radius comparable to Planck length. This is really equivalent to having one of the lobesexceed theelectron’s rotar radiusbyPlanck lengthand theother lobe is less than theRotarradiusbyPlancklength.Thismeansthatthelobesarenotexactlysymmetrical.Actually,theveryconceptofadipoleimpliesthattheremustbetwodifferent opposite propertiesthatare


interacting. With electromagnetic radiation, a dipole oscillator has a positive and negativeelectricalcharge.Similarly,aspacetimedipolehastwolobeswhichproduceanoppositetypeofspatialdistortion bigandsmall ofthepropertiesofspacetimeortheoppositetypeoftemporaldistortion fastandslow ofthepropertiesofspacetime.Plancklengthissosmallthatitishardtoimaginetheverysmalldistortionofspacetimerequiredtomakeanelectronaccordingto theproposedmodel. Wewilluse the followingexample toillustrate this incredibly small difference between the two lobes. Suppose we compare anelectron’stotheradiusofJupiter’sorbit.StretchingspacebyPlancklengthoveradistanceequaltoanelectron’sRotarradiusproducesastrainofabout4.2 10‐23. 1.6 10‐35/3.9 10‐13 410‐23 . StretchingJupiter’sorbitalradius 7.8 1011m by3.3 10‐11mwouldproducea

comparablestraininspace.Toputthisinperspective,theBohrradiusofahydrogenatomis~5.3 10‐11m.NowimagineaspherethesizeofJupiter’sorbit,exceptthatonehemispherehasstrained spacetime such that the radius exceeds theprescribed radiusbyadistance roughlyequaltotheradiusofahydrogenatom.Theotherhemisphereislessthantheprescribedamountby the radius of a hydrogenatom a4 10‐23 volumedifference . Of course, the transitionbetweenthetwolobesisnotanabruptstep.Thissimplifiedexampleismeanttoillustratethevery small distortion of the spacetime field involved in the spacetime‐based model of afundamentalparticle.Continuingwiththeexample,supposethatweweretocomparetherateoftimebetweenthetwolobesofanelectron. Suppose that itwaspossible tostop therotationand insertaperfectlyaccurateclockintothefastlobeofanelectron atdistanceλc andinsertasecondperfectclockintotheslowlobe.Therateoftimedifferenceissosmallthatitwouldtakeabout50,000timeslongerthantheageoftheuniversebeforethetwoclocksdifferedintimebyonesecond.Thisratiointherateoftimeisalsoabout4 10‐23.Inchapter6wewillanalyzethisspacetime‐basedmodeloffundamentalparticlestoseeifthemodelplausiblyyieldsthecorrectenergy,angularmomentum,gravity,etc.However,theaboveexamplesbegintogiveafeelforhowparticlescanappeartobenebulousentitieswhichresultintheir counter intuitive quantum mechanical properties. Rotars made from small amplitudewavesinthespacetimefieldcanbedifficulttolocateexactly.Furthermore,apropertywillbeproposedlaterthatpermitsquantizedwavesinthespacetimefieldtorespondtoaperturbationasasingleunit.Thisgives“particle‐like”propertiestoaquantizedwaveinspacetimeandgivesrisetothefamouswave‐particlepropertiesinnature.Solitons: Whydo a few combinations of frequency and amplitudeproduce resonances thatresult in fundamental particles and all other frequencies and amplitudes not producefundamentalparticles?Theremustbeacombinationofpropertiesofspacetimewhichachievestabilitybycancelinglossatthefewfrequenciesthatformfundamentalparticles.Thereappearstobeasimilaritybetweentheconditionsthatformastablerotarandtheconditionsthatforma


stableopticalsoliton.Anopticalsolitonisformedwhenaveryshortpulseoflaserlightisfocusedintoatransparentmaterialthatexhibitsasetofcomplementaryopticalcharacteristics.OneofthesecharacteristicsistheopticalKerreffect.Aspreviouslymentioned,thisisanonlineareffectinalltransparentmaterialswherethespeedoflightisdependentonintensity.Thisnonlineareffectisalsowavelengthdependent.InsomeopticalmaterialsthedispersionoftheopticalKerreffect can be offset against the optical dispersion of the transparent material. These twopropertiescaninteractinawaythatconfinesratherthandispersestheenergyinthepulseoflaserlight.Thedispersionisalossmechanismforapulseoflaserlight.Thecombinationofthetwodifferenttypesofdispersionplustheintensitydependencetogethercancreateastabilitycondition.Apulseoflaserlightformsapropagatingwavethatfulfillsthisstabilityconditionandthiscombinationofeffectsshapethepulseoflightintoan“opticalsoliton”.Theterm“soliton”isaself‐reinforcingwavethatmaintains itsshapeas itpropagates. Thefirst identifiedsolitonswerewaterwaves propagating in a channel. Optical solitons can exhibitmany particle‐likeproperties.Forexample,twoopticalsolitonspropagatingneareachothercanattractorrepeleachotherdependingontherelativephaseofthelight. Awonderfulvideoisavailableatthefollowingwebsiteshowingparticle‐likeinteractionsofopticalsolitons5.Thecharacteristicsof thespacetime fieldappear to formasimilar losscancellation for the3chargedleptons.Thesearefundamentalparticleswithrestmassthatcanexistinisolation.Thestability of any rotars depends on the existence of vacuum energy, but theremust be a fewfrequenciesandconditionswherethestabilizationisoptimum.Thisisequivalenttoapulseoflightsatisfyingthesolitonconditioninatransparentmaterial.Theanalogytoopticalsolitonscanbeextendedifafundamentalparticleisvisualizedaspropagatingalongthegeodesicatthespeedoflight.DiracEquation:Inhis1933NobelPrizelecture,PaulDiracsaidthefollowing:“Itisfoundthatanelectronwhichseemstoustobemovingslowly,mustactuallyhaveaveryhighfrequencyoscillatorymotionofsmallamplitudesuperposedontheregularmotionwhichappearstous.Asaresultofthisoscillatorymotion,thevelocityoftheelectronatanytimeequalsthevelocityoflight.Thisisapredictionwhichcannotbedirectlyverifiedbyexperimentsincethefrequencyoftheoscillatorymotionissohighandtheamplitudeissosmall.”PerhapswhenDiracgavehislecturehewasvisualizingapointparticleundergoinganoscillatorymotionatthespeedoflightandattheComptonfrequency.However,theDiracequationdoesnotspecifyapointparticle.ItisproposedthattherotarmodelofanelectronactuallyisabetterfittotheDiracequation.Aparticlewithrestmasscannotpropagateatthespeedoflight.Also,suchamovingparticlewouldexceedtheconditionsoftheuncertaintyprincipleanditshouldbepossibletodoanexperimentthatwouldrevealthisinternalstructure.Theproposedmodelisadipolewaveinspacetimewhichnaturallypropagatesatthespeedoflight.ThedisplacementamplitudeofthiswaveisPlancklengthandPlancktimewhichmeets

5 http://www.sfu.ca/~renns/lbullets.html


theconditionofbeinganundetectableamplitudeeventhoughthephysicalsizeofthemotionhastherelativelylargeradiusofλc.Finally,thedipolewaveinspacetimeismodulatingtherateoftime and proper volume which gives it the ability to produce curved spacetime in thesurroundingvolume discussedlater .


Chapter6

AnalysisoftheParticleModelandDerivationofGravity

Inchapter5thespacetime‐basedmodelofafundamentalparticlewaspresentedwithoutanyanalysistoseeifitcansatisfytheknowncharacteristicsoffundamentalparticles.Thischapterwillbedevotedtotestingtherotarmodeloffundamentalparticlesforplausibility.Wewillfirstanalyzewhetherthismodelwillappeartobeapointparticleincollisionexperiments.Thenwewill see if the particlemodel produces the required energy, angularmomentum, and forcesincludingthecorrectgravity.Finallytheimpliedinertiaoftheparticlemodelwillbediscussed. Particle Size Problem:Perhapsthebiggestobjectiontothehypothesisproposedinchapter5isthatexperimentsseemtoindicatethatfundamentalparticlesarepointswithnophysicalsize.Nowwewillexaminewhethertherotarmodeliscompatiblewiththeexperimentsthatseemtoindicateapointparticle.RecallthatthespacetimefieldisaseaofwavespredominantlyatPlanckfrequencybutallotherfrequenciesarealsopresentatlowerdensity.ThewavesareundetectableasindividualwavesbecausetheiramplitudeisaspatialdisplacementofPlancklengthandatemporaldisplacementdifferencebetweenclocks ofPlancktime.Thesewavesarethezeropointenergyandvacuumfluctuations required by quantum electrodynamics and quantum chromodynamics. Theyintroduceuncertaintyintoanymeasurement.Theyalsohavenoangularmomentumandexhibitsuperfluidproperties. Oneofthesepropertiesisthatanyangularmomentumisisolatedintoquantizedunits.Asdiscussedinthelastchapter,asuperfluidBose‐Einsteincondensateisolatesangular momentum into small rotating vortices, each one with ħ angular momentum. It isproposedthatthesuperfluidspacetimefieldalsoisolatesangularmomentumthathasexistedinthespacetimefieldsincetheBigBang discussedinchapters13and14 .A fermion such as an electron or quark is nothing more than a unit of quantized angularmomentumquarantined by the superfluid spacetime field into quantized unitswith angularmomentum of½ ħ or ħ fermions or bosons . An electron is a dipolewave rotating at itsCompton angular frequency 7.7634 x 1020 s‐1 . The point is that an electron, and all otherfundamental particles, arenothingbut anorganized rotationof the spacetime field.There isnothingthereiftheexpectationisaphysicalobjectotherthanspacetime.Thereisnovibratingstring;thereisnoelasticspherethatcanbecharacterizedbyacollisionexperiment.Eventhoughtheradiusoftherotatingdipolewaveisarelativelylarge4.18x10‐13m,thisisjustaslightrotatingdistortionofspacetimethatisthatsize.


Foranelectron, theverysmall strainproduced in thespacetime field isonlyabout4x10‐23dimensionlessstrainslope . Spatially, this isastrainofspacetimecomparabletostretchingJupiter’sorbitbytheradiusofahydrogenatom.Temporallythisiscomparabletoretardingtherateoftimebyonesecondover50,000timestheageoftheuniverse.Itisonlytheincrediblylargeimpedanceofspacetime c3/G 4x1035kg/s andthehighrotationalfrequency ∿1021s‐1 that gives this small strain of spacetime a detectable physical presence. Even then, theoscillationsarenotdetectableaswavesbecausethemaximumdisplacementofspacetimeisonlyPlancklengthandPlancktime.Whenwedodetectthepresenceofanelectron,itexhibitspropertiesthatarenotexplainablefrom classical physics. For example, “finding” an electron interacting with a wave withquantizedangularmomentum isaprobabilisticevent.Anelectroncanseemtojumpfromonelocationtoanotherwithouttraversingthespacebetweenthesetwopoints.Thisisbecausetherotatingdipole in spacetime that is an isolated electron isdistributedover a relatively largevolume with a radius in the range of 4 x 10‐13 m. Interacting with the quantized angularmomentumhappensatapointanywherewithinthisvolumebutevensometimesoutsidethisvolume. Thisgivestheappearanceofapointparticlediscontinuously jumping frompoint topoint.Alsothequantizedangularmomentumcausestheelectron’senergytobequantized.Thelistofcounter‐intuitivepropertiesofanelectronislong,butthenon‐classicalpropertyofinteresthereisthefactthatanelectronseemstohavenophysicalsizeinacollisionexperiment.Eventhoughtherotarmodelgivesaphysicalsizetofundamentalparticles,itisnottheclassical“billiardball”typeofphysicalsize.Forexample,a“collision”betweenanelectronandapositronrotarmodel oftenresultsinthesetworotatingdipolewavesmerelypassingthrougheachotherwiththeonlyinteractionbeingaslightscatteringfromtheoriginaltrajectories.Whenanelectronandpositronannihilateeachotherinaninteractionthatformspositronium notahighspeedcollision ,about10‐10secondsisrequiredforthisannihilation photonemission totakeplace.In a collision at near light speed the overlap time is less than 10‐20 seconds in the frame ofreferencewherethetotalmomentumiszero.Whenthecollisionenergyislessthantheabout1GeV,thenthescatteringcross‐sectionofanelectron‐positroncollisiondecreasesasthecollisionenergyincreases.Athighercollisionenergywherenewfundamentalparticlescanbeformedtheinteractioncross‐sectionbecomescomplexwiththeformationofnewparticles.Inacollisionbetweentwoelectrons,theelectrostaticrepulsioncanbevisualizedasmomentarilybringingthetwocollidingelectronstoahalt.Whathappenstothekineticenergyatthemomentofclosestapproach?Withtherotarmodelthekineticenergyofeachelectronismomentarilyconverted into internal energyof the twoelectrons. This increase in energymeans that thefrequency increases, the wavelength decreases, the circumference decreases, and the rotarradiusdecreases.Thesechangeskeeptheangularmomentumconstantbecausethedecreaseinradiusoffsetstheincreaseinmass/energy.Therotarradiusλcscaleswiththerotar’sinternalenergyEias:λc ħc/Ei.Atthemomentof“closestapproach”thetworotarsareactuallypartially


overlapping. They also have the smaller radius and higher frequency appropriate for theirhigherenergycondition.Howdoesthisradiuscomparewiththeparticlesizeresolutionlimitinacollisionexperiment?Thisresolutionlimit issetbytheuncertaintyprincipleΔxΔp ħ/2. Wehavebeenignoringdimensionlessconstantslike½,sowewilluseΔxΔp ħandthenincludeasingleall‐inclusiveconstantk.Inacollisionbetweentwoelectrons,wehaveanuncertaintyaboutthemomentumtransferredatthemomentofclosestapproach.Isthecollisionheadonoraglancingcollision?Allwereallyknowisthemaximummomentumavailable,sotheuncertaintybecomesΔp mv.Foracollisionbetweenelectronswithultra‐relativisticvelocity v c , thespecial relativitygamma is γ Ek/mc2where Ek is the relativistic kinetic energy. Also,when γ is large, themomentumis:p γmc.WiththisinformationwecansolveforΔx.

Δx ħ ħ

ħ

dimensionlessconstantkincluded

Δx ħ

Therefore, in a collision between ultra‐relativistic rotars, the kinetic energy is momentarilyaddedtotherotar’s internalenergy E mc2energy foratotalenergyofE Ek Ek. Thismeansthattherotarradiusmomentarilyshrinkstoλc ħc/Ekwhichmatchestheuncertaintyresolution limit of the experiment Δx ħc/Ek. It is no coincidence or lucky result that theresolutionoftheexperimentmatchesthemomentarysizeofarotar.Iffundamentalparticlesreallyarerotarswithaprobabilisticinteractionradius,thenthissizemustmatchtheuncertaintyintheinteraction.The combination of the overlap and the reduction in λc results in an expectation separationdistance that is less than the Δx uncertainty resolution of the experiment that attempts tomeasurethesizeoftherotar particle .Thisisanalogoustotheuncertaintyprinciplesayingthatan experiment cannot simultaneouslymeasure the position andmomentum of a particle tobetter than½ħ. Similarly, anexperiment cannotmeasure the sizeof a fundamentalparticlebecausethemeasurementprocessintroducesenergythatmomentarilydecreasesthesizeoftheparticletobelowthemeasurementresolution λc ħc/Ei Δx . This isaclassiccaseof theexperiment distorting the property being measured and invalidating the measurement.Therefore,therotarmodelgivesaplausibleexplanationofwhyfundamentalparticlesalwaysappeartobepointparticlesinexperimentsthatattempttomeasuretheirsize.Thecurrentupperlimitforthesizeofanelectronissetbyanexperimentusingtwoelectronsacceleratedtoakineticenergyofabout50GeV.Whentherotarmodelofanelectronundergoesacollision,the50GeVkineticenergyistemporarilyconvertedtotheelectron’sinternalenergy.Thismomentarilyincreasestheelectron’sinternalenergy dipolewaveenergy byafactorof100,000andreducestherotarradiusbyafactorof100,000.Anisolatedelectronhasarotar


radiusofabout4x10‐13m.Howeverwhen50GeVofkineticenergyisconvertedtoanelectron’sinternalenergyatthemomentofclosestapproach,thisreducesλcbyafactorof100,000toabout4x10‐18m.Combinedwiththeabilitytopartiallyoverlaprotarvolumes,theelectronalwayshasaninstantaneoussizesmallerthantheΔxresolutionlimitoftheexperiment.A50GeVelectronundergoingacollisiontemporarilybecomesmuchsmallerthanaproton ~10‐15m andcanbeusedasaprobeoftheinternalstructureofaproton.Therotarmodelofanelectronalsohasanadvantageoverthepointparticlemodelofanelectronwhenitcomestoexplainingthebehaviorofanelectroninanatom. Anelectronboundinanatomappears tobebigger than the isolated rotar size. Forexample, anelectronbound inahydrogen atomhas a different boundary condition than an isolated electron. This creates adifferentstabilityconditionthatresults in thedipolewaveenergyof theelectrondistributedaround the nucleolus of an atom in a way that enlarges the apparent size and explains thecloud‐likequalityofanelectronboundinanatom.Equations Demand Size:Oneofthestrengthsofthespacetimemodeloffundamentalparticlesisthatitgivesaplausibleexplanationofhowthefundamentalparticle rotar canhaveaphysicalsizeequaltothereducedComptonwavelength equaltoλc andyetalsoalwaysappeartobeapointparticleincollisionexperiments.Oneofthe“mysteries”ofquantummechanicshasbeenthattheequationsofquantummechanicsyieldanunreasonableanswerofinfinitywhentheyincorporatetheassumptionthatfundamentalparticlesarepointparticles.Theseequationsarescreamingthatthisisawrongstartingassumption.Yettheequationsareignoredbecausethephysicalinterpretationofexperimentsisthatthefundamentalparticlesmustbepointparticles.However,thisisafailureofthephysicalinterpretationoftheexperiments,notafailureoftheequations.Theprocessofrenormalizationusedtoeliminatetheinfinityisactuallyadjustingthestartingassumptiontogiveaphysicalvolumetofundamentalparticles.Physicistsbelievethatexperiments are theultimate referee of a theory. Usually experiments are easy to interpretcorrectly. However, the physical interpretation of collision experiments always makes theerroneous assumption that the colliding particles do not change any of their characteristicscomparedtothesameparticlesnotundergoingacollision.Inparticular,theassumptionisthatthe physical size of a fundamental particle remains constant, even if the collision is ultra‐relativistic.However,whereisthekineticenergystoredattheinstantwhenbothparticlesarestopped? It isproposed that thecollisionexperimentsaregivingthecorrectanswer for thisinstantbutthiscollisionmomentcannotbeextrapolatedtodeducethesizeofisolatedparticles.Thisislikeaself‐fulfillingprophecy.Ifyouassumepointparticles,thenyoucaninterprettheexperimentalresultstosupportthismodel.Stability Mechanism:Howexactlydoesthespacetimedipoleachievestability?Whatpreventsthewavesfromsimplypropagatinginastraightlineratherthanformingarotatingdipole?Thisquestionwillbeaddressedlater,butanintroductoryexplanationwillbegivenhere.Chapter5


started by recounting Erwin Schrodinger’s attempted to give a wave based explanation tofundamental particles. Schrodinger eventually abandoned this explanation because he wasunabletoexplainwhatpreventedhis“wavepacket”fromdissipating.Theproposedrotarmodelhasasinglefrequencydipolewaveinspacetimethatformsarotatingclosedloop. Thisdipolewaveisstillpropagatingatthespeedof light.Thismodelachievesalarge energy density thatwill be calculated later. However, it also implies a large pressurerequired to confine this energy. In fact, any concentration of energy density fundamentallyimpliespressure.Therefore,thisproposedrotarmodelrequiressomemeanstocounteractthepressureassociatedwiththeenergydensity. Thisisaccomplishedbyaninteractionwiththevacuumenergydipolewaves in the spacetime field. This vacuumenergypossesses a vastlylarger energydensity than any rotar. Therefore the vacuumenergy exists at a vastly largerpressurethanisrequiredtostabilizetherotar.Thereareonlyafewquarksandleptonsinthestandardmodel.TheserepresentonlyafewComptonfrequenciesthathaveachievedatleastpartialstabilityinteractingwiththesurroundingvacuumenergydipolewavesinspacetime.Thisexplanationwillbeexpandedlater.Rotar Energy Test: Nowwearegoingtosubjecttherotarmodelandtheconceptofdipolewavesinspacetimetoacriticaltest. Wewilluseoneofthe5wave‐amplitudeequationsandattempttocalculatetheenergyofanyrotar.Wearenotattemptingtocalculatetheenergyofspecificparticles.Instead,wearecheckingtoseeiftheconceptofdipolewavesinspacetimethatareconfinedtoaspecificvolumecanproducetheequivalentmass/energyforarotar.Forthisplausibilitytesttobesuccessful, insertingarotar’samplitude,frequencyandvolumeintothewave‐amplitudeequationmustproducethecorrectenergyforarotar ignoringdimensionlessconstantsnear1 .Theequationtobeusedis:E kA2ω2ZV/cwave‐amplitudeequationexpressingenergyEinavolumeVWeknowthattheangularfrequencyωequalstheComptonfrequency:ω ωc c/λc mc2/ħ.Wewillalsosettheamplitudeas:Aβ Lp/ Tpωc.WhereAβ strainamplitudeintherotarvolumeofarotar.Thisamplitudewasobtainedinthelastchapterusingthestartingassumptionaboutthemaximumdisplacementofspacetimeallowedbyquantummechanicsfordipolewavesinthespacetimefield.ThevolumetermVshouldbeequaltothevolumeoftherotarvolume:V kλc3.Itistruethatwearenotaddressingthequestionabouthowuniformlythisvolumeisfilled,butthisisjustaplausibilitytestandweareusingtheconstantkwhichpermitsustobevagueaboutthispoint.Finally,theimpedancetermZissetequaltothepreviouslyobtainedimpedanceofspacetime:Zs c3/G.Wewilllumpalldimensionlessconstantsintoasingleconstantk.E kA2ω2ZV/csetA Aβ Lp/λcω ωc c/λcandZ Zs c3/G


E k k kħ

ħ

E kmc2This important plausibility test is successful. The rotar model establishes the famousrelationshipbetweenenergyandmass inertia .WehaveshownthatanamplitudeofAβ Lp/λc,afrequencyofωcandavolumeofkλc3,togetherproducethecorrectenergyofE mc2 timesapossibleconstant .Themassinthisequationshouldbethoughtofastheinertiaexhibitedbyconfinedenergycirculatingatthespeedoflight.Thecalculationthatwasjustmaderepresentsabridgebetweenthefamiliarconceptofparticlesexhibitingmassandtheunfamiliarconceptofconfinedwavesinthespacetimefieldthatexhibitenergyandinertia.Wearepresumingthatk 1.Weactuallyhavealittlebitofflexibilityinthisregard.Previouslywegaveanexamplewherethedisplacementamplitudewasdefinedasthenormal amplitudeofasinewave.ItwouldalsobepossibletodefinetheamplitudeastheRMSamplitudeorthepeaktopeakamplitude. Thesethreewaysofdefiningamplitudeallapplytothesamewave.Furthermore,theremaybeanotherwayofdefiningamplitude.Iamgoingtopresumethatsomedefinitionofamplitudewillpermitk 1.Itshouldbeemphasizedthattherotarradiusλcisaconvenientmathematicalrepresentationofarotarmodel,buttherotardoesnotabruptlystopatadistanceofλc.Therotarmodelismorecomplexthanthisandpartofthequantumwavethatformstherotarextendsbeyondtherotarradius.Forexample,itwillbeshownlaterthattherotar’selectricfieldandgravityaretheresultoftherotar’swavestructurethatextendsfarbeyondtherotarradius.However,theenergyintheelectricandgravitationalfieldsbeyondλccontainslessthan1%oftherotar’stotalenergy.Theuseofλccanbethoughtofasaconvenientmathematicaltooltoeasilyrepresenttheentirerotarinsimplecalculations.Angular Momentum Test:Thenexttestofthemodelistoseeifthemodelhasapproximatelythecorrectangularmomentum.Wewillbuildontheenergycalculationandtesttoseeiftheangularmomentum ofthisrotarmodelhasthesameangularmomentumforallfundamentalparticlesregardlessofmass/energyandfurthermorewhetherthisangularmomentumisequaltoħwhennumericalfactorsnear1areignored.

prset:p momentum E/c mcandr λc ħ/mc;mc ħ/mc ħ

Mass cancels and allmass/energyhas the same angularmomentum. This solvesone of theproblemsassociatedwith fundamentalparticles. Howis itpossiblethatparticleswithvastlydifferent energy possess the same angularmomentum? The standard approach has been tomerelylabelitas“spin”anddeclarethatthisismerelyoneofthemanymysteriesofquantum


mechanicsbeyondhumanunderstanding.Whenweadoptadifferentmodelofafundamentalparticle,wediscoverthattheanswerisactuallyquitesimple.Westillhavethepeskyproblemthattheangularmomentumshouldbe½ħratherthanħ. Itturnsoutthatthereareactuallytwoconsiderationswehaven’taccountedforandbothhavetheeffectofloweringtheangularmomentumtowards½ħ.First,thisangularmomentumisatthelimitofcausalityanditdoesnothaveasinglewelldefinedaxisofrotation.YoucanvisualizetherotatingdipolewaveasexistinginthespacetimefieldwhichisaseaofPlanckamplitudewavesat all frequencies up to Planck frequencies. This turbulence causes the spin axis to have anexpectationdirection,butallotherrotationaldirectionsarepermittedwith lowerprobabilityexceptfortheoppositeoftheexpectationdirectionwhichhasaprobabilityofzero.Agraphoftheprobabilitiesofspindirectionisshowninfigure10‐8.ThepointisthatthisdistributionofspindirectionslowerstheangularmomentumaroundtheZaxis.Infact,itappearstolowertheZaxisangularmomentumtowardshalfoftheangularmomentumitwouldhaveiftherewasafixed axis and rotational direction. However, this is a simplified analysis, and there is stillanotherconsideration.ThereforeIwillleavetheanalysistoothers.Thesecondconsiderationisthatarotatingwaveisdistributedoveravolume.Ifweonlyhadenergytravelingatthespeedoflightaroundacircle ahoop ofradiusλc, forexample,lightinawaveguide thenweshouldusethemomentofinertiaofahoop mr2).However,thedipolewaveisdiffuseandasshowninfigure5‐2,thereisalsoa“rotatinggravfield”fillingthecenteroftherotarvolume.Inchapter8itwillbeshownthattheenergydensitycontainedinthestrongestpartoftherotatinggravfield thecenter isexactlythesameastheenergydensitycontainedinthestrongestpartoftherotatingdipolewave thecircumference .Infact,therotatinggravfieldisafundamentalpartofthedipolewaveandenergyisjustbeingtransferredbetweenthesetwostates. Thismeans that the energy density is relatively evenly distributed across the rotarvolume.Themomentofinertiaofarotarismostcloselyapproximatelybythemomentofinertiaforadisk: ½mr2). We will calculate the angular momentum of the disk analogy spinning in a plane. ωset: ½mr2 ½mλc2ω c/ λc ħ/mc

½mλc2 ½mc ħ

notethatmasscancels

½ħTherefore,thisisanotherconsiderationwhichalsolowerstheangularmomentumtowards½ħ.Infact,ifwecombinebothconsiderationswewouldhaveangularmomentumlowerthan½ħ.Athirdconsiderationisthatarotarisalsonotaspinningdisk.Itsenergydoesnotendabruptlyattheradialdistanceλc.Infact,theenergydistributionisnotknownbeyondthegeneralizationspresentedhere.Calculating theenergydistributionandangularmomentum is a complicatedproblemthatrequiresamoreadvancedmodelandarigorousanalysis.Thepointisthatthere


arealargenumberofwaysthattheenergycanbedistributedwithinthemodelthatresultinangularmomentumof½ħ.Infact,achievingthisangularmomentumwouldbecomeoneoftherequirements for choosing the “correct” energy distribution. However, this is a successfulplausibility test. The proposed model inherently incorporates angular momentum into thestructureofarotaranditisplausiblethatfurtheranalysiswillconfirm½ħ.Moleculesalsopossessquantizedangularmomentum,butinthecaseofmoleculesitiseasytoprovethatthisquantizedangularmomentumresultsfromthephysicalrotationofthemolecule.Thereareotherexamplesofquantizedangularmomentuminvolvingtherotationofphysicalobjectssuchasthequantizedvorticesthatforminsuperfluidliquidhelium.Thepointisthattheangularmomentum isphysical.There isnoneed to invoke theabstract conceptof anobjectpossessing“intrinsicangularmomentum”inthesecases.Somethingexternalisenforcingthisquantizationofangularmomentum.Thespacetimebasedmodelproposedhereattributesthisenforcementtoallmatter fundamentalparticles,molecules,etc. beingimmersedinaseaofsuperfluid vacuum energy. This spacetime based model also gives a conceptuallyunderstandable explanation of how a fundamental particle such as an electron can possessangularmomentum.Theelectronisarotatingdisturbanceinthespacetimefieldwithaphysicalsize that gives conceptually understandable angular momentum. The concept that a pointparticlecanpossessangularmomentumisanadmissionthatthemodelbeingusedisinadequate.Planck’s Constant Always Implies Angular Momentum: The spacetime based model of theuniverseelevatesangularmomentumto thesinglecharacteristicwhichcreatesallquantizedeffects including particles. There is a good article1 that also makes the point that “spin” offundamentalparticlesimpliesphysicalrotationofawave.Intheconclusionofthisarticle,thestatement is made, “The preceding has great intuitive appeal because it confirms our deepprejudicethatangularmomentumoughttobeduetosomekindofrotationalmotion.Buttherotationalmotionconsistsofacirculationofenergyinthewavefieldsratherthantherotationof somekindof rigidbody.” Another statement from this article is, “A comparisonbetweencalculationsofangularmomentumintheDiracandelectromagneticfieldsshowsthatthespinofan electron is entirely analogous to the angularmomentum carried by a classical circularlypolarizedwave.”All the energy of an electron can be traced to the fact that the electron possesses angularmomentum. Theprevious calculation in this chapter that resulted inE kmc2made somesimplifyingassumptions,butevenwithamoreadvancedmodel, all theobservableenergy iseitherdirectlyattributabletoangularmomentumorassociatedwitheffectssuchasthegravfieldortheelectricfieldwhichareboththeresultofquantizedangularmomentum.Thedimensionalanalysisunits of angularmomentumare:ML2/T. Merelymultiplyingangularmomentumbyfrequency rotation rate with units 1/T yields energywith units ofML2/T2. This is statedbecausethecurrentattitudeistotreathorħasaconstantthathappenstohaveunitswhichare

1 Ohanian, H. "What is spin?" Am. J. Phys. 54, 500 (1986);


thesameasangularmomentumbutangularmomentumisnotakeypartoftheexplanation.FortheenergyofaphotonwesayE ħω. However,thisisnotanabstractequation. Aswillbeshown in chapters 9 and 10, there is real angular momentum associated with a photon.Multiplyingthisangularmomentumbyrotationfrequencygivesthephoton’senergy.Whenanelectronisplacedinanexternalmagneticfieldthatisnotalignedwiththeelectron’smagnetic moment, the electron undergoes Larmor precession. This is analogous to theprecessionexhibitedbyagyroscopewhenatorqueisappliedwithacomponentperpendiculartotheaxisofrotation.Thereisnosuchthingas“intrinsicangularmomentum”.Thistermimpliesanabstractpropertywhichdoesnotincludeanythingthatisphysicallyrotating. Ifdipolewavesinspacetimeandquantizedangularmomentumareelevatedtothestatusofphysicalentities,thenallofphysicswouldbetransformedinto“classicalphysics”.Allofphysicswouldpotentiallybeconceptuallyunderstandableandanewgoldenageofphysicswouldbeborn.720°RotationTest: Oneofthebiggestmysteriesofquantummechanicsisthatfundamentalparticleswithspin½needstwofullrotations 2x360° 720° until theyreturntothesamestate.Sincethereisnothinginourmacroscopicworldwithsymmetrylikethat,thestructureofspin ½ particles such as an electron is often considered to be beyond our conceptualunderstanding. The calculation that demonstrates this 720° effect involves referencing thephaseofanelectronthatisgoingtoundergoarotationagainstahypotheticalstandardelectronthatisnotgoingtoundergoarotation.ThespinofanelectrongivestheelectronamagneticfieldwithNorthandSouthpoles.Itispossibletoimposeanexternalmagneticfieldwhichcanbeusedto rotate the electron since the electron’s North – Southmagnetic poles attempt to remainalignedwiththeexternalmagneticfield.Anyrotationoftheexternalmagneticfieldalsocausesthe electron to undergo precession, but that is another subject. The point is thatwhen theelectronhasundergoneonecomplete360°flip,itisoutofphasewiththereferenceelectron.Ittakesanother360°fliptobringthetestelectronbackintophasewiththereferenceelectron.Thefirstpointisthatthisexperimentisnotexactlythesameasrotatingaclassicalobjectsuchasaball through360°. Wearereferencingthephaseofanelectronnotsomefeatureonthesurfaceofaball.Theelectronhasnophysicalsurface.Next,wewillattempttovisualizewhatwillhappenifwesubjectedtherotarmodelofanelectrontoaforcewhichcausedtheaxisofrotationtobeflipped360°.Thefirstconsiderationisthattherotatingdipolewavethatformstheelectronisalreadypropagatingatthespeedoflight.Itisnotpossibletomaintainitsphysicalsize and frequencywhenwe also flip the axis of rotation becausemaintaining phasewouldrequirepartsofthedipolewavetoexceedthespeedoflight.Theexternalmagneticfieldmustexertaforceontheelectrontocausetheaxisoftheelectrontorotate.Thisforceappliedthroughadistanceisaddingenergytotheelectronwhichcausestherotationalfrequencytoincreaseandtheradius λc todecrease.Thiswouldproduceaphasechangerelativetoastandardwhichis


notundergoingsuchanaxisflip.Icannotprovethatitwouldtakea720°axisfliptoreturntotheoriginalphase,buta360°rotationdefinitelywillnotreturntotheoriginalphase.Dipole Moment:Notonlydoestheproposedrotarmodelgivethesameangularmomentumtoallrotars,therotarmodelalsospecifiesthatallrotarshavethesamedipolemomentdm.Thedipolemomentofarotaristhedipoleamplitudetimestherotarradius.Wewillcalculatethevalueofthedipolemomentsharedbyallrotars.

dm Aβλc ħ

ħ

ħdm dipolemomentAβ‐seeexplanationbelow

dm LpLp dynamicPlancklength

AdipolemadeoftwoelectricallychargedparticleshasadipolemomentwithunitsofCoulombmeters.However,aspacetimedipolehasadipolemomentwithunitsofjustmetersbecauseAβisadimensionlessnumber.RotarswithalargemasshavealargevalueofAβ,butthisisoffsetbyasmallrotarradiusλc. Therefore,allrotarshavethesamedipolemomentofdynamicPlancklengthLp.PlanckUnits:Beforeproceedingfurther,Iwouldliketopauseanddiscuss“Planckunits”whichare physical units ofmeasurement derived from5 physical constantswhich are: ħ, c, G, theCoulombforceconstant1/4πεoandtheBoltzmannconstantkB.These5constantsarecombinedtogive5Planck“baseunits”:PlancklengthLp.PlancktimeTp,Planckmassmp,PlanckchargeqpandPlancktemperature paswellasnumerous“derivedPlanckunits”. Someof themoreimportantderivedPlanckunitsare:PlanckenergyEp,PlanckforceFp,PlanckenergydensityUp,Planckpressureℙp,Planckdensityρp,Planckvoltage p,andPlanckimpedanceZp.WehavealreadyfrequentlymentionedPlancklength,PlancktimeandPlanckmass.However,itwillbeshown that thepropertiesof spacetimearemostnaturallyexpressed inPlanckunits.Goingfurtherwiththisthought,Planckunitsarebasedonthepropertiesofspacetime.Wecanactually learn about the properties of spacetime by examining a particular unit when it isexpressedinPlanckunits.ThetablebelowgivestheconversionsforthesePlanckunits.Also,thistableisrepeatedattheendofthebookinchapter15foreasyreference.Anyseriousanalysisofthespacetimemodeloftheuniversewillmakefrequentreferencetothistable.


Planck Units

Lp PlancklengthLp cTp ħG/c 1.616x10‐35m

mp Planckmassmp ħc/G2.176x10‐8kg

Tp PlancktimeTp Lp/c ħG/c 5.391x10‐44s

qp Planckchargeqp 4πε ħc1.876x10‐18Coulomb

p Plancktemperature p Ep/kB ħ / 1.417 1032°K

Ep PlanckenergyEp mpc2 ħc /G1.956x109J

ωp Planckangularfrequencyωp 1/Tp c /ħG1.855x1043s‐1Fp PlanckforceFp Ep/Lp c4/G1.210x1044NPp PlanckpowerPp Ep/Tp c5/G3.628x1052wUp PlanckenergydensityUp Ep/Lp3 c7/ħG24.636x10113J/m3ℙp Planckpressureℙp Fp/Lp2 c7/ħG24.636x10113N/m2 Up ρp Planckdensityρp mp/Lp3 c5/ħG25.155x1096kg/m3

Ap PlanckaccelerationAp c/Tp c /ħG5.575x1051m/s2

p Planckvoltage p Ep/qp c /4πε G 1.043x1027V

p Planckelectricfield p Fp/qp 4 ħ⁄ 6.450 1061V/m

p Planckmagneticfield p Zs/qp 4 ħ ⁄ 2.152 1053Tesla

p Planckcurrent p qp/Tp 4 ⁄ 3.480 1018ampZp PlanckimpedanceZp ħ/qp2 1/4πεoc29.98Ω

ItisoftensaidthatPlanckunitsarethe“naturalunits”becausetheyarebasedon5constantsofnature. However,thisexplanationdoesnotdojusticetotheimportanceofPlanckunits. Thepropertiesofspacetimearefinite.ThesePlanckunitsrepresentthelimitingvalues maximumof minimum that spacetime can support. For example, it is impossible to make a lengthmeasurementbetweentwopointsmoreaccuratethanPlancklength deviceindependent .ItisalsoimpossibletomakeatimemeasurementmoreaccuratethanPlancktime. Aspreviouslyexplained,thespacetimefieldhasdipolewaveswhicharemodulatingthedistancebetweentwopointsbyPlancklengthandmodulatingtherateoftime thedifferencebetweentwoclocks byPlanck time. Therefore, the inability tomakemeasurements smaller thanPlanck lengthandPlancktimecanbeconsideredtobeduetothe“noise”createdbyvacuumfluctuations.OneofthemanyinsightsobtainedfromEinstein’sfieldequationisthattheuniversehasalimittothemaximumpossibleforcethatcanbeexertedandthislimitingforceisequalto


Planckforce2,3. Thisignoresanumericalfactornear1. Theequationsofgeneralrelativitydeviate from Newtonian gravitational physics in strong gravity partly because of theexistenceofamaximumpossibleforcewhichintroducesnonlinearity. Therefore,generalrelativityandquantummechanicsagreeonthesignificanceofPlanckforce.Iftwoofthesamesizeblackholesareabouttomerge,theforcebetweenthemisPlanckforce.Itdoesnotmatterthesizeoftheblackholes,themaximumisalwaysPlanckforce.PlanckenergyandPlanckmassarealittlehardertoexplain.ItiseasytoexceedPlanckmass∿2x10‐8kg or Planck energy ∿2 109 J , but this assumes a large number of fundamentalparticles. Planckenergy is themaximumenergy that a singlequantizedunit can support.AphotonwithħofquantizedangularmomentumcannothavemorethanPlanckenergy.Afermionwith½ħofquantizedangularmomentumcannothavemoremassthanPlanckmass.TherotarmodelofafundamentalparticlewithPlanckenergywouldhavearadiusequaltoPlancklengthand a Compton frequency equal to Planck frequency. This would form a black hole with aSchwarzschildradiusRs≡Gm/c2equaltoPlancklength.Aspreviouslyexplained,aphotonblackhole rotatingatthespeedoflight hasasmallerSchwarzschildradiusthananon‐rotatingblackhole.Also,therotarmodelofafundamentalparticlehasenergypropagatingatthespeedoflightaround a circle with radius λc. When scaling gravitational effects produced by a rotar, theappropriateSchwarzschildradiusisRs≡Gm/c2 ratherthanrs 2Gm/c2.DimensionlessPlanckUnits: ThepreviousPlanckunitsquotedhaddimensions. Forexample,PlanckforceisFp 1.2x1044NewtonandPlanckenergyisEp 2x109Joule.ThereisanotherformofPlanckunitsthatisadimensionlessratio.Forexample,therearetimeswhenitisveryrevealingtoexpressforceorenergyasaratiorelativetothelargestpossibleforceorenergy.Inthis case we would divide the particular force or energy by Planck force or Planck energyrespectively.WhenweareexpressingforceasthisdimensionlessratiowewillusethesymbolF F/Fp. Note the underline signifying that this is dimensionless Planck units. SimilarlydimensionlessPlanckenergyEwouldbetheactualenergyEinSIunitsdividedbyPlanckenergytherefore:E E/Ep.Whileforceandenergywereusedasexamples,theconceptappliestoallotherunits.Thereforewewouldhavem,ω,etc.Itiscounterintuitivetomixdifferentunitsinanequation.Eventhoughweknowthatwearedealingwithdimensionlessratios,itdoesnotseemcorrecttowriteanequationwhichequatesdimensionless energy todimensionless force. However, on the levelwhereweare reducingeverythingtoadistortionofspacetime,thereisactuallyawayof lookingatforceandenergywhere they are equivalent. A specific amount of energy in the form of fermions requires aspecificamountofforcetostabilizethem.Thisstatementmightnotbeunderstandablenow,but

2 T.Jacobson,“ThermodynamicsofSpacetime:TheEinsteinEquationofState.”Phys.Rev.Lett.75,1260 1995 3 G.W.Gibbons,“TheMaximumTensionPrincipleinGeneralRelativity.”Found.Phys.32,1891 2002 http://arxiv.org/pdf/hep-th/0210109.pdf


inlaterchaptersthiswillbecomeclearer.Thepointisthatifaparticulartermisexpressedasadimensionlessnumbersuchas10‐20,thenitisusing10‐20oftheavailablepropertiesofspacetimeandthereisaconnectiontoanothertermwhichalsoisusing10‐20oftheavailablepropertiesofspacetime.Wewillnextreturntotheparticlepropertiespreviouslybeingexplained.Quantum Amplitude Equalities: ThestrainamplitudeAβofthespacetimewaveinsidetherotar volume of a rotar is an important dimensionless number for a rotar that has beendesignatedwiththesymbolAβ.Thissymbolwaschosenbecauseitisamplitudethataffectstherate of time and proper volume with similarities to gravitational magnitude β. The abovecalculationusedthesubstitutionthatAβ Lp/λc,butthereareseveralotherwaysofexpressingthisstrainamplitude.

Aβ Lp/λc Tpωc /ħ m/mp Ei/Ep ωc/ωp ⁄ / Ei internalenergyofarotar mc2 energy Rs SchwarzschildradiusRs≡Gm/c2Pc circulatingpowerPc Eiωc ωc2ħ explainedlater Thesymbolsmp,ωp,Ep, andPparePlanckmass,PlanckenergyandPlanckpower. Theyaredefinedfurtherinthetablebelow.Tohelpconveythesignificanceofthisstringofequalities,Iwilluseanelectronforanumericalexample.Themassofanelectronis:me 9.1094 10‐31kgElectron’s Characteristics λc ħ/mc c/ωc 3.8616 10‐13mComptonradius rotarradius ωc mc2/ħ c/λc 7.7634 1020s‐1ComptonangularfrequencyEi mc2 ħωc 8.1871 10‐14JinternalenergyPc Eiωc ωc2ħ 6.356 107wcirculatingpower explainedbelow Electron’sCharacteristicsinDimensionlessPlanckUnits

Aβ /ħ 4.185x10‐23strainamplitudeLp/λc 1.616 10‐35/3.8616 10‐13 4.185x10‐23Comptonradius rotarradius ωcTp ωc/ωp 7.76 1020/1.855 1043 4.185x10‐23Comptonfrequencym/mp 9.109 10‐31/2.176 10‐8 4.185x10‐23massEi/Ep 8.187 10‐14/1.956 109 4.185x10‐23energy

⁄ 6.76 10‐58/3.86 10‐13 1/2 4.185x10‐23Schwarzschildradius

P /P 6.36 107/3.63 1052 1/2 4.185x10‐23circulatingpower

Uq/Up 1/4 1.42 1024/4.64 10113 1/4 4.185x10‐23energydensityAq/Ap 1/2 9.74 106/5.58 1051 1/2 4.185x10‐23gravaccelerationgq/Ap 1/3 4.08 10‐16/5.58 1051 1/3 4.185x10‐23gravitationalaccelerationatλc


ItisamazingthatthedimensionlessnumberAβthatrepresentsthestrainamplitudeofarotaris related to so many rotar properties. These include the rotar’s mass, energy, Comptonfrequency, Schwarzschild radius, rotar radius and circulating power defined below . Theseproperties can also be expressed in dimensionless Planck units. As previously explained,symbolswritten inboldandunderlined suchasm andλc will representvaluesexpressed indimensionlessPlanckunits.Usingthisdesignation,Aβhasthefollowingequalities:Aβ m ωc Ei 1/λc Pc1/2 Aq1/2 Uq1/4 Rs1/2dimensionlessPlanckunitsEachfundamentalrotarhasasingledimensionlessnumberthatexpressesallofarotar’suniqueproperties.Angularmomentumandchargearenotuniquetoaspecificfundamentalparticle.Forexample,anelectron,muonand tauoneachhave theirownuniquevaluesofquantumstrainamplitudeAβ shownbelow .Thevaluesofangularmomentumandchargearenotunique.Aβ 4.18x10‐23electron’samplitude,Planckfrequency,mass,energyandinversesizeAβ 8.66x10‐21muon’samplitude,Planckfrequency,mass,energyandinversesizeAβ 1.46x10‐19tauon’samplitude,Planckfrequency,mass,energyandinversesizeMaximum Amplitude Rotar:Outofcuriosity,let’scalculatethemassoftherotarthathasthemaximumpossibleamplitudewhichisaquantumamplitudeofAβ 1.Aβ Gm /ħcsubstituteAβ 1andsquarebothsides1 Gm2/ħcm ħc/G mpAβ 1whenthemassequalsthePlanckmass mp .Therefore,theproposedrotarmodelhasPlanckmassasthenaturalbasis.However,Aβ 1notonly represents a rotar with Planckmass, but because of the above equalities, Aβ 1 alsorepresentsarotarwithPlanckangularfrequencyωp,andarotarwitharotarradiusequaltoPlancklengthlp.Goingevenfurther,arotarwithAβ 1alsohasacirculatingpowerequalingPlanckpowerPp,aninternalenergyequaltoPlanckenergyEp,andanenergydensityequaltoPlanckenergydensityUp.Ifthereweresuchathingasa“Planckrotar”,thisproposedrotarmodelwouldhavethePlanckrotarasthenaturalbasis.Circulating Power: A rotar’s internal energy is confined energymade of dipolewaves inspacetime that are moving at the speed of light. Therefore, there is a specific amount ofcirculating power in any rotar. The circulating power Pc in an isolated rotar is the rotar’sinternal energyEi times the rotar’s Compton angular frequencyωc. This is themomentarypowerthatwouldleavetherotarvolumeifthecirculatingwave rotatingdipole dissipatedbyallpointsinthewavetravelinginstraightlines.Thewavewouldexpandbeyondtherotarradiusinatimeequalto1/ωc.


Pc Eiωc ωc2ħ m2c4/ħ Ei

2/ħPc circulatingpowerAnisolatedelectron’scirculatingpowerisabout63.56millionwatts. 8.2x10‐14J 2/ħThishighcirculating power can be understood when it is realized that the electron’s internal energy8.2x10‐14J ismultipliedbytheelectron’sComptonangularfrequency 7.8x1020s‐1 . Theconceptofcirculatingpowerwillbeimportantwhenweconsiderforces.Forfuturereference,we will calculate the value of circulating power in Dimensionless Planck units by dividingconventionalpowerbyPlanckpower Pp c5/G .Pc Pc/Pp m2c4/ħ G/c5 Gm2/ħcPc circulatingpowerinPlanckunits Characteristics of an Electron:Itisveryusefultohaveasingletableoftherotarcharacteristicsandstandardcharacteristicsofanelectrontotestconcepts. Therefore, the followingtable isprovidedhereandinchapter15whichisacompilationofequationsanddefinitions.

Constants of an Electron Aβ 4.1854x10‐23 electron’sstrainamplitudeλc 3.8616x10‐13m electron’sComptonradius rotarradius ωc 7.7634x1020s‐1 electron’sComptonangularfrequencyυc 1.2356x1020Hz electron’sComptonfrequencyPc 6.356x107w electron’scirculatingpowerFm 0.21201N electron’smaximumforceatdistanceof Rs 6.7635x10‐58m electron’sSchwarzschildradiusRs≡Gm/c2U Ei/λc3 1.42x1024J/m3 electron’senergydensity cubic U 3/4π Ei/ 3.397x1023J/m3 electron’senergydensity spherical V λc3 5.7584x10‐38m3 electron’srotarvolume cubic Ag 9.7413x106m/s2 electron’sgravaccelerationatcenterofrotarvolumeme 9.1094x10‐31kg electron’smassEi 8.1871x10‐14J electron’senergye 1.6022x10‐19Coulomb electron’scharge


Forces

Wearenextgoingtoexaminethestrongforce,theelectromagneticforceandthegravitationalforcebetweentwoofthesamerotars.Theonlyforceexertedbydipolewavesinspacetimeistherelativisticforce Fr Pr/c .Therefore,wewouldexpectthattheforcebetweenparticlesshouldbeasimplefunctionoftherotar’scirculatingpower.Initially,wewillexaminetheforcesexertedunderthesimplestconditionfortherotarmodeloffundamentalparticles.Theforceswillbecalculatedbetweentwoofthesamerotarsseparatedbytherotar’snaturalunitoflength– separated by λc. This is an unrealistic assumption for an actual experiment because aseparation distance of λc corresponds to the point where quantum mechanics becomesdominant.Itisimpossibletopreciselyholdthisseparationdistance.However,thisimportantseparationdistancecanbe rationalizedasmerelyanextrapolation from longerdistance toaseparationdistanceofλc.In later chapters we will examine other distances, but the spacetime based rotar modelpresentedthusfaronlyisabletodefinethecharacteristicsatadistanceequaltotherotarradiusλc. It is reasonable that if thespacetimebasedmodel iscorrect, then thesimplestseparationdistancewouldbeλc,therotar’snaturalunitoflength.Calculationsatarbitrarydistanceinvolveanadditionalconsiderationofhowwavesintheexternalvolumeofarotarfalloffwithdistance.Initially limiting the separation distance just to λc the rotar radius involves the fewestassumptions.Weknowthestrainamplitudeofarotaratthisdistanceis:A Aβ Tpωc Lp/λc.Therefore, this fundamental test conditionwill be used exclusively for the remainder of thischapter.Theoretical Maximum Force: Wewill begin this examination of forces by asking a simplequestion.Isthereatheoreticalmaximumforcethatafundamentalrotarwithaknownenergycangenerateataparticulardistance?Thisquestionconsidersonlytheenergyofarotarandthedistance. Other characteristics of the rotar will determine whether the rotar can actuallyinteract and achieve anything close to the theoreticalmaximum force. At this early stageofdevelopmentofforceswearedealingwiththerelativisticforceinitssimplestform.Sincetherelativisticforceisonlyrepulsive,itfollowsthatasimplifiedmodelofthetheoreticalmaximumforcewilldescribearepulsiveforce.Laterthemodelwillbeexpandedandeventuallyyieldthestrongforcewhichisanattractingforcewithasymptoticfreedomcharacteristics.Fornowwearemerelylogicallyfollowingthenarrowpathassociatedwiththestartingassumption.Thestandardmodeldescribesthestrongforce thestronginteraction astheexchangeofgluonsbetween quarks. There are subtleties in this exchange that are explained by invoking colorchargeandquantumchromodynamics QCD .Wearestartingfromfirstprinciplesandsofarknownothingaboutgluons,etc.Weonlyknowthesimplifiedrotarmodelpresentedsofar.Wehaveadipolewave inspacetime thatpossessesquantizedangularmomentumandaspecificamountofenergy. Thedipolewave ispropagatingat thespeedof light inaclosed loopone


wavelengthincircumference.Thisconceptdefinesaspecificrotationalfrequencyandaspecificamountofcirculatingpower.Maximum Force from Circulating Power:WewillfirstusetheconceptofcirculatingpowerPcofafundamentalrotar.Previously;itwasfoundthattherotarmodelimpliesthateveryrotarcanbeconsideredtohaveacirculatingpowerequalto:Pc Eiωc ωc

2ħ ħc2/λc2Pc circulatingpowerofarotarinside

Previouslyweconcludedthatthestartingassumption theuniverseisonlyspacetime impliedthatthereisonlyonetrulyfundamentalforce–therelativisticforceFr Pr/c. Thestrongestforce thata fundamentalrotarcanexertatdistanceλcwilloccur ifallofarotar’scirculatingpowerisdeflected setPr Pc .Thispresumestwoofthesamerotars,eachwithinternalenergyofEi.Onlyquarksareactuallycapableofexertingsomethingclosetothismaximumforce,butitisstillpossibletocalculatethetheoreticalmaximumforcethatanyrotarcangenerateifalltherotar’scirculatingpowerisdeflected.Thisisequivalenttosayingthatsomerotarscannotdeflectallthecirculatingpowerataseparationdistanceofλc,butitispossibletocalculatethemaximumforcethatwouldbegeneratedifallthecirculatingpowerwasdeflected.WetaketherelativisticforceequationFr Pr/candsetFr FmandPr Pc ħc2/λc2.Fm Pc/c ⁄ m2c3/ħ ωc

2ħ/c Aβ2FpFm rotar’smaximumforceatdistanceλcLaterwewillcomparethisvalueofthemaximumpossibleforceatλctotheelectrostaticforceatthisdistancetosee if it isreasonable. However, first Iwant toexplainaqualificationonthephysical interpretation of this maximum force Fm ħ⁄ . This equation represents themaximumforcethatcanbeexertediftwoofthesamerotarsareheldstationaryatadistanceequaltotheirrotarradiusλc. Iftworotarsarecollidingatrelativisticvelocity,thenagreaterforcecanbegeneratedbecausekineticenergyisconvertedtoincreasetherotar’sinternalenergyEiattheinstantofcollision.ThismomentarilyincreasestheinternalenergyEiofthecollidingrotarswhichalsomomentarilyincreasesFm,ωcandPc.anddecreasestherotarradiusλc.Forexample,anelectronwithrelativisticvelocityequalto50GeVcantemporarilyconvertthiskineticenergyintointernalenergyinacollisionraisingtheelectron’sinternalenergyfrom~0.5MeVto50GeV,afactorofabout100,000.Thiswouldmomentarilydecreasetheelectron’srotarradiusbyafactorof105from3.86x10‐13mto3.86x10‐18m.Thiswouldalsoincreaseboththecirculatingpowerandthemaximumforcebyafactorof1010.Therefore,anelectronundergoinga collision can exhibit a force greater than the theoreticalmaximum force calculated for anisolated electron no collision . Trying to remove a quark from a hadron also changes theinternalenergyofthequarkandcanaffectthebindingforce.Thiswillbediscussedlater.


Maximum Force from the Wave-Amplitude Equation:Wewillalsocalculatethemaximumforceusingthewave‐amplitudeequation:F A2ω2Z /c. Wehavepreviouslyusedasimilarequation to calculate the energy in a rotar. For that calculation we made the followingsubstitutions:A Aβ Lp/λc;ω ωc c/λc;andV kλc3.Thistimewehaveanareaterm .Sincethepresumptionisthatwehavetwoofthesamerotarsseparatedbyλc,thismeansthattheinteractionareawouldbeaconstanttimesλc2. kλc2 .Therearemanyothernumericalfactorscloseto1thathavebeenpreviouslyignored,sowewillalsoignorethisconstant.F A2ω2Z /cset:A Aβ Lp/λc;ω ωc c/λc;Z Zs c3/G;

Fm set:Lp2 ħG/c3;

Fm ħ

ħ

ħ

ħ

Oncemore,IwanttoemphasizethatthiscalculationassumedamplitudeA Aβanddistancer λcwhichwasimpliedwhenwesetarea withtheconstantignored.Thisismentionedbecauseinchapter8wewillextendboththeelectrostaticforceandthegravitationalforcetoarbitrarydistance.Alsoinalaterchapter,twocompetingversionsofthemaximumforcewillbeshowntomakeup themorecomplexstrong forcebetweenquarks. Theconditionknownasasymptotic freedomwill be analyzed and shown to result from competingmaximum forceswhich reach equilibrium. However, a slight displacement from this equilibrium separationdistanceresultsinanetrestoringforcewhichincreaseswithdisplacementandcanalmostreachthemaximumforce.Coulomb Force:ToevaluatethemaximumforceitisnecessarytocompareittotheCoulombforce electromagnetic force thatwouldexistbetween twoelectricallychargedrotarsat theseparation distance of r . There are actually two possible values of charge q that areinterestingandwewillevaluatebothofthem.Oneobviouschoiceistouseelementarychargee.

However,theotherinterestingvalueisPlanckchargeqp 4 ħ e/√ 1.88x10‐18Coulombwhichisabout11.7timeslargerthanelementarychargee.Planckchargeisderivedfromεo,thepermittivityoffreespaceandisthebestchoiceofaunitofchargewhenwearecomparing forces because Planck charge avoids dealingwith the fine structure constantα e2/4πεoħc 1/137.Thefinestructureconstantαisknowntobethecouplingconstantrelatingtothestrengthoftheelectromagneticinteractionbetweenaparticlewithchargeeandaphoton.BychoosingPlanckchargewearesettingthiscouplingconstantequalto1.Byeliminatingthecouplingconstantα,wewouldexpectthatatseparationdistanceofr theelectromagnetic forceshouldbeequal to themaximumforce if theparticleand forcemodelsdescribedthusfararecorrect.TheCoulombforceequationF q2/4πεoħcwillbeusedforthiscriticaltest.WewillusetheforcesymbolFEtospecifythatwearerepresenting


theelectrostaticforcebetweentwoPlanckcharges.ThereforewewillmakethefollowingsubstitutionsintotheCoulombforceequation:

F set:F FE,r λcq qp 4 ħ andFm ħ ⁄

FE ħ Fm

Therefore,thisisaspectacularsuccess.WhenweusePlanckchargetosetthecouplingconstantequal to 1 and r , then we obtain the equation that the electrostatic force equals themaximumforceFE Fm.ThefactthatweobtainanswerswhichimplyPlanckchargeshouldnotbeinterpretedthatsomehowPlanckchargeisarealisticpossibilityofthechargeofarotar.Inthe next chapter itwill be shown that thewave characteristicswhichwould be required toachievePlanckchargewouldalsomakeanunstablerotarwhichwouldradiateawayallitsenergyinatimeperiodof1/ωcwhichforanelectronisonlyabout10‐20s.Planckchargeisatheoreticalideathatimpliesallofaparticle’senergyistransferredtoitsexternalfield.Anytimeintherestofthebookthatwearerepresentingtheelectrostaticforcegeneratedbyparticleswithelementarychargee,wewillusethesymbolFe.Thefollowingisthefirstofthesecalculations.

F set:F Fer λc;q eα e2/4πεoħcandFm ħ ⁄

Fe 2 ħ2 αFm

Therefore, evenusingelementarychargeeweobtaina connectionbetween theelectrostaticforceand themaximum force,but this electrostatic force isdiminishedby the fine structureconstantα 1/137. The finestructureconstanthasneverbeenable tobemathematicallyderivedfromfirstprinciples.Ithasbeenthesourceofmysteryforgenerationsoftheoreticalphysicists. Therefore,we alsowillmerely accept thismysterious number as a couplingconstantofunknownorigin.


Gravity

Nowforthebigquestion:Canwedeveloptheforceofgravityfromfirstprinciplesusingtherotarmodel?Thusfarwehavediscussedthefundamentaldipolewavethatcanexistinthespacetimefield. Ifwearegoing tobeable toexplainall the forcesofnaturewithonlydipolewaves inspacetime,wehavetoexaminethepossibilitythatundersomecircumstancesadipolewaveinspacetimemaynotbeperfectlysinusoidal.Onceagainthestartingassumptionthattheuniverseisonlyspacetimeservesasawonderfulrestriction.Itkeepsusfocusedonexaminingonlythemostbasicpropertiesofspacetime.Thepropertiesofthespacetimefieldarefinite.Thereisamaximumforce,amaximumfrequencyandamaximumstrainamplitude fordipolewaves inspacetime. Thesemaximums can be considered as boundary conditions imposed on dipolewavesinspacetime.Suchboundaryconditionsshouldproducenonlineareffects.If the universe is only spacetime, we do not have many possible explanations for gravity. In fact, the only plausible possibility is that the spacetime field is a nonlinear medium for dipole waves in spacetime. Optical Kerr Effect:IseeasimilaritybetweengravityandanonlinearopticaleffectcalledtheopticalKerreffect.Alltransparentmaterialshaveamaximumintensitylimitwhichisaboundarycondition. Therefore,when light passes through any transparentmaterial, a nonlinear effectoccurs. Even for wavelengths for which thematerial is transparent, there is a limit to themaximumintensity maximumelectricfieldstrength thatcanpropagatethroughthematerial.Thislimitresultsinnonlinearity distortion evenforintensitiesthatarefarbelowthislimit.The oscillating electric field of the light produces a non‐oscillating nonlinear effect whichchangestheindexofrefractionofthetransparentmaterial. Thisnonlineareffectreducesthespeed of light in the transparent material in addition to the normal reduction due to thematerial’sindexofrefractionatzerointensity.AnexpressionoftheopticalKerreffectisgivenbythefollowingsimplifiedequationthatignoreshigherorderterms.nk no k1 ω2simplifiedopticalKerreffectequationnk theindexofrefractionwhichincludestheopticalKerreffectcontributionno thenormalindexofrefractionatzerointensityk1 anonlinearconstantthatdependsonthetransparentmaterialω electricfieldstrengthatfrequencyω

Thismeansthatthespeedoflightinanytransparentmaterialhasafundamentalterm no andasecondorderterm k1 ω2 .Thesecondordertermdependsonthesquareofthealternatingelectricfieldproducedbythelight.


Evensunlightpassingthroughawindowproducesaslightnonlineareffectintheglass.Whenahighpeakpowerpulseof laser light is focused ina transparentmaterial, the light can reachoscillatingelectricfieldstrengthwheretheopticalKerreffectincreasestheindexofrefractiontotheextentthatthelaserbeamisfurtherconcentratedandconfinedtoasmallfilament.Thisconfinementcanbesogreatthatthebeamisnotallowedtodiverge.Thiseffectiseasilyseeninglassandothersolids,butithasevenbeendemonstratedinair.WhiletheanalogybetweentheopticalKerreffectandgravityisfarfromperfect,thepointisthathomogeneousmaterials likeglassorairexhibitanonlinearity thatscalesproportional to thesquare of the electric field strength 2where can be considered awave amplitude. Thissquaring produces an effect that is always positive. In the optical Kerr effect, the index ofrefractionalwaysincreases.This nonlinearity in transparent materials is associated with the fact that any transparentmaterialhasamaximumelectricfieldstrengthlimitation.Exceedingthismaximumelectricfieldstrengthwillphysicallydamagethetransparentmaterial.Themostextremeformofdamageisionizationoftheatoms,butmoleculescanbedecomposedatelectricfieldstrengthlessthantheionizationthreshold.Thismaximumfieldstrengthintroducesanonlinearitythatscaleswith 2andhigherpowersof . However, thehigherpowersonlybecomesignificantasthe limitingintensityisapproached.Thespacetimefieldisalsoamediumthattransmitswaves.Isthereanyevidencethatthespacetimefieldisanonlinearmedium?Gravity – A Nonlinear Effect: Weneedtoconsidertheforceofgravitybetweentwoofthesame rotars at distance . If the universe is only spacetime and if there is only one trulyfundamentalforce therelativisticforce thendipolewavesinspacetimemustalsocausegravity.Wearethereforelookingforamechanismwherebyarotar’scirculatingpowercanbeconvertedintoaforcethathasonlyonepolarityandisvastlyweakerthantheotherforces.Thereisreallyonlyonereasonablechoice.Gravitymustbetheresultofthespacetimefieldbeinganonlinearmediumfordipolewavesinspacetime.FifthStartingAssumption:The spacetime field is a nonlinear medium for dipole waves in spacetime. This nonlinearity ultimately produces gravity.Thegravitationalforcemustbetheresultofthisnonlinearitywhiletheotherforcesareadirectlinear functionofcirculatingpower.Dipolewavesinspacetimehaveatheoreticalmaximumamplitude of Aβ 1. This means that dipole waves in spacetime should also exhibit anonlinearity that scaleswithAβ2evenwhenAβ 1.The strainproducedbydipolewaves inspacetimemusthavealineartermandanonlinearterm.

Strain Aβsinωt Aβsinωt 2 Aβsinωt–½Aβ2cos2ωt ½Aβ2 boldforemphasis


Thelineartermis Aβsinωt andthenonlineartermis Aβsinωt)2.Therearealsohigherordernonlinearterms greaterthansquare butthesecanbe ignoredbecauseAβ isanumberveryclosetozeroandanyhigherpowers cubeorabove ofthisareinsignificant.Thenonlineartermhasbeenfurtherexpandedintoaweakoscillatingterm Aβ2cos2ωt andanon‐oscillatingtermthatisalwayspositive ½Aβ2 .Itisproposedthatthestraininspacetimeproducedbythenon‐oscillating term Aβ2 at distance is responsible for the general relativistic curvature ofspacetimewhichresultsingravity.Ananalysisinchapter8willshowhowthenonlinearterm Aβsinωt 2leadstobothgravitationalattractionaswellasthegravitationaleffectontimeanddistance.Inthischapterwearegoingtostartwithasimplifiedanalysisthatconcentratesonlyonthemagnitudeoftheforceexertedbythegravityofarotar.Withthislimitationweagainaredevelopinganover‐simplifiedrepulsiveforcewiththecorrectmagnitudeofgravity.Thiswilllaterbeimprovedintoanattractingforcethatalsoexhibitsthespatialandtemporalpropertiesofgravity.Theactualforceofgravitywillbeshowninchapter8toresultfromastraininspacetimethatproducesanunbalancedforceonoppositesidesofarotar.Itiseasytodemonstratethatthenonlinearityofthespacetimefieldgivesthecorrectmagnitudeof the gravitational force at distance using a calculation that is somewhat oversimplified.PreviouslywewereusingA Aβasthesubstitutionofthewaveamplitudeforthemaximumforce and other rotar properties. To prove that gravity is caused by the nonlinearity ofspacetime,wewillnowsquare thestrainamplitude termandmake thenonlinearamplitudesubstitutionA Aβ2intotheforceequation:F kA2ω2Z /c.F kA2ω2Z /cforgravityset:A Aβ2 Lp2/ ,ω ωc,Z Zs c3/G, k

Fg Aβ4ωc2Zs /c 4 2

2

ħ

ħ 2

Fg 2 magnitudeofthegravitationalforcebetween2particlesofmassmatdistance

Eventhoughthisisoversimplified,Ifindthiscalculationveryexciting!WeobtaintheNewtoniangravitational force equation starting with rotating dipole waves in spacetime. It was notnecessarytomakeananalogytoacceleration.Thisparticularcalculationwasfortwoofthesamerotarsataseparationdistanceof ,butinchapter8wewillbeabletobroadenthistothemoregeneralcaseofanymass/energyatanydistance.Furthermore,themodelwillbeimprovedandresult in this being an attracting force. To my knowledge, this is the first time that thegravitationalforcehaseverbeencalculatedfromconceptuallyunderstandablefirstprinciples.Therewerenovagueanalogiestorestrainingamassfromfollowingageodesic.Thisimpliesthatgravityisreallyaforceandnottheresultofthegeometryofspacetime.Staticcurvedspacetimeistheresultofdynamic oscillating curvedspacetimeexhibitinganonlinear


effect. In chapter4wedescribedhow thequantummechanicalmodel of the spacetime fieldpossesses elasticity, impedance, energy density, etc. Introducing matter dipole waves withquantizedangularmomentum intothishomogeneousmediumproducesdistortionwhichhasbothalinearandanonlinearcomponent.Gravityisthenonlinearcomponent.Fromtheabovecalculation it is not hard to see that eventually we will obtain the Newtonian equation:F Gm1m2/r2. Eventhoughthisisasuccessfulplausibilitycalculation,inchapter8itwillbeshown to be an oversimplification that gets the magnitude correct but the vector wrong.Additionalstepswillbeintroducedtoobtainthecompletepicture.Itshouldberecognizedthatfor single particles the Newtonian gravitational equation can be considered almost exact.However, there are also small nonlinearities which are being ignored in these simplecalculations. General relativity differs from Newtonian gravity because general relativityincorporatesnonlinearitiesincludingamaximumpossibleforce Planckforce .Whilethisbookdoes not carry this model to the strong gravity limit, it appears that a mature modelincorporatingnonlineartermswouldbecompatiblewithgeneralrelativity.Infact,chapters8and10discusssubtletiesthatgofurtherthangeneralrelativity.Review:WehavejustcalculatedasimplifiedversionofNewton’sgravitationalequationfromasetofstartingassumptions.Thestepsthatbroughtustothispointwillbebrieflyreviewed.Thekeyassumptionsare:

1 Theuniverseisonlyspacetime.2 Dipole waves in spacetime are permitted by quantum mechanics provided that the

displacementofspacetimedoesnotexceedthePlancklength/timelimitation.3 Energyinanyformisfundamentallymadeofdipolewavesinspacetimepropagatingat

thespeedoflight.4 Thereisonlyonefundamentalforce:therelativisticforce.Thisforceoccurswhenwaves

inspacetime,propagatingatthespeedoflight,aredeflected.5 Fundamentalparticlesaredipolewaves in spacetime that forma rotatingdipole,one

wavelengthincircumferencethatpossessescirculatingpower.6 The spacetime field is a nonlinear medium for dipole waves in spacetime. This

nonlinearityultimatelyproducesgravity.Wavesinthespacetimefieldarelikesoundwavespropagatinginthemediumofthespacetimefield.Spacetimehasanimpedance Zs c3/G andtheforcegeneratedbydeflectingwavesinspacetimeis:F A2ω2Zs /cwhereAisamplitudeand isarea.Assumption#6saysthatthespacetimefieldisanonlinearmedium.Thisnonlineareffectcanbeconsideredthesourceofanew nonlinear wave that has strain amplitude that is the square of the amplitude of thefundamental wave amplitude: Aβ2 Aβg Lp2/ . Inserting this amplitude into the force

equation above yields Fg 2⁄ which is a simplified version of Newton’s gravitationalequationthatassumestwoofthesamemassparticlesatdistance dimensionlessconstantsnear1ignored .


Connection Between the Forces and Circulating Power:Iftheforcesofnaturearecausedbytheinteractionofwavesinspacetime,thenthereshouldbeasimplerelationshipbetweentheforce and the circulating power Pc . I previously proposed that there is only one trulyfundamentalforceinnature–therelativisticforce Fr Pr/c .Thisistherepulsiveforceexertedwhenrelativisticpower powerpropagatingatthespeedoflight is“deflected”whichincludesabsorbedandreflected.Forthistoappeartobeanattractingforcethisinteractionmustincludepressure repulsiveforce exertedbyvacuumenergy.Thisisdiscussedinchapter8.However,inallcasesaforcegeneratedbyarotarmustberelatedtoboththecirculatingpowerandtherotar radius of the rotar. Furthermore, since gravity is the result of nonlinearity,wewouldexpectthatthegravitationalforcewouldbeafunctionofPc

2whiletheotherforceswouldbealinearfunctionofPc.Wearethereforegoingtoperformacriticaltestoftherotarmodelandtheconceptofasinglefundamental force.There isnosinglerotarthatexhibitsall threeof the followingproperties:elementarycharge,thestrongforceandgravity.Quarksandthechargedleptonsonlyexhibit2ofthe3properties.However,itispossibletocalculatethemagnitudeofthesethreeforcesasiftheywerepossessedbyasinglepairofthesamerotarsseparatedbyadistanceequaltotheirrotarradius.WewillalsobeassumingthehypotheticalcaseoftwoparticleswithPlanckchargeinsomeofthefollowingcalculations.Thiscriticaltestwillexaminewhetherthereisaneasytounderstandrelationshipbetweenthemagnitudesoftheforces Fm,FE,Fe,Fg andthecirculatingpowerPcoftherotarcausingtheforce.This comparisonwill be done using the natural Planck units of force and power bold andunderlined indicates Dimensionless Planck units – Fm, FE, Fe, Fg, and Pc . Initially, allcomparisonswillbemadeattherotar’snaturalunitoflength,itsrotarradius c/ωc ħ/mc.Thisistheonlydistancethatweknowthecirculatingpower.Substitutionsthatwillbeused:

r ħ;Fm

ħ;Fp ;Pc

ħ;Pp ; ħc, αħc;

Fg 2 Gm2ħ

ħ

Fm ħ

ħ

FE 2 ħcħ

ħ

Fe 2 αħcħ

αħ

Pc ħ

ħ


Since all of these are related to Gm2/ħc,we obtain simple relationships between forces andcirculatingpowerwhenwearedealingwith twoof the samerotarswith chargeeorqpandseparatedbyadistance .Recallthattheconceptofasinglerelativisticforcesaysthatthereshouldbeaneasytounderstandrelationshipbetweenforceandcirculatingpower.Fm PcFm maximumforceindimensionlessPlanckunits closelyrelatedtothestrongforce FE PcFE electromagneticforceindimensionlessPlanckunitswhereq qp Planckcharge Fe αPcFe electromagneticforceindimensionlessPlanckunitswhereq e elementarycharge Fg Pc2Fg gravitationalforceindimensionlessPlanckunitsThisisaspectacularsuccessthatstronglysupportsthespacetimebasedmodelofforces.Thissimplificationofrelationshipsoccursatthespacetimebasedmodel’sfundamentalunitoflengththereducedComptonwavelength . ThemaximumforcedeflectsallofthecirculatingpowerFm Pc .TheelectromagneticforcealsodeflectsallthecirculatingpowerifweassumePlanckcharge FE Pcbutelementarychargeeonlydeflectsabout1/137ofthecirculatingpowerandisthereforeabout137timesweaker Fe αPc).However,forcesFmFEandFearesimilarbecausetheyallscalelinearlywithcirculatingpower scalewithPc1 .Gravityisdifferentbecauseitisthe result of a nonlinear effect and scales proportional to Pc2. When circulating power isexpressed in dimensionless Planck units it is always a dimensionless number close to zero.Thereforesquaringthisnumberproducesanumberevenclosertozero.TheweaknessofgravitycomparedtotheotherforcesisduetothedifferencebetweenPcandPc2.Theanalysisofgravitywill continue in chapter 8, but gravity is the result of the spacetime field being a nonlinearmedium that scales with amplitude squared which in turn results in a Pc2 scaling of thegravitationalforce.Wecanalsorelatetherotar’s internalenergyEi,energydensityUq,SchwarzschildradiusRs,reducedComptonwavelengthλcandstrainamplitudeAβtotheforcesbetweentwoofthesamerotarsseparatedbydistance whentermsareexpressedindimensionlessPlanckunits.AgainweshowtherelationshiptoGm2/ħc.

Ei Ei/Ep ħ 5

ħ

Uq Uq/Up ħ

ħ

ħ

Rs Rs/Lp ħ

ħ

λc λc/Lp ħ

ħ

ħ2

Aβ Lp/ ħ

ħ

ħ


Combiningalloftheseweobtain:

Fg Fm2 FE2 Fe/α 2 Pc2 Ei4 Uq Rs4 1/λc4 Aβ4

Thisisoneofthemostimportantfindingsinthisbook. It isaseriesofequalitiesthatcanberewrittenas54individualequations.Thesimplicityofthisseriesofequationsisjawdropping.Itshowshowthegravitationalforceiscloselyrelatedtonotonlythemaximumforceandtheelectromagnetic force but also to a rotar’s energy, circulating power, energy density,Schwarzschildradius,Comptonwavelengthandstrainamplitude.Itwillbeshownlaterthatthestrongforceisalsorelatedtotheseforces.Just to help internalize these relationships, we will use two electrons separated by 3.86 x 10‐13 meters as illustration. All the other values are for electrons expressed in

dimensionlessPlanckunits.ReadersareinvitedtosubstitutethefollowingvaluesofFm, Fe, Fg, Pc, Ei andAβintotheaboveequations.Fg Fg/Fp Gm2/ / c4/G 3.07x10‐90Fm Fs/Fp ħc/ / c4/G 1.75x10‐45Fe Fe/Fp e2/4πεo / c4/G 1.28x10‐47Pc Pc/Pp ħc2/ / c5/G 1.75x10‐45Ei Ei/Ep mc2 G/ħc5 1/2 4.19x10‐23Uq Uq/Up m4c5/ħ3 ħG2/c7 3.07x10‐90Rs Rs/Lp Gm/c2 c3/ħG 1/2 4.19x10‐23

λc λc/Lp ħ/mc c3/ħG 1/2 2.39x1022Aβ Lp/ Gm2/ħc 1/2 4.19x10‐23Alternative Derivation:Whiletheabovecalculationsthatrelatetheforcestocirculatingpowerisonewayofderivingtheseequations,thereisanotherderivationthatissimilarbutmakessomedifferentpoints.ItisbasedontheideathatweshouldbeabletocalculatethemagnitudeoftheforceexertedbywavesinspacetimewithamplitudeAβatfrequencyωcandseparationdistanceusingthefollowingwaveamplitudeequation:F A2ω2Z /c.Wewillstartbymakingthe

substitutionofA AβforthemaximumforceFmattheseparationdistanceequaltotherotarradiusRqofarotar.F A2ω2Z /cset:F Fm,A Aβ Lp/ Ei/Ep Ei, ω ωc c/ ,Z Zs, k Fm Aβ2ω2cZs /c Ei2 c/ 2 c3/G /c Fm Ei

2 c4/G set:Fm FmFp Fm c4/G Fm Ei2Note:EiisbothparticleenergyindimensionlessPlanckunitsandwaveamplitude:Ei Lp/


Whentwoparticleswithelementarychargeeareseparatedbyadistance ,thentheequationbecomesFe αEi2.However,thefinestructureconstantisknowntobeacouplingconstant.IfwesetchargeequaltoPlanckchargeq qp,wearesettingthiscouplingconstantequalto1.Planckchargeisbasedon1/4πεoandPlanckchargeisoneofthe“basicPlanckunits”.consideredtobemorefundamentalthanchargee.DesignatingtheelectrostaticforcebetweentwoPlanckcharges qp asFEtheequationbecomes:FE Ei2Gravityisproposedtobetheresultofthespacetimefieldbeinganonlinearmediumfordipolewavesinspacetime.Thisnonlineareffectscaleswithwaveamplitudesquaredandhigherordertermscanbeignoredbecausetheyaretoosmall.Therefore,tocalculatethegravitationalforceFgatthisseparationdistance r wesubstituteA Aβ2intothesameforceequation. F A2ω2Z /cset:F Fg,andA Aβ2Fg Aβ2 2ωc2Zs /c Ei4 c/ 2 c3/G /c Fg Ei

4 c4/G Ei4/Fp

Fg Ei4Therefore,wecancombineFE Ei2and Fg Ei4intothefollowing:

Fg FE2

Thisequationneedstobestatedinwordsforfulleffect.Assumingaseparationdistanceofr λcandq qp,thegravitationalforceequalsthesquareoftheelectrostaticforcewhentermsarestated indimensionlessPlanckunits. Anumerical examplehelps to internalize this concept.Supposeweassumetwoparticles,eachwiththemassofanelectronme 9.1x10‐31kgthereforer λc 4.185x10‐13mbutwithPlanckchargeqp 1.88x10‐19CHerearethevalues:Fg 3.7x10‐46NandFg 3.07x10‐90dimensionlessPlanckunitsFE 0.212NandFE 1.75x10‐45dimensionlessPlanckunitsEi 8.19x10‐14Jand Ei 4.18x10‐23dimensionlessPlanckunitsThefollowingequationmakesthesameassumptionsofseparationdistanceandcharge,butitdoesnotusedimensionlessPlanckunits.

Inwords,Fg/FE FE/Fpsaysthatatthepreviouslystatedconditions,theratioofthegravitationalforce to the electrostatic force equals the ratio of the electrostatic force to Planck force.Thereforeatr λc.thereisasymmetrybetweenthegravitationalforce,theelectrostaticforceand Planck force Continuing with the numerical example previously given, Planck force is:


Fp c4/G 1.2x1044N. ThereforebothFg/FEandFE/Fpequal theabovedimensionlessratio:1.75x10‐45.TheaboveequationsusedPlanckchargebecausePlanckchargehasacouplingconstantof1andthissimplifiestheequations.However,itiseasytoconverttheaboveequationstoelementarychargeebysubstitutingFE Feα‐1whereFedesignateschargeeandFEdesignatesPlanckcharge.The fine structure constant isα, thereforeα‐1 137. In chapter8wewill extend the forceanalysistoarbitraryseparationdistance.Gravitational Wave Calculation:Wewillnowmoveontoanotherplausibilitytestthatcanbeperformedontheproposedrotarmodel. Recall thatdipolewaves inspacetimehaveenoughsimilarities togravitationalwaves thatwecanusegravitationalwaveequations foranalysis.However,wehavetoignoredimensionlessconstantsandinterprettheresultsappropriatelyfordipolewavesinspacetime.ThereisanequationusedtoestimategravitationalwaveamplitudeAgw atthelowintensitylimitwherenonlinearitiescanbeignored.Thisequationcanbeappliedtotheproposedrotarmodel.Agw kGω2 ε/c4r momentofinertia,ε asymmetryofarotatingobjectThissimplifiedequation,wouldnormallybeusedtoestimatethegravitationalwaveamplitudeofarotatingrodorarotatingbinarystarsystem.Itcontainsanangularfrequencytermω,themoment of inertia of the rotating object, the radius r of the rotating object, and amassasymmetry term ε. For example, a spherically symmetric objectwould have no asymmetryε 0 andtwoequalpointmassesseparatedby2rwouldhaveanasymmetryofε 1.Wearegoing toassume that ε 0and thedimensionlessasymmetry termεwillbe included in theall‐inclusiveconstantk.Wewillnextconvertthemomentofinertiatermtoangularmomentum.Agw kGω2 ε/c4rset: /ω, angularmomentum,εincludedinkAgw kGω /c4rThereasonforconvertingtoangularmomentumisbecausewewanttoapplythisequationtorotars.Weknowtheangularmomentumofparticlesas ½ħ.However,the½inthisangularmomentum is subject to interpretation as previously discussed. The constant, whatever itsvalue,willbeincludedinthegeneralconstantk.Therotarmodelimpliesthatε 0becausethedipolecoreofarotaristwolobesrotatingatthespeedoflight.Wewillalsolumptheeccentricitytermε,intothegeneralconstantk.Wewillnowcalculatethehypotheticalgravitationalwaveamplitudeforafundamentalrotar.


Agw kGω /c4rsubstitute:ω ωc c/ ; kħandr

Agw kGħ

Agw k ħG/c3 / Agw kLp2/ kAβ2wewillignoretheconstantkThisisanothersurprisingconnection.Wetakeagravitationalwaveequationusedincosmologyandinsertarotar’sangularmomentum½ħandComptonfrequencyωc.Wethendeterminethegravitationalwave amplitude excluding constant thatwould exist at adistanceof . WeobtaintheamplitudeAgw Lp2/ Aβ2.Wepreviouslydeterminedthatthenonlinearityofthespacetime field creates a non‐oscillating strain amplitude equal toAβ2 Lp2/ in the rotarvolume of a rotar. The same amplitude expression is obtained using an entirely differentapproach.Previouslyweemployedreasoningbasedonthequantummechanicalpropertiesofspacetimeandalsoonthespacetimefieldbeingnonlinear.Nowweobtainthesameanswerbyinsertingrotarsproperties angularmomentumandCompton frequency intoagravitationalwaveequationfromgeneralrelativity.Theabovecalculationisasuccess,butitalsoseemstoimplyaproblem. Areallfundamentalrotarscontinuouslyradiatingawaytheirenergyasgravitationalwaves? It is true that ifanyarbitraryvalueofComptonfrequencyorrotarradius c/ωc isassumed,thentherewouldprobablyberadiatedpowerboth for the fundamentalwaveand for thenonlinearwavewithamplitudeLp2/ .However,itisproposedthatthefundamentalrotarsthatdoexistcorrespondtospecialfrequency‐amplitudecombinationswhereawaveinteractionoccursthatcancelsthisradiatedpower.Evenshortlivedfundamentalrotars,suchasthetauon lifetime 3x10‐13s ,arelonglivedcomparedtothelifetimetheywouldhaveiftheircirculatingpowerwasradiated.SincePc Eiωc,thetimerequiredtoradiatetherotar’sinternalenergyEiwouldbetheinverseoftherotar’sComptonfrequency.Foratauonthiswouldbe:1/ωc 3x10‐25second.Thetauon’slifetime is about1012 times longer than its1/ωc timeand is considered tobea “semi‐stableparticle”.Thereareaninfinitenumberofpossiblefrequencies,amplitudesandconfigurationsthatcouldhypotheticallyexist,butdonotexistlongenoughtobeconsideredfundamentalrotars.Thefewfrequenciesthatexistlongenoughtobeconsideredfundamentalrotarshavesomesortofwaveinteraction that constructively interferes to reinforce the rotating dipole rotar model anddestructively interferes inaway thateliminates radiatedenergy. There isananalogy to thestabilityconditionachievedbyelectronsinatoms.NormallyanacceleratedelectronradiatesEMradiation. However, theelectrons in thegroundstateofatomsachievestabilityby findingacombination of conditions which eliminates energy loss. The bottom line is that the fewfundamentalrotarsthatexistbelongtothesmallgroupoffrequency‐amplitude‐configurationcombinations that do not radiate either the fundamental dipole wave in spacetime or thenonlinearwaveassociatedwithgravity. Even though there isnotcontinuous lossofenergy,


someoftherotar’senergydoesextendbeyondtherotarvolume.Itwillbeshownthataparticle’sgravityandelectricfieldareboththeresultofstandingwavesgeneratedbyarotarinteractingwiththevacuumenergythatsurroundsarotar.Morewillbesaidaboutthisinchapter8.Gravitational Wave Radiation:Outofcuriosity,wewillcalculatehowlongitwouldtakeforanelectrontoradiateawayitsinternalenergyifgravitationalwaveswerebeingcontinuouslyradiated.WewillassumeagravitationalwaveamplitudeofAgw Lp2/ andaradiationareaequaltotheelectron’srotarradiussquared.Wewilluseoneofthe5wave‐amplitudeequationsthatrelatethepowerinawave.P A2ω2Z setA Agw Lp2/ ω ωc c/ ;Z c3/G;

P set / PlanckPower

P Aβ4Pp

Toobtainaphysicalfeelforthemagnitudeofthispower,wewillexaminethegravitationalwavepowerthatwouldberadiatedusingthepropertiesofanelectron. Weknowthatanelectronmusthaveamechanismthatcancelsthisradiatedpower.P 4.18 10‐23 4x3.63x1052w 1.1x10‐37wt Ei/P 8.18x10‐14J/1.1x10‐37wsetEi 8.18x10‐14J electron’senergyt 7x1023s 2x1016yearsTherefore,sincetheuniverseis1.38x1010yearsold,itwouldtakemorethanonemilliontimestheageoftheuniverse 2x1016years foranelectrontoradiateawayitsenergyingravitationalwaves.Whilethisisalongtime,itwouldbedetectablebecauseelectronsintheearlyuniversewould havemore energy measured locally than today’s electrons. If up and downquarksradiatedawaypowerasgravitationalwaves,theywouldhaveradiateawayalltheirenergyinatimeshorterthantheageoftheuniversebecauseoftheirmuchlargerComptonfrequencyωc.Thepowerradiatedasgravitationalwavesscaleswithωc4.If rotars radiated energy with amplitude equivalent to the fundamental wave amplitudeAf Lp/r with no cancelation mechanism from vacuum energy, then the picture changescompletely.IfanelectronradiatedawayitsenergyinawaveatitsComptonfrequencyanditsfundamentalamplitude,anelectronwouldradiateabout63millionwattsanditwouldsurviveforatimeofonly1/ωc ~10‐20second .Thisismentionedbecauseitisproposedthatthefewfundamentalrotarsthatexisthaveacancelationmechanism discussedlater thatpreventsthistypeofenergyloss.Standingwavesremainintherotar’sexternalvolumeafterthiscancelation.Standingwaveshaveequalpowerflowinginoppositedirectionsandthereforedonottransferenergy.Onlytravelingwavesarecontinuouslytransferringenergy.


Inertia:Inchapter1,wefoundthatlightcirculatingaroundacircularfiberopticloopsatisfiestheconditionofp 0becauseallmomentumvectorscancel.Therefore this circulating lightexhibitsinertiaeventhoughthelightisnotsuperimposedlikethereflectingboxexample.Forexample,supposethatweaccelerateafiberopticloopinadirectionparalleltotheplaneoftheloop. Light indifferentpartsof the loopreceivesdifferentDopplershifts.Forexample, lighttravelingwithavectorcomponentinthedirectionoftheaccelerationwillincreaseinfrequencyandlighttravelingwithavectorcomponentintheoppositedirectionwilldecreaseslightlyinfrequency.Alsolightpropagatingperpendiculartotheaccelerationvectorwouldbedeflectedandexertpressureon thewall of the fiberoptic loop tokeep the lightwithin the fiber.Theresultantdistributionoffrequenciesanddeflectionsaroundtheloopproducethesameinertialforcesonthefiberopticloopasthereflectingboxreceivedfromacceleratingconfinedlight.Eveniftheloopisonlyonewavelengthincircumference,therewouldstillbethesameinertialforces.Iftheloopisacceleratedperpendiculartotherotationalplane,thelightwouldexhibitinertiabyexertingapressuredifferenceonoppositesidesofthefiberoptic.Therotarmodelhasadipolewavetravelingatthespeedoflightinaclosedloop.Thisissimilartotheexampleoflightinafiberopticloop,exceptthatthereisnophysicalwallconfiningtheenergycirculatingatthespeedoflight.Theinteractionwiththedipolewavesinthespacetimefieldaccomplishestheconfinement. Thisisalsoconfinedenergypropagatingatthespeedoflight.Thetranslationalmomentumvectorsinastationaryrotaradduptozero p 0 . Thissatisfiestheconditiontogenerateinertiabythesamemechanismpreviouslyexplainedfortheinertiageneratedbyconfinedlightinchapter1.Theinertiaoftherotarmodelexactlymatchestheinertiaofanequalamountofenergyintheformofconfinedphotons.EquivalencePrinciple: Einsteinneededtopostulate the“equivalenceprinciple”whichstatesthat inertialmass isequivalenttogravitationalmass. Forexample, theequivalenceprincipleimpliesthattheforcerequiredtoaccelerateanelectronatarateof9.8m/s2isexactlythesameastheforcerequiredtoholdanelectronstationaryinagravitationalfieldwithaccelerationof9.8m/s2.IftheHiggsmechanismispresumedtobethemechanismbywhichanelectrongainsinertial mass, then there is no obvious connection to an electron’s gravitational mass. It isthereforenecessarytopostulatetheequivalenceprinciple.However,theproposedmodeloftheuniversereducesinertialmassandgravitationalmasstotheirunderlyingenergy.Energydoesnotneedtobecategorizedasinertialconfinedenergyorgravitationalconfinedenergy.Energycanbemathematicallydefinedataspecificlocationandinaspecificgravitationalpotentialwithoutreferencingacceleration.Ifparticlescanbereducedtoconfined energy propagating at the speed of light, then there is no need to postulate theequivalenceprincipleortheHiggsmechanism.


InertiafromtheHiggsField:TheHiggsmechanismwasoriginallydevisedtoimpartinertiatoWandZbosons.Aphotonisabosonandithasnorestmasswhenitisfreelypropagating.WandZbosons need to interact with something to give them inertia and prevent them from beingmasslessparticleslikephotons.Thespacetimemodeloftheuniversehasnotbeendevelopedsufficientlytoaddressthisquestion.ThereforeWandZbosonsmightinteractwiththeHiggsbosontoacquireinertia.HoweverWandZbosonsareextremelyrareinnature.Virtuallyallthemass in the universe is associated with the quarks and leptons that form ordinary matter.Therefore,themostimportantquestionconcerninginertiaishowdoleptonsandquarksacquireinertia?IstheinertiaimposedbyanexternalHiggsfieldoristheinertiatheresultofaninternalstructurethathasenergypropagatingatthespeedoflightinaconfinedvolume.Therotarmodelmatches theway that confinedphotons acquire inertia.A rotar is energypropagating at thespeedoflightinaconfinedvolumeresultinginzeromomentum p 0 inarestframe.Thereforetheinertiaofarotarexactlyachievesthecorrectamountofinertiaforagivenamountofinternalenergy.Therotarmodelachievesaconnectiontogravity.Itwillalsobeshownthatthisstructureproducescurvedspacetime.Thepictureoftheuniverseproposedhereisverysimple. Manyofthemysteriesofquantummechanicsandgeneralrelativitybecomeconceptuallyunderstandableintheproposedmodel.Itispredictedthatinthefuture,Occam’srazorwillprevailandthesimplestexplanationwillturnouttobethecorrectexplanation.Atthemoment,thespacetimemodeloftheuniversewithonefieldandonebuildingblockofeverythingintheuniverseappearstobebyfarthesimplestmodel.


Chapter7

VirtualParticles,VacuumEnergyandUnity

Introduction:Inthelastchapterwewereonarollcalculatingasimplifiedversionofthestrongforce,theelectromagneticforceandthegravitationalforcebetweentwoofthesamerotarsatafixedseparationdistanceequaltotherotar’srotarradius .Inchapter8wewillimprovethemodel so that these forces exhibit attraction. We will also extend the calculation of thegravitationalforcetolongerdistancesandincludemultiplerotars.However,beforedoingthisitisnecessarytolaysomeadditionalgroundwork.Thisincludesadescriptionofvirtualparticlepairs,vacuumenergy,asymptoticfreedomandaproposedpropertycalled“unity”thatpermitsquantizedwavestoexhibitparticle‐likeproperties.Incosmologytheterms“vacuumenergy”and“darkenergy”areoftenconsideredsynonymous.Thisbookmakesadistinctionbetweenthesetwoterms.Darkenergyisahypotheticalconceptthat is required to fill the gapbetween theobserved energydensityof theuniverse and thetheoretical“criticalenergydensity”.Theapparentaccelerationoftheexpansionoftheuniverseseemstorequireasourceofdiffuseenergydensity ~6 10‐10J/m3 distributedthroughouttheuniversethatcounteractsgravity. Thisiscompletelydifferentthantheverylargeenergydensity ~ 10113 J/m3 of the spacetime field and implied by the terms “vacuum energy” or“vacuum fluctuations”. Darkenergyand thecosmological constant shouldnotbeequated tovacuumenergyandvacuumfluctuations.Probabilistic Nature of Rotars: In chapter 5 figures 5‐1 and 5‐2 show the distortion ofspacetimebelievedtobepresentintherotarvolumeofarotar.Figure5‐1showsadipolewaveinspacetimethathasformedintoaclosedloop,onewavelengthincircumference.Thiswaveistraveling at the speed of light around the closed loop.We previously calculated the angularmomentumofthismodel.Thismotionisnotinasingleplaneasdepicted;insteaditisachaoticdistortionofspacetime.Placingarotarinamagneticfieldcanpartiallyalignthespindirectiongivingprecessionaroundanexpectationspindirection.However,eventhenalmostallrotationdirectionsarepossiblewithdifferentprobabilities.Theexceptionistheoppositespindirectiontotheexpectationdirectionwhichhasaprobabilityofzero.Thechaoticnatureofarotarisduetothefactthatthelobesareaslightdistortionofenergeticspacetimeatthelimitofcausality.Thissmallstrainisbelowthequantummechanicallimitofdetection.Foranelectronthespatialandtemporaldistortionproducedbytherotatingdipolewaveislessthan1partin1022.Toafirstapproximation,therotarmodelofanelectronisan“empty”vacuum.Thedipolelobesofanelectronaresoclosetobeinghomogeneousspacetimethattherateoftimeinthetwolobesonlydiffersbyonesecondin50,000timestheageofthe


universe. The spatial properties of the lobes are so homogeneous that the distortion isequivalenttodistortingaspherethesizeofJupiter’sorbitbytheradiusofahydrogenatom.ThereasonthattherotarmodelcanachievetheE mc2energyofthefundamentalparticlesistheincrediblylargeimpedanceofspacetime c3/G andthelargeComptonfrequency ~1020to1025Hz ofthefundamentalrotars.Theselobesarepropagatinginaclosedloopatthespeedoflight and interacting with vacuum energy so they are approximately confined to a volume.However, “finding the particle” means interacting with this incredibly weak distortion ofspacetimeinawaythatresultsinameasurablemomentumistransferred.Thisisaprobabilisticeventthatcanhappenoverasubstantialvolumethatscaleswith .Furthermore,thechaoticnatureoftherotarstructurepermitstherotatingdipolewavetodisappearfromonevolumeandreform in an adjacent location thatwas previously part of the rotar’s external volume. Therotatingdipolecanalsobevisualizedasarotatingrateoftimegradientasdepictedinfigure5‐2.Virtual Particle Pairs:Theterm“virtualparticle”iscommonlyappliedintwodifferentways.First,therearethevirtualparticlesthataccordingtothecommonlyacceptedphysicstheoryarethecarriersofforces.Forexample,virtualphotonssupposedlycarrytheelectromagneticforce.Theothertypeofvirtualparticlesisthevirtualparticlepairsthatarecontinuouslybeingcreatedfromthevacuumandannihilatedbackintothevacuum.Thesevirtualparticlepairsareproposedtobeanothermanifestationofspacetime.Theassumptionthattheuniverseisonlyspacetimeimpliesthatthesevirtualparticlepairsarejustanotherformthatthespacetimefieldtakes.Theyhave no angularmomentum and therefore do not have a long term life. Rotars are also amanifestation of spacetime, but the difference is that rotars possess quantized angularmomentumofħ/2.Evenwhenanunstablefundamentalparticledecays,theangularmomentumsurvives.Avirtualparticlepairisacounterrotatingmatter/antimatterpair.Counterrotatingmeansthatthequantizedangularmomentumiseliminated zerospin .Foraninstanttheproposedvirtualparticlemodelisstrainedspacetimethatlooksgenerallysimilartothetwodipolelobesdepictedin figures 5‐1 and 5‐2. However, the virtual particle lobes are not rotating. Also, thedisplacement amplitudes of thesewaves forming the virtual particlepairsmaymomentarilyexceedthedisplacementamplituderequiredforasingleparticle.However,itisnotclearifthisisnecessarybecausetheshortlifetimemakestheexactamplitudenebulous.Suchlobepairswouldformrandomlyoutofthedipolewavesthatareresponsibleforvacuumenergy.Ifwemakeananalogytowavesonwater,thenavirtualparticlepairisawavemaximumandminimumthatmomentarily looks like figures5‐1and5‐2 inchapter5. These lobesareseparatedbyadistancecomparabletotwicetherotarradius diameter 2 oftherotarbeingsimulated.Apparentlyspacetimehasaresonanceatconditionsthatcorrespondtotheformationof virtual particle pairs that correspond to real matter‐antimatter pairs such aselectron/positronpairsormuon/antimuonpairs. Therefore these frequenciesarepreferred


overrandomfrequencies.Thesewavestructuresformanddisappearfromthedipolewavesthatformvacuumenergy.Whensuchashapeforms,itmomentarilycanlooklikeaparticle/antiparticlepairsuchasanelectron/positronpairoramuon/antimuonpair.However,thisdeceptionisquicklyrevealed.Forexample,witharealelectron/positronpair,thetworotarsshouldcounterrotate½radianeach 1radiantotal inatimeof1/2ωc ħ/2mc2 6.4 10‐22s.Thetworandomlyformedlobeswoulddissipateintowaveletsinasimilartimeperiod.Thisisthesamelifetimegiventovirtualparticlepairsbytheuncertaintyprinciple.ItcanbeshownthatwhentheenergyuncertaintyissetasΔE Ei,thenΔt 1/ωcandΔx λc.Therefore,theuncertaintyprincipleisdescribingthetimeanddistancerequiredforthismodelofavirtualparticlepairtorevealitselfanddissipateintootherrandomdipolewaves.Ifwehadassumedtheshortesttimeperiodpossible 1unitofPlanck time , then ΔE Ep Planck energy and Δx Lp Planck length. When themaximumfrequency is set equal to Planck frequency inverse of Planck time then the implied energydensityequalsPlanckenergydensityUp 10113 J/m3. It isproposedthatallaspectsof theuncertaintyprinciplecorrespondtothespacetimefield.Virtualparticlepairsandrealparticlesare both obtained from the spacetime field. The only difference is that real particles are aquantizedunitofangularmomentum ½ħ whilevirtualparticleshavenoangularmomentum.Rotar Model Requires Vacuum Pressure:Recallinchapter4thepointwasmadethatenergydensity U andpressure ℙ bothhaveunitsofM/T2Landonafundamentalleveltheyareboththesame U kℙ .Iarguedthattheimplicationisthatenergydensityalwaysimpliespressure.Ifamodelofafundamentalparticlewithfiniteenergyhasnovolumebecauseiteither is a point particle or a one dimensional vibrating string, then thesemodels haveinfiniteenergydensityandinfiniteinternalpressure.Evenifaparticleisconsideredtobean“excitation”ofafield,theexcitationimpliesvolume,energyandfinitepressure.Evenifotherparticlemodelsdonotaddressthequestionoftheneedtooffsettheimpliedpressure,thisisaveryimportantpartofthespacetime‐basedmodeloftheuniverse.Infact,theexplanationsofalltheforcesgeneratedbyaparticleincorporatetheparticle’sinternalpressureandtheoffsettingpressure force generatedbythespacetimefield.Thereforeanexplanationof forcesmust incorporate rotar’s internal energydensity Uq and its internalpressureℙq.Ignoringnumericalfactorsnear1,theenergydensityandpressureoftherotar’sinternal volume can be determined using one of the 5wave‐amplitude equations: U ℙ

kA2ω2Z/c.Usingthesubstitution:A Aβ Tpωc) ħ / ωcandprevioussubstitutions:

ℙq ħ

ħ

ħ

Up ℙp


Inparticular,Ei/ hastheformofenergydensityandFm/ hastheformofpressurewhereFmistheparticle’smaximumforce. For example, an electron has: Ei 8.19 10‐14 J, Fm 0.212N and 3.86 10‐13m.Therefore, anelectronhasenergydensityand internalpressureof about1024 J/m3and1024N/m2 respectively. Vacuum energy is required to stabilize and confine this energydensity/pressure. Therefore,vacuumenergymustexceedthisenergydensity/pressure. If ittakes1024N/m2tostabilizeanelectronwithenergyof8 10‐14J,howmuchpressuredoesittake to stabilize the highest energy particle? The most energetic particle that has beenexperimentallyobservedisthetopquarkwithenergyof:Ei 3 10‐8JandtheHiggsbosoniscloseatEi 2 10‐8J.UsingU Ei4/c3ħ3theenergydensityofvacuumenergymustexceedabout 1045 J/m3 and the pressuremust exceed 1045 N/m2 to support these particles. Thisrepresentsalowerlimitfortheenergydensityofvacuumenergy. Thesepressuresareeasilyaccommodatedbythespacetimebasedmodelofvacuumenergy.Thehighenergydensityofvacuumenergyrequiredbythespacetimebasedmodelproposedhere should not be surprising since a large vacuum energy density is also required for theformationofvirtualparticlepairsandmanyotheroperationsofQEDandQCD.Theuniversalspacetimefieldhasvariousresonanceswhichgiverisetothevariousvirtualparticlepairsofthestandardmodel. Therefore,oneoftheseresonancescouldbecalledthe“topquarkfield”andanothercouldbecalledthe“Higgsfield”.SomeestimatesoftheHiggsfieldplacetherequiredenergydensityatabout1046J/m3.ItispossiblethattheHiggsfieldresonancemighthavesomestabilizingeffectonWandZbosonsbecausethismodelhasnotbeendevelopedtothepointofunderstandingWandZbosons.However,itcanbesaidthatallfermionsachievetheirinertiathroughthesamemechanismasthepreviouslydiscussedconfinedphotonsachieveinertia.AHiggsfieldisnotrequiredtoimpartinertiatoeitherconfinedphotonsorfermions.Astronomicalmeasurementsindicatethattheuniversehasaverageenergydensityofonlyabout10‐9 J/m3 the “critical density” . About 70% of this is attributed to “dark energy” whichsupposedlyhomogeneouslyfillsallofspace.Theother30%isordinarymatteranddarkmatterwhichareinhomogeneouslydistributedthroughouttheuniversebutcanbeaveragedoutifalargeenoughvolumeofspacetimeisassumed.Theexistenceofdarkenergywillbeexaminedlater.Theremainingobservableenergydensityintheuniverseisfermionsandbosonswhicharedipolewavesinspacetimethatpossessesquantizedangularmomentum.Thevastlylargerportionoftheuniverse’senergy dipolewaves doesnotpossessangularmomentumandonlyinteracts with our observable universe through quantum mechanics. This vacuum energydensityisusuallyignored,butitgivesspacetimeitspropertiessuchasconstantsZs,c,G,εo,ħ,etc. and is essential for EQD and QCD calculations. This vacuum energy density is ashomogeneousandisotropicasquantummechanicsallows.Gravitationaleffectsareadistortionofthishomogeneousspacetimefieldproducedbythe1partin10120oftheenergyintheuniversethatpossessesquantizedangularmomentum.However,concentrationsoffermionssuchasa


neutronstarcanproduceaconditionwhichcomesclosetomatchingthemaximumconditionsofthespacetimefieldforaparticularwavelengthandvolume.Theseconditionswhichcreateablackholewillbediscussedlater.Theenergydensityofarotarfundamentalparticleimpliesapressurethatmustbecontainedtoachievestability.Vacuumenergycanexertthispressurewithoutitselfneedingtobecontainedbyastilllargerpressurevessel.Wedonotknowwhethertheuniverseisinfinitelylargeorjustvastly larger than our observable portion of the universe. In either case, the vacuumenergy/pressureinourobservableportionoftheuniversehasnowheretogo.Itisinequilibriumwiththerestoftheobservableuniverse.Oneinadequacyofpointparticlesandonedimensionalstringsisthattheyhaveenergybutnovolume.Whatmechanismcontainstheinfinitepressureofaparticlewithnovolume?Rotars in Superfluid Vacuum Energy:IfIwavemyhandthroughspacetime,Iamnotawareofanyinteractionwiththevastenergydensityofspacetime.Thereisnoresistance;thereforeitis hard to visualize spacetime as having a large energy density or being a very stiff elasticmedium.However,itisnecessaryformetorememberthatthefundamentalparticlesthatmakeupmyhandaremerelyunitsofquantizededdies.Theytemporarilyorganizeavolumeofchaoticdipolewaves so that a Planck length distortion rotates around a circle that is one Comptonwavelength is circumference. For example, for an electron the Compton wavelengthcircumferenceisabout1022timeslargerthanthePlancklengthdistortion.Thisquantizedangularmomentumcaneffortlesslypassthroughtheseaofsuperfluidvacuumfluctuations dipolewaves withoutencounteringanyresistanceorleavingawake.Noparticulardipolewaveismovingandnodipolewavesarebeingcompressed.Onlywhenweintroduceanewwave,suchasagravitationalwave,arewetrulyinteractingwiththespacetimefieldinawaythatexposes its impedanceandenergydensity. Spacetimehasabulkmodulusbut thisbulkmodulusonlyrevealsitselftoawaveinspacetimethatisphysicallyintroducingacompressionandexpansionofspacetime.However, if a rotar possessing quantized angularmomentumencounters another rotarwithquantizedangularmomentum,thenthisisentirelydifferent.Eventhoughthesetworotarsarealsojustdistortionsofspacetime,thequantizedangularmomentumpermitsthemtoexceedthehomogeneousenergydensityofvacuumenergy.Thisstartsachainofinteractionswithvacuumenergy/pressurethatultimatelyresultintheforcesofnature.Forexample,rotarscancoalesceintomassivebodiesrangingfromhadronstogalaxies.Theseareislandsofconcentratedenergyinaseaofsuperfluidvacuumenergythatwaspreviouslyhomogeneous.Eachrotarincreasestheenergydensityataspecificlocationcausingadisturbanceweknowascurvedspacetime.Allothernearbyrotarsnowexperienceanenergydensitygradientwhichresultsinagravitationalinteractionbetweenrotars particles .Theotherforcesaretheresultofsimilarinteractionsas


will be explained later. Chapters 13 and 14will discuss furtherwhy the energy density ofspacetimedoesnotformablackhole.Is the Spacetime Field the New Aether? If theuniverse isonlyspacetime, itshouldnotbesurprising that spacetime is ultimately responsible for all of physics. The description ofspacetime offered here is a combination of the energetic vacuum fluctuations described byquantummechanicsandthegeneralrelativisticdescriptionwherespacetimecanbecurvedandtime is the fourth dimension. Ultimately energetic spacetime even performs the functionspreviouslyattributedtotheaether.Forexample,inchapters9and11amathematicalanalysiswillindicatethatphotonsarequantizedwavespropagatinginthespacetimefield.Thissoundsalotliketheaether.Whatarethesimilaritiesanddifferences?There is a short book titled “Einstein and the Ether” by Ludwik Kostro that I found veryinterestingandinformative.Beforereadingthisbook,IassumedthatEinsteindidnotbelieveintheaetherandalsoassumedthathewasakeyreasonthattheaetherfelloutoffavorinmodernphysics.Tomysurprise,Idiscoveredthatonlybetweentheyears1905and1916didheentirelyrejecttheaether.After1916heoftenreferredtohisrelativisticviewoftheaetheranddefinedthe aether as “physical space endowedwith physical attributes”. According to Kostro, overEinstein’slifehehad3differentconceptsoftheaether.Afterabout1934hebegantosubstitutethe terms “physicalspace”or “total field” for “theaether”,buthewasreferring for thesamephysicalconcept.HerearesomeEinsteinquotesobtainedfromtheKostrobook,buttheoriginalsourceoftheEinsteinquotesarealsogiven.

“In1905Iwasoftheopinionthat itwasnolongerallowedtospeakabouttheaetherinphysics.Thisopinion,however,wastooradicalaswewillseelaterwhenwediscuss thegeneral theoryof relativity. It is stillpermissible, asbefore, tointroduceamediumfillingallspaceandtoassumeelectromagnetic fields andmatteraswell areitsstates.”1“Physicalspaceandaetherareonlydifferenttermsforthesamething;fieldsarephysicalstatesofspace.”2“Accordingtogeneralrelativity,theconceptofspacedetachedfromanyphysicalcontent doesnot exist. Thephysical reality of space is representedby a fieldwhosecomponentsarecontinuousfunctionsoffourindependentvariables–thecoordinatesofspaceandtime.”3

1 A. Einstein, (Morgan Manuscript) Morgan Library, New York, section 13 2 A. Einstein Mein Weltbild (Amsterdam: Querido, 1934), p. 237 3 A. Einstein, “Relativity and the Problem of Space,” pp. 375-376


“According to the general theory of relativity, space is endowedwith physicalqualities;inthissense,therefore,thereexistsanaether.”4“The ether includes all objects of physics… Matter and the elementary particles from which matter is built also have to be regarded as “fields” of a particular kind or as particular “states” of space.”5

ThislastEinsteinquoteshowsthatheevolvedtoaconceptofarelativisticaetherwhichincludedallparticlesandforces.Hisconceptoftheaetherwasthatitincludedeverythingintheuniverse.Einstein’sconceptcanperhapsbesummarizedasfollows:Theuniverseisonlyaether.Theproblemwiththeterm“aether”isthatithassomanydifferentmeaningsthatnowitistooimprecise a word for scientific discussion. Kostro estimates there were about 14 differentdescriptionsof theaether.Themost importantoneswere theLorentz aether, theEddingtonaether,theWeylaetherandEinstein’srelativisticaether.Thespacetimefielddescribedinthisbookbroadlycanbeconsideredamodelof“theaether”sinceitdescribesauniversalfieldthatfillsallofspace.Inchapter9thespacetimefieldwillalsobe shown to be associated with the propagation of light. However, the spacetime field isdescribedandquantifiedtoadegreenotachievedbyanyearlieraetherconcepts.Experimentswithin current technology cannot detect motion relative to the spacetime field becausespacetimeisaseaofenergeticdipolewaveswhicharealwaysformingnewwaveletsandallofthisispropagatingchaoticallyatthespeedoflight.Aspreviouslyexplained,gravitationalwavespropagateinthespacetimefieldandahypotheticalMichaelson‐Morleyexperimentconductedusinggravitationalwaveswouldnotdetectmotionrelativetothespacetimefield.Inchapter4itwasmentionedthattheharmonicoscillatorsofzeropointenergyhavespectralenergydensityof:U ω dω k ħ ⁄ .ThequotefromPuthoff6is:

“Thisspectrumwithitsω3dependenceofspectralenergydensityisuniqueinasmuchasmotionthroughthisspectraldistributiondoesnotproduceadetectableDopplershift.It is a Lorentz invariant random field. All inertial observers are equivalent. Anyparticular spectral component undergoes a Doppler shift, but other componentscompensate so that all components taken together do not exhibit a Doppler shift.Thereforethisspectralenergydistributionsatisfiestherequirementthatitshouldnotbepossibletodetectanydifferenceinthelawsofphysicsinanyframeofreference.”

4 A. Einstein, Ather und Relativitatsheorie (Berlin: Springer, 1920) p.15 5 A. Einstein, “Uber den Aether,” VSNG, 105, 1924 pp. 85-93 6 Puthoff, H.E. Phys. Rev. A Volume 40, p.4857, 1989 Errata in Phys. Rev A volume 44, p. 3385, 1991 See also New Scientist, volume 124, p.36, Dec. 2, 1989


Thisquoteisonlyaccurateifweassumethatfrequencyrangeoftheharmonicoscillatorsofzeropointenergyextendstoinfinitefrequency.However,theactualzeropointenergyhasanupperfrequencylimitequaltoPlanckfrequency.Withthislimit,itisnolongerpossibletoassumethatamotionthroughZPEisLorentzinvariantinallconceivableframesofreference.Lorentz Invariance:ItisproposedthatLorentzwasrightwhenheassumedapreferredframeofreferenceforhiscalculationsandthispreferredframeofreferenceisthecosmicmicrowavebackground CMB restframe.ThisistheframewhichtheCMBlooksisotropicinalldirections.Wearecurrentlymovingatabout269km/srelativetothelocalCMBrestframe,thereforetheCMBappearsslightlyanisotropicfromourframeofreference redandblueshifts .IfwecouldseethePlanckfrequencydipolewavesthatformthespacetimefield,theyalsowouldappeartobe slightly anisotropic because of ourmotion relative to the CMB rest frame.We can neverdirectly observe dipole waves in spacetime because their Planck length and Planck timedisplacementofspacetimesetsadetectablelimitaspreviouslydiscussed.However,therearehypotheticalexperimentsthatwecanimaginewhichwouldrevealananisotropyifanextremelylarge frame of reference is assumed relative to the CMB rest frame. For example, it waspreviously shown that all spacetimeparticles rotars have energydensity and therefore aninternalpressure.Thisinternalpressureisoffsetbypressureexertedbytheinteractionwiththe surrounding vacuum energy dipolewaves that stabilized the particle. To achieve thisstabilizationthevacuummustbeabletoexerttherequiredpressureonallsidesofaparticle.Imagineaparticlemovingrelative to theCMBrest frameataspeedwithanextremely large

special relativity gamma 1 ⁄ . It isproposed that thereareexotic framesofreferencewhichwouldexpose theanisotropy in the frequencies thatmakeup thespacetimefield.Thiswouldhappenifthespecialrelativityγissolargethatthevacuumcannotexerttherequiredpressureonallsidesoftheparticlebecauseonesideexperiencestoolargearedshift.Thenthattypeofparticlecouldnotexistinthatframeofreference.Forexample,atopquarkhasenergyofEi 2.77x10‐8J,frequencyofωc 3x1026s‐1,energydensityofabout1046J/m3andinternalpressureofabout1046N/m2.Tooffsetthisinternalpressureatopquarkcannotexistina frameof referencebeyond aboutγ 7x1016 relative to theCMB rest frame. Similarly, anelectroncannotexistataframebeyondaboutγ 3x1022TheselimitsaresetbecauseintheredshiftdirectionthedipolewavethatformthespacetimefieldwouldbeDopplershiftedtoafrequencythatisunabletoexerttherequiredpressuretostabilizethetopquarkorelectron.AnotherwayoflookingatthislimitisthatatopquarkorelectronpropagatingattheirlimitingframeofreferencewouldhaveadeBrogliewavelengthlessthanPlancklengthwhenviewedfromtheCMBrestframewhichisanimpossibility. Thelawsofphysicswouldbedifferentintheseexoticframesofreferencewherefundamentalparticlesbeyondacriticalenergyarenotallowedtoexist.Therefore,Lorentzinvarianceiscorrectforordinaryframesofreference,butLorentzinvarianceisnotcorrectinthelimitofexoticframesofreferencewheresomeparticles


cannotexist. Lorentzassumedapreferredframeofreferenceasthebasisofhiscalculations.EinsteinclaimedthatthetheoryofrelativitydidnotallowforapreferredframeofreferenceandEinsteinproposedwhathecalledthe“relativisticaether”.ItisnowproposedthatEinsteinwascorrectforaccessibleframesofreferencebutLorentzwascorrectwhenexotic,highγframesofreferenceareconsidered.ItisironicthattheconditionwhichviolatesLorentzinvarianceisalsotheconditionthatprovesthatLorentz’sassumptionaboutapreferredframeofreference.Thisdiscussiondoeshaveone important implication for theoreticalphysics. String theory isbasedonthreemathematicalassumptions.OneofthesethreeassumptionsisLorentzinvariancewhichisexpressedasanequation.IfthepreviousdiscussionaboutlimitstoLorentzinvarianceis correct, thenoneof the three foundationsof string theorywouldbeprovablywrong.Thiswouldunderminethefoundationofstringtheory.ReturningtoEinsteinandLorentz,itmustbeunderstoodthatbothEinsteinandLorentzweredealingwithmathematicalanalysiswhichincorporatedassumptions,butneitherofthemhadanactual mechanistic model of the aether such as the dipole wave model with quantifiablepropertiessuchasproposedinthisbook.Therefore,Einsteincouldsimplydecidethattheaethermustpossessrelativisticpropertiesandformulateequationsaccordingly. Hedidnotwrestlewiththedifficultproblemofdevelopingaphysicalmodeloftheuniversewhichachievedthisgoal.Similarly,LorentzdevelopedequationsforthetransformationsrequiredtokeepMaxwell’sequations unchanged when viewed from different frames of reference. Lorentz assumed apreferredframeofreferenceforhisanalysis,butsincethelengthandtimetransformationsmadeallframesofreferencelookthesame,therewouldbenowayofexperimentallyidentifyingthepreferredframeofreference.Again,Lorentzdidnotactuallydevelopaphysicalmodeloftheuniversewhichachievedtheresultshecalculated.

StabilityofaParticleMadeofWavesSchrodinger’s Wave Packet: Previously it was mentioned that about 1926 Schrodingerattemptedtoexplainparticlesasconsistingonlyofa“wavepacket”.Schrodinger’swavepackethadmanyfrequenciesthat,whenaddedtogether Fouriertransform ,producedaconcentratedwave.ThiswasSchrodinger’swavebasedmodelofaparticle.Hewasattackedforthisideabyotherscientists.Theproblemwasthatthesemanydifferentfrequenciescouldonlytemporarilyaddtogethertoformaconcentratedwaveatasinglelocationthatactslikeaparticle.AnotherwayofsayingthisisthatSchrodinger’sconfluenceofwavescanmomentarilycreatetheenergydensity of a particle, but this implies a pressure. Schrodinger was unable to explain whatpreventedthewavepacketfromdissipatingandheeventuallyabandonedthisidea.Radiated Power by Unstable Rotars:Theamplitudeoftherotarwavewithintherotarvolumeat distance has been given as Aβ Lp/ . A simple extrapolation of this amplitude to


distancesbeyond wouldresultinafundamentalwaveamplitudeofAf Lp/rwheredistancerisgreaterthan .Rotatingdipolesofanytypeattempttoradiateawaytheirenergy.Angularmomentum cannot be destroyed but the volume over which the angular momentum isdistributedcanexpand.ItisproposedthatatthefewComptonfrequenciesthatactuallyformrotars,atypeofresonanceisformedthatoffsetsthedipoleradiation.Foranyfundamentalrotarthat achieves sufficient stability to be a named particle, theremust be amechanismwhichcancelsthetravelingwavewithamplitudeAf Lp/randleavesonlyresidualstandingwavesasevidenceof thebattle that is takingplace. Withoutsome formofcancelation in theexternalvolume,wewouldexpectarotartoradiateenergyintotheexternalvolumewithamplitudethatdecreases with 1/r at a frequency equal to the rotar’s Compton angular frequency ωc. TocalculatethehypotheticalradiatedpowerthatwouldoccurfromamplitudeAfatfrequencyωcwewilluseoneofthe5wave‐amplitudeequations:P A2ω2Z .Thisequationcontains“ ”whichistheradiatingarea.Itisnotnecessarytoassumeadistanceof forthiscalculation.Wecanimagineasphericalshellwitharbitraryradiusr.Therefore,weonlyneedtocalculatethepower that passes through this shell. At distance r the surface area “ ” of this imaginarysphericalshellwithradiusris: kr2.P A2ω2Z setA Af Lp/r,Z c3/Gand kr2 ignorek

P ħ

ωc2ħ Pc

P Pcradiatedpower rotar’scirculatingpowerPc ωc2ħ EiωcTherefore,a1/ramplitudedistributionmeansthattheradiatedpowerisequaltotherotar’sfullcirculatingpowerPc Eiωc.Atthisradiatedpower,alltherotar’sinternalenergyEiisradiatedaway in a time period of only 1/ωc. If an electron radiated power at this rate, itwould beradiatingabout63millionwattsandhavealifetimeoflessthan10‐20seconds.Anystructurethatisradiatingawayitsinternalenergyinatimeperiodofonly1/ωhasabsolutelynostability.Infact,itlastsaslongastheuncertaintyprinciplepredictsforenergyuncertaintyΔE.Ifarotarsurvivesforatimeperiodlongerthan1/ω,thismeansthattheremustbesomemechanismforreducingthewaveamplitudeintheexternalvolumefromAf Lp/r.Wave Cancelation:HereisthepicturethatIhaveforthestabilityofarotar.Itisnotacompletepicture,butitissufficientlycompletethatIfinditplausiblewhencombinedwiththebodyofotherinformationcontainedinthisbook.Imaginearotatingdipolewaveinspacetimethatisonewavelengthincircumference.Itisasinglefrequency,soradiationfromthiswaveattemptstofilltheuniverse.Powerwouldhavetobecontinuouslysuppliedtothisrotatingdipole.Inthiscase, the outgoingwave is acting exactly aswould be expected for a single frequencywaveexpandingfromasource.Thiswouldproduceperfectmonochromaticradiation,limitedonlybythe Fourier transform of the finite emission time. Since a stable rotar is not continuouslyemittingenergy,theremustbeanewsourceofoffsettingwaves.


Thiscancelationofwavesintheexternalvolumedoesnotmeanthatalltracesofwaveenergyhave been eliminated. A very important part of the rotar model is that the destructiveinterference is incomplete. Standing waves oscillations where nodes and antinodes arestationary areleftbehind.Thesestandingwavesinteractwithvacuumenergyinawaythatalsoproducesnon‐oscillatingstrainsinspacetime.Twoexamplesoftheseresidualnon‐oscillatingstrains are electric fields chapter9 and curved spacetime. Inparticular, curved spacetimeresults in a static rate of time gradient and a non‐Euclidian spatial distortion discussed inchapter8 .Travelingwavesimplythatpowerisbeingtransferredinthedirectionofthewavepropagation.Standingwaves or a static rate of time gradient implies that no power is being transferred.Therefore,theproposeddestructiveinterferencehaseliminatedthepowerdrainfromtherotar,buttheremainingstandingwavesandgradientsaretheevidencethatadestructiveinterferencebattleisgoingon.Standingwaveshaveenergy,sothispictureimpliesthatasmallportionoftherotar’senergyisdistributedoutsidetherotarvolume.Thisenergyisresponsiblefortherotar’selectricandgravitationalfields.Thevacuumenergywavespropagatingtowardsthecore wavelets arereturningtheradiatedpower to the rotating dipole core. These returning waves must have the correct phase toconstructively interferewith the rotatingdipole.Out of the infinitepossible combinationsoffrequency, amplitude and angular momentum, only the electron, muon and tauon havefrequency/amplitudecombinations tosurviveas isolatedchargedrotars impliesrestmass .Thequarksonlyfindstabilityinpairsortriplets.Aspreviouslystated,eachchargedleptonhasa single dimensionless number that expresses all its unique characteristics in dimensionlessPlanckunits.Neutrinoswillbediscussedlater.Attraction and Repulsion: Theconventionalexplanationforactionatadistanceisthattheforces of nature are the result of the exchange of virtual particles. This explanation isconceptuallyunderstandablewhenitisappliedtotwoparticleswhichrepeleachothersuchastwoelectrons.Itispossibletoimaginevirtualphotonspropagatingbetweentwoelectrons.Eachvirtualphotoncarriesasmallamountofmomentumthereforemultiplevirtualphotonstogetherproducewhatappearstobeacontinuousrepulsiveforce.However,evenforrepulsionthereisthequestion:Howdovirtualphotonsfindadistantpointparticle?Isthereahomingmechanismoraretherealmostaninfinitenumberofvirtualphotonsexploringeverypossiblelocation?Whentheconceptofvirtualphotonexchangeisfirstintroducedtostudents,thenextquestionisusually “How does the exchange of virtual photons create attraction?” The answer usuallyincludes mention of the uncertainty principle, Feynman diagrams, and mathematicalabstractions. Theseanswersstillareunsatisfying,butthestudentreluctantlyadoptstheideathat it is necessary to move beyond classical physics with its conceptually understandableanswers and accept the counter intuitive explanations of quantum mechanics. This book


attempts to bring conceptually understandable ideas to quantummechanics. The subject ofactionatadistance,especiallyattraction,isaprimeexampleofanareathatneedsanimprovedexplanation.Thereisverylittle“wiggleroom”foractionatadistanceifwestartwiththeassumptionthattheuniverseisonlyspacetime.Thisrestrictionleadstotheconceptthatthereisonlyoneforce:therelativisticforceFr Pr/c.Thisistheforceimpartedbypowertravelingatthespeedoflight.Thisleadstoasurprisingrealizationthattherelativisticforceisonlyrepulsive.Thesamewaythatphotonpressureisonlyrepulsive,wavesinspacetimetravelingatthespeedoflightcanonlyproducearepulsiveforce.Whatappearstobeanattractingforceisactuallyarepulsiveforceexertedbythevacuumenergy/pressure.Eachrotarrequiresvacuumenergytoexertalargepressuretostabilizetherotar.Previouslywecalculatedthepressurerequiredtostabilizearotaris:ℙ m4c5/ħ3andapplyingthispressureoveranareaofk 2producesaforceequaltotherotar’smaximumforce Fm m2c3/ħ ignoringdimensionlessconstantsnear1.Ifwe mentally divide a rotar into two hemispheres, vacuum energy is exerting the rotar’smaximumforceFmtokeepthosetwohemispherestogether.Thisisthesameforcerequiredtodeflectarotar’scirculatingpowerPc/c ℙ Fm.EvenleptonswhichdonotfeelthestrongforcestillexperienceaforceequaltothemaximumforceFmexertedbythepressureassociatedwithvacuumenergy.Inchapter8itwillbeshownlaterthatthisforceexertedbyvacuumenergycanbeunbalancedandcanappeartobeattraction.This maximum force was first calculated assuming that the rotar’s full circulating power isdeflected.Theagentthatisaccomplishingthisdeflectionmustbeanexternalrepulsiveforce.Nowweseethatthevacuumenergy thespacetimefield isexertingthisrequiredforceontherotar. Inequilibrium,thecompressionforceexertedbyvacuumenergyneedstobalancetheoutwardforceexertedwhenarotar’scirculatingpowerisconfined deflected .Therefore,itisreasonablethattheforceexertedbyvacuumenergyneedstoequaltherotar’smaximumforce.Asymptotic Freedom:Thestrongforceisanattractingforcewhichhasthepropertyofallowingquarksboundinhadronstofreelymigratewithinthenaturaldimensionsofthehadronasifthereisnoforceactingonthem.However,ifthereisanattempttoremoveaquarkfromthehadronincreasethenaturalseparation ,thenaforceofattractionappearsandresistsincreasingthisseparationdistance.Furthermore,thisattractingforceincreaseswithdistance.Anattempttoremoveaquarkfromahadronagainstthisincreasingforceofattractionproducesanewmesonratherthanafreequark.Oncethenewmesonisformedtheattractingforcedropstonearzeroandthemesoncanberemoved.Thereforethestrongforcehasaforcecharacteristicthatseemscounterintuitive.Thestrongforcealsoisresponsibleforbindingprotonsandneutronstogetherinthenucleusofanatom.Theattractionbetweennucleonscausedbythestrongforceissubstantiallylargerthan


the electromagnetic force generated by the protons attempting to repel each other. It isestimatedthatthestrongforceisatleast100timesgreater perhaps1/αtimesgreater thantheelectromagneticforceatadistancecomparabletotheradiusofaproton ~10‐15m .Previouslyinchapter6wecalculatedthattheproposedwavemodelindicatedthattwoquarksshouldrepeleachotherwithaforceequaltotherotar’smaximumforceataseparationdistanceequalto .However,thisrepulsionisonlyoneoftwoforcesactingonquarkswhentheyarebound together inahadron.Thequark isalso interactingwithvacuumenergy inaway thatvacuumenergyisexertingalargepressureonthequark.Anisolatedelectronhassymmetricalvacuumenergypressureexertedonthesphericalrotarvolume.However,aquarkboundinahadrondoesnothavesymmetricalpressure.Afeaturethatmakesprotonsandneutronsstableis that there is an interactionbetweenadjacentquarkswhich cancels thepressurenormallyexerted by vacuumenergy on the part of the quark that is nearest its neighbor quark. Theremainingpressureappliedovertheremainingportionofthequarkexertsaforceequaltothequark’smaximumforceFm previouslycalculatedFm ℙq .Thisunbalancedpressurepushesthequarkstogethersoitappearstobeaforceofattractionpseudo‐attraction .UltimatelyequilibriumisreachedwheretherepulsiveforcebetweenthetwoquarksisequaltothemaximumforceFmandthisalsoequalsthevacuumenergyforcethatpushesthequarkstogether.Anyattempttoeither increaseordecreasetheseparationwouldresult ina large forceattempting to return thequarks to theseparationwhere theopposingforces balance. This equilibrium is proposed to create the condition known as “asymptoticfreedom”.Acollisionthatattempts toremoveaquark fromahadron increases theseparationbetweenquarksbeyondtheequilibriumposition.Therepulsiveforceexertedbytheotherrotarrapidlydecreasesastheseparationisincreased.Workisbeingdoneanditappearsasifthepseudo‐attractionexertedbyvacuumenergy/pressureremainsconstantasthequarksareseparated.The decrease in the repulsive force exerted by the other rotar combined with a relativelyconstantpseudo‐attractionforceresults inanet forcethatappearsto increasewithdistance.Thestrongforceisproposedtobethenetforcethatresultsfromthetwoopposingforces.Thisnet force the strong force approaches themaximum forceas the separation increases.Thework done separating quarks increases the energy frequency of the quarks rotars andeventuallytheextraenergyformsanewmeson.Thissubjectwillbediscussedfurtherinchapter12.Allthatisimportantforacomparisonofforcesisthatthemagnitudeofthestrongforceapproachesthemaximumforceasquarksareseparated. Forexample, theupanddownquarksthat forman isolatedprotonwouldhaveamaximumforceofroughly80,000N.ThismaximumforceisobtainedfromFm m2c3/ħwherethemass isapproximately 1/3 theproton’smass.Thespacetimebasedmodelexplains forceswithoutexchangeparticles. Gluonsareakeypartofthestandardmodelbuttheyarevirtual


particlesthathavenotandcannotbedirectlyobserved.Theneedtoreplacethegluonvirtualparticlemodelwithawavebasedmodelofforceswillbediscussedfurtherinchapter12.Casimir Effect Similarity:Thisexplanationforattraction unbalancedpressurefromvacuumenergy has some similarities to the explanation for the Casimir effect. Previously it wasmentionedthattherandomwavesinvacuumenergyarecreatingallcombinationsandtheseinclude spacetimewaves that appear tobe zeropoint electromagnetic radiation. When twometalplatesarebroughtclosetogether,theseconductiveplatesexcludeelectromagneticwaveswith wavelengths larger than the gap between the metal plates. These excludedwavelengths/frequenciesarestillpresentontheoppositesideofthemetalplates.Thisslightlylowersthepressureexertedbythedipolewavesinspacetime vacuumenergy betweenthetwoplatescomparedtothepressureexertedontheoutsideofthemetalplateswherenowavesareexcluded. Practical considerations such as surface smoothness, electrical conductivity andmetalliccutofffrequencyallservetodegradetheeffectfromthetheoreticalperformance.TheCasimireffecthasbeenexperimentallyverifiedtowithinabout5%accuracy.Assuminganidealelectricallyconductivesurface,thetheoreticalpressureℙgeneratedbytheCasimireffectwithgapsizeof“r”is:ℙ k ħc/r4CasimirPressureℙforparallelmetalplatesseparatedby“r”This shouldbecompared to thepressureℙexertedbyvacuumenergyona rotarwithrotarradius :ℙ k ħc/ ℙ pressureexertedbyvacuumenergyonrotarwithradius Itcanbeseenthatthesearethesameformifgapsize“r”isequatedtorotarradius andtheconstantisignored.The point of this is that even electrostatic attraction or the strong force has a similarity to the Casimir effect. The reasoning is that all of these attractions are the result of reducing the pressure exerted by vacuum energy on one side of an object more than the pressure exerted on the opposite side of the object.Thisproposalmakesattractionconceptuallyunderstandable. There isonlyone fundamentalforceandthisforceisonlyrepulsive.Weliveinaseaofvacuumenergy.Itislikeafishthatlivesatgreatdepthintheocean.Thefishissubjecttogreatpressure,butthefishhappilygoesaboutits lifewithout realizing that there is anypressure. Only if somethinghappens to create animbalanceofpressuredoesthegreatpressurebecomeevident.Eventhen,anythingthatlowersthepressureononesideofanobjectappearstobecreatinganattraction.Theforceisdeliveredbywhat appears to be a featureless environment water for the fishes and vacuum for us .Gravitationalattractionwillbediscussedinthenextchapter.


“Unity”HypothesisThe wave‐particle duality is perhaps the most basic mystery of quantum mechanics. Bothphotons and particles exhibit properties that sometimes require a wave explanation andsometimesrequireaparticleexplanation. It ispossible to imagineapointparticle thathasapercentageofitsenergyasawavesurroundingthepointparticle. However,theexperimentsseemtoindicatethatsometimesthereare100%particlepropertiesandothertimesthereare100%waveproperties.Thesearesuchdifferentconceptsthattheyseemmutuallyexclusive.Today’s physics puts the primary emphasis on the particle interpretation. The waves areconsideredtobeapropertyofparticlesratherthanparticle‐likeinteractionsbeingthepropertyof quantizedwaves. Not only are the leptons and quarks viewed as particles, but photons,gravitonsandgluonsarealsoconsideredparticles. Theforcesofnatureareconsideredtobecarriedby “exchangeparticles”. Thewavepropertiesof all particles are recognized,but theparticlepropertiesareconsideredparamount.Mybackground is lasersandoptics. In this field, thewavepropertiesof lightareconsideredparamount.Theparticlepropertiesofphotonsareimportant,buttheseparticlepropertiesaresecondary to thewavepropertieswhendesigningopticsor lasers. It is easiest to thinkof aphotonasaquantizedwaveratherthanaparticlethatpossesseswaveproperties.Inthispicture,aphoton is aquantizedwave that isdistributedovera volumewhen thephoton is in flight.Absorptionofaphotonbyanatomiseasiesttopictureasthequantizedwavecollapsingintotheabsorbingatom.Fromthisbackground,thereisapredispositiontoquantizedwavesratherthanparticles. Having admittedmy predisposition towards waves, I will start my attack on theconceptthatphotonshaveparticlepropertiesbyassertingthefollowing:There are no experiments that prove that photons have particle properties. All the experiments like the photoelectric effect and atomic photon absorption merely prove that a photon possesses quantized energy. Even Compton scattering will be shown in chapter 11 to have a wave explanation.Itisacommonmisconceptiontoequatequantizationwithaparticle.However,ifspacetimeisvisualizedastheenergeticspacetimeofquantummechanics,andifthesevacuumfluctuationshave superfluid properties, then angular momentummust appear as quantized units. Thisquantizedangularmomentumhasasabyproductthatenergypossessingangularmomentumalsocomesinquantizedunits.Ithasbeenproposedearlierthatcurrentlyonlyabout1partin10120of all theenergy in theuniversepossessesquantizedangularmomentum. Energy thatpossessesquantizedangularmomentumistheonlyenergywithwhichweandourinstrumentscan interact. A photon can carry any energyup toPlanck energy, but it always carries ħ of


quantizedangularmomentum orbitalangularmomentumcanaddmultiplesofħ,butthisisaspecialcase . Ifwecanonlyinteractwithquantizedangularmomentum,theneverythingweinteract with will be forced to possess quantized energy. Waves with quantized angularmomentumwillappeartohaveparticle‐likeproperties.Weareamazedbytheapparentmysteryofthequantummechanicalpropertiesofparticlesandphotons.However,wemustrememberthatweareonlyinteractingwiththeminutepartoftheenergyintheuniversethatpossessesangularmomentum.Thisminutepartofthetotalenergyoftheuniversemustfollowtherulesofquantizedenergytransfer.Theserulesareenforcedbythevastseaofvacuumenergyinthesuperfluidstatethatsurroundsusandfillstheuniverse.Forexample,amoleculeisolatedinavacuumcanonlyrotateatafundamentalrotationalrateoratintegermultiplesofthisfundamentalrotationalrate.Thesequantizedchangesinenergyareassociatedwithquantizedchangesinangularmomentum.Thismysteryofquantummechanicsbecomesconceptuallyunderstandablewhenitisrealizedthatthemoleculereallyisnotisolated.Itlivesinaseaofsuperfluidvacuumenergythatmustisolatepocketsofangularmomentum.Enforcingthisquantizationofangularmomentumrequiresthataunitofenergywithquantizedangular momentum must be able to collapse faster than the speed of light. Is there anyexperimentalproofthatfasterthanlightactioncanoccur?Next,wewillattempttoexplainhowquantizedwavesinspacetimecanexhibitparticle‐likeproperties.Thisexplanationstartswithentanglement.Entanglement – Unity Connection: Entanglement occurs when two or more photons orparticlesinteractinawaythattheirquantumstatescanonlybedescribedwithreferencetoeachother.Separatingtheseentangledphotonsorparticlesdoesnotbreakthequantumconnection.Therefore,measuringaquantumpropertyofoneobjectaffectsthequantizedstateofthesecondentangledobject.Thiseffecthappensinstantly,evenatalargeseparationdistance.Theexistenceofentanglementhasbeenproveninmanydifferentexperiments.If entanglement provides an instantaneous response between two entangled particles orphotons,isitnotreasonablethatthereshouldalsobeasimilareffectwithinasingledipolewavewith quantized angular momentum? Chapters 11 and 14 will offer additional insights intoentanglement and the super luminal communication. For now we will merely acceptentanglementasanexperimentallyproveneffectandexaminetheimplicationsofitsproposedcloserelative,unity.Apurelyspacetimewavemodeloffundamentalparticlesmustexplainhowawavethatisdistributedoveravolumecanexhibitparticle‐likepropertiessomeofthetime.Ifawaveisenvisionedasbeingdivisibleintosmallerpartslikeasoundwave,thenitisimpossibleforsuchawavetoexhibitparticle‐likeproperties.However,arotarisadipolewaveinspacetimethatiscarryingaquantizedamountofangularmomentuminaseaofvacuumenergythatlacksangularmomentum.Thistypeofwavecanchangeitsenergyinacollision,butitalwaysmustcarrytheassignedquantizedangularmomentumof½ħorħ,forarotarorphotonrespectively.


It is true that Iamnotgivingaconceptuallyunderstandableexplanationofwhyasuperfluidcannot possess angular momentum andwhy any angular momentum that is present in thesuperfluid is broken into quantized units. This is an experimentally observed property ofsuperfluidliquidheliumandIbelievethatthereisatheoreticalexplanationfortheeffectinliquidhelium.However,ImustadmitthatIdonothaveaconceptuallyunderstandableexplanationforthis when it is reduced towaves in spacetime. This is a good project for someone else. However,ifweassumequantizedangularmomentumexists,thenitiseasytoseethatawavecarryingquantizedangularmomentummustrespondasaunittoaperturbation.Inacollisionwithanotherquantizedwave,thewavewithquantizedangularmomentummustinteractasaunittopreciselypreservetheangularmomentum.The preservation of quantized angular momentum requires that the quantized wave possess faster than speed of light internal communication. This is the proposed property called “unity”. The property of unity gives particle-like properties to a wave carrying quantized angular momentum.Thepropertiesofspacetimedeterminethesize ½ħ ofthequantizedangularmomentum.Ifwe accept this as a given, then the property of unitymust be a component of anymodel ofparticlesbasedonlyonwaves. Someeventssuchas theemissionofaphoton fromanatomoccursoveralongenoughperiodoftimethatthereisenoughtimeforthequantizedwavetorespondwithouttheneedtoinvokesuperluminalcommunication discussedlater .However,other events such as the collision of two rotars at relativistic speed requires that the rotarrespond in a time period faster than required for speed of light communication across thephysicalsizeoftherotar’srotarvolume.Theexternalvolumeofarotarrespondsdifferentlyandwillbediscussedlater.Natureiscapableofsuperluminalcommunicationasdemonstratedbythemanyexperimentsthatprovetheexistenceofentanglement.Thesamewaythatitisnotpossibletosendamessagefasterthanthespeedoflightusingentanglement,italsoisnotpossibletosendamessagefasterthanthespeedoflightwhenaquantizedwaverespondstoaperturbationasasingleunit.Thisismerelyaninternalhousekeepingfunction.Theentirequantizedwave withquantizedangularmomentum mustrespondasifitisoneentangledunit.Assumethatarotaristhedipolewavemodelpreviouslydescribed.Itisnotpossibletointeractwithjust1%ofaquantizeddipole.Itisnotpossibletotransferlessthanħofangularmomentum.Either 100% of the rotar volume responds to the interaction or none of the rotar volumeresponds.Ifthereisatransferofangularmomentum,italwaysoccursinquantizedunitsofħ.Thecommunicationwithinasinglequantizedrotarvolumewouldbeinstantaneous,justliketheresponseinvolvingtwoentangledparticles.Infact,theresponsewithinasinglequantizedwaveshouldbebetterthanwhentwophotonsortwoparticlesareentangled.


SixthStartingAssumption: A wave in spacetime with quantized angular momentum responds to a perturbation as a single unit. This superluminal internal communication gives the quantized wave, particle-like properties.Unity is proposed to be the property responsible for the mysterious wave‐particle dualitypresenteverywhereinnature.Everyphysicalentityintheuniverseismadeofdipolewavesinspacetime.Unitypermitsthesewavestorespondwithparticle‐likeproperties,buttheresponseexhibitsaprobabilisticcharacteristics.Recalltheincrediblysmalldistortionofspacetimethatformsafundamentalrotar.“Finding”aparticlesomewherewithinaquantizeddipolewaveisreallyunitycausingthequantizedwavetointeractwithaprobe anotherwave inawaythatappearstoexhibitparticle‐likepropertiesatasinglelocation.Theparticle‐likepropertiesofaquantizedwavecanexhibitdiscontinuousjumpsbecauseinteractingwiththequantizedwavecanhappenatanypartofthevolumecontainingthequantizedwave.Theinteractionandtheapparentlocationoftheinteractionisaprobabilisticevent.Collapse of the Wave Function: A“collapseofthewavefunction”inquantummechanicsisproposedtoberelatedtothepropertyofunity.However,thisconnectioniscomplicatedbythefactthatoftenthemathematicalexpressionofawavefunctionincludesboundaryconditionsnotencounteredbyisolatedrotars. Forexample,a“particleinabox”oranelectronboundinanatombothhaverestrictiveboundaryconditionsthatchangethedistributionofspacetimewavescomparedtoanisolatedrotar. Thesearemorecomplicatedconditionsthatwillbediscussedlater.Inquantummechanics,thephysicalinterpretationofthecollapseofthewavefunctionisliterallythattheprobabilisticwavepropertiesofapointparticledisappear collapse whentheparticleis“found”.Thephysicalinterpretationofunityisthatarotar’swavepropertiesremainafteritis “found”. The distributed wave of a rotar just responds to a probe another wave as aquantizedunit.Sincetherotarisdistributedoveravolume,thereisinternalcommunicationwithintherotarthatoccursfasterthanthespeedoflight.Therefore,therotarrespondstoaperturbationasifitwasconcentratedatasinglelocation.Unityallowsfundamentalrotarstorespondtoarelativisticcollisionbymomentarilyshrinkingtheradiusoftherotatingdipoleasasinglecohesiveentity.Thisreductioninrotarradiushappensfasterthanthespeedoflight,soitisimpossibletodetecta fundamental particle’s size using inferences from collisions. In a collision, the angularmomentumremainsconstant,butthefrequencyandenergyincreaseastheradiusdecreases.Thequantizedwaveappearstobeapointconcentrationofmass/energythatdiscontinuouslychangeslocation.Thereisjustnoliteralcollapseofwavesintoapointparticle.


Thecharacteristicofunity is the finalpieceof thepuzzle required for fundamental rotars toappeartobepointparticles.Inexperimentsthatattempttomeasurethesizeofthefundamentalrotars, theresolutionof theexperimentdependsontheenergyof thecollision. Imaginetworotars colliding at relativistic velocity. In the interaction, the kinetic energy is temporarilyconvertedtointernalenergyofthetworotars.Inordertopreservetheangularmomentumoftherotar,itisnecessaryfortherotartoreduceitsrotarradiusfromthesizecharacteristicofanisolatedrotartothesizeappropriateforarotarthathasabsorbedextraenergy.Thistemporarysizereductiongivestheenergeticrotararotarradiuscomparabletotheresolutionlimitofthecollision experiment. Not only does the rotar reduce the size of in a collision, but thisreductionhappensfasterthanthespeedoflight.Theentireenergyintherotarvolumereactsasaunit,sotheinertiaappearstooriginatefromapoint.Thelocationofthatpointisprobabilistic,soitcanappearthatarotarmovesindiscontinuousjumps.Laterwewilladdressthequestionof the small amount of a rotar’s energy that is external to the rotar volume and respondsdifferently.Partial Explanation of Unity:Thefollowingpartialexplanationofunityisofferedforrotarsthatexhibitrestmass.Unitywithinphotonswillbediscussedlater.Itishopedthatotherscanimproveonthispartialexplanation.Allrotarswithrestmassareproposedtobequantizedwavescirculatingatthespeedoflightinaconfinedvolume.Eventhoughthecirculationhappensinalimitedvolume,thefactremainsthatthesewavesdonotexperiencetimeordistance.Thereisafundamentaldifferencebetweenthewayweperceivetheuniverse 3spatialdimensionsplustime andthewayquantizedwavestravelingatthespeedoflightperceivetheuniverse.Theyliveinazerodimensionaluniverse.Dipolewavesinspacetimeconsidertheuniversetobeasinglepoint.Itshouldnotbesurprisingthatwefindmanyaliencharacteristicswhenwetransferfromthe4dimensionalmacroscopicperspectiveintothezerodimensionalquantumperspective.Withinaquantizedwavecirculatingatthespeedoflightthereisnotimeandnodistance.Thisgivesrisetoboththeproposedpropertyofunityandtoentanglement.Sincetherotarperceivesthatthereare no spatial dimensions, an interactionwith the rotar cannot take placewith only a smallportionoftherotar.Itisallornothing.In this book I have attempted to make quantum mechanical operations conceptuallyunderstandable. The above explanation of unity and entanglement is really only a partialexplanation. Inchapter11amodelof twoentangledphotonswillmakeentanglementmoreunderstandable. In the cosmologychapters13and14anewpictureof theuniversewillbeofferedwhichwillfurtherimprovetheexplanationofunityandentanglement.


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Chapter8

GravitationalAttractionandUnificationofForcesInchapter6,thereaderwasaskedtotemporarilyconsiderallforcestoberepulsive.Thiswasasimplificationwhichallowedthecalculations inchapter6 toproceedwithoutaddressing themorecomplicatedsubjectofattraction.Inchapter7,vacuumenergy/pressurewasintroducedasanessentialconsiderationinthegenerationofallforces,butespeciallyforcesthatproduceattraction. In this chapter we are going to attempt to give a conceptually understandableexplanationfortheforceofattractionexertedbygravitywhenabodyisheldstationaryrelativetoanotherbody.Physical Interpretation of General Relativity: Einstein’s general relativity has passednumerousexperimentalandmathematicaltests.Thismathematicalsuccesshasconvincedmostphysicists to accept the physical interpretation usually associated with these equations.However, the most obvious problem with the physical interpretation is examined in thefollowing quotes. The first is from B. Haisch of the California Institute of Physics andAstrophysics.“The mathematical formulation of general relativity represents spacetime as

curvedduetothepresenceofmatter….Geometrodynamicsmerelytellsyouwhatgeodesicafreelymovingobjectwillfollow.Butifyouconstrainanobjecttofollowsomedifferentpath ornot tomoveatall ,geometrodynamicsdoesnot tellyouhoworwhyaforcearises….Logicallyyouwinduphavingtoassumethataforcearisesbecausewhenyoudeviatefromageodesicyouareaccelerating,butthatisexactlywhatyouaretryingtoexplaininthefirstplace:Whydoesaforcearisewhenyouaccelerate?…Thismerelytakesyouinalogicalfullcircle.”

Talkingaboutcurvedspacetime,thebookPushingGravity M.R.Edwards states:“Logically,asmallparticleatrestonacurvedmanifoldwouldhavenoreasontoend

itsrestunlessaforceactedonit.Howeversuccessfulthisgeometricinterpretationmaybeasamathematicalmodel,itlacksphysicsandacausalmechanism.”

Generalrelativitydoesnotexplainwhymass/energycurvesspacetimeorwhythereisaforcewhenanobjectispreventedfromfallingfreelyinagravitationalfield.Ifrestraininganobjectfrom followingageodesic is theequivalentof acceleration, thenapparently thegravitationalforceisintimatelytiedtothepseudoforcegeneratedwhenamassisaccelerated.Inthestandardmodel,particlespossessnointrinsicinertia.TheygaininertiafromaninteractionwiththeHiggsfield.IstheHiggsfieldalsonecessarytogenerateagravitationalforcewhenaparticlewithout


intrinsicinertiaispreventedfromfollowingthegeodesic?IstheHiggsfieldalwaysacceleratingtowards amass in an endless flow that attempts to sweep along a stationary particle? Thestandardmodeldoesnotincludegravity.Isgravityaforce?Thepointofthesequestionsistoshowthattherearelogicalproblemswiththephysicalmodelnormally associated with general relativity. The equations of general relativity accuratelydescribegravityonamacroscopicscale.However,theseequationsaresilentastothephysicalinterpretation,especiallyatascale spatialortemporal wherequantummechanicstakesover.Gravitational Nonlinearity Examined: Previouslywe reasoned that spacetimemust be anonlinearmedium for waves in spacetime and gravity is the result of this nonlinearity. Atdistance fromarotarwecalculatedthegravitationalforceusingoneofthe5wave‐amplitudeequationsF A2ω2Z /c.InthiscalculationwesubstitutedA Aβ2 Lp/ 2 Tpωc 2.Alsotheangularfrequencyωisequaltotherotar’sComptonfrequencyω ωc.Atdistance oftwoofthesamerotarsweobtainedF Gm2/ .Thisisthecorrectmagnitudeofthegravitationalforce between two rotars of mass m, but there are two problems. First: the equationF A2ω2Z /c is for a traveling wave striking a surface. This traveling wave implies theradiationofpowerthatisnothappening.Second:awaveinspacetimetravelingatthespeedoflightgeneratesarepulsiveforceifitinteractsinawaythatthewaveisdeflectedorabsorbed.Gravity isobviouslyanattractive force. Wehave themagnitudeof the forcecorrect,but themodelmustberefinedsothatthereisnolossofpowerandsothattheforceisanattraction.Thereareseveralstepsinvolved,anditisprobablydesirabletobeginwithabriefreview.Recallthatwearedealingwithdipolewavesinspacetimewhichmodulateboththerateoftimeandvolume. There are twoways thatwe can express the amplitude of thedipole in spacetime:displacement amplitude and strain amplitude. The maximum displacement of spacetimeallowed by quantum mechanics is a spatial displacement of Planck length or a temporaldisplacementofPlancktime.Sincetheseareoscillationamplitudes,wesometimesusetheterm“dynamic Planck length Lp” or “dynamic Planck time Tp”. As previously explained, thesedistortionsofspacetimeproduceastraininspacetime.ThestrainisadimensionlessnumberequivalenttoΔl/lorΔt/tInthiscaseΔl/l Lp/ andΔt/t Tpωc.Inchapter5weimaginedahypotheticalperfectclockplacedatapointonthe“Comptoncircle”ofarotarasillustratedinfigure5‐1.Thisistheimaginarycirclewithradiusequaltotherotarradius .Thisclock hereaftercalledthe“dipoleclock”withtimeτd wascomparedtothetimeonanotherclockthatwecalledthe“coordinateclock” withtimetc . Thiscoordinateclockismeasuringtherateoftimeiftherewasnospacetimedipolepresent.Itisalsopossibletothinkofthecoordinateclockaslocatedfarenoughfromtherotatingdipolethatitdoesnotfeelany

significanttimefluctuations.Figure5‐3showsthedifferenceintheindicatedtimeΔt τd‐tc.Thedipoleclockspeedsupandslowsdownrelativetothecoordinateclockandthemaximumdifference is dynamicPlanck timeTp. ThereforeTp is the temporaldisplacement amplitude.


ThereisalsospatialdisplacementamplitudethatisequaltodynamicPlancklengthLp.Thestrainofspacetimeproducedbythesedisplacementsofspacetimeisdepictedinfigure5‐4.Withintherotarvolume,thestrainofspacetimeisdesignatedbythestrainamplitudeAβ Tpωc Lp/ .Thisisequivalenttothemaximumslopeofthesinewavewhichoccursatthezerocrossingpointsinfigure5‐3.Nonlinear Effects: The above review now has brought us to the point where we can askinterestingquestions:Doesthedipoleclockalwaysreturntoperfectsynchronizationwiththecoordinate clock at the completion of each cycle? Does the volume oscillation of spacetimeproduce a net change in the average volume near a rotar? Since we are going to initiallyconcentrateonexplainingthegravitational forcebetweentwofundamentalparticles,wewillinitiallyconcentrateontheeffectontime.Therefore,doestherateoftimeoscillationcausethedipoleclocktoshowanetlossoftimecomparedtothecoordinateclock?Ifspacetimehasnononlinearity then the clockswould remain substantially synchronized. However, if there isnonlinearity,thedipoleclockwouldslowlylosetime.Aspreviouslyexplained,thestrainofspacetime instantaneousslopeinfigure5‐3 hasalinearcomponentandanonlinearcomponent.Theproposedspacetimestrainequationforapointontheedgeoftherotatingdipoleis:Strain Aβsinωt Aβsinωt 2… higherordertermsignored Thelinearcomponentis“Aβsinωt”andthefirstterminthenonlinearcomponentis Aβsinωt 2.TherewouldalsobehigherordertermswhereAβisraisedtohigherpowers,butthesewouldbesosmallthattheywouldbeundetectableandwillbeignored.Thisnonlinearcomponentcanbeexpanded:Aβsinωt 2 Aβ2sin2ωt ½Aβ2–½Aβ2cos2ωt


Figure 8‐1 plots the linear component Aβ sinωt and the nonlinear component Aβ sinωt 2separately.ItcanbeseenthatthenonlinearcomponentisasmalleramplitudebecauseAβ 1andsquaring thisproducesa smallernumber. Also thenonlinearcomponent isat twice thefrequency of the linear component. Most importantly, the nonlinear component is alwayspositive.Makinganelectricalanalogy,thiscanbethoughtofasifthenonlinearwavehasanACcomponentandaDCcomponent. It isobviousthatwhenthe linearandnonlinearwavesareadded together, the sumwill produce an unsymmetricalwave that is biased in the positivedirection.Itwasnecessarytousesomeartisticlicenseinordertoillustratetheseconceptsinfigure8‐1.ForfundamentalrotarsthevalueofAβisroughlyintherangeof10‐20.ThismeansthatAβ2 10‐40

andthereforeAβisapproximately1020timeslargerthanAβ2.ItwouldbeimpossibletoseetheplotofAβ2sin2ωtwithoutartificiallyincreasingthisrelativeamplitude.Therefore,theassumedvalueinthisfigureisAβ 0.2.InthiscasethedifferencebetweenAβandAβ2isonlyafactorof5ratherthanafactorofroughly1020.Therefore,forfundamentalrotarsitisnecessarytomentallydecreasetheamplitudeofthenonlinearwavebyroughlyafactorofroughly1020.


Whenweaddthetwowavestogetherweobtaintheplotinfigure8‐2.Becauseoftheartisticlicense,itisvisuallyobviousthatthisisanunsymmetricalwave.Thereisalargerareaunderthepositiveportionofthewavethantheareaunderthenegativeportionofthewave. Thepeakamplitude for the positive portion is Aβ Aβ2while the negative portion has peak negative


amplitude of: – Aβ Aβ2. If this was a plot of electrical current, we would say that thisunsymmetricalwavehadaDCbiasonanACcurrent.Tousetheanalogyfurther,itisasifthenonlinearitycausesspacetimetohavetheequivalentofasmallDCbiasinitsstress.Thedipoleclockdoesnotreturntosynchronizationwiththecoordinateclockeachcycle.Figure8‐3 attempts to illustrate thiswith a greatly exaggerated plot of the difference between thecoordinate clockand thedipole clock. The “X” axisof this figure is timeas indicatedon thecoordinateclockwhilethe“Y”axisisthedifferencebetweenthecoordinateclockandthedipoleclock t–τ .Ifthetwoclocksranatexactlythesamerateoftime,theplotwouldbeastraightlinealongthe“X”axis.Normallythisplotforafewcyclesshouldlooklikeasinewavesimilartofigure5‐3.However,thepurposeoffigure8‐3istoillustratethatovertimethecoordinateclockpulls ahead of the dipole clock or the dipole clock loses time . Therefore, for purposes ofillustration,thiseffectoftheaccumulatedtimedifferencehasbeenexaggeratedbyafactorofroughly1022.The unsymmetrical strain plot in figure 8‐2 produces a net loss of time on the dipole clockrelativetothecoordinateclock.Inthefirstquartercycleoffigure8‐3,thecoordinateclockfallsbehindthedipoleclockbyanamountapproximatelyequaltoPlancktime.Thisoccurswhenthefast lobe of the rotating dipole passes the dipole clock first. However, with each cycle, thecoordinateclockgainsasmallamountoftimeonthedipoleclock.Theamountoftimegainedpercycleisillustratedbythegaplabeled“Singlecycletimeloss”.ThisisequaltoTp2ωcwhichisabout2.2 10‐66sforanelectron.Thepointof figure8‐3 is to illustratethecontributionof thenonlineareffect. ThenonlinearwavewithstrainofAβ2sin2ωtatdistance producesthecontributionthatcausesthenetlossoftimeforthedipoleclockrelativetothecoordinateclock.Thisnettimedifferencebetweenthetwoclocks aftersubtractingAβsinωt isshownasthewavylinelabeled“nonlinearcomponent”.TheaverageslopeofthislineisequaltothegravitationalmagnitudefortherotarvolumewhichhasbeendesignatedasAβ2 βq.Foranelectronthisslopeisabout1.75x10‐45whichmeansthatittakesabout30secondsforthecoordinateclocktohaveanettimegainofPlancktimeoverthedipoleclock.Thistakesabout4x1021cyclesratherthan4cyclesasillustratedinfigure8‐3.Ifwesubtractedthenonlinearwavecomponentfromfigure8‐3,wewouldbeleftwithasinewavewithamplitudeofTp.TheslopeonthisnonlinearwavecomponentisAβ2atdistance whichisobtainedfromthestrainequation–theimportantpartishighlightedbold

Aβsinωt Aβsinωt 2 Aβsinωt–½Aβ2cos2ωt ½ Aβ2

Derivation of Curved Spacetime: TheDCequivalentterm non‐oscillatingterm isAβ2.Thisisthenonlinearstrain inspacetimeproducedby therotaratdistance . This isan important


conceptsinceitrelatestocurvedspacetime.Beforeperusingthisthoughtfurtheritisnecessarytointroduceanewsymbol: . Thenaturalunitoflengthforarotarisλc. Thereforewewilldesignatetheradialdistancefromarotarnotinunitsoflengthsuchasmeters,butasthenumberofreducedComptonwavelengths numberofrotarradiusunits .

≡ ħ

For example, the non‐oscillating strain in spacetime produced by a rotar should decreaseproportionalto1/ .Wecantestthisideasincetheproposalisthatthenon‐oscillatingstraininspacetimeatdistance λc where 1 isequal toAβ2andthisdecreasesas1/ . WewillevaluateAβ2/ andcallthisthegravitationalamplitudeAg.

Ag

Therefore we have succeeded in producing the previously discussed weak gravity gravitational magnitude β ≈ Gm/c2r. This is the curvature of spacetime that we associate with gravity.ThissimpleevaluationisanothersuccessofthismodelbecausethetermGm/c2r istheweakgravitycurvatureofspacetimeproducedbymassmatdistancedr.Forexample,thepreviouslydefinedgravitationalmagnitude isβ≡1– dτ/dt . Theweakgravity temporaldistortionofspacetimeis:dt/dτ 1 Gm/c2r .Inflatspacetimedt/dτ 1,sotheweakgravitycurvaturetermisβ Gm/c2r .Forfundamentalparticles rotars atdistance thistermisintherangeof10‐40,sothisisvirtuallyexact.The question of how matter “causes” curved spacetime has been a major topic in generalrelativityandquantumgravity.Nowweseethemechanismofhowdipolewavesinspacetimeproducebothmatterandcurvedspacetime.Thisusesequationsfromquantummechanicstoderiveanequationfromgeneralrelativity. Thisisnotonlyasuccessfultestofthespacetimebasedmodel,but it isalsoapredictionof thismodelof themechanismthatachievescurvedspacetime.Oscillating Component of Gravity: There is proposed to be another residual gravitationaleffectthathasnotbeenobservedbecauseitisaveryweakoscillationatafrequencyinexcessof1020Hz.Infigure8‐3thenon‐linearwavecomponentisshownasawavylinelabeled“nonlinearcomponent”.Wecaninteractwiththenon‐oscillatingpartofthislineresponsibleforgravity,but there is also a residual nonlinear oscillating component. At distance this oscillatingcomponenthasamplitudeAβ2andfrequency2ωc.Whathappenstothisoscillatingcomponentbeyond intheexternalvolume?Weknowthatthefewfrequenciesthatformstableandsemi


stable rotars exist at resonanceswith the vacuum fluctuationsof spacetimewhich eliminateenergy loss. If the amplitude of the oscillating componentwasAβ2/ , then therewould becontinuous radiation of energy. Energetic composite particles such as protons or neutronswouldradiateawayalltheirenergyinafewmillionyears.Inthechapter10ananalogywillbemade to theenergydensityof a rotar’s electric field. Theamplitude term for theoscillatingcomponentofgravitywillthenbeproposedtoscaleasAβ2/ 2.Thiswouldbeanextremelysmallamplitudeand furthermore it isastandingwave thatdoesnotradiateenergy.However, thisoscillatingcomponentwouldtheoreticallygiveenergydensitytoagravitationalfield.Theenergydensity of a gravitational field and its contribution to producing curved spacetime will bediscussedattheendofchapter10.Theoscillatingcomponentofagravitationalfieldmayalsobeimportantintheevolutionoftheuniverse.Thiswillbediscussedinchapters13and14.Summary: Since we need to bring together several different components to achievegravitational attraction,wewill doanother review. This timewewill emphasize the roleofvacuumenergy,circulatingpower,thecancelingwaveandnon‐oscillatingstraininspacetime.Arotarisarotatingspacetimedipoleimmersedinaseaofvacuumenergywhichisequivalenttoa vacuum pressure. This vacuum energy/pressure is made up of very high energy density1046J/m3 dipolewavesinspacetimethatlackangularmomentum.Therotaralsohasahigh

energydensitythat isattemptingtoradiateawayenergyat therateof therotar’scirculatingpower.Therotarsurvivesbecauseitexistsatoneofthefewfrequenciesthatachievearesonancewiththevacuumenergy/pressure.Thisresonancecreatesanewwavethathasacomponentthatpropagatesradiallyawayfromtherotating dipole and a component that propagates radially towards the rotating dipole.Tangential wave components are also created, but these add incoherently and effectivelydisappear. The resonant wave that is propagating away from the dipole cancels out thefundamental radiation from the dipole. Besides having the correct frequency and phase toproducedestructive interference, the cancelingwave alsomustmatch the rotar’s circulatingpower. Thismeansthatthecorrectpressureisgeneratedfromthevacuumenergy/pressurethatisrequiredtocontaintheenergydensityoftherotar.Onlyafewfrequenciesthatformstablerotarscompletelysatisfytheseconditions.Forexample,anelectronhasacirculatingpowerofabout64millionwatts.Inordertocancelthismuchpowerfrombeingradiatedfromtherotarvolume,thecancelationwavegeneratedinthe vacuum energy must have an outward propagating component of 64 million wattsattemptingtoleavetherotar’svolumeandaninwardpropagatingcomponentofthesamepower.Therecoilfromtheoutwardpropagatingcomponentprovidesthepressurerequiredtostabilizetherotatingdipole that is therotar theelectron . Thispressurecanbe thoughtofasbeingcarriedbytheinwardpropagatingcomponentthatreplenishestherotatingdipole.


Ifitwaspossibletoseethisprocess,wewouldnotseeoutwardorinwardpropagatingwaves.Wewouldonlyseethesumofthesetwowaveswhichisastandingwavewhichdecreasesinamplitudewithdistancefromthecentralrotar.Astandingwaveintherotar’sexternalvolumemeans thatnopower isbeing radiated.These standingwaves cause the rotar’s electric fielddiscussedlater . Wewouldalsoseethattherewasaslightnon‐oscillatingresidualstraininspacetimewithstrainamplitudeofAβ2/ gm/c2r. Newtonian Gravitational Force Equation:Therearestilltwomorestepsbeforewearriveattheexplanationthatgivesthecorrectattractingforceatarbitrarydistancebetweentworotars.Wewill startbyassuming twoof the same rotars massm separatedbydistance r. Itwaspreviously explained that deflecting all of a rotar’s circulating power generates the rotar’smaximumforceFm.Arotaralwaysdependsonthepressureofthespacetimefieldtocontainitscirculatingpower.Whentherotarisisolated,theforcerequiredtodeflectthecirculatingpowerisbalanced.However,agravitationalfieldproducesagradientinthegravitationalmagnitudedβ/dr.When a first rotar is in the gravitational field of a second rotar, there is a gradient dβ/dr that exists across the rotar radius of the first rotar. This means that there is a slight difference in the force exerted by vacuum energy/pressure on opposite sides of the first rotar. This difference in force produces a net force that we know as the force of gravity.Thiswillberestatedinadifferentwaybecauseofitsimportance.Imaginemassm1beingarotarrotatingdipole attemptingtodispersebutbeingcontainedbypressuregeneratedwithinthevacuumenergy/pressurepreviouslydiscussed.Thispressureexactlyequalsthedispersiveforceof the dipole wave rotating at the speed of light. However, if there is a gradient in thegravitationalmagnitudedβ/drthenthereisagradientacrosstherotarwhichwewillcallΔβ.Thisaffectsthenormalizedspeedoflightandthenormalizedunitofforceonoppositesidesoftherotar.Recallfromchapter3wehad:Co ГCgnormalizedspeedoflighttransformationFo ГFgnormalizedforcetransformationГ 1 βapproximationconsideredexactforrotarsTherefore,becauseofthestraininspacetime,thetwosidesoftherotar separatedby arelivingunderwhatmightbeconsideredtobedifferentstandardsforthenormalizedspeedoflightandnormalizedforce.Onanabsolutescale,ittakesadifferentamountofpressuretostabilizetheoppositesidesoftherotarbecauseofthegradientΔβacrosstherotar.Thenetdifferenceinthisforceistheforceofgravityexertedontherotar.WewillfirstcalculatethechangeingravitationalmagnitudeΔβacrosstherotarradius ofarotarwhen it is in thegravitational fieldof another similar rotar another rotarof the same


mass .Inotherwords,wewillcalculatethedifferenceinβatdistanceranddistancer fromarotarofmassm.

Δβ approximationvalidif r

Theforceexertedbyvacuumenergy/pressureonoppositehemispheresoftherotarisequaltothe maximum force Fm m2c3/ħ. The difference in the force absolute value exerted onoppositesidesoftherotaristhemaximumforcetimesΔβ.Therefore,theforcegeneratedbytworotarsofmassmseparatedbydistanceris:

F ΔβFm ħ

ħ

ħ

Ifwehavetwodifferentmassrotars massm1andmassm2 ,thenwecanconsidermassm1inthegravitationalfieldofmassm2.Inthiscase, andFmareformassm1andΔβischangeinthegravitationalmagnitudefrommassm2acrosstherotarradius frommassm1.

Fg ΔβFm

ħ

ħ

ħ

Fg Newtoniangravitationalforceequationderivedfromadipolewavemodel

Gravitational Attraction:WehavederivedtheNewtoniangravitationalequationfromstartingassumptions,butwestillhavenotshownthatthisisaforceofattraction.However,fromthepreviousconsiderations,thislaststepiseasy.ThereisaslightlydifferentpressurerequiredtostabilizetherotardependingonthelocalvalueofβorГ inweakgravityГ 1 β .Thiscanbeconsideredasadifferenceinnetforceexertedbyvacuumenergyonthehemisphereoftherotarthatisfurthestfromtheotherrotarcomparedtothehemispherethatisnearesttheotherrotar.The furthesthemispherehasasmalleraveragevalueofГ than thenearesthemisphere. Thenormalizedspeedoflightisgreaterandthenormalizedforceexertedonthefarthesthemispheremustbegreatertostabilizetherotar.ThisproducesanetforceinthedirectionofincreasingГ.ThemagnitudeofthisforceisF Gm1m2/r2andthevectorofthisforceisinthedirectionofincreasingГ towardstheothermass .Weconsiderthistobeaforceofattractionbecausethetworotarswanttomigratetowardseachother increasingГ . However,theforceisreallycomingfromthevacuumenergyexertingarepulsivepressure. There isgreaternormalizedpressurebeingexertedonthesidewith thelower Г. The two rotars are really being pushed together by a force of repulsion that isunbalanced.


CorollaryAssumption:The force of gravity is the result of unsymmetrical pressure exerted on a rotar by vacuum energy. This is unbalanced repulsive force that appears to be an attractive force.Example: Electron in Earth’s Gravity:Wewilldoaplausibilitycalculationtoseeifweobtainroughlythecorrectgravitationalforceforanelectronintheearth’sgravitationalfieldbasedonthe above explanation. We will be using values for the electron’s energy density and theelectron’smaximumforcethatwerepreviouslycalculatedbyignoringdimensionlessconstants.Therefore, we will continue with this plausibility calculation that ignores dimensionlessconstants. An electron has internal energy of Ei 8.19x10‐14 J and a rotar radius of 3.86x10‐13m. Ignoring dimensionless constants, this gives an energy density of about

Ei/ 1.4x1024J/m3.Thisrotarmodelofanelectronisexertingapressureofroughly1.4x1024N/m2.Thispressureoverareaλc2producestherotar’smaximumforcewhichforanelectronisFm 0.212N obtainedfromℙq 1.4x1024N/m2 3.86x10‐13m 2 0.212N .Theweakgravitygravitationalmagnitudeis:β Gm/c2r.Fortheearthm 5.96x1024kgandtheequatorialradiusis:r 6.37x106m.Therefore,atthesurfaceoftheearththegravitationalmagnitudeis:β 6.95x10‐10.Toobtainthegradientinthismagnitudewedividebytheearth’sequatorial radius6.37x106m toobtainagradientofdβ/dr 1.091x10‐16/m. Thechange ingravitationalmagnitudeΔβacrosstherotarradius 3.862x10‐13m ofanelectronis:Δβ 1.091x10‐16/m 3.862x10‐13m 4.213x10‐29Δβacrossanelectron’s The electron’s internal pressure is being stabilized by the pressure being exerted by thespacetimefield.However,thehomogeneousspacetimefieldinzerogravityismodifiedbytheearth’sgravitationalfield.Aspreviouslycalculated,gravityaffectsnotonlytherateoftimeandpropervolume,butalsotheunitofforce,energy,etc.Thepreviouslycalculatednormalizedforcetransformationis:Fo ГFg.Thegradientintheearth’sgravitationalfieldmeansthataslightlydifferentvalueofΓexistsonoppositesidesoftheelectron.ThisismoreconvenientlyexpressedasadifferenceinthegravitationalmagnitudeΔβthatexistsacrosstheelectron’sradiusλc.Therewill be a slight difference in the force exerted by the spacetime field exerted on oppositehemispheresoftheelectron rotar .Calculatingthisdifferenceshouldequalthemagnitudeofthegravitationalforceontheelectron.F ΔβFm 4.213x10‐29x0.212N 8.89x10‐30NWewillnowcheckthisbycalculationtheforceexertedonanelectronbytheearth’sgravityusingF mgwheretheearth’sgravitationalaccelerationis:g 9.78m/s2F mg 9.1x10‐31kgx9.78m/s2 8.89x10‐30N


Success!TheanswerobtainedfromthecalculationusingF ΔβFmisexactlycorrect.Apparentlytheignoreddimensionlessconstantscancel.Thisisanothersuccessfulplausibilitytest.At the beginning of this chapter two quotes were presented that pointed out that generalrelativitydoesnotidentifythesourceoftheforcethatoccurswhenaparticleisrestrainedfromfollowingthegeodesic.M.R.Edwardsstates:“Howeversuccessfulthisgeometricinterpretationmaybeasamathematicalmodel,itlacksphysicsandacausalmechanism.”Theideasproposedin thisbookgivea conceptuallyunderstandable explanation forboth themagnitudeand thevectordirectionofthegravitationalforce.Thegravitationalforcewasobtainedfromthestartingassumptionswithoutusinganalogyofacceleration.Electrostatic Force at Arbitrary Distance:ThestrainamplitudeofthespacetimewaveinsidetherotarvolumehasbeendesignatedwiththesymbolAβ Lp/λc Tpωc.WehavejustshownthatthegravitationaleffectexternaltotherotarvolumescaleswithAg Aβ2/ gm/c2r.Thisis the gravitational curvature of spacetime produced by a rotar with radius λc and angularfrequencyωc.NowwewillexaminetheelectromagneticeffectonspacetimeproducedbytheeffectofthefundamentalwavewithamplitudeAβ notsquared .Fromchapter6weknowthatthisamplitude isassociatedwiththeelectrostatic force.Nowwewillextendthistoarbitrarydistance.Asbefore,weneedtomatchtheknownamplitudeatdistanceλc.Thisisachievedbyscalingdistanceusing ≡r/λcbecause 1atdistanceλc.Wewillagainassumethattheelectrostatic amplitude AE decreases as 1/ for the electrostatic force we assumeAE Aβ/ Lp/λc λc/r .WewillusetheequationF = kA2ω2Z /c and also insert the following: F = FE, ω = ωc, Z = Zs = c3/G, r/λc, = kλc

2 and ħc = 4⁄

= /c = ħ

Therefore,wehavegenerated theCoulomb lawequationwhere thecharge isq qp Planckcharge . It should not be surprising that the charge obtained is Planck charge rather thanelementarychargee.Planckchargeisqp 4 ħ about11.7timeschargee andisbasedon the permittivity of free space εo. Planck charge is known to have a coupling constant tophotonsof1whileelementarychargeehasacouplingconstanttophotonsofα,thefinestructureconstant. This calculation is actually themaximumpossible electrostatic forcewhichwouldrequirea coupling constantof1. The symbolFE implies theelectrostatic forcebetween twoPlanckchargeswhileFeimpliestheelectrostaticforcebetweentwoelementarychargese.TheconversionisFE Feα‐1.Wewillcontinuetousetheequation:F = kA2ωc

2Zs /c even though it implies the emission of power which is striking area andexertingarepulsiveforce.ThisisnothappeningbuttheuseofF = kA2ωc

2Zs /c gives the correct magnitude of forces. This simplified equation allows a lot of quick calculations to be made which give correct magnitude.


Calculation with Two Different Mass Particles:Untilnowwehaveassumedtwoofthesamemass/energyparticleswhenwecalculatedFgandFE.Nowwewillassumetwodifferentmassparticles m1andm2 ,butwewillkeeptheassumptionthatbothparticleshavePlanckcharge.Whenwe have two differentmass particles, thismeans thatwe have two different reducedComptonwavelengths λc1 ħ/ c andλc1 ħ/ .The single radial separation r nowbecomestwodifferentvalues 1 r/λc1and 2 r/λc2. Also, therewouldbe twodifferentstrainamplitudesAβ1 Lp/λc1andAβ2 Lp/λc2aswellasacompositearea kλc1λc2.

k

Note that the only difference between the intermediate portions of these two equations is that the gravitational force Fg has the strain amplitude terms squared ( and the electrostatic force FE has the strain amplitude terms not squared ( . The tremendous difference between the gravitational force and the electrostatic force is due to a simple difference in exponents.Unification of Forces: Inchapter6wefoundthatthereisa logicalconnectionbetweenthegravitationalforceandtheelectromagneticforce.However,thosecalculationsweredoneonlyfor separationdistance equal to λc. Nowwewill generate somemore general equations forarbitraryseparationdistanceexpressedas , thenumberof reducedComptonwavelengths.Thefollowingequationscouldbemadeassumingtwodifferentmassparticles,butitiseasiertoreturntotheassumptionofbothparticleshavingthesamemassbecausethenwecandesignateasinglevalueof separatingtheparticles.SomeofthefollowingequationsassumePlanckcharge qpwith force designation FE rather than charge e designatedwith force Fe. TheconversionisFE Feα–1.WewillstartbyconvertingtheNewtongravitationalequationandtheCoulomblawequationsothattheyarebothexpressedinnaturalunits.Thismeansthatboththeforcesandtheparticle’senergywillbeinPlanckunits Fg Fg/Fp,FE FE/Fp,Ei Ei/Ep, .Alsoseparationdistancewillbeexpressedintheparticlesnaturalunitoflength,thenumber r/λcofreducedComptonwavelengths. Also,weassumetwoparticleseachhavethesamemass/energyandtheybothhavePlanckcharge.Convertbothequations:FE qp2/4πεor2andFg Gm2/r2intoequationsusingFg;FEand .

Substitutions:r ħc/Ei;m Ei/c2;Ei EiEp Ei ħ ⁄

FE ħ

ħ ħ Ei

2/ 2

Fg = ħ

ħ

Ei4/ 2


(Fg2) FE 2 2 Ei4

Theequation(Fg

2) FE 2 2clearlyshowsthatevenwitharbitraryseparationdistancethesquarerelationshipbetweenFgandFEstillexists.ItisinformativetostatethesesameequationsintermsofpowerbecauseafundamentalassumptionofthisbookisthatthereisonlyonetrulyfundamentalforceFr Pr/c.Ifthisiscorrect,thenwewouldexpectthattheforcerelationshipbetweenrotarswouldalsobeasimple functionof therotar’scirculatingpower:

ħ⁄ .ToconvertPctodimensionlessPlanckunits Pc Pc/PpwedividebyPlanckpowerPp c5/G.Notethesimplicityoftheresult.FE Pc/ 2Fg Pc

2/ 2So farwe have used dimensionless Planck units because they show the square relationshipbetween forcesmostclearly. However,wewillnowswitchanduseequationswithstandardunits.Thenextequationwillfirstbeexplainedwithanexample.Wewillassumeeithertwoelectronsor twoprotons bothchargee andwehold themapartatanarbitraryseparationdistancer.Asbefore,thisseparationdistancewillbedesignatedusingthenumber ofreducedComptonwavelengths,thereforer Nλc.Protonsarecompositeparticles,butwecanstillusetheminthisexampleifweusetheproton’stotalmasswhencalculating . Nowweimaginealogscaleofforce.AtoneendofthisforcescaleweplacethelargestpossibleforcewhichisPlanckforceFp c4/G. Attheotherendofthislogscaleofforceweplacethegravitational force Fg which is weakest possible force between the two particles either 2electronsor2protons .Nowforthemagicalpart!ExactlyhalfwaybetweenthesetwoextremesonthelogscaleofforceisthecompositeforceFe α–1.Inwords,thisistheelectrostaticforceFebetween the two particles times the number of reduced Comptonwavelengths times theinverseofthefinestructureconstant α‐1 137 .Particlephysicistsliketotalkaboutvarioussymmetries.Iamclaimingthatthereisaforcesymmetrybetweenthegravitationalforce,PlanckforceandthecompositeforceFe α–1.Theequationforthisis:

Itisinformativetogiveanumericalexamplewhichillustratesthisequation.Supposethattwoelectronsareseparatedby68nanometers thisdistancesimplifiesexplanations .Theelectronsexperience both a gravitational force Fg and an electrostatic force Fe. The electrons haveλc 3.86x10‐13m, therefore this separation is equivalent to 1.76x105 reducedComptonwavelengths. The gravitational force between the two electrons at this distance would beFg 1.2x10‐56NandtheelectrostaticforcewouldbeFe 5x10‐14N.Alsoα –1 137socombining


Fe, andα –1wehave:Fe α–1 1.2x10‐6N.AlsoPlanckforceisFp c4/G 1.2x1044N.Tosummarizeandseethesymmetrybetweentheseforces,wewillwritetheforcesasfollows:Fg 1.2 10 NFgfortwoelectronsat6.8x10‐8mis1050timessmallerthanFe /αFe α –1 1.2 10 NFe /αfortwoelectronsat6.8x10‐8mFp 1.2 10 NFp Planckforce is1050timeslargerthanpreviousFe /αAnotherwayofstatingthisrelationshipwouldsetPlanckforceequalto1.ThereforewhenFp 1thenFe α –1 10‐50andFg 10‐100.Thenumericalvaluesarenotimportantbecausedifferentmass particles or different separation distance could be used. The important point is thesymmetrybetweenFp,FeandFgwhenweinclude andα‐1inthecompositeforceFe α –1.Force Ratios Fg/FE and Fg/Fe α-1:Nextwewillshowhowthewavestructureofparticlesandforcesdirectlyleadstoequationswhichconnecttheelectrostaticforceandgravity.Previouslywestartedwiththewave‐amplitudeequation F = kA2ωc

2Zs /cwhichisapplicabletowavesinspacetime. In thisequationA isstrainamplitude,ωc isComptonangular frequency,Zs is theimpedanceofspacetimeZs c3/Gand isparticlearea.WehaveshownthattheinsertingthestrainamplitudetermA Aβ/ intoF = kA2ωc

2Zs /cgivestheelectrostaticforcebetweentwoPlanckchargesFE,.Wehavealsoshownthatgravityisanonlineareffectwhichscaleswithstrainamplitudesquared Aβ2 .InsertingA Aβ2/ intothisequationgivesthegravitationalforceFgbetweentwoequalmassparticles.SincewehaveequationswhichgenerateFEandFg,,weshouldbe able to generate new equations which give the ratio of forces Fg/FE. In the followingAβ Lp/λc Tpωc.Forgravity,A Ag Aβ2/ andforelectrostaticforceA AE Aβ/ .Fg = k(Aβ2/ )2ωc2Zs /c Fg = gravitationalforcebetweentwoofthesamemassparticles FE = k(Aβ/ )2ωc

2Zs /c FE theelectrostaticforcebetweentwoparticleswithPlanckcharge Setcommontermsequaltoeachother: (kωc

2Zs /c) = (kωc2Zs /c)

2

where: Aβ = Lp/λc = Tpωc = rotar strain amplitude

22 2

TheequationFg/FE Aβ2showsmostclearlythevalidityofthespacetimebasedmodeloftheuniverseproposedhere.Recallthatallfermionsandbosonsarequantizedwaveswhichproducethesamedisplacementofspacetime.ThespatialdisplacementisequaltoPlancklengthLpandthe temporal displacement is Planck time Tp. Even though all waves produce the samedisplacementofspacetime,differentparticleshavedifferentwavestrainamplitudesbecausethestrainamplitudeisthemaximumslope maximumstrain producedbythewave.Thereforearotar’sstrainamplitude isAβ Lp/λc Tpωc. Now we discover that the force produced by particles with strain amplitude Aβ reveal their connection to the underlying physics because Fg/FE = Aβ

2.


All theprevious equations relating Fg and FE either specified either a specific separationorspecifiedseparationdistanceusing .However,Fg/FE = Aβ2 doesnotspecifyseparation.Thisispossiblebecausetheratioofthegravitationalforcetotheelectrostaticforceisindependentofdistance.Forexample,theratioforanelectronisFg/Fe 2.4x10‐43.However,whenweadjustforthecouplingconstantassociatedwithchargee,theratiobecomes:Fg/Feα‐1 1.75x10‐45.TherotarstrainamplitudeforanelectronisAβ Lp/λc 4.185x10‐23.ThereforeAβ2 1.75x10‐45Clearly,thederivationofthisequationandthephysicsbehinditgivestrongproofofthewave‐based structure of particles and forces. Nextwewill extend the relationshipbetween theseforcesonemoresteptobringanewperspective.

ħ

ħ

ħ

or:

Theequation Fg/FE = Rs/λc isveryinterestingbecause Fg/Fe isaratioofforcesand Rs/λc isaratioofradii.Recallthat Rs≡ Gm/c2 istheSchwarzschildradiusofarotarbecauseitwouldformablackholerotatingatthespeedoflight.SuchablackholehashalftheSchwarzschildradiusofanon‐rotatingblackholetherefore Rs Gm/c2.Also,λc = ħ/mc istheradiusoftherotarmodelofafundamentalparticle.Forexample,foranelectronRs 1.24x10‐54mandanelectron’srotarradiusis:λc 3.85x10‐13m.Thesetwonumbersseemcompletelyunrelated,yettogethertheyequaltheelectron’sforceratioFg/Feα‐1 1.75x10‐45 Rs/λc.However,asthefollowingequationsshow,therearetwoamazingconnectionsbetweenRsandλc.First,Rsλc Lp2.ThesecondisRs λc‐1Inwords,thissaysthattherotar’sSchwarzschildradius Rs≡Gm/c2)istheinverseofthereducedComptonradiuswhenbothareexpressedinthenaturalunitsofspacetimewhicharedimensionlessPlanckunits Rsandλcunderlined .

Rsλc ħ

ħ

Rs λc‐1equivalentto: ⁄ ⁄ The Schwarzschild radius comes from general relativity and is considered to be completelyunconnected to quantummechanics. A particle’s reduced Comptonwavelength comes fromquantum mechanics and is considered to be completely unconnected to general relativity.However,whentheyareexpressedinnaturalunits dimensionlessPlanckunits thetworadiiarejusttheinverseofeachother Rs 1/λc.Also .TherelationshipsbetweenRsand


λc are compatible with the wave‐based rotar model of fundamental particles but they areincompatiblewiththemessengerparticlehypothesisofforcetransmission.TosummarizealltheequationsequaltoFg/FE,plusafewmorewehave:

⁄ ⁄ ⁄ Aβ2 Rs

2 λc–2 ωc2= Ei

2 Pc All the previous force equations also work with composite particles such as protons if theproton’stotalmassisusedtocalculatethevarioustermssuchasλc ħ/mc.Forexample,twoprotonshaveFg/Feα‐1 5.9x10‐39atanyseparationdistance.AlsoforprotonsRs/λc 5.9x10‐39and Lp/λc 2 5.9x10‐39. Iwanttopauseforamomentandreflectontheimplicationsofallthepreviousequationswhichshowtherelationshipbetweentheelectrostaticforceandthegravitationalforce.Tome,theyclearlyimplyseveralthings.Theseare:

1 Gravitycanbeexpressedasthesquareoftheelectrostaticforcewhenseparationdistanceisexpressedas multiplesofthereducedComptonwavelengthλc.

2 The equations relating the gravitational and electrostatic forces imply that they bothscaleasafundamentalfunctionofaparticle’squantummechanicalpropertiessuchasComptonwavelengthorComptonfrequency.

3 All the connections between the gravitational force and the electrostatic force areproposedtobetraceabletoarotargeneratingaComptonfrequencystandingwaveinthesurroundingvolume.Spacetime isanonlinearmedium,soasinglestandingwavehasbothafundamentalcomponent electrostatic andanonlinearcomponent gravity .

4 Theelectromagneticforceisuniversallyrecognizedasbeingarealforce.Theequationsshowthatgravityiscloselyrelated.Therefore,gravityisalsoarealforce.

5 These equations are incompatible with virtual photons transferring the electrostaticforceorgravitonstransferringthegravitationalforce.Theequationsalsoappeartobeincompatiblewithstringtheory.

6 There is a quantummechanical connection between a particle’s rotar radius and itsSchwarzschildradius.

DerivationoftheEquations:IwanttorelateabriefstoryaboutthefirsttimethatIprovedarelationshipbetweenthegravitationalandelectrostaticforces.Aspreviouslystated,theinitialideathatledtothisbookwasthatlightconfinedinahypothetical100%reflectingboxwouldexhibitthesameinertiaasaparticlewiththesameenergy.Theotherideasinchapter1followedquickly and I was struck with the idea that these connections between confined light andparticleswere probably not a coincidence. Iwill now skip ahead several yearswhen Iwasmethodically inventing a model of the universe using only the properties of 4 dimensionspacetime. I had the idea of dipole waves in spacetime and quantized angular momentumforming a “rotar” that was one Compton wavelength in circumference and rotating at the


particle’sComptonfrequency.Thisimpliedasphericalvolumewithradiusofλc 3.86x10‐13.IhadindependentlyderivedZs c3/Gwhichwaskeytoalltheotherequations.IhaddevelopedthedimensionlesswaveamplitudeforanelectronwhichwasAβ 4.185x10‐23. AnimportantsuccesswastosubstitutethesevaluesintoE A2ω2ZV/candIobtainedEi 8.19x10‐14Jforanelectron.Thenextlogicalstepwastofindtheforcethatwouldexistbetweentwoelectronsatadistanceequaltoλc.ThisdistancewasimpliedbecauseIwasusingtheonlyamplitudethatIknewwhichcorrespondedtoadistanceequaltoλc.WhenImadenumericalsubstitutionsforanelectronintotheequationF A2ω2Z /cIobtainedF 0.212N.Thiswasabout137timesgreaterthantheelectrostatic force between two electrons at the separation distance λc. I was quite happybecauseIrealizedthatthiswouldbethecorrectforceifthechargewasPlanckchargeqpratherthanelementarychargee. Thiswasactuallyapreferableresultbecauseitcorrespondedtoacouplingconstantof1,whichwasreasonableforthiscalculation.Thiswasunderstandableanditwasquiteexciting.NextIthoughtaboutgravity.Iknewthatgravitywasvastlyweakerthantheotherforcesandonlyhadasinglepolarity. Iwasremindedabout theopticalKerreffectdiscussed in the lastchapter.Thisnonlineareffectscaleswithamplitudesquared electricfieldsquared andalwaysproducesan increase in the indexof refractionof the transparentmaterial a singlepolarityeffect .Gravityhasasinglepolarityandisvastlyweakerthantheelectrostaticforce.Therefore,IdecidedtocalculatetheforceusingF A2ω2Z /c.andset:A Aβ2 1.75x10‐45,ωc 7.76x1020s‐1,Zs 4.04x1035kg/sand λc2 1.49x10‐25m2.Usingapocketcalculator,therewasaEurekamoment when I got the answer which exactly equaled the gravitational force between twoelectronsatthisseparation F 3.71x10‐46N .ThisstoryistoldbecauseIwanttosupporttheclaimthatthepredictioncamefirst.Itisoftensaidthattheproofoftheaccuracyofanewidealiesinwhetheritcanmakeapredictionrevealingsomepreviouslyunknownfact.Usuallytheproofrequiresanexperiment,butinthiscaseitwasasimplecalculation.Tomyknowledgethisisthefirsttimethatthegravitationalforcehasbeencalculatedfromfirstprincipleswithoutreferencingacceleration.Gravitational Rate of Time Gradient: Intheweakfieldlimit,itisquiteeasytoextrapolatefromthegravitationalmagnitudeβproducedbyasinglerotarataparticularpointinspacetothetotalgravitationalmagnitudeproducedbymanyrotars.Naturemerelysumsthemagnitudesof all the rotarsat apoint in spacewithout regard to thedirectionof individual rotars. Thegravitationalaccelerationgwaspreviouslydeterminedtobe:g c2dβ/dr –c2d dτ/dt /dr.


Agravitationalaccelerationof1m/s2requiresarateoftimegradientof1.11x10‐17secondspersecondpermeter.Theearth’sgravitationalaccelerationof9.8m/s2impliesaverticalrateoftimegradientof1.09x10‐16meter‐1. Thismeansthattwoclocks,withaverticalseparationofonemeterattheearth’ssurface,willdifferintimeby1.09x10‐16seconds/second.Similarly,thereis also a spatial gradient. A gravitational acceleration also implies that there is a differencebetweencircumferentialradiusRandradiallengthL.Intheearth’sgravitythisspatialdifferenceis1.09x10‐16meters/meter β 1–dR/dL .Outofcuriosity,wecancalculatehowmuchrelativevelocitywouldberequiredtoproduceatimedilationequivalenttoaonemeterelevationchangeintheearth’sgravity.Ifelevation2is1meterhigherthanelevation1,then:dt1/dt2 1–1.09x10‐16.Usingspecialrelativity:

v 1 setdt1/dt2 1–1.09x10‐16

v 4.4m/sThis4.4m/svelocityisexactlythesamevelocityasafallingobjectachievesafterfallingthroughadistanceof1meterintheearth’sgravity.Carryingthisonestepfurther,anobserveringravityperceivesthataclockinaspaceshipinzerogravityhasthesamerateoftimeasaclockingravity,ifthespaceshipismovingatarelativevelocityofve,thegravity’sescapevelocity.Forexample,anobserveronearthwouldperceive that a spaceship in zerogravitymoving tangentially atabout40,000km/hrhasthesamerateoftimeasaclockontheearth. Ontheotherhand,anobserverinthespaceshipperceivesthataclockontheearthisslowedtwiceasmuchasiftherewasonlygravityoronlyrelativemotion.Equivalence of Acceleration and Gravity Examined: AlbertEinsteinassumedthatgravitycouldbeconsideredequivalenttoacceleration.Thisassumptionobviouslyleadstothecorrectmathematicalequations.However,onaquantummechanicallevel,isthisassumptioncorrect?Todayit iscommonlybelievedthatanacceleratingframeofreferenceisthesameasgravity.This is associatedwith thegeometric interpretationof gravity. A corollary to this is thataninertialframeofreferenceeliminatesgravity.Theimplicationisthatgravityisnotasrealastheelectromagneticforcewhichcannotbemadetodisappearmerelybychoosingaparticularframeofreference.Itiscommonforexpertsingeneralrelativitytoconsidergravitytobeageometriceffectratherthanatrueforce.Theconceptspresentedinthisbookfundamentallydisagreewiththisphysicalinterpretationofgeneral relativity. There is no disagreement with the equations of general relativity. Theprevious pages have shown that it is possible to derive the gravitational force using waveproperties and the impedance of spacetime. This is completely different than acceleration.Numerous equations in this book have shown that there is a close connection between thegravitationalforceandtheelectrostaticforce.Inparticular,Iwillreferencethetwoequations


whichdealtwithtwodifferentmassparticles m1andm2 . Theconcludingstatement inthisanalysiswas:“Note that the only difference between the intermediate portions of these two equations is that the gravitational force Fg has the strain amplitude terms squared ( and the electrostatic force FE has the strain amplitude terms not squared ( . The tremendous difference between the gravitational force and the electrostatic force is due to a difference in exponents.”

Surely this qualifies as proof that gravity is closely related to the electrostatic force andthereforegravityisalsoarealforce.Theimplicationisthatwhenwegetdowntoanalyzingwavesinspacetimethereisadifferencebetweengravityandacceleration.Anargumentnotinvolvingdipolewavesinspacetimegoesasfollows:Aparticleinfreefallinagravitational field has not eliminated the effect of gravity. The particle is just experiencingoffsettingforces.Thegravitationalforceisstillpresentinfreefallbutitisbeingoffsetbytheinertialpseudo‐forcecausedbytheacceleratingframeofreference.Thesetwoopposing“forces”just offset each other and give the impression that there is no force. The acceleration alsoproducesanoffsettingrateoftimegradientandanoffsettingspatialeffect.Thepseudo‐forceofinertiahasnotbeeneliminatedandthegravitational forcehasnotbeeneliminated. Einsteinobtainedthecorrectequationsofgeneralrelativitybyassumingthatgravitywasthesameofacceleration.Sincetheyexactlyoffseteachotherinfreefall,thisassumptiongavethecorrectequationsbutthephysicalinterpretationiswrong.Onthequantummechanicalscaleinvolvingwaves,gravityisdifferentthanacceleration.Onewaytoprovethatgravityisatrueforceistoshowthatagravitationalfieldpossessenergydensity.Thiswillbediscussedinchapter10. “Grav” Field in the Rotar volume:Theabovediscussionofgravitationalaccelerationfromarateof timegradientpreparesus to return to the subjectof the “grav field” inside the rotarvolumeofarotar.Recallthattherotatingdipolethatformstherotarvolumeofanisolatedrotarwasshowninfigure5‐1. Thisrotatingdipolewavehastwolobesthathavedifferentratesoftimeanddifferenteffectsonpropervolume.Thedifferenceintherateoftimebetweenthetwolobesproducesarotatingrateoftimegradientthatwasdepictedinfigure5‐2.Arotarisverysensitivetoarateoftimegradient.Arateoftimegradientof1.11x10‐17seconds/second/metercauses a rotar to accelerate at 1m/s2 and the acceleration scales linearlywith rate of timegradient. Therefore, the rotating rateof timegradient in the centerof a rotarmodel canbeconsideredtobearotatingaccelerationfieldthathassimilaritiestoarotatinggravitationalfield.Wenormallyencountertherateoftimegradientinagravitationalfield.Thisistheresultofanonlinearity that produces a static stress in spacetime. A static rate of time gradient hasfrequencyofω 0andnoenergydensity.However,inanactualgravitationalfieldthereisalsotheoscillatingcomponentofagravitationalfieldandthisdoeshaveenergydensitythatwillbediscussedlater.Ifarotatinggravitationalfieldissomehowgenerated,thensucharotatingfield


wouldalsohaveenergydensity.Therotatingrateoftimegradient rotatinggravfield thatispresentnearthecenterofarotardoeshaveenergydensitythatwillbecalculatednext.In a time period of 1/ωc, the fast time lobe of the dipole gains Planck time displacementamplitudeTp andtheslowtimelobelosesPlancktimeTp.Theselobesareseparatedby2 .Therefore,inatimeof1/ωcthereisatotaltimedifferenceof2Tpacrossadistanceof2 .Therateoftimegradientpermeteris:

–

Theaccelerationproducedbythisrateoftimegradient gravacceleration g istherateoftimegradienttimesc2.Thefollowingareseveralequalitiesforgravacceleration g:

g –

c2 Lpωc2 Aβ2 pħ

g gravaccelerationand p c/tp /ħ PlanckaccelerationComparison of Grav Acceleration and Gravitational Acceleration: Howdoestherotatinggravaccelerationat thecenterofa rotar’s rotarvolume g comparewith thenon‐rotating,gravitationalacceleration gq attheedgeofthesamerotar’srotarvolume?.gq Aβ2ωccrotar’snon‐rotatinggravitationalaccelerationatdistance fromthecenterg Aβωccrotar’srotating ωc gravaccelerationatthecenterofarotar

Aβratioofgq staticgravitationalaccelerationat torotatinggravacceleration g

ForanelectronAβ 4.18x10‐23,sotherotatinggravfieldatthecenteroftheelectronisabout2 1022timesstrongerthanthenon‐rotatinggravitationalfieldatdistance .Thisresultsinan electron having a rotating grav acceleration of: g 9.73 x 106m/s2. The gravitationalacceleration notrotating ofanelectronatdistance is:gq 4.07x10‐16m/s2.Thereforethegravacceleration intherotarvolumeofanelectron isaboutamilliontimesgreaterthanthegravitationalaccelerationatthesurfaceoftheearth.Theearth’sgravityisnotrotatingandisanonlineareffect.Theelectron’sgravfieldisrotatingandisafirstordereffectresultingfromtherateoftimegradientestablishedintheelectron’srotatingdipolewave.Recall the incredibly small difference in the rate of time that exists between the lobes of anelectron. Itwould take50,000 times the ageof the universe for thehypothetical lobe clockrunningattherateoftimeinsidetheslowlobetoloseonesecondcomparedtothecoordinate


clock.Thedifferencebetweentherateoftimeontheslowlobeclockandthecoordinateclockiscomparabletothedifferenceintherateoftimeexhibitedbyanelevationchangeofabout4x10‐7m in the earth’s gravity. The reason that the rotating grav field has amillion times largeracceleration than the earth is because this difference in the rate of time occurs overapproximatelyamilliontimesshorterdistance ~4x10‐13m .Therotatingrateoftimegradientinside a rotar is a first order effect related to Aβ while the non‐rotating gravitational fieldproducedbytherotarisasecondordereffectrelatedtoAβ2.Conservation of Momentum in the Grav Field:Itwouldappearthattheconceptofagravfieldmustviolatetheconservationofmomentum.Anexamplewillillustratethispoint.Supposethatasmallneutralparticle suchasaneutralmeson wandersintothecenterofanelectron’srotarvolume.Evenifthemassofthemesonis1000timeslargerthantheelectron,therotatinggravfieldoftheelectronshouldproducethesameaccelerationoftheneutralparticle.ThiswouldbeaviolationoftheconservationofmomentumunlessthedisplacementproducedbytherotatinggravfieldisequaltoorlessthanPlancklength theuncertaintyprincipledetectablelimit .Wewillcalculatethemaximumdisplacement thattakesplaceinatimeperiodof:t 1/ωc.Wechoosethistimeperiodbecausetherotatingvectorofthegravfieldischangingbyoneradianinatimeperiodof1/ωc.Hypotheticallytheneutralparticlewouldnutateinacirclewitharadiusrelatedto ignoringdimensionlessconstants .

½at2

Lp

Lp maximumradialdisplacementproducedbyarotar’srotatinggravfieldThereforeanymass/energyrotaralwaysproducesthesamedisplacementequaltoPlancklengthignoringdimensionlessconstants inthetimerequiredforthegravfieldtorotateoneradian.Thisdisplacementispermittedbyquantummechanicsandisnotaviolationoftheconservationofmomentum.Thisisanothersuccessfulplausibilitytest.Energy Density in the Rotating Grav Field: Anacceleratingfieldthat isrotatingpossessesenergydensity.Itwouldhypotheticallybepossibletoextractenergyfromsuchafieldifthefieldproducedanutationthatwaslargerthanthequantummechanical limitofPlancklength. Noenergycanbeextractedfromarotar’srotatinggravfieldbecausethenutationisatthequantummechanicallimitofdetection.However,thisfieldstillpossessesenergydensity.PreviouslywedesignatedthestrainamplitudeofarotarasAβ Lp/ Tpωc ωc/ωp.Thesewereoriginallydefinedintermsofthestrainamplitudeofadipolewavethatisonewavelengthin circumference. This definition tended to imply that the energy density of a rotar wasdistributedaroundthecircumference.However,itisproposedthattherotatinggradientthatispresentat thecenterof therotarmodelcanalsobecharacterizedashavingadimensionless


amplitudeofAβ Lp/ Tpωc.ThisamplitudeAβisjustintheformofarotatingrateoftimegradientandarotatingspatialgradient.ThespatialgradientfromthelobetothecenterisstillLp/ .TherateoftimegradientisstillrelatedtoTpωc,althoughthisishardertosee.PreviouslywesubstitutedAβ,ωcandZsintoU A2ω2Z/candobtainedarotar’senergydensityintherotarvolumeUq kmc2/. Ifwe ignorethedimensionlessconstantk, this is therotar’s internalenergyinthevolumeofacubethatis onaside.HerearesomeotherequalitiesforUq.

Uq ħ

Aβ4Upsetħ

g2

Uq

Thereforetherotating“gravfield”hasthesameenergydensityastheenergydensityoftheentirerotar Ei/λc3 . Ifwe broaden the definition of Aβ so that it also defines rotating rate of timegradientsandrotatingspatialgradients, then theenergydensityof therotarmodelbecomeshomogeneous.Theenergydensitynearthecenterofarotaristhesameastheenergydensityneartheedge.Thisenergydensityisjustintwodifferentforms.Inchapter6weattemptedtocalculatetheangularmomentumofarotar.Ifweassumedthatalltheenergywasconcentratedneartheedgeofahoopwithradius ,thenweobtainedananswerofangularmomentumofħ.However,ifweassumedthattheenergywasdistributedmoreuniformly likeadisk andalsorotationinasingleplane,thentherotarmodelwouldhaveangularmomentumof½ħ.Thefactthat energy is contained in the grav field does smooth out the energy distribution, therebytendingtowardstheanswerof½ħ.However,aspreviouslyexplained,thereisalsoachaoticrotationwherethereisanexpectationrotationalaxisbutotherrotationaldirectionsareallowedwithlessprobability.Also.theenergydensitydistributionwithinarotardoesnotendabruptlyataradialdistanceofλc.Thedetailsthatresultinnetangularmomentumof½ħwillhavetobeworkedoutbyothers.If a rotating grav field has energy density, does a static gravitational field also have energydensity? This question will be examined in chapter 10 after some additional concepts areintroducedabouttheoscillatingcomponentofgravityEnergy Density in Dipole Waves:TheaboveinsightsintothegravfieldalsohaveimplicationsforanyPlanckamplitudedipolewaveinspacetime,notjusttherotatingdipolesthatformrotars.Iamgoingtotalkaboutdipolewavesinspacetimebutstartoffbymakingananalogytosoundwaves in a gas. Sound waves can be depicted with a sinusoidal graph of pressure. Thecompression regions have pressure above the local norm and the rarefaction regions havepressure below the local norm. These can be represented as a sine wave maximum andminimum.However,ifagraphwastobedrawnshowingthekineticenergyofthemoleculesinthegas,themaximumkineticenergyoccursinthenoderegionsbetweenthepressuremaximumandminimum.Akineticenergygraphdepictingmotion velocity leftandrightwouldhavea90°phaseshifttothepressuregraph.Theenergyinthesoundwaveisbeingconvertedfrom


kineticenergy particlemotion toenergyintheformofhighorlowpressuregas.Whenthesetwoformsofenergyareaddedtogether,thenasoundwavewithaplanewavefronthasauniformtotalenergydensity sin2θ cos2θ 1 .Theenergyisjustbeingexchangedbetweentwoforms.This concept of energy being exchanged between two different forms also applies to dipolewaves in spacetime. In one form, energy exists because the vacuumenergy of spacetime isdistortedsothatthereareregionswheretherateoftimeisfasterorslowerthanthelocalnorm.Perhapsthisisanalogoustothecompressionandrarefactionrepresentationofasoundwave.Theregionsbetweenthemaximumandminimumratesoftimehavethegreatestgradientintherateoftime. Thesearethegravfieldregionsandtheyareanalogoustoregionsinthesoundwavewherethegasmoleculeshavethegreatestkineticenergy.Addingtogetherthetwoformsofenergydensitypresentineithersoundwavesordipolewavesinspacetimeproducesatotalenergydensitywithoutthecharacteristicwaveundulations.The waves in spacetime have sometimes been discussed emphasizing either the temporalcharacteristics rate of time gradients, etc. or emphasizing the spatial characteristics forexampleLp/r .Actuallybothcharacteristicsarealwayspresent;itissometimeseasiertoexplainusingjustonecharacteristic.Therefore,thegravfieldcouldhavebeenexplainedemphasizingthepropervolumegradientratherthantherateoftimegradient.Gravitational Potential Energy Storage:Whenwelookatthegravitationaleffectthatarotarhasonspacetime,weconcludethattheslowingoftherateoftimealsoproducesaslowingofthenormalizedspeedoflight Co ГCgfromchapter3 .Toreachthisconclusionwemustassumethatproperlengthisconstant,evenwhenthereisachangeinГ.Thisisanunspokenassumptionforphysicsthatdoesnotinvolvegeneralrelativity.Theeffectontherateof timeandonthenormalizedspeedof lightultimatelyeffectsenergy,force,mass,etc.aspreviouslydiscussed.ThereasonforbringingthisupnowisthatIwanttoaddressgravitationalpotentialenergy.Gravitationalpotentialenergyisconsideredanegativeenergythathasitsmaximumvalueinzerogravityanddecreaseswhenamassisloweredintogravity.Whatphysicallychangeswhenarotariselevatedorloweredingravity?Inchapter3itwasfoundthatsubstitutingthenormalizedspeedoflightCgandthenormalizedmassMgintotheequationE mc2givesenergythatscalesinverselywithgravitationalgammaΓ rest frame of reference . We illustrated this concept by calculating the difference in theinternalenergyofa1kgmassforanelevationofsealevelandonemeterabovesealevel.Thecalculateddifferenceinthenormalizedinternalenergywas9.8Jouleswhichisexactlythesameasthegravitationalpotentialenergy.ThischangeinenergyisduetothechangeinthenormalizedspeedoflightaffectingtheComptonfrequencyoftherotarasseenfromzerogravity.Forexample,afreeelectroninzerogravityhas


aComptonangularfrequencyof7.76x1020s‐1.Earth’sgravityhasβ 7x10‐10.Afreeelectroninearth’sgravityhasanormalizedComptonangularfrequencythatisslowerthanazerogravityelectronbyabout5.4x1011radianspernormalizedsecond 7x10‐10x7.76x1020s‐1 . Thislower Compton frequency decreases the normalized internal energy of an electron anddecreases the gravity non‐oscillating strain generated by an electron. The non‐oscillatingstrainisresponsiblefortherotar’sgravity,soarotaratrestingravitycontributeslessgravitytothetotalgravitythanthesamerotaratrest sametemperature inzerogravity.Foranotherexample,wewillcalculatethechangeintheinternalenergyofanelectronwhenitiselevated1meterintheearth’sgravitationalfield.Inchapter3thereisasectiontitled “Energy Transformation and Calculation”wherethedifferenceinthegravitationalgammawascalculatedfor a 1 meter elevation change near the earth’s surface. This was expressed as Γ2 – Γ1 1.091x10‐16 dt2/dt1.SincethereducedComptonfrequencyofanelectronisabout7.76x1020s‐1,thismeansa1meterelevationchangewillproduceafrequencychange:Δωc 7.7634x1020s‐1x1.0915x10‐16 84,737s‐1ΔE Δωcħ 84,690s‐1xħJs 8.936x10‐30JWewillcomparethistothegravitationalpotentialenergystoredwhenanelectroniselevated1meterintheearth’sgravitationalfieldwithaccelerationofg 9.81m/s2.ΔE meΔhg 9.109x10‐31kgx1mx9.81m/s2 8.936 10‐30JTheseconceptsalsoleadtoaphysicalexplanationforpotentialenergy.Inchapter3theconceptof potential energywas related to a reduction in thenormalized speedof light reducing theE mc2internalenergy.Nowwegoonestepfurtherandtracethegravitationalpotentialenergytoachangeintherotationalfrequencyofarotarinagravitationalfield.Thisnotonlyaffectstheinternalenergyoftherotar,butitalsoaffectstheamountofgravitygeneratedbytherotar.Whenwehavelookedatmass,energy,inertiaandgravityfromthepointofviewofazerogravityobserver,wehaveseenadifferencenotobviouslocally. SinceachangeinГaffectsmassandenergydifferently zerogravityobserverperspective andsincegravityscaleswithenergy notrestmass ,tobetechnicallycorrectthegravitationalequationsshouldbewrittenintermsofenergy,notmass.Thetransformationsandinsightsprovidedherehaveforcedustorecognizethattheterm“mass”isaquantificationofinertia.Massisnotsynonymouswithmatterandmassscalesdifferentlythanenergywhenviewedbyanobserverusingthezerogravitycoordinaterateoftime.


Personal Note: Iwanttotelltwoshortpersonalstoriesthatrelatetothischapter.Thefirststoryhastodowiththegravitationaleffectonrotarfrequency.Inormallyrunalmosteveryday.Someofthebestideasinthisbookcametomeduringthesedailyruns.ManyyearsagoIusedtorunonflatgroundbuttopreservemykneesInowrunupasteepsectionofahillandwalkdowntothestartingpoint.Irepeatthisfor½hourwhichtypicallyisabout15roundtrips.AsIrunupthehillIoftenamawarethattheworkthatIamdoingisultimatelyresultinginanincreaseintheComptonfrequencyofalltheelectronsandquarksinmybody.Locallythereisnomeasurablechangeinthe Compton frequencies of these rotars, but using the absolute time scale of a zero gravityobserver, I am increasing the frequency of these particles. It somehow is comforting tounderstandthephysicsofrunningupahill.Theconceptof“gravitationalpotentialenergy”hasbeendemystified.Inowunderstandwhyitisdifficulttorunupahill.The second story is about the experienceof resolvingamystery about gravity. When Iwasinitiallywritingthisbook,IthoughtthatIhadunlockedthekeytounderstandinggravitywhenIhaddevelopedtheconceptspresentedinchapter6 theconceptsthatarenowregardedasbeingoversimplified .ThemagnitudeofthegravitationalforcewascorrectandIthoughtIcouldeasilyextrapolate to largerdistancesand largermass. Then it occurred tome that thevectorwaswronganditwasobviousthatIwasmissingothermajorconcepts.Iwasfarfromfinishingmyquesttoexplainimportantaspectsofgravity.My initial reaction was to try to rationalize changes that wouldmake the simplifiedmodelexplain the correct vector attraction rather than repulsion . This thought process wassomethingliketryingtoreverseengineergravity.Iwasattemptingtoworkbackwardsfromthedesiredresult attraction tofindthechangestothemodelthatwouldgivethedesiredresult.Ispent a long time working backwards from result to cause, but it was getting nowhere.Therefore,infrustrationIreturnedtotheapproachthatIhadpreviouslyusedtodevelopthemodeltothatpoint.Thatapproachmerelymovedforwardfromthestartingassumption theuniverseisonlyspacetime andacceptedthelogicalextensionsofthisassumption.OnceIgotbackonthistrack,Irealizedthattheenergydensityoftherotarmodelimpliedpressure.WhenItookthelogicalstepstocontainthispressure,Ieventuallyobtainednotonlythegravitationalforcewith the correct vector but also obtained improved insights into the strong force, theelectromagneticforceandthestabilityoffundamentalparticles.While otherpeople are attempting to adjustmodels to explain specific physical effects, I amfinding that logically extending the starting assumption gives unexpected explanations. Theexpandedmodelexplainsdiverseeffectsnotinitiallyunderconsideration. Theseexperienceshavegivenmeagreatdealofconfidenceinthismodelandapproach.


Chapter9

ElectromagneticFields&SpacetimeUnits

Introduction: In the last chapterwe analyzed gravitational attraction and established thenecessityofvacuumenergy/pressureingeneratingthegravitationalforce.Thischapterisanintroductiontotheelectromagnetic EM radiationandelectricalcharge.Wewillstartwithapreliminaryexaminationofspacetimecharacteristicsresponsiblefortheelectricfieldofchargedparticles. After some insights aredevelopedwith chargedparticleswewill then switchandexaminethepropertiesofspacetimethatareaffectedwhenelectromagneticradiationispresent.Anexperimentwillbeproposed.Wewillthenswitchbackandusesomeoftheinsightsgainedfromelectromagneticradiationtodevelopfurtherthemodelofachargedparticle’selectricfield.Finally,wewilldevelopasystemoffundamentalunitsbasedonlyonthepropertiesofspacetime.Theseunitsdesignatechargeutilizingonlythepropertiesofspacetime.WetendtothinkoftheelectricandmagneticfieldsassociatedwithEMradiationasbeingverysimilartotheelectricandmagneticfieldsassociatedwithchargedparticles.However,thereareobviousdifferences. The fieldsassociatedwithEMradiationarealwaysoscillatingand theypropagateatthespeedoflight.Thefieldsassociatedwithchargedparticlesarenotoscillatingandtheyarenot freelypropagating. Withthesedifferences, itshouldbeexpectedthat thereshouldalsobeconsiderabledifferencesintheexplanationsoftheelectricfieldassociatedwithphotonscomparedtotheelectricfieldassociatedwithelectrons.Developingamodelofelectricandmagnetic fields has been themost difficult task of this entire book. While considerableprogresshasbeenmade,themodelisnotcomplete.Furthermore,themodeloftheelectricfieldassociatedwithchargedparticlesislesscompletethanthemodelofEMradiation.Thischapterbeginsby examining themagnitude and typeof distortionof spacetime required toproduceelementarychargee. Spacetime Interpretation of Charge: Iftheuniverseisonlyspacetime,thereshouldbeaninterpretationofelectricalcharge,permeability,electricfield,etc.thatconvertstheseelectricalcharacteristics into properties of spacetime. Like gravity, we might be looking for both anoscillatingcomponentandanon‐oscillatingstraininthespacetimefieldcausedbytheoscillatingcomponent wavesinspacetime .Itwillberecognizedthattherearetwotypesofelectricfield:1 Theelectricfieldassociatedwithachargedparticlewhichappearstobeonlystaticbutmustalso have an oscillating component and 2 the oscillating electric field of electromagneticradiation.Thisdistinctionismadebecauseitwillbeshownthattheelectricfieldassociatedwithachargedparticleismorecomplex.


To obtain an insight into the electrical properties of nature,wewill start by expressing theelectricalpotential thevoltagerelativetoneutrality forasingleparticlewhichhasPlanckcharge 4p oq c atdistanceris:

4

4 4p o

Eo o

q c

r r

.

The symbol E willbeused tosignify thatweareassumingPlanckchargeqp.NextwewillexpresselectricalpotentialofPlanckchargeindimensionlessPlanckunits.Thisisdonebecausedimensionless Planck units are fundamentally basedon the properties of spacetime.We areattemptingtogainaninsightintothedistortionofspacetimecausedbyelectriccharge.IfweexpressthiselectricalpotentialindimensionlessPlanckunits E noteunderlined ,thenweareconvertingtodimensionlessPlanckunitswhichinthiscasemakesaratiorelativetothelargest

possibleelectricalpotential,Planckvoltage 1/24 274 1.043 10p oc G Volts

E E p 4

4 4

4o o

o

c G

r c

3

1 pLG

r rc

Before commenting,wewill next calculate the electric field indimensionlessPlanckunits atdistancerfromPlanckchargeqp.Thesymbolusedwillbe: E E p where 24E p oq r and

Planckelectricfieldis 7 24p oc G

22

2 7 3 2 2

4 4

4po o

E E po

Lc G G

r c c r r

Whatisthephysicalinterpretationof E pL r and 2 2

E pL r ?Expressingelectricalpotential

and electric field in dimensionless Planck units is expressing these in the natural units ofspacetime.Therefore,wewillminethistoextracthintsabouttheeffectthatPlanckchargehasonspacetime.

1 Sincethereisnotimetermineitherequation,theimplicationisthatanelectricalchargeonlyaffectsthespatialpropertiesofspacetime.Thisisalsoreasonablesinceagradientintherateoftimealwaysimpliesgravitationalacceleration.Ifelectricalchargeproducedarateoftimegradient,thenneutralobjectssuchasaneutronwouldbeacceleratedbyanelectric field. Therefore, it is reasonable that an electric field affects only the spatialdimension theLpterm

2 ThepresenceofPlancklengthterm Lp inthevoltageandelectricfieldequationsimpliesthat Planck length is somehow associatedwith the electric field produced by Planckcharge.

3 Thedimensionless ratio Lp/r implies a slope. Thiswill have to be a spatial strain ofspacetimethatcanbeexpressedasaslope.

4 Onlytheradialspatialdimension r seemstobeaffected.


Charge Conversion Constant η:Toquantifythemagnitudeoftheeffectonspacetimeproducedbyacharge,wewillattempttofindanewconstantofnaturewhichconvertsunitsofelectricalcharge Coulomb intoapropertyofspacetime.Consideringtheprevious4points,wewouldexpectthatchargemightproduceapolarizedstrainofspace radiallengthdimension withoutaffecting the rate of time. The validity of this approachwill be established only if it passesnumeroustests.There aremanyways to derive this charge conversion term.Wewill use one of the aboveequations:VE/Vp Lp/rwhereVEistheelectricalpotentialgeneratedbyaPlanckchargeqpatdistancer.IfwesolveforPlanckchargeqp,whileretainingPlancklengthLp,wewillbeabletodeduce a charge conversion constant that converts charge into a spatial distortion of thespacetimefield.

4

p p pE

o

q L

r r

The proposed conversion between charge and a spatial distortion of spacetime will bedesignatedasthe“chargeconversionconstant”anddesignatedaseta .

≡ 8.61 10 m/Cchargeconversionconstant

Eta hasunitsofmetersperCoulomb.Thisunitisreasonablebecauseweareexpectingtoconvertchargeintoapropertyofspacetimeandthetimedimensionhasbeeneliminated.The“meters”referencedinmeters/coulombaretheradialdirectionifwearereferringtotheeffectachargedparticlehasonthesurroundingspacetimeorifwearereferringtoanelectricfield,thenthe“meters”are inthedirectionof theelectric field.Weknowthatthisconstant iscompatiblewiththedimensionlessvoltageequationforPlanckcharge E Lp/r sincethatequationwasusedtodefinetheconstant.However,totestthisfurther,wewilluseeta toeliminateCoulombfromotherconstantsandequations.Wewillstartwithtwoconstants:1 theCoulombforceconstant1 4 o withunitsofm3kg/s2C2and2 themagneticpermeabilityconstantµo/4πwithunits of kgm/C2. Bothof thesehave1/C2. To eliminate1/C2 requiresmultiplyingbothoftheseby1/η2.

Fp 1.21 1044N

Coulombforceconstant convertstoPlanckforce withunitsofNewton.

1.35 1027kg/m


Therefore,vacuumpermeabilityμ4 convertsto withunitsofkg/m.

ItisquitereasonablethattheCoulombforceconstant1 4 o getsconvertedtoFpPlanckforce.Ifelectricchargeandanelectricfieldaredistortionsofthespacetimefield,thenitisreasonablethattheCoulombforceconstantconvertstothelargestforcethatthespacetimefieldcanexertwhichisPlanckforceFp c4/G.Infact,thisconversionstronglysupportsthevalidityofeta . If the universe is only spacetime, then it is reasonable that all forces, including theelectromagneticforce,shouldreferencePlanckforce.Thevacuumpermeabilityalsoconvertstoanimportantpropertyofspacetimewhichisc2/Gwithunitsofkg/m.Thisistheconstantthatconvertsmassintolength theradiusofablackhole .Thenexttestistoseeifc2 1/εoµoisstillcorrectwhenwesubstituteεo G/4πc4andµo 4πc2/G.Weobtain: 4 ⁄ 4⁄ .NextwewillalsoconvertelementarychargeeandPlanckchargeqptoadistortionofspacetimebymultiplyingbyη.

eη √ Lpconversionofelementarychargeewhere:ħ

qpη LpconversionofPlanckchargeqpWecannowdoamorerevealingtest.Wewillcalculatetheforcebetweentwoelectrons chargee twodifferentways.FirstwewillusetheconventionalCoulomblawequationtocalculatethisforce,thenwewillcalculatetheforceusingtheaboveconversions.

ħCoulomblawforcebetweentwoelectrons:chargee 1.6 10‐19C

ħ ħ

ForceequationwhichconvertseandεotospacetimestrainThisisasuccessfultest.Itisinterestingtoseehow1/4πεoconvertstoPlanckforceandstillgivesthe same answer as the Coulomb law equation which utilizes electrical charge and thepermittivityoffreespace.Itisinterestingtomakeothertestsofη.Italwaysworkscorrectly.

ImpedanceCalculationBeforeproceedingwith the following test calculation, Iwant to tell a story. There are twocalculationsinthisbookthatgavemethebiggestthrill. OneofthemwaswhenIwasabletoderive Newton’s gravitational equation from my starting assumptions. The second is thefollowingcalculationthatconvertstheimpedanceoffreespaceZointoadistortionofspacetime.Thisdoesnotseemlikeaparticularlyimportantrelationship,whichisperhapsthereasonthatitwassosurprising.


Impedance of Free Space and Planck Impedance:TheimpedanceoffreespaceZowithunitsofohmsisaphysicalconstantthatrelatesthemagnitudesoftheelectricfield andthemagneticfieldstrength ofelectromagneticradiationpropagatinginavacuum.

Zo≡ / μoc 376.7Ωimpedanceoffreespace

PlanckimpedanceZpis:

Zp 4 29.98ΩPlanckimpedance

The units of the impedance of free space Zo and Planck impedance Zp are both L2M/Q2T.Thereforetoeliminate1/Q2wemustmultiplyZpandZsby1/η2.

Zp Zs

Zo 4π 4πZs

Zp ZsandZo 4πZs

PlanckimpedanceZpcorrespondstotheimpedanceofspacetimeZsandtheimpedanceoffreespaceZocorrespondsto4πZs–Fantastic!Impedance of free Space Converts to the Impedance of Spacetime: Whenweconverttheimpedance of free space Zo ≡ / using the charge conversion constant, Zo becomes theimpedanceof spacetimeZs timesa constant 4π . We can ignore the4πand in factPlanckimpedancedoeseliminatethisnumericalfactor.Thisisafantasticoutcomebecauseitimpliesthatelectromagneticradiationissomeformofwaveinthespacetimefield.Hereisthereasoning.From gravitational wave equations we know that waves propagating in the medium of thespacetimefieldexperienceimpedanceofZs c3/G 4 1035kg/s.Nowwediscoverthatnotonlydoeselectromagneticradiationpropagateatthesamespeedasgravitationalwaves,andhastransversewaveslikegravitationalwaves,butelectromagneticradiationalsoexperiencesthesameimpedanceasgravitationalwaves theimpedanceofspacetime .Theconclusionis:Electromagnetic radiation must be a wave propagating in the medium of the spacetime field.TheequationZs c3/Gisonlyapplicablewhenwavesusethespacetimefieldasthepropagationmedium.Thisisunderstandableandfullyexpectedforgravitationalwaves,butnowwefindthatelectromagnetic radiationmust also use the spacetime field as a propagationmedium. The


impedanceoffreespaceZo fundamentaltoeverythingelectromagnetic is:Zo/η2 4πZswhenexpressed using a conversion constant η that converts charge to a strain of spacetimewithdimensionsoflength ignore4π .ThissaysthatphotonsarenotpacketsofenergythattravelTHROUGH theemptyvoidofspacetime.PhotonsarewavesIN themediumofthespacetimefield.Theyappeartoalsohaveparticlepropertiesbecausephotonspossessquantizedangularmomentum. The superfluid spacetime field quarantines angularmomentum into quantizedunitsofħforbosonsor½ħforfermions.Itisnotpossibletobreakapartaunitofquantizedangularmomentum.Thetransferofquantizedangularmomentumisallornothing 100%or0% . The photon’s energy also becomes quantized because the energy of a photon isfundamentally associatedwith the quantized angularmomentum. The proposed property ofunity makes the energy in the distributed waves collapse transfer their quantized angularmomentum at faster than the speed of light. This apparently localized interaction givesparticle‐likepropertiestophotons.TheequationZo/η2 4πZs alsoimpliesthatphotonsarefirstcousinstogravitationalwaves.Photonsandgravitationalwavesdisturbthespacetimefield’sseaofsuperfluiddipolewavesindifferentways, but they both are transverse disturbances in the spacetime field that do notmodulatetherateoftimeandpropagateatthespeedof light. Recallfromchapter4thatthespacetimefieldisanelasticmediumwithimpedanceandtheabilitytostoreenergyandreturnenergytoawavepropagatinginthismedium.Theimplicationisthatgravitationalwavesarealsoquantizedandcarryquantizedangularmomentum.Lightwavesarenotdipolewavesinspacetimesincewecandetectlightwavesasdiscretewaves.Recall that it is impossible to detect dipole waves in spacetime. Dipole waves in spacetimemodulate the rate of time and proper volume. Their amplitude must be limited to Planckamplitude Lp and Tp . A larger amplitude would produce effects that violate theconservationofmomentumandenergy.Also,alightwavecannotcauseanoscillationintherateoftimebecauseagradientintherateoftimeproducesgravitation‐likeeffects.Forexample,astrong light wave would cause a neutron to undergo a detectable transverse oscillatorydisplacement similar to the effect on an electron. Therefore, lightwaves are a spacialwavedistortionofthehomogeneousspacetimefield.Eachindividualphotonhasquantizedangularmomentumandotherspecialpropertieswhichwillbediscussedlater.ConstantSpeedofLight: Is itreasonablethatlightwavesarepropagatinginthemediumofspacetime?Forexample,ifthepropagationmediumisthespacetimefield,wouldweexpectthatitshouldbeimpossibletodetectmotionrelativetothismedium?First,weknowthatεoandµoarepropertiesofthespacetimefield.Wealsoknowthatεoandµoremainconstantinallframes

ofreference.Sincec 1 μ⁄ itfollowsthatawavepropagationspeedthatscaleswithεoandµoshouldalsobeindependentoftheframeofreference.


Second,gravitationalwavesaredefinitelypropagatinginthemediumofthespacetimefieldandtheyalwayspropagateatthespeedoflightinallframesofreference.IfitwaspossibletodoaMichelson‐Morleyexperimentusinggravitationalwaves,thisexperimentwouldobtainthesameresultastheexperimentusinglight.Inbothcasestherewouldbeanullresult–nomotionwouldbedetectedrelativetothepropagationmedium.Thespacetimefield isaseaofdipolewavespropagatingatthespeedoflightbutalsostronglyinteracting.Nomotionisabletobedetectedrelativetothismedium.Alsoallobservableobjects allparticles,fieldsandforces areobtainedfromthesinglebuildingblockof4dimensional spacetime.Therefore,allparticles, fieldsandforcescompensatewithaLorentztransformationwhichmakesthelocallymeasuredspeedoflight constant. Therefore, thespacetime fieldpossesses theproperties required tomake thespeedoflightconstantinallaccessibleframesofreference. Attheendofchapter14wewillexplorethelimitsofextremeframesofreferencewherethisbreaksdown. CosmicSpeedLimit:Thefollowingquestionhaspuzzledphysicists:Whyisthereacosmicspeedlimit? Now we can answer this question. A photon is not a quantized packet of energypropagatingthroughtheemptyvoidofthevacuum.Atrulyemptyvacuumwouldnotdictateanyspeedlimittoa“packetofenergy”thatismerelypropagatingthroughit.However,ifaphotonisawave propagating in the spacetime fieldmedium, then the impedance and energy densitycharacteristicsofthismediumdictateaspecificspeedofwavepropagation.Themediumnamedthe“spacetimefield”consistsofstronglyinteractingdipolewavesinspacetimepropagatingatthespeedof lightatall frequenciesup toPlanckangular frequency ωp 1043s‐1 . Photonsdisturbthehomogeneityofthismediumandthisdisturbancealsopropagatesatthespeedoflight.Photonsappeartobeparticlesbecausephotonspossessangularmomentumandangularmomentumisquarantinedintoħor½ħunitsbythesuperfluidspacetimefield.Sinceangularmomentumisonlytransferredinquantizedunits,thisgivesparticle‐likepropertiestoquantizedwaves. Thephrase“wave–particleduality”cannotbeconceptuallyunderstoodsince“wave”impliesaperiodicvariationoveravolumeand“particle”impliesapointdiscontinuity.Theyaremutuallyexclusiveterms.Iproposethatbosonsandfermionsarefundamentallywaveswhichactlikeparticlesbecausetheypossessangularmomentumwhichisquantizedbythesuperfluidspacetimefield.Allparticles,fieldsandforcesarealsodistortionsofthespacetimefield.ThereisasinglespeedlimitforallframesofreferencebecauseallparticlesfieldsandforcescompensatetheirsizeandothercharacteristicsinawaythatachievesaLorentztransformationwhichkeepsthelawsofphysics constant. Even the forces between particles compensate to maintain the distancebetweenparticlesthatpreservesthelocallymeasuredconstantspeedoflight.Accuracy Check: ThereisanothermoresubtleimplicationfromtheequationZo/η2 4πZs.andthesuccessofthechargeconversionconstant.Theseequationsimplytomethatthemanysteps that startedwith gravitationalwave equations and endedwith these relationships arecorrect.Theimpedanceofspacetimec3/Gwasdeducedfromgravitationalwaveequations.The


impedanceoffreespaceZowasderivedfromMaxwell’sequationsofelectromagnetism.Startingfrom the assumption that the universe is only spacetime,wewent looking for a constant toconvertchargeintoadistortionofspacetime: ≡ 4⁄ / .ThisresultedinZo/4πη2 Zs.Theseapparentlydissimilarimpedancesarethesamewhenweassumethattheuniverseisonlyspacetime.Thisunificationofimpedancessupportsthestartingassumption.

SpacetimeModelofanElectricFieldElectric Field Conversion:Nextwewillattempttogainadditionalinsightsintoelectricfields.We had some success describing the voltage and electrical field produced by Planck charge

E pL r and 2 2E pL r . However the underlyingmechanismbywhich a charged particle

producesanelectricfieldissomewhatmorecomplexthananelectricfieldproducedbyaphoton.Thereforewewillswitchtoelectromagneticradiationtoexamineelectricandmagneticfieldsthencomebacktochargedparticleslater.In chapter 4 itwas stated that thewave‐amplitude equation for intensity kA2ω2Z is auniversalclassicalequationthatisapplicabletowavesofanykindprovidedthattheamplitudeand impedance is stated in units that are compatible with this equation. Electromagneticradiationcommonlyuseselectricfield toquantifywaveamplitudeofEMradiation.Also,theimpedanceoffreespace Zo 377ohms isusedtoquantifytheimpedanceencounteredbyEMradiation.Thiswayofstatingamplitudeandimpedanceisnotcompatiblewiththeotherunitsoftheabovewave‐amplitudeequation.Therefore,thiscreatestheimpressionthatthiswave‐amplitudeequationislessthanuniversal.However,itispossibletoconvertelectricfieldofEMradiationintoanamplitudetermthatiscompatiblewiththe5wave‐amplitudeequations.Forexample, amplitude of EM radiation can be expressed using strain amplitude A, angularfrequencyωand the impedanceof spacetimeZs. It is informative to analyze the connectionbetweenelectricfield ormagneticfield andthestraininspacetime.Thiscanbeeasilydoneusingthetwodifferentwaysofexpressingintensity: kA2ω2Zsand ½ 2/Zo.Wewillequatetheseandsolvefor and .SincetheseassumeEMradiationandnotnecessarilytheelectricfield of a charged particle,wewill designate the electric field of EM radiation as γ and themagneticfieldofEMradiationas γ.γ2/Zo kA2ω2Zsintensityequationsignoringnumericalfactorsnear1

γ kAω γ ElectricfieldofEMradiationexpressedutilizingA,ω,ZsandZo

γ kAω γ magneticfieldofEMradiationobtainedfrom γ k γ/Zo


Thisisinteresting,butitstillusesZowhichimpliescharge.NextwewillconverttheelectricfieldandmagneticfieldofEMradiationintoadistortionofspacetimeusingη.ThisresultsinZobeingconvertedtoZs.Applyingthisto γ kAω and γ kAω ⁄ weobtain:γ/η kAωZs γ electricfieldstrengthofEMradiationγη kAω γ magneticfieldstrengthofEMradiation

Theseequationsexpress and intermsofA,ωandZs.However,theremainingtaskistobeabletodefinetheelectromagneticwaveamplitudeasastraininspacetime defineamplitudeA .

Also,thereshouldbeafactorof√4 intheseequations,butthereareotherunknownnumericalfactorsnear1whichhavebeenignored,sotheseequationsarestatedusingktorepresenttheunknownnumericalfactornear1.What Is an Electric Field?Ifindthecurrentlyacceptedexplanationofanelectricfieldastotallyinadequate. The electromagnetic force that exists between charged particles is supposedlycarriedbyexchangingvirtualphotons.However,anelectricfieldhasenergydensity,thereforeithasatangiblequality.Theenergydensityimpliesthatanelectricfieldmustalsohaveinertiaandgenerateitsowngravitationalfield.Thisimpliesthatanelectricfieldhasmoreofaphysicalpresencethanimpliedbytheconceptofvirtualphotons.Oneofthecharacteristicsofavirtualphotonisthatitisundetectable;thereforeitisnot“falsifiable”.Sometimesanelectricfieldisdescribedasacloudofvirtualphotons.Moreoften,physicistsprefertodescribetheeffectsofanelectric field and avoid addressing the question ofwhat an electric field is. Still, the closestanswerusuallyinvolvesavaguereferencetovirtualphotons.If we require virtual photons to describe an electric field, this leaves many unansweredquestions. Forexample,dorealphotonsgenerateanelectric fieldbysendingout transversevirtualphotons?Whydovirtualphotonstransferforcebutnotangularmomentum?Whydoesanelectricfieldhaveenergydensityiftheenergyofvirtualphotonsmustaverageouttoequalzero?Howexactlydovirtualphotonsachieveattraction?Whatpreventstheimpliedpressureassociatedwith theenergydensityofa staticelectric field fromdissipating? Ifanelectron issmallerthantheclassicalelectronradius ∿10‐15m ,thenwhatpreventstheenergyinitselectricfieldfromexceedingthetotalenergyoftheelectron?Somephysicists freelyadmit thatvirtualphotonsaremerelyamathematical tool. Thepath‐integralformulationofquantummechanicsassumesthatvirtualphotonstakeallpossiblepathsbetweentwochargedparticles.Inthecalculation,pathsareallowedwherethevirtualphotonisgoingfasterthanthespeedof lightorviolatingenergyconservation. Thesecalculationsgivecorrectanswers,butthephysicalmodelislacking.

Icouldgoonwithmoreexamples,butthetruthisthattherearesomanyconceptualmysteriesinthequantummechanicsthatphysicistslearntoembracethelackofconceptualunderstanding.


Thedesireforconceptualunderstandingisoftencriticizedasaruminantofclassicalphysicsthatcannotbefulfilledbyquantummechanics.Theequationsofquantummechanicsobviouslyarecorrectbuttheconceptualmodelsarelacking. Theobjectiveofthisbookistopresentanewmodel that is conceptually understandable yet is compatible with the equations andexperimental verifications of quantummechanics and general relativity. We start with theelectricfieldassociatedwithEMradiation.Maximum Confinement of a Photon: Wewill first look at the simplified case of a photonconfinedinareflectingchamber.Ifwehad100%reflectingwalls,whatisthesmallestvolumethatwouldconfineasinglephoton?Combiningthetransmissioncharacteristicsofwaveguideswiththeresonancecharacteristicsoflasers,itispossibletoanswerthisquestion.Inwaveguides,asharpcutoffoccurswhenthewidthofthewaveguideinthepolarizationdirectionisequaltoor less than ½ wavelength. However, the width needs to only be slightly larger than ½wavelengthtoachievegoodtransmission.Forcircularlypolarizedelectromagneticradiation,acylindricalwaveguideslightlymorethan½wavelengthindiameteristhesmallestwaveguidewhichhasgoodtransmission.Makingawaveguideintoaresonatorrequiresaddingtwoflatandparallel reflectors separated by½wavelength and oriented perpendicular to the axis of thecylinder. Thiscylindricalwaveguideresonatoristheminimumevacuatedvolume maximumconfinement that we can achieve for coherent circularly polarized light of a particularwavelength.Thisconfigurationwillbecalledthe“maximumconfinementresonator”andwillbeutilizedinbothcalculationsandaproposedexperimentlater.When electromagnetic radiation is freely propagating, the electric and magnetic fields areperpendiculartoeachotherandinphase.However,whenelectromagneticradiationisconfinedinaresonatorsuchasthemaximumconfinementwaveguideresonatordescribedhere orevenina laser resonator theradiation formsstandingwaves thathave theelectricandmagneticfields90°outofphase.Thisiseasiesttoseebyimaginingelectromagneticradiationreflectingoffametalmirror.Theelectricfieldisaminimumatthesurfaceofeachmirrorbutthemagneticfieldisatamaximumatthemirrorsurfaces.Theelectronsinthemetalmirrorareundergoingamotion thatminimizes the electric field but this creates an oscillatingmagnetic field. If thereflectorsareseparatedby½wavelength,thestandingwavecreatedbetweenthetwomirrorshasmaximumelectricfieldoscillationsinthecentralplaneofthe½wavecavity antinode andtheminimumelectricfieldatthemirrorsurfaces nodes .Conversely,themagneticfieldisataminimum node inthecentralplaneandatamaximumatthemirrorsurfaces antinodes .Displacement of Spacetime Produced by a Single Photon: Next we will calculate thedisplacementofthespacetimefieldproducedbyasinglephotoninthismaximumconfinementresonator. Byspecifying the “maximumconfinementcondition”,wecanavoidspecifying thecharacteristicsofafreelypropagatingphotonthatwillbediscussedinchapter11.Sinceweareignoringnumericalfactorsnear1,wewillmodelthedisplacementrequiredtoproduceauniformoscillating electric field over the central volume of λ3 and assume zero electric field in the


remainder of themaximumconfinement cavity. Alsowe are not specifyingwhetherwe aredesignatingthepeakelectricfieldstrengthortheRMSelectricfieldstrength.Therefore,ignoringthesefactors assumingavolumeofλ3 ,theenergydensityofasinglephotonwithenergyħωanduniformenergydensityinavolumeofλ3wehave:U ħω/λ3 ħc/λ4 ħω4/c3 Lp/λ 4Up.Tofindthedisplacementamplitudeofspacetimerequiredtoproducethisenergydensity,wewillequateU ħω4/c3withU A2ω2Zs/candsolveforΔLinA ΔL/λ.

U ħ

solveforΔL

ΔL ħ Lpdisplacementamplitudeofasinglephotonin“maximumconfinement”

A Lp/λthestrainamplitudeAofasinglephotonin“maximumconfinement”TheequationΔL Lpisforasinglephotoninvolumeλ3.ThereforethiscalculationpresumesthatΔLisoveratotalpathlengthofλ ignoringnumericalfactorsnear1 .Atotalpathlengthofπλ the diameter of the maximum confinement waveguide is within the allowed range,especiallysincetheelectricfieldismaximizedoverthecentralλ.Sincephotonsarebosonsandmanyphotonscanoccupythesamevolume,wewillnextcalculatethedisplacementofspacetimerequiredifmanycoherentphotons samefrequencyandphase areintroducedintothevolumeλ3.Inthiscalculationwewilluse“n ”asthenumberofphotonsoccupyingthevolume.ThereforeU n ħω/λ3 n ħω4/c3.

U ħ

SolveforΔL

ΔL Lpdisplacementamplitudeofn photonsin“maximumconfinement”TheequationΔL Lp istheoscillatingdisplacementofspacetimefor“n ”photonsinthemaximumconfinementpreviouslydiscussed.ThisvalueofΔLisoverdistanceλ,thereforethestraininspacetimeproducedbyn coherentphotonsinmaximumconfinementis:

A strainproducedbynγcoherentphotonsinmaximumconfinementTheequationΔL Lpisveryrevealing.First,itincorporatesPlancklengthLp.Previously

wefoundthatPlancklengthwasalsoassociatedwithPlanckcharge E pL r and 2 2E pL r


Therefore,thefactthataphotonexhibitsthisamplitudeinmaximumconfinementsupportsthemodel.Also,itisnotpossibletoactuallymeasuretheelectricfieldormagneticfieldproducedbyasinglephoton.NowwecanunderstandthisbecausemeasuringtheelectricfieldproducedbyasinglephotonwouldbeattemptingtomeasureadisplacementofPlancklength.Referencesinchapter4showedthatitisfundamentallyimpossible deviceindependent todetectadisplacementofspacetimeequaltoorlessthanPlancklength.Itistheoreticallypossibletodetectandmeasurethe oscillating electric field produced by many photons n 1 photons because manycoherentphotonsproducesΔL Lp.Nowwecanconceptuallyunderstandthiseffect.Similarity to Gravitational Wave: WepreviouslylearnedthatPlanckimpedanceZpisthesameas the impedance of spacetime Zs when we use the charge conversion constant η. Thisconversionconstantandawave‐amplitudeequationalsogivethatn photonsinthemaximumconfinementconditiongivestheoscillatingstrainamplitude inthespacetimefieldAequal toA ⁄ .Thisimpliesthatelectromagneticwavesareverysimilartogravitationalwaves.

Whilegravitationalwavesappear tobecompletelydissimilar toelectromagneticwaves, theymustbefirstcousins.Botharetransversewavesthatpropagateatthespeedoflightthroughthemedium of the spacetime field. They both experience the same impedance thereforeelectromagneticwavesmustalsobewavesinspacetime.Thequantummechanicaldescriptionof the spacetime field is vacuum fluctuations with Planck length/time displacements at allfrequenciesuptoPlanckfrequency. Thisdescriptionofthespacetimefieldhasahighenergydensity.Thespacetimefieldhaselasticityandverylargeimpedance. Wavesinthespacetimefieldpropagateatthespeedof lightbutthedisplacementofthespacetimefield isverysmallbecausethespacetimefieldalsohasanincrediblylargeimpedanceandbulkmodulus.Asinglephotoninmaximumconfinement volumeλ3 onlyaffectsthespacetimefieldbyaPlancklengthdisplacementoveradistanceofλ.One of the biggest differences is that positive and negative electrically chargedparticles areavailabletogenerateelectromagneticradiation.Gravitationalwavescanonlybegeneratedbyparticlesthathaveasinglepolarity onlypositivemass thereforeonlyquadrupolegravitationalwavesarepossible.However,ifweareattemptingtounderstandthephysicsofelectromagneticradiationpropagatinginthespacetimefieldthedifferencesingenerationarenottooimportant.Agravitationalwaveisatransversewavethatcausesasphericalvolumetobecomeatransverseoscillatingellipsoid.Ifwefreezethisellipsoidforamomentthereisanaxisthatincreasesthedistancebetweenpointsandanorthogonalaxisthatdecreasesthedistancebetweenpoints.Thepoints themselves are not accelerated and physicallymoved by the gravitationalwave. Thespatialpropertiesofthespacetimefieldareaffectedinawaythatincreasesordecreasesproperdistance as might be measured by a tape measure. With gravitational waves there is nopolarization vector that distinguishes between opposite directions along either of these two


axes.Theeffectonthespacetimefieldbygravitationalwavesissymmetrical reciprocal .Thiseffectcanbethoughtofasadifferencebetweenthecoordinatespeedof lightandtheproperspeed of light along the two axes. We interpret this difference as a change in the distancebetweenpointsbecauseweassumethattheproperspeedoflightisconstant.Also,allphysicalobjects metersticks,protonradius,etc. scaletheirsizewithproperlengthwhichinturnscaleswiththeproperspeedoflight.Thisisunderstandablefromtheproposedspacetimebasedmodeloftheuniversebecauseallmatterandforcesareultimatelywavesinthespacetimefieldwhichscalewiththeproperspeedoflight.Proposed Model of an Electric Field: Electromagnetic radiation has transverse oscillatingelectricandmagnetic fields. Ifwe imagine freezing thewave, theelectric fieldhasaspecificvectordirectionwhichbyconventionwesaypointsawayfrompositiveandtowardsnegative.ThemagneticfieldbyconventionpointsfromNorthtoSouth.Thereforeonedifferencebetweenanelectromagneticwaveandagravitationalwaveisthattheelectromagneticwaveproducestransverse vectorswhich are not reciprocal electric andmagnetic fields vectors while thegravitationalwaveisareciprocaltransversewave.Reversingdirectionalongeitherthelongorshort axis distortion produced by a gravitationalwave produces the samedistance betweenpoints. Wewill designate the unsymmetrical non‐reciprocal effect on the spacetime fieldproducedbyEMradiationas“polarizedspacetime”.BothEMradiationandgravitationalwavesdonotmodulateeithertherateoftimeorpropervolume.Ifwearegoing toexplainelectromagnetic fieldsusingonly thepropertiesof spacetime, it isnecessarytoincorporateintotheexplanationanonreciprocal polarized effectthatdoesnotproduceanoscillationofeitherpropervolumeortherateoftime.Explaininganelectricfieldandlateramagneticfield usingonlythepropertiesofspacetimehasbeenthemostdifficulttask invention ofanycreativenewideadescribedinthisbook.Inparticular,itwasdifficultto1 initiallyrecognizethatthesolutionmustinvolvepolarizedspacetime,2 tofindamodelthatwould generate the correct force 3 not modulate proper volume and 4 result in thenon‐reciprocal polarized characteristicsrequiredforanelectricfield.Forexample,theremustbeaphysicaldifferencebetweenthepositiveelectric fielddirectionandthenegativeelectricfielddirection theoppositedirection .A gravitationalwave does not produce a net change in proper volume. An increase in onedimension is offset by a decrease in the orthogonal transverse dimension to keep the totalvolumeconstant.Iftherewasadetectablechangeinvolume,therewouldalsobeadetectablechangeintherateoftimeandthiswouldcreateconditionswheretherecanbeaviolationoftheconservationofmomentum seedipolewavediscussioninchapter4 .Anoscillatingelectricfieldmustmodulatedistancebetweenpointswithoutmodulatingpropervolume.Thewaythisis accomplished is even simpler than the mechanism used by a gravitational wave. In anoscillatingelectricfieldthedimensionalincreaseanddecreasehappensinonlyonedimension.Onepropagationdirectionexperiencestheincreasewhiletheoppositepropagationdimension


experiencesthedecrease.Theroundtrippropagationtime roundtripdistance isunchangedthereforethereisnochangeinpropervolume.Thisisthesimplestpossiblewaythatproducesanetpolarizationofthespacetimefieldwithoutalsoproducinganetchangeinvolumeoranetchangeintherateoftime.Thisproposedmodelofanelectricfieldwillbefurthersupportedinchapter10byacalculationwhichshowsthatthismodelproducesthecorrectelectrostaticforcebetweenrotarswithelementarychargee.Thecalculationcannotbepresentedherebecauseitrequirestheintroductionofadditionalconcepts.It is proposed that an electric field is an asymmetric distortion of the spatial dimension of the spacetime field parallel to the electric field direction. This results in a slight asymmetric distance between points propagating in the positive electric field direction compared to propagating in the negative electric field direction. There is neither a net volume change nor a rate of time change because the round trip time between the points is unchanged (except for the slight gravitational effect). The proposed model of an electric field implies that n photons in maximum confinementproduceaneffectonspacethatresultsinadistortionofΔL Lpoverdistanceλ.Thevectordistortion implies that if therotatingelectric is imaginedasbeingmomentarily frozen, therewould be a difference in the distance of ΔL Lp progressing from to – compared to

progressing in theoppositedirection – to . It is not knownwhichpolarizationdirectionproducesthelongerdistance,buthypotheticallythisisexperimentallymeasurable.Thisnon‐reciprocalpropertyseemsstrange,butanotherexampleofanon‐reciprocaleffectisaFaraday rotator. When a magnetic field is imposed on any transparent material, circularlypolarizedlightexperiencesadifferentpathlengthdependingonwhetheritispropagatingintheNorth magnetic direction compared to propagating in the South magnetic direction. ThisdifferenceinpathlengthbetweenthetwooppositedirectionscanbeexpressedasΔL/L.Italsocauses linearly polarized light to exhibit a different rotation direction when propagating inoppositedirections. Thiseffectiscommonlyusedtoformanopticalisolatorthatonlyallowslaserbeamstopropagateonedirection.ChargedParticles:SinceEMradiationhasanoscillationatfrequencyω,itiseasytoimaginethatthattheoscillatingelectricfield oscillatingdistortionofthespacetimefield producedbyEMradiationhasenergydensity.However,theelectricfieldproducedbyachargedparticleappearstobestaticandyetitalsopossessesenergydensity.Ofweinsertω 0intotheenergydensityequationU kA2ω2Z,weobtainthatthereshouldbenoenergydensityifthereisnooscillation.Yetweknowthattheelectricfieldproducedbychargedparticlesalsohasenergydensity.Itwillbe proposed in chapter 10 that all particles charged and neutral produce an oscillatingdisturbance in the surrounding volume of spacetime. This will be shown to be oscillatingstandingwavesat theparticle’sCompton frequency.Thesestandingwavesproducenotonly


particle’sdeBrogliewavesbutalso theyare required for theproductionof anon‐oscillatingportionoftheelectricfieldandalsogravitationaleffects.Electric Field Produced by Charged Particles:Earlierinthischapterwederivedtheelectricalpotential for Planck charge in dimensionless Planck units as: E pL r . When we assume

elementarychargee designatedwithsubscript“e” theequationbecomes:

ee

p

√

ThereforethevoltageexpressedindimensionlessPlanckunitsisadimensionlessnumberthatisdescribingtheslopeofthepolarizedstrainofspacetime.Aspreviouslyexplained,anelectricfield produces polarized spacetime where there is a difference in distance progressing inoppositedirections.ThedifferenceindistanceisdesignatedΔLsotheratioΔL/ΔrcanbethoughtofasatypeofslopedesignatingthedifferenceindistancebetweenoppositedirectionsΔLoverradiallinesegmentΔr.Forexample,wewillstartwithasingleradialvectorpointingawayfromachargedparticle.Wewilldesignatedistancesr1andr2alongthisradialvectorandspecifythatr2 r1.Thedistancebetweenr1andr2isslightlydifferentifatimeofflightdistancemeasurementis made progressing from r1 to r2 compared to progressing in the opposite direction. ThisdifferenceΔLproducedbyaparticlewithelementarychargee,isequalto:ΔL √ Lpln r2/r1 distortionproducedbytheelectron’schargebetweenr2andr1Forexample,ifr2 1meterandr1 10‐12meter,thenΔL √ Lp 27.6 3.26 10‐36m.Whilethisseemslikeaverysmallnetdistance,itmustberememberedthatthestrainisaffectingtheenormouslylargeenergydensity,impedanceandbulkmodulusofthespacetimefield.Lateritwillbeshownthatthistypeofpolarizedstrainofspacetimecanproducethemagnitudeoftheforceweexpectofanelectricfieldactingonanelectron rotar .Iftherearemultiplecharges,the strain produced by each elementary charge is a vectorwhich adds to the vector strainsproducedbyall theothercharges. Forexample, if therearea largenumberofelectrons neelectrons onachargedsphere,thenthevalueofΔLbecomes:ΔL √ Lpln r2/r1 .Thus farwe have dealt with the distortion produced by a spherical electric field such as isproduced by a single electron of a charged sphere. However, what about the distortion ofspacetimeproducedbyaparallelplatecapacitor?WewillassumeavacuumcapacitormadewithparallelplatesofdimensionsDxDandseparatedbydistanceD. This is an idealizedvacuumcapacitor which forms a cube. We will ignore the effects of fringing electric fields in thisexploratory discussion. The same way that expressing voltage electrical potential indimensionlessPlanckunits forasinglechargedparticlegavestrainΔL/r,soalsoexpressingthe voltage on a vacuum capacitorwith separation distance D gives the strain of spacetimeproducedbytheelectricfieldgeneratedbyavacuumcapacitor.


p

L

D

L D

Forexample,atimeofflightdistancemeasurementalongtheelectricfieldofavacuumcapacitormightbemadeifasmallholeisprovidedinthecenterofbothplates.Thepredictionisthatthetimeofflightdistanceproceedingfrompositivetonegativeshoulddifferby L D compared

toproceedingtheoppositedirection.However,thiswouldbehardtomeasure.Forexample,ifD 1mandvoltagewas1,000,000volts,thensince 2710p volts, ≈ 10-22 and ΔL ≈ 10-22 m.

This is much smaller than current interferometer technology can measure. However, there is also another problem which will be discussed later relating to whether light can ever be used to measure the ΔL effect. Comparison of Electric Fields: TheequationΔL Dforacubicvacuumcapacitorspecifiesthelengthdifferenceintermsof andD.Sinceanelectricfield /LandinthecaseofavacuumcapacitorL D,wecanalsoexpressΔLoverdistanceLinauniformelectricfieldas:

ΔL 2

p

L

Withphotonswewereable tospecifyΔL in termsof thenumberofphotonsnγ inmaximumconfinement ΔL Lp .NextwewillcalculateΔLinacubicvacuumcapacitorintermsofthenumberofelectronsneonthevacuumcapacitorwithdimensionsDxDxD.Thiscalculationwillmakeuseofthefollowingsubstitutions:

q/C, q = nee, C = kεoD2/D = kεoD, knee/εoD, p c /4πε G, Lp ħ ⁄ , ,

e α4 ħ , ne = numberofelectrons

L D = 4

4 4 44e o e o o

e pp o p o

n D c n cD Gn L

D c

L √ neLp Theaboveequationwillbeexpressedinwords.Thenon‐reciprocaldistortionofspacetimeΔLproducedacrossthewidthofacubicvacuumcapacitor D isequalto1.38 10‐36metersperelectron.Thisholdsforallsizecubicvacuumcapacitors allvaluesofD . Convertingelementarychargeetocoulomb,thestrainofspacetimeproducedbyacoulombofchargeonacubicvacuumcapacitoris:


ΔL/q = √ /e = 8.62×10-18 meters/coulomb η thechargeconversionconstant

Thereforeoneofthephysicalinterpretationsofthechargeconversionconstantηisthatthisisthe distortion of spacetime produced over the width D of a cubic vacuum capacitor by acoulombofcharge.

Previously itwascalculated ΔL Lp forn photons in themaximumconfinementcavity

approximatelyofvolumeλ3.NowwehaveΔL √ neLp foravacuumcapacitorwithdimensionsDxDxD ignoringnumerical factorsnear1 .Theelectric fielddistortionproducedbyphotonsandelectricchargebecomeverysimilarifwehavesimilarsize(λ ≈ D). Thecalculationisnotshownhere,butanelectricfieldgeneratedbyphotonsproducesthesamepolarizeddistortionofspacetime ΔL asanequalelectricfieldstrengthproducedbychargedparticlesonavacuumcapacitor. Inotherwords, 2

pL L holds for electric fields generated by either photons or

electrons. Thephotonfieldisoscillatingandhasanassociatedmagneticfield,sotherearealsodifferencesinthevalueofkwhichisbeingignored.However,thepointisthateventhoughthephotonequationscaleswiththesquarerootofthenumberofphotons andthecapacitor

equationscaleswiththenumberofelectronsne nosquareroot ,stillwhentheeffectisreducedtoelectricfieldofequalstrengthinequalvolumes,thevaluesofΔLarethesame ignoringk . ExperimentsUsingLight: Inearlierversionsofthisbookandinanearliertechnicalpaper,Idiscussed possible experiments using light in an attempt to measure the predicted non‐reciprocal path length difference between opposite propagating directions. However, now Idoubt that light canmeasure this effect. The doubt has nothing to dowith the fact that allpracticalexperimentsproduceavalueofnon‐reciprocallengthchangeΔLintherangeof10‐20to10‐18m.Thebestinterferometerscurrentlycanseeamodulatedlengthchangeontheorderof10‐18m.Therefore,iftheeffectsweremeasurablebyinterferometers,theywouldbenearthedetectable limit. However, the real problem is that it now appears as if the light from aninterferometerwillnotbeabletoseeanyvalueofΔLproducedbyanelectricfieldnomatterthemagnitude.Theproblemhastodowiththewaythatlightpropagatesthroughspacetime.BeforeIexplaintheproblem,Iwanttogivesomehistoryabouttheunderstandingoflight.In the late1600’soneof thebiggestmysteriesof sciencewasdouble refraction. When lightpassesthroughacrystalknownasIcelandspar acrystalofCaCO3 ,thelightisbrokenintotwobeamsanddoubleimagesappear.IsaacNewtonproposedacorpusculartheoryoflightbuthiscorpusculartheorycouldnotexplaindoublerefraction.In1690,ChristianHuygensproposedawave‐basedtheoryoflight.Huygensalsoderivedapartialexplanationofdoublerefraction.TheHuygensPrincipleconsiderseachpointonawavefronttobethesourceofanewwave.Huygensrealizedthatifthevelocityoflightvariedwiththedirectioninthecrustal,thenthesphereswoulddeformintoellipsoidsandbeabletopartiallyexplaindoublerefractionofIcelandspar.Eventhoughhisexplanationcouldbereducedtoanequationwhichcorrespondedtoexperiment,the


explanationdidnotgiveconceptualunderstandingofthefundamentaldifferencebetweenthetwo beams. He, and everyone else for the next 100 years, assumed that light waves werelongitudinalwaveslikesoundwavesinair.Theproblemwasthatlongitudinalwavescouldnotexplaindoublerefraction.Thepuzzleofdoublerefractionwassoiconic,thatin1807theFrenchInstituteofferedacashprize to anyone who could explain this phenomena. The prize was initially awarded to aFrenchman,EtienneMalus,in1810andasecondcashprizewasawardedtoaFrenchwoman,Sophie Germain, in 1816. However, these were only partial solutions. Themost importantinsightwasmadebyThomasYoungseveralyearslaterwhenheshowedthatlightisatransversewave. Thetransversewavesofunpolarized lightwerebeingsplit intotwoorthogonal linearpolarizationsbythecrystal. Thisgaveanexplanationofdoublerefractionoflightwhichwasbothconceptuallyunderstandableandmathematically rigorous.However,ThomasYoung,anEnglishman,wasnotawardedathirdcashprizebytheFrenchInstitute.Thisstoryaboutdoublerefractionandpolarizedlightistoldtopreparethereaderforamorecomplexmodelofthedistortionofspacetimeproducedbyaphoton.InearlierdraftsofthisbookIproposedthatthedistortionofspacetimecouldbemeasuredbyaninterferometerwherethetwo beams propagate in opposite directions across the rotating electric field produced bymicrowaveradiation inamaximumconfinementcavity. One laserbeamwouldexperienceadecreaseinpathlengthwhiletheoppositepropagationbeamwouldexperienceanincreaseinpathlength.Ahalfcyclelatertherotatingmicrowaveelectricfieldwouldhavereversedpolarity.IftheintensityoftherotatingEMradiation probablymicrowaves couldbemadehighenough,themodulatedpathlengthmightbedetectableasanintensitymodulationwhenthetwolaserbeamsarecomparedinaninterferometer.The problemwith this experiment is that it does not properly address the transversewavestructure of light. As previously stated, a photon is not a packet of energy that propagateslinearly through the empty voidof spacetime. Instead, aphoton is a transversewave that isinteractingwith thepropertiesof spacetime transverse to thedirectionofpropagation. Thespeedoflightforlinearlypolarizedlightissetbythespeedofwavepropagationintheplaneofpolarization.Therefore,lightcannotbeusedtoaccuratelymeasuredistanceinthedirectionofpropagationifthemediumbeingprobedisnothomogeneous.Thistransversewavepropertycreatesaproblemforanyexperiment thatattempts touse light tomeasure thedistortionofspacetimeproducedbyanelectricfield.When light propagates through homogeneous glass, it has a speed of light less than 1. Therelative permittivity ε of the glass and the index of refraction are both homogeneous in alldirections.However,supposethattheglassisstressedbyauniformcompressionalongwhatwewillconsidertobetheZaxisoftheglass.ThepermittivityεintheZdirectionbecomesdifferentthanthepermittivitymeasuredintheXandYaxis.Linearlypolarizedlightwithitselectricfield


vectoralongtheZaxispropagatesthroughtheglassatadifferentspeedthanlightofthesamewavelengthbutpolarizedalonganyorthogonaldirection.Whenlightispropagatingthroughauniformmedium,thenwecanignorethetransversewavenatureoflight.However,whenthemediumisnotuniforminalldirections,wehavetobecarefulabouttheactualmechanicsofwavepropagationthroughthemediumwhenattemptingtomakeadistancemeasurement.Normally,thevacuumisconsideredtobeperfectlyhomogeneous.However,theproposalisthatboth gravitationalwaves and electric fields distort the vacuum spacetime so that it is nothomogeneousinalldirections.Theinhomogeneityisextremelysmall,butweareattemptingtodeviseexperimentstomeasurethiseffect.Therefore,theexperimentisspecificallydesignedtoenhance the inhomogeneity to ameasurable level. However, there is aproblem thatwill beillustratedwithanexample.Supposethatavacuumcapacitorcontainstwosmallholesinthemiddleofeachflatplateofthecapacitor.Sendingalaserbeamthroughtheholeswouldhavethelaserbeampropagatingonewayalongtheelectricfieldofthecapacitor.Thepredictionisthatthe distance along the electric field should be different for a “time of flight measurement”proceeding in the two opposite directions positive to negative compared to negative topositive .However, the speed of propagation of the laser beam is determined by the properties ofspacetimeperpendiculartothepropagationdirection.Evenwithanextremelystrongelectricfield,itshouldbeimpossibleforabeamofEMradiationtomeasureanyeffect.Evenifthebeamdirectionischangedsothatitpropagatesperpendiculartotheelectricfieldwiththepolarizationdirectionparalleltotheprobedelectricfield,therestillshouldbealmostnodetectableeffect.Recall that the round tripdistance in anelectric field shouldbeunchanged compared to theroundtripdistancewithnoelectricfield.TheEMradiationhasanoscillating reversing electricfieldsoitisprobingtheroundtrippropertiesofspacetime.Onecompletecycleoftheoscillatingelectric fieldshouldproducenofirstordereffect. Therewouldbeaverysmallsecondordernonlineareffectassociatedwith thegravityproducedby theelectric field,but thatwouldbevastlysmallerthanthefirstordereffect.Itwouldalsobesymmetrical.ThereisanotherhypotheticalwaytoprobethedistortionofspacetimeproducedbyanelectricfieldthatgetsaroundtoproblemsofEMradiationbuthasitsownpracticalproblems.Supposethataneutralparticlesuchasaneutronorneutrinocouldbesentalongtheelectricfield.Ifthespeedwasaccuratelyknown,thendistancecouldtheoreticallybemeasuredbyatimeofflightmeasurement. This is not subject to the transverse wave problems of light and it shouldhypothetically give the oneway distance. The obvious problem is that neutron or neutrinobeamsarecompletelyimpracticaltoexperimentallymeasuredistanceaccurately.Implied Maximum Energy Density for Photons: Even though experimental verificationappears impractical, there are other ways of checking the validity of the predictions. Thefollowingcalculationstestthemodelatthehighestenergydensity.WhenIfirstdevelopedthe


equations for the distortion of the spacetime field produced by electromagnetic radiation, Iquickly realized that therewas an implied limit. The finite properties of the spacetime fieldimpliedthatthereshouldbeamaximumintensitylimitforEMradiation.ThislimitissetbecauseEMradiationproducesadistortionofthespacetimefieldwhichhasfiniteproperties.Itshouldbeimpossibletoexceed100%distortionofanyportionofthespacetimefield if themodel iscorrect.Initially,thispredictionwascounterintuitiveandappearedtoberevealingaflawinthemodel.Thisimpliedlimitwillbeexplainedusingthemaximumconfinementcavitybutitapplieseventoalaserbeamfocusedinavacuum.Forexample,theequationΔL appliestoacavity

ofvolumeλ3.ThisequationimpliesthatthereisamaximumpossiblevalueofΔLthatcanbeachieve.WhenΔL λ,thenthedistortionΔLequalsthesize λ ofthemaximumconfinementcavity ignoringnumericalfactorsnear1 .Thereforeifthemodelofthedistortionofspacetimeproduced by the electric field generated by photonswas correct, it should be impossible toexceed the condition where ΔL λ in a volume of λ3 because this would require 100%modulationofthepropertiesofthespacetimefieldwithinthisvolume.As you probably have guessed, experimentally testing this prediction would require a peakpower that is totally beyond human capability. For example, suppose that we attempted toexceedthe100%modulationofspacetimelimitusingapulsedlaserwithawavelengthof10‐6m1micron . Ifweassumeapulselength1wavelengthlong 3 10‐15spulsehasalengthof1micron andthebeamfocusedtoaspot1wavelengthindiameter,thenthevolumeofthefocusedpulsewould be about 1micron in diameter. To exceed the ability of the spacetime field totransmit thisradiation, thepeakpower inthepulsewouldhavetobeabout1053watts. Theenergy in the 3 10‐15 s pulsewould have to be about 1038 Joules to reach this theoreticaltransmission limit for a 1 micron diameter volume. To put this energy requirement inperspective,thisrequiredenergyisequivalenttotheannihilationenergyofabout1021kgwhichisaboutthemassofalltheoceansonearth.Anexperimentisobviouslynotpossible,butitisquiteeasytocheckthispredictiontheoretically.We will return to the maximum confinement resonator previously described which had adiameterof½wavelengthandalengthof½wavelength.Sincetheenergydensityinsideisnotuniform and maximum near the center, we loosely defined this volume as λ3 by ignoringnumerical factors near 1. For this condition, we previously determined the displacementamplitudeΔLas:ΔL Lp.wherenγequalsthenumberofphotonsintheresonator.Wewillnowdesignate nc as the critical number of photons at frequencyω required to theoreticallyachieve100%modulationatwavelengthλ.ThisconditionoccursatΔL λ.Thiscriticalnumberofphotons involumeλ3hasacriticalamountofenergyofEc. Thereforewewillanalyze thecriticalenergyEcinvolumeλ3toseeifthereisanyobviousreasonpreventingelectromagneticradiation from exceeding the implied limit that would achieve 100% modulation achieveΔL λ.


nc ħ

ħsetΔL2 ncLp2

ΔL2 ncLp2 ħ

ħsetΔL λ

λ RswhereRs≡Gm/c2theSchwarzschildradiusforenergyof

Thisisafantasticresult!Itisnotnecessarytodoanexperimenttoprovethatthispredictioniscorrect. The intensity energy density that would achieve 100%modulation of spacetimewouldalsomakeablackhole!Themaximumconfinementresonatorcanbeconsideredtohavearadiusequaltoλ.Forthecriticalcondition,λ Rs,whereRs≡Gm/c2 isthepreviouslydefinedSchwarzschildradiusofablackhole.Therefore,itisindeedimpossibletoexceedtheimpliedlimitthatwouldachieve100%modulationofspacetimebecausethisistheconditionthatcreatesablackhole.Ifmoreenergythanthetransmissionlimitwasprovided,theenergydensitywouldformablackholewhichwouldblocktransmissionthroughthevolume.Whilethecalculationassumedamaximumconfinementcavity,thissamelimitwouldapplyifthelaserbeamachievedthissizeandenergydensitybymerelybeingfocusedinavacuum.Thissuccessfultesthasseveralprofoundimplications.

1 Theproposedquantummechanicalmodelofspacetime dipolewavesinthespacetimefield isstronglysupported.

2 The concept that EM radiation is awave propagating in themedium of spacetime issupported.

3 Thedisplacementamplitudeofn photonsin“maximumconfinement”is:ΔL Lp4 Theconditionthatcreatesablackholecannowbeunderstoodmorecompletelythan

merelydiscussingvaguetermssuchas“curvedspacetime”.Nowtheinternalworkingsofspacetimethatcreatescurvedspacetimecanbequantified.

We cannot exceed the intensity that would demand more than 100% modulation of thespacetimefield.ThisfieldhasallfrequenciesuptoPlanckfrequencyandatotalenergydensityofabout10113J/m3.However,itisnotnecessarytoexceed10113J/m3toachieveablackhole.ThatistheintensityrequiredtoachieveablackholewithwavelengthequaltoPlancklength.Toachieveablackholeusing1micronlight,itisonlynecessarytoachieve100%modulationoftheportionofthespacetimefieldwithfrequencyofabout3 1014Hzinavolumeabout1micronindiameter thisexampleignoresnumericalfactorsnear1 .MaximumVoltageonaVacuumCapacitor:Thereisanotherpredictionthatwilltesttheproposedmodelofthedistortionofspacetimeproducedbyanelectricfield.Inthecaseofthecubicvacuumcapacitorcalculatedearlier,wegeneratedtheequationΔL D foravacuumcapacitorwithdimensionsDxDxD. In otherwords, the distance between the parallel plates of the vacuumcapacitorisD.ThereforethemaximumdistortionofthespacetimefieldoccurwhenΔL D.Thiswould also be 100% distortion of the properties of the spacetime field within the vacuum


capacitorvolume DxDxD .Thisconditionisreachedwhen 1.Sincethedefinitionof is:p ,itshouldbeimpossibletoexceedtheconditionwherethevoltageonthecubicvacuum

capacitorequalsPlanckvoltage p becauseatthisvoltage 1.Thislimitseemsstrange

becausethisfixedmaximumvoltageappliestoanysizecubicvacuumcapacitor.Therefore,thereisnotamaximumelectricfieldbutamaximumvoltage.WewillcalculatetheenergycontainedinacubicvacuumcapacitorwhenitischargedtoPlanckvoltage.Substitutionstobeused: p c /4πε G, E = kC 2,C = kεoD2/D = kεoD where:C capacitance

E = kC p2 kεoD)

D Rs

ThereforetheequationD Rssaysthattheconditionwhichachieves100%modulationofthepropertiesofspacetimewithinthevacuumcapacitoralsoformsablackholewithradiusRs.ItisimpossibletoexceedPlanckvoltageonacubicvacuumcapacitorbecausethisrequiresenergywhichwouldformablackhole. Thisstatementignoresnumericalfactorsnear1andignoresfringingoftheelectricfieldaroundtheedgesoftheplates. Again,thisanalysisofthemaximumvoltage on any size vacuum capacitor is revealing something about the mechanics of theformationofablackholebeyondthestandardgravitationalexplanations.GravitationalWaveDetection:Wearegoingtopausefromthediscussionofelectricfieldsandbriefly examine the implications for the detection of gravitational waves. For review, agravitationalwaveisatransversewavethatpropagatesthroughthespacetimefieldatthespeedof light. It causes a spherical volume of spacetime to become an oscillating ellipsoid. If thegravitationalwavepropagatesintheZaxisdirection,theellipticalelongationandcontractionwill occurperpendicular to theZ axis. We candefine theXandYaxis to correspond to thetransversedirectionsofmaximumoscillation.ThereisnoeffectontherateoftimeandnochangeinpropervolumebecauseanexpansionoftheXaxisisoffsetbyacontractionoftheYaxisandviceversa.TheLIGOexperimentwillbeusedasanexampleofexperimentsaroundtheworldcurrentlyattempting to detect gravitational waves. This experiment is a power recycled Michelsoninterferometer. Massive mirrors suspended by wires are located at each of the corners ofL‐shapedevacuatedtubeswhichare4kmlong.Supposethatalargegravitationalwavepasseswithpropertieswouldproduce themaximumdifference inpath lengthbetween themirrors.What isphysicallyhappening? Is therea forceexertedon themirrorscausing themirrors tophysicallymove accelerateanddecelerate ?Alternatively,doesthechangeintheseparationdistanceresultfromachangeinthepropertiesofthespacetimefieldwithnoforceexertedonthemirrors noaccelerationanddecelerationofthemirrors ?


Thissecondalternativeisproposedtobecorrect.Thepropertiesofspacetimechangeinwaythataffectsthesizeofrotars,atoms,etc.Thiswouldmakephysicalobjectssuchasmetersticksandtapemeasuredchangetheirsize.Thedistancebetweenthemirrorswouldchangeaccordingtophysicalmeasurements,buttherewouldbenophysicalmotionofthemirrors.Therewouldbenoforce,andnoaccelerationofthemirrors.Herearetworeasonsforbelievingthis.

1 Supposethatoneofthetwomirrorshadvastmass,equivalenttothemassdensityofaneutronstar.Theforcethatmustbeexertedtoachievetherequiredphysicalmotionofthismirrorwouldalsohavetobevast.Yetthereisnooffsettingmassacceleratingintheoppositedirectionrequiredtoachievetheconservationofmomentum.Assumingthatthemirrorsphysicallymoveisaviolationoftheconservationofmomentum.

2 Agravitationalwavedoesnotaffecttherateoftime.Ifagravitationalwaveachievestheaccelerationofmatterusinggravity,thegravitationalwavewouldhavetoaffecttherateoftimeandpossessarateoftimegradient.Asdiscussedinchapter2,allgravitationalaccelerationimpliesarateoftimegradient.Sinceagravitationalwavedoesnotexertagravitationalforce,isthereanyotherforcewhichmightacceleratethemirrors?

Therefore, a gravitational wave affects the spatial properties of two of the three spatialdimensions.Thereisnoeffectonpropervolumebutthethreeorthogonaldirections X,YandZaxis ofthespacetimefieldhavereceiveddifferentdistortions.Nowthequestionis:whateffectdoesthisspatialchangehaveonEMradiation?Ifyouthinkofaphotonasbeingabullet‐likepacket of energy, then only one dimension the propagation direction is being probed anddistanceasmeasuredbytimeofflightofphotonsinthatonedimensionshouldagreewiththemeterstickmeasurement.However,ifthemodelofphotonsisatransversewavepropagatingin themedium of the spacetime field, then it is not obviouswhat effect the two transversedimensionsmightexertonthedistancemeasurement.Toexplorethisquestion,wewilltemporarilyshiftfromagravitationalwaveaffectingspacetimetoagravitationalwaveaffectingatransparentcubemadeofahomogeneousmaterialsuchasglass.ThefacesofthecubearealignedwiththeX,YandXaxispreviouslydefined.Agravitationalwave passing through this transparent cube will cause the cube to become an oscillatingrectangle which exhibits both modulated birefringence and a modulation of the relativepermeabilityε.Themodulatedbirefringenceandmodulatedpermittivityinglasswouldcauseasinglebeamofcircularlypolarizedlightpropagatinginanydirectionintheglasscubetobecomeanoscillatingellipticalpolarizedbeamoflight.ThebiggesteffectonpolarizationofcircularlypolarizedlightwouldbeforthebeampropagatingalongtheZaxis.Thisisthedirectionthatdoesnotexperienceanychangeinthetimeofflightpathlength.However,sincelightisatransversewave,changesinεoorthespeedoflightforlightpolarizedintheXandYdirectionsconvertcircularlypolarizedlightpropagatingintheZdirectionintooscillatingellipticallypolarizedlight.IfweconsiderthespacetimefieldtobeamediuminwhichEMradiationpropagates,thenitshouldreacttothe


passageofagravitationalwavesimilartothewaythatatransparenthomogeneousmaterialsuchasglasswouldreact.Inparticular,thepropagationspeed inverseindexofrefraction dependsonthepolarizationdirection,notthedirectionofpropagation.Thereissomeuncertaintyinthis,butifgravitationalwavesturnthespacetimefieldintoanoscillatingbirefringentmaterial,thentherewouldbesimplerandmoresensitivewaysofdetectinggravitationalwaves.Presently,thebiggestsourceofnoiseusinginterferometersistheseismicnoise.Interferometersare attempting to detect very small difference in optical path length between the two arms.Anything that produces even a very small vibration is a source of noise. Even the photonpressureofthelightmakingthemeasurementisasourceofnoise.Ifthespacetimefieldbecomesabirefringentmedium,thenitwouldbepossibletodetectgravitationalwaveswithoutusinganinterferometer. Forexample,circularlypolarized lightpropagating in theZdirection,shouldencounterabirefringence differenceinε intheXandYdirections.Thiswouldcauseevenasingle beamof circularly polarized light to become an oscillating ellipsoid of polarized light.Otherexperimentscanalsobedevisedbutthepointisthatitismucheasiertodetectchangesinasinglepolarizedbeamthanchangesinthepathlengthoftwobeamspropagatingintwowidelyseparatedarmsofaninterferometer.Also,thisproposedbirefringencecanalsocreateproblemsforinterferometers.Ifthetwobeamsofaninterferometerhavethesamepolarizationdirection,thenthiswouldcausethesamepathlengthchangetooccurinbotharmsoftheinterferometer.Eveniftherewasalargesignal,botharms would experience the same path length change. The interferometer would detect nodifferenceinpathlengthandtherewouldbenosignal.

MagneticFieldAnalysisComparison between Electric and Magnetic Fields:Whatistheeffectonspacetimeproducedbyamagneticfield?Weknowthatanelectricfieldandamagneticfieldareintimatelyconnectedbythespeedoflight c .Amagneticfieldinoneframeofreferencecanappeartobeanelectricfieldandmagneticfieldinanotherframeofreference.Therearetwocommonwaysofproducingamagneticfield1 anelectriccurrentinawireand2 themagnetic field associatedwith the spin of subatomic particles. There is a very goodexplanationofamagneticfieldgeneratedbycurrentinawire.ThisexplanationbyEdLowry1isbasedonthespecialrelativitytransformationofanelectricfield.Whencurrentflowsinawire,the negatively charged electrons in the wire are moving relative to the positively charged

1 http://users.rcn.com/eslowry/elmag.htm


protonsinthewire.Therefore,thepositiveandnegativechargesinthewireareintwodifferentaverageframesofreference.Whenanexternalelectronmovesparalleltothewire,italsoisinadifferentframeofreference. Themovingelectronexperiencesatransverseelectric fieldthatexertsa transverse forceon theelectron. This forcewillbe towards thewire if theexternalelectronismovinginthesamedirectionastheelectronsinthewire.Theforcewillbeawayfromthewireifthemovementsareinoppositedirections.Whilethisexplanationgivesimportantinsights,itdoesnotexplainthedistortionofspacetimeproducedbyamagneticfield.Theforceontheexternalelectronincreaseswithrelativespeed.Whentheelectronistravelingnear the speedof light, theenergydensity of themagnetic fieldhasbeenalmost completelytransformed into a transverse electric field. A photon is propagating at the speed of light,thereforeifitispropagatinginatransversemagneticfieldthephotonexperiencesatransverseelectricfield.Thedirectionoftheelectricfieldis90°relativetothetransversemagneticfield.Whateffectwouldthishaveonaphoton?Unfortunately,atransverseelectricfieldproducesasubtle effect. A transverse electric field is polarized spacetime exhibiting the asymmetrypreviouslydiscussed. Thetransversedirectionoftheasymmetrywouldproducenoeffectondistanceandnoeffectonthepolarizationof the light. Instead, Iproposethat theonlyeffectwouldbethatthedirectionofpropagationofthelightwouldnotbepreciselyperpendiculartothewavefront.Aconvergingbeamoflaserlightwouldcometoafocusatonespotwhenthereisno transversemagnetic fieldand focusataslightlydifferentspotwhen there isa transversemagneticfield.Inatypicalexperimentthesespotswouldoverlaptoadegreethatitwouldnotbepossibletomeasurethedifference.Thesignaltonoiseratiowouldbetoolow.An Electron’s Magnetic Field:Thespinofanelectronproducesamagneticfieldthatisalignedwiththespinaxis. Therefore,perhapsitispossibletoobtainaninsightintothedistortionofspacetimeproducedbyastaticmagneticfieldbylookingattherotarmodelofanelectron.Sinceamagneticfieldiscloselyrelatedtoanelectricfield,amagneticfieldmustalsobeassociatedwithastrainofthelengthdimensionswithnodistortionoftherateoftimeandnochangeintotalvolume.Anelectronismostwell‐knownforitselectricfield,butthespinoftheelectronalsoproducesamagnetic field. Theenergydensityofanelectron’smagnetic fielddecreasesmore quickly with distance than the electron’s electric field. Therefore the electric fielddominateswhendistancerismuchgreaterthananelectron’srotarradius .However,whenr ,theelectron’smagneticfieldshouldhaveacomparableenergydensitytotheelectron’selectricfield.We will begin the quest to deduce the spacetimemodel of a magnetic field by doing someplausibilitycalculationstoseeiftherotarmodelofanelectroncangiveplausibleagreementwiththemagneticpropertiesofanelectron.Iftheuniverseisonlyspacetime,thenamagneticfieldmust be a distortion of the spacetime field. We are going to attempt tomake a connectionbetweenanelectricalcurrentflowingaroundaloopofwireandtherotarmodelofafundamentalparticle.Weknowthatacurrentflowingaroundaloopofwireproducesamagneticfield.Ifthe


properconnectionismade,thenweshouldbeabletoseehowtherotatingdipolewaveoftherotarmodelproducesanelectron’smagneticfield.Therotarmodelofafundamentalparticlehasadipolewaveinthespacetimefieldchaoticallypropagatingatthespeedoflightaroundavolumeofspace.Thisvolumecanbemathematicallyapproximated by the rotar model with a circumference equal to the particle’s Comptonwavelength.Thereforetheradiusofthisvolumeisequalto .Eventhoughthepropagationischaotic,thereisadefinableexpectationrotationdirectionandrotationaxis.Supposethatwetestthepostulatethatthemagneticfieldproducedbyanelectronisequivalenttoapointparticlewithchargeepropagatingatthespeedoflightaroundaloopwithradiusequaltotherotar’srotarradius . Thisradiushasacircumferenceequaltotherotar’sComptonwavelengthλc.Sincethepropagationspeedequalsthespeedoflight,therotationfrequencyaroundtheloopwouldbeequaltotheparticle’sComptonfrequencyνc ωc/2π.Wewillassumeafixedaxisofrotationratherthanthechaoticaxisofthedipolewave.Withtheseassumptions,itispossibletodeterminetheelectron’scirculatingcurrent designatedIe thatwouldbeflowingaroundthishypotheticalloop.Wewillassumetheconstantsassociatedwithanelectron.Therefore 3.86 10‐13mandanelectron’sComptonfrequencyisequalto:νc c/λc 1.23 1020Hz.Theelectron’sequivalentcirculatingcurrent symbolIe issimplyelementary charge e 1.602 10‐19 Coulomb times the electron’s Compton frequencyνc 1.236 1020Hz.e eνc 19.796amps e electron’sequivalentcirculatingcurrent~19.8ampsWecanchecktoseeifthiscurrentproducesthecorrectmagneticeffectsifweimaginealoopofwirewithradiusequaltoanelectron’srotarradius .Specifically,wewillseeifthiscurrentinthissize loopofwirewouldproducethesamemagneticmomentasanelectron beforeQEDinteractions . Themagneticmomentμmofaloopofwirewitharea πr2andcurrent is:μm .Tosimulateanelectron,wewilluse: π and c eνc.μm π c π 3.8616 10‐13m 2 19.796amp 9.274 10‐24J/TeslaThis is a successful test because 9.274 10‐24 J/Tesla equals an electron’s Bohr magneton

μB eħ/2me 9.274 10‐24 J/Tesla. A more rigorous calculation not shown here

incorporating ,e,νcetc.showsthattherotarmodelofanelectrongivesμB eħ/2me.Theterm“Bohr magneton” is used to express a simplified version of an electron’s magnetic dipolemoment.Theexperimentallymeasuredvalueofanelectron’sdipolemomentdiffersfromthisBohrmagnetronnumberbyabout0.1%.ThissmallcorrectionfactorobtainedfromQED theanomalousmagneticmoment will not bediscussed further because the objective here is todevelop a spacetime basedmodel of amagnetic field. To achieve this goal,wewill test thepostulatethattherotarmodelofanelectron onaxis producesthesamemagneticfieldasif


therewasapointparticlewithchargeepropagatingatthespeedof lightaroundaloopwithradiusequaltotheelectron’srotarradius .Fromthispostulate,theelectron’smagneticfieldcalculatedfrom cand willbedesignated e.Theequationforthemagneticfieldatthecenterofasinglecircularloopofwirewithradiusrandcurrentis:

μo /2rset candr foranelectron 3.86 10‐13m e 3.22 107Tesla e electron’sequivalentmagneticfield

Thislargemagneticfieldwouldhaveenergydensityof3.22 1022J/m3.Totestthepostulate,wemustaskthequestion:Isthisreasonable?Howdoestheimpliedenergyinthemagneticfieldcompare to the energy in the electron’s electric field external to the electron’s rotar volumeexternalto ?Thecalculationisnotshownhere,buttheenergyintheelectricfieldexternaltoanelectron’sradius is ½ αEi 3 10‐16J.Wewanttotestwhetherthelargemagneticfieldof3.22 107Teslaisreasonablebymakingacomparisontotheenergyintheelectron’sexternalelectricfield.Wewillmaketheassumptionthatthemagneticfieldfillsavolumeequaltotheelectron’srotarvolume Vr 4π/3 2.41 10‐37m3 .Italsoproducesanexternalmagneticfield,sothisestimateshouldbelow.Istheenergyintheelectron’sexternalelectricfield ~3 10‐16J comparableto,butmorethan,theenergyina3.22 107Teslamagneticfieldfillingtheelectron’srotarvolume?E UVr B2/2μo 4π/3 10‐16JenergyinthecalculatedmagneticfieldinvolumeVrThereforethissimplecalculationshowsthatamagneticfieldof3.22 107Teslaisreasonable.Wecanalsoperformamoreexacttest.Recallthatforanyfundamentalrotarsuchasanelectron,theslopeofthestraincurveatarbitrarydistancewasalwaysimpliedaspatialdisplacementatthecenterofanelectronof√ Lpwhichis1.38 10‐36m.Outofcuriosity,wewilllookattheimpliedΔL fora3.22 107Teslamagnetic fieldoveradistanceequal to theelectron’srotarradius 3.86 10‐13m.EventhoughwehavenotyetproposedthephysicalinterpretationofΔLproducedbyamagneticfield,itisstillpossibletocalculatethemagnitudeofΔLforthismagneticfieldof3.22 107Teslaoveradistanceequalto .ForauniformelectricfieldoverdistanceL,wepreviouslyhadΔL 2

pL which converts to 2L L (note underlines implying

dimensionless Planck units). NowwewillreplaceLwith ,whichistheradiusoftherotarmodelofafundamentalparticle.Wewillnotproveithere,butaLorentztransformationofanelectricfield expressed in dimensionlessPlanckunits / p canbe considered equivalent to amagnetic field expressed in dimensionless Planck units / p . In this equation p isPlanckmagnetic fieldwhich is p Zs/qp 2.15 1053 Tesla. Thismeans that, ignoringnumericalfactorsnear1,itisreasonabletoconvert 2L L to 2

cL whenwearedealingwithafundamentalparticlewithradius .

2cL dimensionless Planck units

ΔL = / pLpconvertedfromdimensionlessPlanckunitstoSIunitsSet 3.22 107Tesla, p 2.15 1053Tesla,Lp 1.6 10‐35m, 3.86 10‐13m


ΔL ≈ 1.38 × 10-36 m ≈ √ LpAmoregeneralcalculation notusingnumbers canbemadeanditgivesthesameanswerthatΔL = √ Lp.Therefore,thisisasuccessfulplausibilitycalculationontwofronts.First,itsaysthatthecalculatedmagneticfieldproducesthesamemagnitudeofΔLoverthesamedistance astheelectricfieldfortherotarmodelofanelectron.However,thephysicalinterpretationofΔLisdifferentforamagneticfieldthanforanelectricfield.Secondly,itgivesaddedassurancethatwecanlooktotherotarmodelofanelectrontounderstandthedistortionofthespacetimefieldthatproducesamagneticfield.Aspreviouslyexplained,therotationofanelectron’sextremelysmalldistortionofthespacetimefield ischaoticbecause it isat the limitofcausality. However foranalysis,wecan imagineastabilizedelectronrotationwithafixedaxisofrotation.Proceedingalongthisaxisofrotation,wewouldexperiencethedistortionofthespacetimefieldthatisproducingamagneticfieldofabout3 107Tesla. This is a very strongmagnetic field compared to anything that canbegeneratedbyman.However,itisalsoveryweakcomparedtoPlanckmagneticfieldofabout1053 Tesla. What is different about this rotar volume compared to a typical volume of thespacetimefieldthatisnotinsideanelectronanddoesnothaveamagneticfield?The“typical”volumeofthespacetimefieldcontainschaoticdipolewaveswithPlanckenergydensity about10113J/m3 ,butthedipolewavedistortionaveragesouttobeingashomogeneousasquantummechanicsallows Theobviousdifferenceisthattheelectron’saxialvolumealsocontainsanorganized spatial and temporal distortion of the spacetime field that has a small rotatingcomponentwithstrainamplitudeequaltoAβ 4.2 10‐23.Forreview,seefigures5‐1and5‐2from chapter 5. This would produce a rotational path length difference that is difficult toarticulate.Recallthatanelectricfieldisproposedtobea distortion of one spatial dimension of the spacetime field (parallel to the electric field direction) that results in a slight asymmetric distance proceeding in opposite propagation directions. The round trip distance is unchanged. A magnetic field is similar except that the plane transverse to the magnetic field has a rotational distance asymmetry. The time required for a particle with known speed to proceed around the circumference of a circle proceeding clockwise in this plane would be slightly different than the time required to proceed counterclockwise. Since light is a transverse wave, it is not accurate to imply that a pulse of light propagating around a circle would be able to measure this subtle effect. Therefore terms like “speed of light” would be misleading. However, since light is a transverse wave, the implication is that a total vacuum should exhibit a Faraday effect for light propagating parallel to the magnetic field direction. Ifaphotonwasabullet‐likeenergypacket,thenwewouldnotexpectanyeffectifitpropagatesthroughavolumeof space that is experiencing a rotational asymmetry of the spacetime field. However, atransverse wave propagating through a rotationally strained volume of spacetime would


experience a polarization effect. Linearly polarized light propagating along amagnetic fieldshouldexperienceaslightrotationoftheplaneofpolarization.Circularlypolarizedlightwouldexperienceapathlengthchange.ThevalueΔLcanbethoughtofasthedifferenceinpathlengthexhibitedbyoppositecircularpolarizationspropagatinginamediumwithamagneticfield.ThisisessentiallyapredictionthatacompletevacuumcontainingamagneticfieldshouldexhibitaFaradayeffect. Themagnitudeoftheeffectisdifficulttopredictsinceitisnotclearhowtotreatthetransversesizeofaphotonpropagatingalongthemagneticfield.Ifitispossibletomakeananalogytoanelectricfield,thenhereisoneattemptatacalculation. ThemagnitudeofΔLcanperhapsbecalculated from 2

cL which converts to 2 2 2p pL L L L when we set L . Also

p 2.15 1053TeslaandLp 1.6 10‐35m.

2

2p p p

L L

L L

2

53 352 10 1.6 10

LL

tesla meter

= 3x10-19 L2meter

Thereisaspecialwaythatthelengthtermmustbecalculatedexplainedintheanalogouselectricfield experiment calculation. Without going into the details, the experiment is marginallydetectable ΔL 10‐18or10‐19m forsomehypotheticalvalues.Therearesomepossibleeffectswhichmight hide the proposed vacuumeffect. For example, any residual gas in the vacuumwouldexhibitacompetingFaradayeffect.However,themagnitudeoftheeffectduetoresidualgascouldbeapproximatelyoffsetbychangingthegaspressureandattemptingtosubtractoutthiseffect.Comparison of Models: As a parting gesture, I justwant to stand back and compare theproposedspacetimebasedmodelofastaticelectricormagneticfieldtothecurrentlyacceptedstandardmodel. In the standardmodel all force is conveyedby “messengerparticles”. Theelectromagneticforceisconveyedbyvirtualphotonswhicharenottobeconfusedwithvirtualphotonpairs seechapter7 .Astaticelectricfieldandastaticmagneticfieldbothhaveenergydensity.Thisisarealeffectthat implies some tangible difference between a volume of spacetime that has anelectric/magnetic field and a volume of spacetime that has no electric/magnetic field. The“Reissner‐Nordstrom"solutiontoEinstein'sfieldequationgivesthegravitationaleffectofthisenergy density, but there is no theoreticalmodel of an electric ormagnetic field itself. Theelectromagnetic force issupposedlytransferredbymessengerparticles. This impliesthatanelectricormagneticfieldmustgeneratevirtualphotonswithoutanycontactwiththematterthatisgeneratingthefield.Twoexampleswillillustratethis.First,supposethatafreeneutronisstationaryinamagneticfield.Whenitdecaysitgeneratesarapidlymovingelectron,aproton


andelectronantineutrino.Therapidlymovingelectronandtheslowerprotonimmediatelyfeela Lorentz force exerted by the magnetic field. The force happens before speed of lightcommunicationexerts anopposing forceon the sourceof themagnetic field. Therefore, theLorentzforceonthemovingelectronisbeinggeneratedbyvirtualphotonswhichmustoriginatefromwithinthemagneticfielditself.Istherealimittotheamountofforcethatcanbegeneratedbyaweakmagneticfieldwithoutcommunicationbacktothesource?Thesecondexamplewillexaminethisquestion.Supposethatthemagneticfieldofastarequalstheearth’smagneticfieldstrength ~5 10‐5Tesla atadistanceof3 109mfromthestar.Therefore,atthisdistanceanyforceexertedbythemagneticfieldtakesabout10secondstobecommunicatedbacktothestar.Nowatthisdistance,supposethatthereisasquareloopofwirethatisonemeteroneachside.Furthermore,supposethattwoofthe4sidesareparalleltothemagneticfieldandtwoofthe4sidesareperpendiculartothemagneticfield.Ifacurrentflowsinthiswire,aLorentzforcewillbeexertedonthetwoperpendicularsectionsofwire onemetereach andanettorquewillbeexertedontheloopofwire.Theoretically,anycurrentcanbemadetoflowintheloopofwireuptoPlanckcurrentwhichisabout3.5 1025amps.Forexample,acurrentof2millionampswouldexerta100Newtonforceoneachofthetwowiresectionsthatareperpendiculartothemagneticfield.Ifthecurrentflowisstartedquickly,thenallthetorqueexertedonthewireloopcomesfromaninteractionwithalimitedvolumeofthemagneticfield.Theenergydensityofa5 10‐5Teslamagneticfieldisonlyabout10‐3 J/m3and it takes10 seconds to transfer this torque to the star. Therefore, if thecurrentstartedover3 10‐8seconds,themaximumvolumethatcouldbeaccessedatspeedoflightcommunicationwouldbeabout1,000m3.Thislimitedvolumehasonlyabout1Jouleofenergyinitsmagneticfield.Whilenoenergyisbeingextractedfromthestar’smagneticfield,itwouldtake3 1010wattsofrealphotonstogeneratea100Nforceifthisforcewasgeneratedbyphotonpressure.A100Newtonforcelasting10‐8swouldrequireabout300Joulesifitwasgeneratedbydeflectingrealphotons.Howdovirtualphotonswithnorealenergyaccomplishthis?Thisexampledoesnotimplyaviolationoftheconservationofmomentum.Apowerfulmagneticfieldisbeingestablishedandthestar’sweakmagneticfieldisdistortingtheformationofthenewmagnetic field. However, thequestion remains:Howexactlydoes thevirtualphotonmodelexplain the forcemagnitude 100N and forcevectorsexertedon thesewires?Carrying thisthoughtexperimenttoanextreme;Planckcurrentwouldgenerateaforceofabout1021Noneachofthetwowiresectionswithoutcommunicatinganytorquebacktothestar.Thereisnocommonlyacceptedexplanationinthestandardmodelforanelectric/magneticfieldintermsofsomethingmorefundamental.Ontheotherhand,thespacetimebasedmodeloftheuniversecaneasilyexplainanelectric/magneticfieldintermsofadistortionofthespacetimefield.Eventheinstantaneousgenerationofa1021Newtonforcecanbeexplained.Themagnetic


fieldisadistortionofthespacetimefieldwithitsseaofdipolewaves.ThemaximumforcethatthespacetimefieldcanexertisequaltoPlanckforcewhichisabout1044N.Thespacetimemodeloftheuniversehastheabilitytoexplainmanyofthemysteriesofquantummechanics.

SpacetimeUnitsIn chapter 10 we will try to combine the insights gained from charged particles and fromelectromagneticradiationtogiveaconceptuallyunderstandablemodeloftheexternalvolumeofchargedrotars.Thelaststepinthischapteristobuildontheinsightsthatweregainedintheexercisethateliminatedchargeasaunit.Eliminatingchargeandreplacingthiswithastraininspacetimeisasteptowardsdevelopingunitsbasedonthepropertiesofspacetime.However,thisstepdoesnottaketherealplunge.Itisnecessarytoeliminatemassasthefundamentalunitanddevelopunitsbasedonlyonthepropertiesofspacetime.Eventhoughlengthandtimearerelatedtospacetime,themeterandsecondarehumanconstructs.Planckunitshavealwaysbeenconsideredthemostfundamentalofunitssincetheyarenothumanconstructs.Ithasbeensaidthatifaliensattemptedtocommunicatewithus,theywouldusePlanckunitsbecausetheseunitsarederivedfromtheconstantsofnature.Whatcouldbemorefundamentalthanasystemofunitsbasedonħ,Gandc?Iftheuniverseisonlyspacetime,itshouldbepossibletoexpressconstantsandunitssuchaskilogram, Newton and Coulomb using only the fundamental properties of spacetime. Thisspacetimeconversion isnotparticularlyconvenient touse,and it iscloselyrelatedtoPlanckunits.However,itisveryinformativetoseehowcommonunitscanbeconstructedoutoftheproperties of spacetime. In particular, it is important to grasp the idea that mass is not afundamental unit when we look at the universe from the standpoint of spacetime beingfundamental.Massisameasurementofinertiaandinertiaisacharacteristicofenergytravelingatthespeedoflightinaconfinedvolume.Deflectingenergytravelingatthespeedoflightcausesmomentumtransfer.Thisisthesourceofall forcesincludingthepseudo‐forceof inertia.Thegoalistoexpresseverything,includingmass,intermsofthepropertiesofspacetime.Wewillstartthesearchforspacetimeunitsbylookingatoneofthe5wave‐amplitudeequationspreviouslydescribed.U A2ω2Z/cequationgivingtheenergydensityinawaveWhenweapply thiswave‐amplitudeequation tospacetime, it shouldbeeasy toexpress thisequationifweusethefundamentalpropertiesofspacetime.Thefirstobviouscandidateforafundamental property of spacetime is “Z” the impedance of spacetime Zs c3/G . Anothercandidateis“c”,thespeedoflight,butthisisnotascertainasZs.


OthercandidatesforbeingfundamentalunitsofspacetimemustbecontainedintheamplitudetermA.Weknowthatageneralexpressionofthemaximumpermitteddipolewaveinspacetimeis:Amax Lp/λ Tpω.Thisisthemaximumstrainamplitudewhichinturnisdictatedbythemaximumdisplacementamplitudeofspacetime:dynamicPlancklengthLpanddynamicPlancktimeTp. It is proposed thatdynamicPlanck lengthLp anddynamicPlanck timeTp arebothfundamental properties of spacetime. They are added to our list making a total of fourcandidates.Therefore,thefourcandidatesareZs,Lp,Tpandc.Wereallyonlyneedthreetermstoexpresseverythingintheuniverse,thereforetherearethreepossiblecombinationsofthreetermsthatcouldserveasthebasicunitsofspacetime.Theseare:1 Tp,Lp,Zs;2 c,Lp,Zs;and3 c,Tp,ZsAllofthesecombinationshaveadvantagesanddisadvantages.Iwillusethecombinationof:c,Tp,Zs, as the units of spacetime. After working with the different combinations I find thiscombinationthemostintuitive.Forexample,thespeedoflightandtheimpedanceofspacetimeseem to belong together. The unit of Planck time becomes the quantized heartbeat of theuniverse.Whileworkingtodevelopthemodelofelectricfieldandcharge,thiscombinationissomehoweasiertovisualize.RecallthattheimpedanceofspacetimeisPlanckmassdividedbyPlanck time. Zs Mp/Tp . Thereforeall conventionalunits canbeexpressedusing these3propertiesof spacetime. However, theuseof c,Tp, andZs gives answers that correspond toPlanckunitswhicharenotconvenientforeverydayuse.Toillustratehowthesespacetimeunitswork,theunitofforcehasdimensionalanalysisunitsofML/T2andconventionalunitsofkgm/s2.ThespacetimeunitsofforcearecZs.However,theseunitsspecifyPlanckforce ~1.2 1044N whichisthelargestforcespacetimecanexert.Foranotherexample,tospecifythegravitationalconstantGusingconventionalunitsitisnecessarytoincludeaconstant 6.673 10‐11 andtheunitsofm3/kgs2.Withspacetimeunitsthegravitationalconstantisequalto1andtheunitsofthegravitationalconstantarec3/Zs.Thespacetimeunitsonthenextpagetreatchargeasastrainofspacetimewithunitsoflength.Recallthatthechargeconversionconstantηis:Wehave longago found theoptimumwaysof expressingconversionconstants that simplifycalculations.Instead,thisexerciseisintendedtoillustratehowthepropertiesofspacetimecanbemanipulatedtoproducefamiliarconstantsandunitsofphysics. Itmaybedifficultforthereader to imagine physicswithoutmass or energy being a fundamental unit. However, themaximum force that spacetimecansupport is cTpand themaximumquantizedmass isTpZs.Massandenergyareaquantificationofpropertiesofspacetime.Thischangeinperspectivehasagreatdealofappealonceitisinternalized.Onthefollowingtable:

η≡ √

8.617 10‐18meter/Coulomb


Spacetime Units NameSpacetimeConversionelementarychargee 1/η √ cTpimpedanceoffreespaceZo η24π Zsspeedoflightc cPlanck’sconstantħ c2Tp2ZsgravitationalconstantG c3/Zs

Coulombforceconstant 1/4πεo η2cZspermeabilityoffreespace μo/4π η2Zs/cTransformationofPlanckUnitsintoSpacetimeUnitsPlanckUnitsStandardConversionSpacetimeConversion

PlancklengthLp ħ ⁄ Lp cTp

Planckmassmp ħ / mp TpZs

Planckfrequencyωp /ħ ωp 1/Tp

PlanckenergyEp ħc /GEp c2TpZsPlanckforceFp c4/GFp cZsPlanckpowerPp c5/GPp c2ZsPlanckenergydensityUp c7/ħG2Up Zs/cTp2PlanckimpedanceZp 1/4πεocZp η2ZsPlanckchargeqp 4πε ħcqp 1/η cTp

Planckelectricfield p c4/G 4πε ħ p ηZs/Tp

Planckmagneticfield p c3/G 4πε ħc p ηZs/cTp

Planckvoltage p c /4πε G p ηcZsWhileitispossibletoexpressalltheunitsofphysicsusingonlythepropertiesofspacetime c,TpandZs ,itwillbeshowninchapter14thatitisnecessarytoaddoneadditionaldimensionlessdesignation Гu to quantify the changing properties of spacetime. Aswill be explained inchapter14,thespacetimefieldisundergoingatransformationthatischangingalltheunitsofphysicsrelativetoanabsolutestandardthatisunchangedsincetheBigBang.


Chapter10

Rotar’sExternalVolume

External Volume of an Electron (Conventional Model): Beforepresenting the spacetimebased model of the external volume of a fundamental particle, we will first look at thecompetition.Theconventionalmodelofanelectronisapointparticle orvibratingstringwithnovolume surroundedbyanelectricandmagneticfield.Theenergydensityofamacroscopicelectricfieldfromelementarychargeeis:U 1/8π αħc/r4 .Theenergyintheelectricfieldexternaltoagivenradialdistanceris:Eext 1/8πεo e2/r ½ αħc/r .Whenr λc,thentheelectricfieldenergyexternaltothequantumradiusλcisEext ½ αEi whereEiistheinternalenergyoftheelectron.Thisenergydensityshowsthatanelectron’selectricfieldisarealphysicalentity.Aninteractionwithanelectron’selectricfielddoesnotexhibitanydelaythatwouldoccurifmessengerparticleshadtobesentoutbyanelectron. Inchapter9anexamplewasgiveninvolving themagnetic fieldof a star. This example clearly illustrates the inadequacyof theexchangeof virtualphotonmessengerparticles to explain theelectromagnetic force. In thischapterwewilldevelopfurtherthespacetimebasedexplanationofelectricandmagneticfields.Iftheelectron’sradiusislessthantheclassicalradiusofanelectron ~10‐15m thenthereisanadditionalproblemwiththepointparticlemodelbecauseasmallerradiusmakestheenergyintheelectricfieldexceedthetotalenergyoftheelectron.Forexample,ifaparticleorthevibratingstringwasconsideredtobecontainedinavolumewitharadiusofPlancklength,thentheenergyinthesurroundingelectricfieldwouldbeabout1020timeslargerthantheelectron’sinternalenergy 107Jcomparedto∿10‐13J .Thisproblemisusuallyignoredbysayingthattheelectronhasan“intrinsic”electricfieldassociatedwithelementarychargee.Ifweareattemptingtogiveconceptually understandable explanations of quantum mechanics using the properties ofspacetime, thenwedonothave the luxuryofbeingable to ignoresuchproblems. It isevennecessarytodescribechargeandelectricfieldintermsofthepropertiesofspacetime.External Volume of a Rotar:Wewillnowlookatthespacetimemodelofthe“fields”associatedwithafundamentalparticle.Arotarhaspreviouslybeendescribedasaunitofquantizedangularmomentum in a sea of vacuum fluctuations. These vacuum fluctuations have superfluidpropertiesaspreviouslydescribed.Thevacuumfluctuationscannotpossessangularmomentumandthereforeanyangularmomentummustbeisolatedintoquantizedunitsjustlikesuperfluidliquidheliumisolatesangularmomentumintoquantizedvortices.The“rotarvolume”ofarotarpossesses angular momentum so this volume is in a different state than the surroundingspacetimefield. Whilethevacuumfluctuationssurroundingarotaravoidpossessingangularmomentum, the surrounding volume is still affected by the presence of a rotar quantizedangularmomentum initsmidst.Therotarproducesdisturbancesinthevolumeexternaltothe


rotarwhichslightlyaffecttheseaofvacuumfluctuationsthatsurroundarotar.Thischapterwillexaminethestandingwavesandstaticstrainproducedinthevolumesurroundingarotar.Theseeffectsareresponsiblefornotonlytherotar’sgravitationalandelectromagneticfieldsbutalsonumerousothereffectsincludingdeBrogliewavesandComptonscattering.Theprobabilityofinteractingwitharotar findingaparticle doesnotendattheedgeoftherotarvolume.Theedgeoftherotarvolumeismathematicallysignificantbecauseitallowsustocharacterizepropertiesanddimensions,buttheproposedquantummechanicalnatureofarotarisnotboundbyourconvention.Thereispartofarotarthatextendsfarbeyondtherotarradius.Theseexternaleffectswillbeshowntobebothoscillatingstandingwavesandstaticstrains

distributedacrosstheseaofvacuumfluctuationsthatarepartofspacetime.Thevolumebeyondtherotarradiuswillbecalledthe“externalvolume”.Thisexternalvolumestillpossessespartoftherotar’squantizedangularmomentum.Thereforetheexternalvolumeisstillconsideredtobepartoftherotarbuttheexternalvolumehasdifferentcharacteristicsthantherotar’squantumvolume.ThedipolewaveinspacetimeresponsibleforarotarhaspreviouslybeendescribedasrotatingatitsComptonangularfrequencyandpossessingamplitudeof:Aβ Lp/ Tpωc.Itwasalsoproposedthattherotarisattemptingtoradiateawayitsenergyintotheexternalvolume theseaofvacuumfluctuations .Theamplitudeofthisattemptedradiationhasbeendesignatedthe“fundamentalamplitude”Af.Thisfundamentalamplitudedecreaseswithdistancersuchthatthehypotheticalamplitudewouldbe:Af Lp/r cTp/r.Iftherewerenooffsettingeffects,thisamplitudewouldradiateawayarotar’s fullenergyinatimeof1/ωcwhichistypically intherangeof10‐21to10‐25s.Thisisthesameashavingnostability.Thefewrotarfrequenciesthatarestableorsemi‐stablemustproduceaninteractionwithvacuumenergythatgeneratesanewwavethatcancelsenergylossbutleavesoscillatingstandingwaves.Forexample,anelectronhaslongtermstabilitythereforetheprobabilityofenergylossiszero.However,thisdoesnotmeanthatalloftheenergyofanelectronisconfinedtoitsrotar’squantumvolume.Thereisabattle going on in the external volumebetween the attempted emission and the cancelationwaves.Theresidualeffectsthatexistintheelectron’sexternalvolumeareresponsiblefortheelectron’sgravity,theelectron’selectric/magneticfieldsandtheelectron’sdeBrogliewaves.Gravitational and Electromagnetic Strain Amplitudes:Inchapters6and8itwasshownthatgravityistheresultofspacetimebeinganonlinearmediumfordipolewavesinspacetime.Whilethereiscancelationofthefundamentalwaveemission,thenonlineareffectsremainfromthebattle.Thisresultsinanon‐oscillatingstraininspacetimewhichhaspreviouslybeendesignated“gravitational magnitude β”. In this chapter we are dealing with the various amplitudesassociated with rotars. Therefore, the gravitational non‐oscillating strain amplitude inspacetimeproducedbyarotarwillbedesignatedbythesymbolAG β Gm/c2r Aβ2/ wherehaspreviouslybeendesignatedasthenumberofreducedComptonwavelengths: r/ .


TherewasalsoaproposedoscillatingcomponentofgravitywithstrainamplitudedesignatedAg.Itwillbeshownthatthisoscillatinggravitationalcomponentintheexternalvolumeresultsinenergyexternal to that is in therangeof1040 timessmaller thanarotar’s internalenergythereforeitisundetectablewhendealingwithindividualrotars.However,theimplicationisthatthe gravitational field of massive bodies has an oscillating component that can achievesubstantialenergydensity.Thiswillbediscussedattheendofthischapter.As before, the simplest example used for illustration in this chapter is a single isolatedfundamentalparticlewithelementarychargee. Onlyelectrons,muonsand tauonsmeet thiscriterion.Tofurthersimplifythesemanticsitiseasiesttouseelectronsinexamples.Therefore,thischapterwillattempttodescribetheexternalvolumeofanelectron.Oncethisisdonetherewillbesomediscussionoftheexternalvolumeofprotons,neutrons,etc.Aspreviouslystated,understandingtheconnectionbetweenelectricfields magneticfields anddipolewavesinspacetimehasbeenthemostdifficulttaskindevelopingthespacetimebasedmodeloftheuniverse.Furthermore,themostdifficultcomponentofthisexplanationhasbeenmodeling the electric fieldof a rotar. In chapter9we concluded that an electron andotherchargedleptonswithchargeeproduceanon‐oscillatingstraininspacetime.AtdistancerthisstraincorrespondstothedimensionlessPlanckelectricalpotential √ Lp/r.Sofarithasnotbeenexplainedhowthisnon‐oscillatingstrainisproduced.Thebreakthroughoccurredwiththerealizationthatgravityhasanoscillatingcomponent figure8‐1 andastaticcomponent. Oncloseexaminationitwasfoundthattheelectricfieldproducedbyachargedparticlesuchasanelectron must also have an oscillating component and a static component. Therefore agravitational field has two strain components one oscillating and one static while theelectric/magnetic field alsohas two strain components oneoscillatingandone static . Thestaticcomponentofagravitationalfieldhasalreadybeendiscussedinchapters6and8.ThischapterwillconcentrateontheremainingthreecomponentsThemodelassumesthattheelectricfieldproducedbyanisolatedelectronpossessestheclassicalenergydensityexternaltotherotarvolumewherer . Thereisnocontinuouslossoftheelectron’senergy,sotheseexternaloscillationsmustbestandingwavesthatremainaftertheproposedcancelationthatmusttakeplacetoeliminateemissionofenergyatfrequencyωcandamplitudeAf Lp/r.Inorderforthestandingwavestoachievetheenergydensityoftheelectricfield,itisnecessaryforsomepartoftheelectron’senergytoresideoutsidedistance .Wewillcalculatetheoscillating“standingwave”amplitudedistributionrequiredtoachievethisenergydensity.

U ½ εo 2energydensityinanelectricfield ofasingleelectron

1/4πεo e/r2electricfieldproducedbyaparticlewithchargee


U ħsubstitutionincludingαħc

4

WealsohaveU A2ω2Zs/cfromthe5wave‐amplitudeequations.Thereforewecansetthesetwoenergydensityequationsequaltoeachotherandignoredimensionlessconstants.A2ω2Zs/c αħc/r4substituteZs c3/Gandω c/ ,thensolveforAA2 α ħG/c3 /r4setħG/c3 Lp2;A Aeand r/

Ae √

√

√

Ae oscillating standingwave amplitudecomponentoftheelectricfieldatfrequencyωcNotethatthedifferentComptonfrequenciesofanelectronandamuonareabsorbedintotheAβand terms which both have a frequency dependence Aβ Lp/ ωc/ωp and r/ rωc/c WhilethisoscillatingamplitudeAegivesthecorrectenergydensity,these

standing waves do not directly convey force between charged rotars. If the oscillatingcomponentwasresponsibleforelectrostaticforce,thiswouldimplythatoscillatingenergyintheexternalvolumewaspropagatingandenergywouldbecontinuouslyradiated.Thestandingwavesinarotar’sexternalvolumedonotdirectlygenerateforces.However,theyareindirectlyresponsiblefortheforcesbetweenrotars. Hereisthepicturethathasemergedafterlengthyexamination.Arotarisattemptingtoradiateawayitsenergytothesurroundingseaofvacuumenergy.Thefewfundamentalparticlesthatarestableexistatoneofthefewspecialfrequenciesthatgeneratecancelingwavesinvacuumenergyeliminatingthelossofenergy.Eventhoughthelossofenergyiseliminated,therearefourresidualeffectsthatshowthatabattlehastakenplace. These4residual effects are really combined into a distortion of spacetimewith oscillating and non‐oscillatingcomponents,butforanalysisitisconvenienttoseparatethemintocomponentparts.

1 There is non‐oscillating strain in spacetime responsible for gravity and previouslydiscussed in chapters 6 and 8. This strain has been designated as the gravitationalmagnitudeβ,buttomakeadesignationAGinkeepingwithotheramplitudetermswewillalsodesignatethenon‐oscillatingtermasAG β Aβ2/ .

2 Thereisanoscillatingnonlineareffectassociatedwithgravityandillustratedinfigure8‐3asthesmallamplitudewavesonthelinedesignated“nonlinearcomponent”.ThisoscillatingcomponentofgravityhaspreviouslybeenshowntohaveamplitudeofAβ2atdistance .Itwillbeproposedthatthisgravitationaloscillatingtermexternalto hasamplitude Ag Aβ2/ 2. This gives energy density to a gravitational field calculatedlater


3 Therearestandingwaves associatedwiththeelectricfield remaininginthevacuumenergy that surrounds the rotar. These standing waves are at the rotar’s ComptonfrequencyωcandhavetheoscillatingamplitudeAe √ Aβ/ 2.

4 Itisproposedthatthereisanon‐oscillatingtermassociatedwiththeelectricfieldwithamplitude A √ Aβ/ . This non‐oscillating component is what we usuallyconsidertobeanelectron’selectricfield.Hereisthereasoning.

Thefollowingisasummaryoftheelectromagneticandgravitationalamplitudesgeneratedbyafundamentalparticlewithchargee.Forexample,anelectronhasenergyEi 8.2 10‐14J.Thismeans that it has dimensionless strain amplitude Aβ 4.18 10‐23 and reduced Comptonwavelength λc 3.86 10‐13m. Distance from the electron is specified as the number ofreducedComptonwavelengths.

Ae √ √ electromagneticstandingwaveamplitudeoscillatingatωc chargee

A √ √ electromagneticnon‐oscillatingstrainamplitude chargee

Ag gravitationalstandingwaveamplitudeoscillatingat2ωc

AG 2 β gravitationalnon‐oscillatingstrainamplitude spacetimecurvature

TheintroductionofAe,A ,AgandAGismerelyacaseofgivingnewsymboldesignationstotheconceptspreviouslydiscussed. We justderivedAe √ Aβ/ 2as theamplituderequired toproduce the energydensity of an electric field associatedwith energydensityU ½ εo 2numericalconstant½isignored .AG Aβ2/ βisthenon‐oscillatingamplituderequiredtoproducethegravitational fieldofarotar. AE is thesymbolgiventothenon‐oscillatingstraindevelopedinchapter9.Agisthesymbolgiventotheoscillatingcomponentofgravitypreviouslydiscussedanddepictedinfigures8‐1and8‐3.TheoscillatingcomponentofgravityAgwillbeexamined in more detail at the end of this chapter and shown to give energy density to agravitational field. Thepicture that emerges is thatboth theelectric/magnetic field and thegravitational fieldofarotarsuchasanelectronpossessanoscillatingcomponentandanon‐oscillatingcomponent.TheoscillatingcomponentsgiveenergydensitytothesefieldsbutthePlanck amplitude oscillations are undetectable. However, the oscillating components areessentialbecausetheycreatethenon‐oscillatingstrainsthatweeasilydetect.Non-Oscillating Strain Amplitude A : Theproposedelectromagneticnon‐oscillatingstrainamplitudeAEisresponsibleforwhatweconsidertobetheelectricfieldofchargedleptonssuchasanelectronormuon.ThestrainA √ Aβ/ lookssimilartothefundamentalamplitudeAf Aβ/ whichisthetheoreticaloscillatingstrainamplitudethatisbeingcanceledintherotarsthatarestableenoughtobeconsideredfundamentalparticles.However,thereareseveralkey


differences. First istheobviousdifferenceof √ whichreducestheamplitudebyafactorofabout 11.7 at any given value of . Second, A is a non‐oscillating strain while Af is ahypotheticaloscillatingamplitudeatfrequencyωcthatisattemptingtoradiateawaytherotar’senergy.Third,thetypeofdisplacementofspacetimeproducedbyanelectricfieldisdifferentfromthetypeofdisplacementofspacetimeproducedbydipolewavesinspacetime.RecallthatdipolewavesmodulatetherateoftimeandthedistancebetweenpointsaswellashavingthePlancklength/timeamplitudelimitation.TheelectricfieldcomponentdesignatedA isthenon‐oscillating strain which produces the non‐reciprocal characteristic previously discusseddifference in theoneway travel time fora timeof flight . Themagnitudeof thiseffect cangreatlyexceedthePlancklengthlimitationofoscillatingcomponents.Electrostatic Force Calculation: We will next check to see if the non‐oscillating strainamplitudeAEgivesthecorrectelectrostaticforcebetweentwochargedleptons twoelectrons .Whenwecalculatedthegravitationalforceonarotarproducedbyanotherrotarcreatingthenon‐oscillatingstrainAG Aβ2/ ,wecalculatedtheimplieddifferenceintherateoftimeacrosstheradiusofarotarandconvertedthistothedifferenceingravitationalmagnitudeΔβacrosstherotar.Thiswasnecessarybecauseforforcegeneration,itisonlythegradientinβorГthatisimportant,nottheabsolutevalue.Theproposedcharacteristicof anelectric field isdifferent in the sense that thedifference isbetweenoppositedirections,notthegradientacrossarotar.Infact,thedifferenceacrossthewidthofarotarisinsignificantcomparedtothedifferencebetweenoppositedirectionsattheaveragelocationoftherotarfeelingtheforce.Imaginetherotarmodelpresentedinchapter5rotating inotherwisehomogeneous spacetime. Now imagine this rotar rotating inpolarizedspacetimewhere there is a difference in the time required for speedof light propagation inoppositedirectionsacrosstherotar.Thestraininspacetimeproducingthiseffectisstatic,buttheinteractionwiththerotarisdynamic.Theinteractionoccursatafrequencyequaltoωcand

theinteractionmodulatestherotaratamplitudeofA √ Aβ/ .Thisinturnaffectstheinteraction with vacuum energy/pressure surrounding the rotar. This creates a pressureimbalancethatcanappeartobeeitherattractionorrepulsion.Themagnitudeoftheforceis:

F

setA2 α α αħ

;ω ;Zs ; ;α ħ

F αħ

αħ

ħ

ħ Success!

Thisexplanationisfarfromcomplete.Forexample,thereisnoexplanationforthedifferencebetween a force of attraction and repulsion. Also, there is no explanation for thedifferencebetweenanelectronandapositron.However,theexplanationsofferedheredomakeafirststep


towardsdevelopingaphysicallyunderstandableexplanationofelectricfieldsandelectrostaticforce.Thestandardapproachistomerelyaccepttheexistenceofmysterious“fields”withnoattempttofindanunderlyingcausality.Electric Field Cancelation:Thenon‐oscillatingcomponentoftheelectricfieldproducedbyanelectrondoesnothaveanyinherentenergydensity.Onlytheoscillatingcomponentproducingstandingwavesoscillatingatωcpossessenergydensity.Thesetwocomponentsoftheelectricfieldareconnectedsoitisnotpossibletohaveanelectricfield themeasurablenon‐oscillatingcomponent withoutalsohavingtheenergydensityprovidedbytheoscillatingcomponent.Theoscillatingcomponentproducesthenon‐oscillatingeffect.It is possible to cancel out the non‐oscillating strain of an electric field when two oppositechargesarebroughttogether.However,exactlywhathappenstotheoscillatingportionwhenoppositechargesarebroughttogetherhasnotbeenexactlydetermined.Forexample,bringinganelectronandprotontogethertoformahydrogenatominitslowestenergystatereleases13.6eVofenergy.ThisenergyprobablycamefromareductionintheComptonfrequenciesofthefundamentalparticles.However,theremighthavealsobeenareductionintheamplitudesAeforeachparticlecomparedtothevalueofAeifeachoppositelychargedparticlewasisolated.Thisispartoftheunknown.However,evenaneutronstillhasdeBrogliewavesasdemonstratedbyadoubleslitexperimentusingneutrons.Therefore,notalltheoscillatingportionofenergyintheexternalvolumehasbeeneliminated.Also,therehasclearlybeensomeadditionalchangebecausethe3quarksthatformaneutronhavelosttheirindividualComptonfrequenciesandinsteadaneutronwithrestenergy of 939.6 MeV exhibits de Broglie wave characteristics of the composite frequency.Thereforethebondingprocessmustincludesomeunknownmechanismforfrequencyadditionofcomponentparts. Wewillnotspeculateonthisanyfurthersincethischapterisprimarilyabouttheexternalvolumeofthe3chargedleptonsandabouttheelectron’sexternalvolumeinparticular.Thisisasubjectthatneedsfurtheranalysis.Energy in the External Volume: Aspreviouslycalculated,theenergyinachargedlepton’sexternalvolumecausedbytheoscillatingcomponentoftheelectricfieldisequaltoα/2 roughly0.4% of the charged lepton’s total energy.Hadrons do not have a fixed percentage of theirenergyexternaltotheirradiusbutevenaneutronhassomeofitsenergyexternaltoitsradius.Thethreequarksthatformaneutronhaveaddition/subtractionofthestaticstraincomponentsof the quark’s electric fields. A short distance from the neutron there is effectively chargecancelation.However,thisisthenon‐oscillatingcomponentoftheelectricfieldthatresultsinvectoradditionorsubtraction.Theoscillatingpartisproposedtoremainandproducestandingwavesintheneutron’sexternalvolume.Thesestandingwavesbecometheneutron’sdeBrogliewaveswhentheneutronisobservedinamovingframeofreference.Forexample,aneutronproducesadiffractionpatternwhenitispassedthroughadoubleslitexperiment.Theimplied


frequencyisequaltothesumofthefrequenciesofthethreequarks.Noefforthasbeenexpendedtodevelopamodeloffrequencyadditioninhadronsandothercompositeparticles.Amuonandanelectronbothhavethesamechargeandsamefractionoftheirtotalenergyintheirexternalvolumes.However,amuonhasabout200totaltimesmoreenergyand200timessmallerradius.Almostallofthemuon’sextraexternalenergyiscontainedinthesmalldifferencebetweentherotarvolumeofamuonandtherotarvolumeofanelectron.Atadistancelargerthantheelectron’srotarradius, theybothgeneratethesameelectrostatic forcebecausetheybothgeneratethesamenonoscillatingstraininspace.Thiscanbeshownbythefollowing:

A √ √ √ thisisthesameforallrotarswithchargee.

TheequationA √ Lp/rhaslostalltermswhichrelatetoaspecificComptonfrequencyoraspecific rotar radius size. Therefore thenon‐oscillating strain thedetectable electric field producedbyamuonisexactlythesameasthenon‐oscillatingstrainproducedbyanelectron.The oscillating components of an electron andmuon retain their frequency dependence butthese oscillating components are only detectable as de Broglie waves and other quantummechanicalwavecharacteristics.Internal Electric Field:Eventhoughthischapterisabouttheexternalvolumeofarotar,wearegoingtotakeabriefdiversionandtalkabouttheextensionofthenon‐oscillatingstrainoftheelectric field into the interior of a lepton’s rotar volume. If rotars are slight distortions ofspacetimethatcanpartiallyoverlap,doestheelectricfield thenon‐oscillatingstrain continueto increase inside the rotar’s rotar radius? The problem is thatwhen two electrons colliderelativistically,therepulsionforceexceedstheforcethatcouldbegeneratediftheelectricfieldended at the surface of the rotar volume. In chapter 9 itwas determined that the strain inspacetimeproducedbychargeeatdistanceris:ΔL/L √ Lp/r.Isuspectthatthisequationchangesgraduallywhenr ifonlybecauseoftheuncertaintyofdesignatingthelocationtospecifyasthecentrallocationtoserveasthepointwherer 0.However,thepointisthatthestraincancontinuetoincreaseintotheinternalvolumeofarotarsuchasanelectron.Therefore,thereisnoreasonwhycollidingtwoelectronstogethershouldrevealanyinternalstructure.Thestraininspacetime,andthereforetherepulsiveforcecontinuestoincreaseasthetwoelectronsoverlap.Furthermore,thecollisionoftworotarscausesbothrotarstoconvertthekineticenergytointernalenergy. Forexample,collidingtwoelectronswithkineticenergyof50GeVcausesbothelectronstomomentarilygain100,000timestheiroriginalenergyandshrinkbyafactorof100,000atthepointofclosestapproach.ThisdecreaseinsizecombinedwiththestrainequationΔL/L √ Lp/r ,permitsanelectrontoappeartobeapointparticleinrelativisticcollisions.Accelerating Charged Leptons: Wewillnowreturnto theexternalvolumediscussionandfocus on the electromagnetic properties, temporarily ignoring the gravitational effects. The


externalvolumeofanon‐accelerating,isolatedelectronisacombinationofstandingwavesatfrequencyωcwithoscillatingamplitudeAe √ Aβ/ 2andanon‐oscillatingstrainofspacetimewithamplitudeA √ Aβ/ .Previouslytheelectricfieldenergyinarotar’sexternalvolumewascalculatedatEext ½αEi.Tobeprecise,thisisthelimitingcaseofanisolatedrotarthathasnotinteractedwithanythingforaninfinitelylongtime.Inthisidealizedcasearotar’sexternalvolumehashadsufficienttimeatspeedoflightcommunicationtobecomefullyestablished.Inpractice,theextentofarotar’sexternalvolume itsundisturbedelectricfield islimitedbythelengthoftimeanisolatedelectronhasremainedundisturbedinaconditionthatdoesnotradiateelectromagneticradiation.Thereforewhatiscommonlyconsideredtobetheelectricfieldofaparticlecanbeconsideredpartoftherotar’sexternalvolume.However,inalargersenseboththeelectric“field”producedbytherotarandindeedtherotaritselfarejuststrainsintheseaofvacuumenergyofspacetime.Inchapter4itwassaidthattherewasonlyonetrulyfundamentalfieldsinceallfieldsarejustdifferentdistortionsofvacuumenergy/fluctuations.Keepingthisinmind,wewillusetheterm“electricfield”toindicatethedisturbanceinvacuumenergythatresultsinelectromagnetism.Thepartofarotar’sexternalvolumethatproducestheelectromagneticdisturbancehasspeedof light communicationback to the rotar’s rotarvolume. In chapter9a thoughtexperimentinvolvingthemagneticfieldofastarillustratedthefactthatamagneticfieldhastheabilitytoexert a force before communication is established back to the source of the field the star .Similarly, an electron’s electric field spacetime disturbance can interact “before”communicationisestablishedbacktotheelectron frameofreferencedependent .Thereisanobviousobjectiontotheconceptthatanelectron’sexternalvolumecanextendmanymeters from the rotar volume. This objection is that accelerating an electronwould breakcontact atspeedoflight withadistantpartoftheexternalvolumetherebyabandoningasmallpartofanelectron’sstructureandenergy.However,ratherthanbeingadefect,thisisactuallyastrengthofthisconceptbecauseitprovidesamechanismfortheemissionofelectromagneticradiation a photon with a quantized unit of angular momentum when an electron isaccelerated.Theaccelerationintroducesamodulation ω 0 intowhatwaspreviouslyastaticstrain ω 0 .Also,aportionofthestandingwaveenergyintheexternalvolumelosescontactwhentherotarvolume.Theenergyabandonedbytheaccelerationisconvertedtotheenergyofthephotonthatisformedwhenanelectronisaccelerated.Theaccelerationinitiallyintroducesadistortionintothewavesoftherotar’sexternalvolume.Thisdistortionbothlaunchesaphotonandgivesenergytoreestablishthelostportionoftheexternalvolume.Chapter11willdiscussfreelypropagatingphotonsinmoredetail.Another test of an electron’s distributed energy is whether a test can be devised thatdistinguishesbetweentherotarmodelwithitsdistributedenergyandthecurrentlyacceptedmodelofachargedpointparticle.Ittakestimetomeasureenergyorinertia.Themoreaccuratethemeasurementneedstobe,themoretimeisrequired.Thisallowstimefortheenergyinthe


externalvolumetocommunicateitspresenceatthespeedoflightandaddtothetotalenergyorinertiaoftheelectron. It ispossibletodoaplausibilitycalculationtoseeif itwouldeverbepossibletodoanexperimentthatwouldgiveadifferentanswerfortherotarmodelofanelectroncomparedtoapointparticlemodelofanelectronwhichhasallofitsenergylocalizedbutalsoexperiencesaretardationaspartoftheemissionofaphoton.Theenergyinanelectron’selectricfieldexternaltoradiusris:Eext αħc/2r.Thisisenergythatisintheformofstandingwavesexternaltotheelectron’srotarvolume.Fromtheuncertaintyprinciple ħ/2 ΔEΔt ,itispossibletocalculatetheintegrationtimeΔtthatwouldberequiredtodetectanenergyuncertaintyofΔE Eext.

Eext ħset

ħ ΔEΔt

Eext ΔEΔtsetEext ΔE

Δt 137

Therefore,itwouldtakeanintegrationtime137timeslongerthanthetimer/ctodetectthisenergydiscrepancy.However,thisis137timeslongerthanittakesforthediscrepancytobecorrectedtoadegreethatisundetectable.Therefore,amodelofanelectronwiththeproposeddistributionofenergyintheexternalvolumeisindistinguishablefromapointparticle.Chaotic Waves: The transition from the rotar volume to the external volume of a rotar isactuallyilldefinedbecausetherotarvolumehasachaotic,probabilisticquality. Therotatingdipoleinspacetimeisatthelimitofcausality.Therotarvolumeisattemptingtoradiateitsfullenergyinatimeof1/ωc.AfundamentalamplitudeofAf /risactuallyattemptingtocarryingawaythefullenergy.Itisonlythereturnwavegeneratedinvacuumenergythatissomehowcancelingthisemission.However,thechaoticprocessatthelimitofcausalitycanreconstructtherotar reconstructthequantizedangularmomentum atadifferentlocationdescribedbytheuncertaintyprinciple.Alsoadoubleslitexperimentcaninterferewiththenormalreconstructionand result in the rotarbeing reconstructedon theother sideof thedouble slit. Thiswillbediscussedlater.The point is that all of the quantum mechanical properties which seem mysterious for a point particle become conceptually understandable if a particle is a vortex of quantized angular momentum in a sea of vacuum fluctuations.Wehavepreviouslydiscussedthatallrotarsmustsatisfyasolitonconditioninspacetime.Thismeansthatthefewfundamentalparticlesthatexistmustexhibitacombinationofcharacteristicsthatoffsetthetremendousemissionofdipolewavesinspacetimethatleavethevicinityoftherotar volume. The few leptons and hadrons that exist somehowachieve a soliton condition


wheretheemissionisoffsetbythegenerationofwavesinvacuumenergythateffectivelycanceltheemissionfromtherotar’srotatingdipolecore.Model of the External Volume: Figure10‐1 is a simplified representationof the standingdipolewavesthatsurroundarotar.RecallthattherotarisattemptingtoradiateawayitsenergyandemitsdipolewaveswithfrequencyωcandamplitudeAf Lp/r.Thefewrotarsthatarestableorsemi‐stablemustformaresonancewiththesurroundingvacuumenergythateliminatestheenergy loss but leaves both standing waves and non‐oscillating strains in spacetime aspreviouslydiscussed.Allthefiguresinthischapterdealwiththestandingwavesassociatedwiththeoscillatingpartoftheelectricfield.ThesestandingwaveshaveamplitudeAe Lp/ 2andangularfrequencyωc.Figure10‐1isthefirstinthisseriesoffiguresandthisfigurehasbeengreatlysimplifiedcomparedtoanactualrotar.Therotatingdipolehasbeenreplacedbyasimplemonopolesourceofwaves.Infact,wewilluseamonopoleemitterforthefirstseriesoffiguresbecause the initial illustrations are easier to understandwithout the added complexity of arotatingdipolesource.Thefigureswilllaterbeillustratedusingadipolesourcewhenthissourcebecomesimportanttotheillustration.Initiallywewillimaginethatfigure10‐1representssoundwavesbeingemittedbyamonopoleemitterofsoundwavesatthecentercircleandbeingreflectedbyasphericalreflectoroutsideoftheareashowninthefigure.Theinteractionbetweentheemittedandreflectedwavesformsthestandingwavesdepictedinfigure10‐1andsubsequentfigures.Anacousticmonopoleemittercanbethoughtofasaspherethatexpandsandcontractsitsradiusatanacousticfrequency.


Figure10‐1showsstandingwavesinanacousticmediumdepictedatamomentintime.Theblueregionscanrepresentregionsofmaximumacousticpressureandtheyellowregionscanrepresent regions ofminimum acoustic pressure. A half cycle later the standingwaveswillreverseandregionsthatpreviouslyhadmaximumpressurewillhaveminimumpressure.Theblack regions between the yellow and blue regions would be the wave nulls in thisrepresentation,butthereisanotherwayofdepictingthisstandingsoundwave.Theblackregionshavethemaximumpressuregradient.Thismeansthattheblackregionshavethe maximum kinetic energy of the acoustic medium. Therefore, there is another way ofrepresentingthestandingacousticwavewhereweemphasizethekineticenergyofmolecules.Inthistypeofrepresentationtheblackregionswouldbedepictedasregionsofmaximumkineticenergy,notthenullsshownabove.Infacttheenergyinthestandingacousticwaveisjustbeingtransferredbetweenenergyincompression/rarefactionandkineticenergy.Wewillnowswitchtoconsideringfigure10‐1asrepresentingstandingwavesinspacetime.Thevacuumfluctuationsthat formspacetimehaveavastly largerenergydensitythantheenergydensityofarotar.Aspreviouslydiscussed,thepressureofvacuumenergyisstabilizingtherotarand exerting thenecessarypressure to confine the energydensity of the rotar. Figure10‐1representsamomentintimewherethedisturbancecausedbythepresenceoftherotarresultsinstandingwavesinthesurroundingvacuumenergy.Thesestandingwavesfluctuateboththerateoftimeandpropervolume.Regionsoffasttimeareshowninblueandregionsofslowtimeareshowninyellow.Ahalfcyclelaterthefastandslowtimeregionswillreverse.Theblackregionsbetweenyellowandbluehavethemaximumgradientintherateoftime.Theseregionsareequivalenttothegravfieldpreviouslyexplained.Justlikethestandingsoundwave,therereallyarenonullsinthestandingwaveinspacetime.Theregionsoffluctuatingrateoftimehavethesameenergydensityastheregionsofmaximumgravfield maximumrateoftimegradient .Thetotalwaveenergyisconstant sin2θ cos2θ 1 .Wavelets:AlldipolewavesinspacetimeareproposedtohavepropagationcharacteristicsthataresimilartotheHuygensPrincipleinoptics.TheHuygensprincipleassumesthateverypointonanadvancingwavefrontofanelectromagneticwaveisthesourceofanewdisturbance.Theelectromagneticwavemayberegardedasthesumofthesesecondarywaves called“wavelets” .Reflection, diffraction and refraction are explained by assuming that all parts of anelectromagneticwavearethesourceofthesenewwavelets.Thesurfacethatistangenttoanylocusofconstantphaseofwaveletscanbeusedtodeterminethefuturepositionofthewave.


AsoriginallyformulatedbyChristiaanHuygens,theHuygensPrinciplerequiresthatthewaveletsarehemisphericalandonlyradiateintotheforwardhemisphericaldirectionofthepropagationvector.AmodificationofthiswasmadebyGustavKirchhoffwherethewaveletsemitintoanamplitudedistributionofcos2 θ/2 .Thisdistributionhasmaximumamplitudeintheforwarddirection and zero amplitude in the reverse direction. The result is the classical Huygens‐Fresnel‐Kirchhoffprinciplethataccuratelydescribesdiffraction,reflectionandrefraction.Thiswillbediscussedinmoredetailinchapter11.ItisproposedthatthefewfrequenciesthatformrotarsinteractwithvacuumenergyinawaythatallowsthemtoemitwaveletsthatpropagateintoacompletesphericalpatternasshowninFigure10‐2.Withthishypothesisthecos2 θ/2 amplitudedistributionofthewaveletsoflightisnotsharedbythewaveletsofvacuumenergythatstabilizesrotars.Theconditionsthatstabilizerotarsrequirethatbothaforwardpropagatingwaveandanequalbackwardspropagatingwavebe formed in the external volume. This is accomplished if each wavelet propagates into asphericaldisturbancepatternasshowninfigure10‐2.Thesesphericalwaveletsaddtogethertoproduce the next generation of dipole waves in spacetime. This results in wavefrontspropagating in both the forward and backward radial directions. These newwavefronts arelabeled inward propagating and outward propagating. In the tangential direction there isincoherentadditionthatproducescancellation.Iftheenergyflowisequalinbothdirections,theresultisstandingwavesintheexternalvolumeofarotar.Standingwavesareoscillatingwaves


thathavefixedregionsofnodesandantinodes.Theypossessenergy,butthereisnocontinuousenergydrain.PathIntegral: Akeypointhere isthat thewaveletsofdipolewavesinspacetimeexploreallpossiblepathsbetweentwopoints.Furthermore,theamplitudeatanypointisthecoherentsumamplitudeandphase ofthesewaves.Theintensityatanypointisthesquareoftheamplitudesum. This concept gives a physical interpretation to the path integral operation of quantumelectrodynamics.Itisastretchtoexplainhowpointparticlesexploreallpossiblepathsbetweenevents,butwavesinspacetimethatformnewwaveletsintrinsicallyaccomplishthistask.Again,thisproposedspacetimebasedexplanationmakesquantummechanicaloperationsconceptuallyunderstandablewhilethepointparticlemodelhasnumerousmysteries.This explanation that involves backwards propagating waves sounds good, but there is aproblem.Ifthiswastheonlymechanismstabilizingarotar,theresidualstandingwaveswouldbemuchlargerthanthecalculatedamplitudeofAe √ Lp/ 2requiredforthestandingwavepartoftheelectricfield.Thereappearstobeanadditionalunknownmechanismgeneratedinvacuumenergythatformsawavethatprovidesadditionalcancelation.Thesestandingwavesremainevenwhenthenon‐oscillatingcomponentoftheelectricfieldhasbeencanceled.Ifotherschoosetomodelthesestandingwaves,itshouldbenotedthataccuratemodelingoftheHuygensPrincipleinopticsrequiresthatthemodelingmustbedoneinthreespatialdimensions1.IftheHuygens’sPrincipleismodeledinonly2spatialdimensions,thereisincompletecancelationofwavesthatdonotcontributetoawavefront.de Broglie Waves: Inchapter1itwasshownthatalasercontainslighttravelinginoppositedirections.Whenthewavesinthislaserareobservedfromastationaryframeofreference,thebidirectional light forms standingwaves. In otherwords, the standingwaves are stationaryrelativetothelasermirrors.Whenthelaseristranslatedrelativetoanobserver,thestandinglightwavesarestillstationaryrelativetothemovingmirrors,butthemovingframeofreferencemeansthattheobserverseesthelightbeingDopplershiftedupinfrequencyinthedirectionoftravelandbeingDopplershifteddowninfrequencyintheoppositedirection.ThesuperpositionofthesetwoDopplershiftedbeamsoflightproduceswhatappearstobeamovingenvelopeofwaves.

1 http://www.mathpages.com/home/kmath242/kmath242.htm


Figure 1‐1 shows the moving envelope of waves and moving laser mirrors. An analogy isproposedtobepresentwhenarotarisobservedinamovingframeofreference.ItisdesirabletoexaminethedeBrogliewavesofarotaringreaterdetail.Figure10‐3issimilartoFigure10‐1,buttherearetwodifferences.First,Figure10‐3showswavespropagatingonlyawayfromthemonopole source arrows pointing away from the source . Second, Figure 10‐3 shows themonopolesourcemovingdownwardrelativetotheobserver. ThecombinationofthesetwofactorsproducestheDopplerwavepatternshown.Figure10‐4alsohasadownwardsmoving


frameofreference,butthedifferenceisthatonlywavespropagatingtowardsthesource inwardpropagation areshownwitharrowspointingtowardsthesource.Thewaveletspreviouslyshowninfigure10‐2meansthatwavesaresimultaneouslypropagatingboth towards the source and away from the source. Thismeans that amoving sourcewillproduceawavepatternthatisasuperpositionoffigures10‐3and10‐4.WhenweaddthesetwopatternstogetherweobtaintheresultshowninFigure10‐5.Itissurprisingtoseethatweobtainalinearwavepatternfromthesuperpositionofsphericalwavesinamovingframeofreference.Thesearetherotar’sdeBrogliewaves. Theyhaveall thecorrectcharacteristics–correctdeBrogliewavelength,correctdeBrogliephasevelocityandthecorrectdeBrogliegroupvelocity.MovingtherotarmodelproducestherotarequivalentofdeBrogliewaves.This figure is not static. Not only is there translation relative to the observer, but the darkinterferencefringesaremovingataspeedfasterthanthespeedoflight.Forexample,iftherotarmodelismovingat1%ofthespeedoflightrelativetoanobserver,thentheinterferencepatternis moving in the same direction as the relativemotion, but at 100 times the speed of lightwd c2/v .Alsonoticethatthereisaphaseshiftgoingacrossthedarkinterferencepattern.ThisisrepresentedbyareversalofcolorfollowingawaveacrossthedarkdeBroglienull.

Figure10‐5makesnoattempttoshowthattheamplitudedecreaseswithradialdistancefromthe source. Figure10‐6 is a 3‐dimensional graphical representationof Figure10‐5with the


addedfeatureofa1/ramplitudedependence.Theactualamplitudeshouldfalloffproportionalto1/r2,butthissharpdecreaseinamplitudemakesitdifficulttoseethedeBrogliemodulationwave.

Strain Amplitude Graph:ItiseasiesttoexplainthedeBrogliemodulationwaveusingFigure10‐7.Thisfigureisagraphofthewavesinfigure10‐5 radialcross‐section .Infigure10‐7,thehighfrequencywavesaredesignatedas“dipolewavesinspacetime”.Whenarotarisstationaryrelativetotheobserver,thenallthedipolewavesinspacetimehaveequalamplitude. Atthisstationaryconditionthefrequencyofthesewavesistherotar’sComptonfrequency ~1020Hzforanelectron andthewavelengthofthesewavesistherotar’sComptonwavelengthλc.Whentherotarismovingrelativetotheobserver,thentherotar’sdeBrogliewaveappears.Thisisjustthemodulationenvelope that results from thedifferentDoppler shift forwavespropagatingawayfromtherotar’srotarvolumeandtowardstherotarvolume.Figure10‐7showsagraphicalrepresentationoftherotar’sdeBrogliewavelengthλd.Thisgraphplotsstrainamplitudeversus radialdistancer.Onlya short radial segment is shown. Thereshouldbearadialdecreaseinamplitude,buttheshortradialdistancedepicteddoesnotshowthisdecreaseinamplitude. IntheexternalvolumeofarotarthefundamentaltravelingwavewithamplitudeLp/rhasbeencanceled leavingbehind thestandingwaveresponsible for the


rotar’selectricchargewithamplitudeofAe √ Aβ/ 2.Thenonlineareffectresponsiblefortheoscillatingportionofgravityistoosmalltobeshown.Therefore,theYaxisofthisgraphisthe strain amplitude Ae. The maximum value of Ae is the value given by the equationAe √ Aβ/ 2 √ Lp /r2.To give an idea of scale, the approximate Compton wavelength λc is shown. An electron’sComptonwavelengthisabout2.43 10‐12m.ThedeBrogliewavelengthλddependsonrelativevelocity v and is illustrated as being approximately 20 times longer than the Comptonwavelength in this example. Therefore, the de Brogliewavelengthwould be approximately5 10‐11minthisexample.TheComptonwavelengthλcandthedeBrogliewavelengthλdarerelatedasfollows:λc v/c λd approximationv c .Therefore,thisfigureillustratesthedeBrogliewavepatternifanelectronistravelingatabout5%thespeedoflight λd 20λcinthisfigure .TheYaxisof thisgraph isstrainamplitudewhichcanbeexpressedeitherasaspatialstrainmeters/meter orasatemporalstrain seconds/second .Bothwaysofexpressingthisgivethesame dimensionless number for a specific point in space and instant in time. Suppose ourobservationpointataparticularinstantisonemicrometer 10‐6meters fromanelectronthatismovingpastusat5%thespeedoflight.Wecanthenquantifythestrainamplitudedepictedin Figure 10‐7. Using Ae √ Aβ/ 2 √ Lp /r2 and substituting r 10‐6 m and 3.86 10‐13mforanelectron,weobtain:Ae 5.3 10‐37.ThisisthemaximumvalueofAe

aboveandbelowthezerostrainline the“x”axis .It is possible to calculate thedisplacement of spacetime required to produce this amount ofdimensionlessstrain.Thisstrainexistsoverapproximatelyoneradianofthewavewhichisadistance equal to . For an electron 3.86 10‐13m thereforeAe 2 10‐49m.Therefore, thespatialdisplacementof spacetime displacementamplitude whichcauses thestrainamplitudeillustratedhereissmallerthanPlancklengthbyafactorofabout1014.Ifwewouldhavechosentoworkinthetemporaldomainwewouldobtainthesamedimensionlessstrainwhich could be thought of as seconds/second. The temporal displacement amplitudecausingthisstrainwouldthenbeAe/ωc 6.8 10‐58s.ThisissmallerthanPlancktimeandthedifferenceisagainafactorofabout1014.Ψ Function: It isnotobvious in figure10‐7but there isa180degree πradian phaseshiftbetweeneachdeBroglielobe.ThereforeonecompletedeBrogliewavelengthincludestwolobesasshown. This180degreephaseshiftbetween lobes isa fundamentalpropertyofstandingwavesviewedfromamovingframeofreference. ThisphaseshiftgivesrisetothedeBrogliewave interpretationshownin figure10‐7. Perhapsmost important,the wave designated de Broglie wave envelope is really the moving rotar’s Ψ function.ItissometimessaidthattheΨfunction has no physical interpretation. However, figure 10‐7 is the proposed physicalinterpretation of the quantummechanical Ψ function. This is another example of how the


proposed spacetime based model makes quantum mechanical mysteries conceptuallyunderstandable.Relativistic Contraction: Above it was stated that λc illustrated in figure 10‐7 is“approximately”equaltotherotar’sComptonwavelength.ThereasonthatthisisnotexactisthatthisfiguredepictsamovingrotarandthereisrelativisticlengthcontractionintheComptonwavelength.Iftherotarwasstationary,therewouldbenodeBrogliewavemodulationenvelopeandthedipolewaveswouldbeexactlyequaltotheComptonwavelength.Thereasonforbringingupthispointisthatitispossibletoseethephysicalcauseofrelativisticlengthcontractionwhenthereisrelativemotion. AmovingrotarhaswavesthatareDopplershifterupinfrequencyandDopplershifteddowninfrequencyaspreviouslyexplained.WhentheseDopplershiftedwavesarecombined,theresultantwavehasashorterwavelengththantheoriginalwavelengthwithoutDopplershifts.ThiswasprovenmathematicallyinAppendixAattheendofchapter1.Thisanalysisappliesequallytostandingwavesinamovinglasercavityortostandingwavesintheexternalvolumeofarotar.ThecombinationofthetwoDopplershiftsinthetwooppositelypropagatingwavesproducesanetdecreaseintheComptonwavelengthby

afactorof: 1–v /c .Thisisproposedtobethesourceofrelativisticcontraction.Amovingmeterstickwillappeartodecreaseinlengthbecauseallthewavesthatmakeupthemeterstickdecreasetheirwavelengthbecauseofthiseffect.Rotating Dipole Model: Wewillnowattempt togiveacrudemodelof thestandingwavespresentintheexternalvolumeofrotatingspacetimedipoles.Anaccuratemodelofthisrequireshighlevelcomputermodeling.Itinvolvesmodelingalargenumberofwaveletsthatareaddedtogether and then become the source of newwavelets that form the next generation. Thisprocess is repeated a large number of times. This task is beyond the scope of this book.Furthermore,thesimplestmodelingwouldprobablymakenodistinctionbetweentheinfinitenumberoffrequenciesthatdonotformstablerotarsandthefewfrequencieswiththeunusualcharacteristics that combine to form stable rotars. Also, howdowehandle the chaotic spindistributioncharacteristicsofspin½particles?Thefollowingmodelingisabestguessmodeloftheexternalvolumeofanelectron.Thiscanthenserveasastartingpointforothersthatcanimproveonthismodel.Beforeattempting tomodel theexternalvolumeof rotars, it isdesirable to firstdescribe thechaoticspinpropertiesexhibitedbyisolatedrotars.Becausearotarisatthelimitofcausality,itshould not be a surprise that a rotar has probabilistic characteristics. The displacement ofspacetime is so small that dipole waves in spacetime do not violate the conservation ofmomentum.Recalltheexamplesgivenpreviouslycomparingtheminutevolumeandrateoftimechangesrequiredtoformanelectron’sspacetimedipole. expandingtheradiusofJupiter’sorbitbyahydrogenatomorslowing therateof timebyseveralmicrosecondsover theageof theuniverse.


Figure10‐8showsaplotinsphericalcoordinatesoftheprobabilityofspinbeingorientedinthedirection θ. This spin direction probability is proportional to cos2 θ/2 . Suppose that weconcentrate not on the spin direction, but on the axis of spin. Figure 10‐8 specifies theexpectation axis and an arbitrary axis. The point of this is to show that a rotar can have aspherical distribution averaged over time even though the rotar has an expectation spindirection.Therotatingdipoleshowninfigure5‐1wouldnothaveasphericaldistributioniftheaxisofrotationwerefixed.However,figure10‐8showsthattheprobabilityofspindirectionissuchthatallaxisorientationsareequallyprobable.Inotherwords,theexpectationaxisshowninfigure10‐8isthesamelengthasthearbitraryaxiswhenoppositespinprobabilitiesareaddedtogether.Ifweconsiderthelengthofthespinaxisprobabilityasζ,thenwehave:ζ cos2 θ/2 cos2 θ π /2 cos2 θ/2 sin2 θ/2 1With this being said,wewill simplify themodelingby looking at theequatorial planeof theexpectationrotationaxis.Forexample,theexpectationaxiscanbesetbyplacinganelectroninamagneticfield.Thisisthesimplesttomodelandonestepbetterthanassumingamonopoleemitter.


Figure10‐9showsarotatingdipoledesignated“rotatingdipolelobes”.Thetwolobesdepictedrepresent the two dipole lobes discussed and depicted in chapter 5. When the lobesmovearoundtheimaginarycircleatthespeedoflight,anydisturbancepropagatesawayfromtheselobesatthespeedoflightandformstheoutwardpropagatingArchimedesspiralsshown.Thissimplifieddescriptiondoesignorethefactthateverypartofthewaveformsnewwavelets,butwewillproceedwiththisdescriptionandattempttoincludewaveletslater.Fordiscussion,wewillinitiallyassumethatthesolidlinesrepresentregionswheretherateoftimeisfasterthannormalandthedashedlinesrepresentregionswheretherateoftimeisslowerthannormal.Sincetherateoftimeaffectsall3spatialdirectionsequally,thesewavesareneitherlongitudinalnortransverse.Theyaresimplytimewaves.Besidesaffectingtherateoftime,thesewavesalsorepresentaspatialdistortion.Sofar,wehaveignoredthefactthatthemodelcallsforeverypointonthewavefronttobethesourceofanewwavelet.Anextremelysimplifiedmodeltakestheoutgoingwavepatternshownin figure10‐9andgenerate thebackwardspropagatingwavesbyassumingthat theoutwardpropagatingwavesarereflectedoffaconcentricsphericalreflector.Thesereflectedwavesthenpropagatebacktowardstherotatingdipole.ThismakesanotherpairofArchimedesspiralsthatinastaticimagearethemirrorimageoffigure10‐9.Theyhavethesamerotationaldirectionwhenviewedfromtheperspectiveoffigure10‐9,buttheyarepropagatinginward.


WhentheoutgoingwavesandincomingwavesareaddedtogetherweobtainaninterferencepatternshowninFigure10‐10.Thisisacross‐sectionviewoftheequatorialplaneoftheexternalvolumeofarotar.Thisalsodepictsaninstantintime.Theactualpatternisrotatingatthesamerateastherotatingdipole theComptonfrequency .Foranelectron,thisimagewouldrotateabout1.24 1020revolutionspersecond.Thisiscurrentlythebestrepresentationofanisolatedrotarsuchasanisolatedelectron.WeareassumingastationaryframeofreferenceforFigure10‐10 nodeBrogliewavessuperimposed .Notethatthereisa180degreephasechangeatthedestructive interferencebar blackbar thatgoesacross thecenterof this figure.Thisphasechangecanbeseenbecausethereisareversalofcolor yellowtoblueorbluetoyellow atthisregionofdestructiveinterference.Eventhoughthisfigurewasmadeusingsomequestionablesimplifications,itisareasonablefirstattempt.Theamplitudeofthewavesshoulddropoffwitha1/r2decreaseinamplitudefromthecenter.Forillustrationpurposes,thisfiguredepictsuniformradialwaveamplitude.Thereadershouldmentallyadjustfortheradialdecreaseinamplitude.


Conservation of Angular Momentum:Figure10‐10lookssimilartoarotatingdisk,butthisisthewronginterpretation.Thereisnoviolationoftheconservationofangularmomentumasthispatternenlarges.Thisisbestexplainedbyreturningtofigure10‐9.Wewillassumethatalloftheenergyandangularmomentumstartsoffintheregiondesignated“rotatingdipolelobes”.Ifsomeenergyleavesthisregion,itsoutwardpropagationfollowstheArchimedesspiralpatternshowninfigure10‐9.Thisisthepatternthatmaintainsconstanttotalangularmomentum.Provingthisstatementwouldrepresentasubstantialdiversion,butonebriefpointsupportsthiscontention.Figure10‐9showsanarrowdrawnperpendiculartotheArchimedesspiralinthefarfieldofthespiralpattern.Inthelimitofthefarfieldtheprojectionofthisperpendicularlinebacktowardsthecenterresultsintheprojectedlinebeingtangenttotheimaginarycirclewithradiusof .Thisimpliesaconservationoftheangularmomentumforenergythatleavesthiscircle.Thepatternshowninfigure10‐10isasuperpositionoftwoArchimedesspiralpatterns,bothofwhichcanbeshownto individuallyexhibitconservationofangularmomentum. Therotatingpatterninfigure10‐10hasatangentialspeedfasterthanthespeedoflightforanyradialdistancegreaterthanr .Thisispermittedbecausethisisjustaninterferenceeffectthatcanmovefasterthanthespeedoflight.InFigure10‐11weare lookingat this samepatternof figure10‐10 fromamoving frameofreference.ToobtainthispictureweaddedtogethertheDopplershifteroutgoingandincomingwavepatterns.ThisissimilartoaddingtogetherFigures10‐3and10‐4togetfigure10‐5.Toobtain figure 10‐11 we added together Doppler shifted outgoing and incoming Archimedes


spirals rather than the concentric circles. Like figure 10‐5, Figure 10‐11 is a snap‐shot of amovinginterferencepattern.TheinterferencepatternwouldbemovingfasterthanthespeedoflightaspreviouslyexplainedforFigure10‐5.Infact,thedeBrogliewavelengthandtranslationspeedisthesamewhetherwemodelamonopolesourceoradipolesourceprovidedthatweassumethesameComptonfrequency.Figure10‐11isactuallyalargespiralpattern,butthespiralcharacteristicisonlyobviousnearthecenterofthefigure.Theexactcenteroffigure10‐11thatistheinitiationofthespiralisnotanaccurateillustrationofthepatternthatwouldbeproducedbyarotar.Figure10‐9illustratesthat thespiraldoesnotextend to theexactcenter. Instead, there isa transition to therotarvolume that is representedbya circlewith radius . Figure10‐11wasdrawnwith the twospirals inwardandoutwardpropagating extendingallthewaytothecenter.Therefore,amorerealistic version of figure 10‐11 would have a transition to black over the center ~ 1/πwavelength.Compton Scattering:In1905AlbertEinstein’spublishedapaperonthephotoelectriceffect.Thispapersuggestedthatlightexhibitsparticle‐likeproperties.Atthetimelightwasconsideredtobeonlyawavephenomenon.Intheearly1920’s,theparticle‐likepropertiesoflightwasstillbeingdebated.However,debateeffectivelyendedwiththeobservationbyArthurComptonofthescatteringofx‐rayphotonsbyelectrons calledComptonscattering . Thescatteredx‐rayphotonsexhibitadecrease in frequencythat isa functionofscatteringangle. The individualelectronsalsoexhibitrecoilwhentheyscatteranx‐rayphoton.Thedecreaseinthefrequencyofthescatteredphotoncorrespondstotheenergytransferredtotherecoilingelectron.AsimpleDoppler shift ofwaves reflectingoff themovingelectrondoesnot correspond to the correctfrequencyshift. Allofthisisperfectlyexplainedbythemodelthatassumesthatphotonsareparticleswithenergythatisafunctionoffrequency. Whenaphoton pointparticle collideswithanelectron pointparticle thereismomentumtransferandenergytransferbetweentheseparticles.TheinteractionisnicelydescribedbyCompton’sequationsandhereceivedthe1927NobelPrizeinphysicsforthiswork.The physics community has universally adopted the wave‐particle photon model. Photonsclearly have wave properties, but Compton scattering, the photoelectric effect and otherexperimentsalsoseemtorequireaparticleexplanation.Thewave‐particledescriptionofbothphotons and particles works well, but the conceptual understanding of this has puzzledgenerationsofphysicsstudents.Schrodinger Article on the Compton Effect: Since this book proposes that fundamentalparticlesaredipolewavesinspacetime,itwouldbehelpfultosupportthiscontentionbyofferingaplausibleexplanationforComptonscatteringusingtheproposedspacetimebasedmodelofbothelectronsandphotons.AsIwasworkingonthisexplanation,IassumedthatnoonehadbeensuccessfulinproposingapurelywavebasedexplanationforComptonscattering.Tomy


surprise,Idiscoveredthatin1927ErwinSchrodingerhadpublishedatechnicalpapertitled“TheComptonEffect”2.ThisarticleisavailableinSchrodinger’sbook“WaveMechanics”whichhashadmultipleeditionsinEnglish.Schrodingerdidnotconceiveofwavesinspacetimeorarotarmodel,buthedidproposeaplausiblewaveexplanationforComptonscattering.Hisproposedexplanationinvolvedanelectron’sdeBrogliewavesinteractingwithlightwavestoproducethecorrectscattercharacteristicsforboththelightandtheelectron.Inthisarticle,Schrodingerusedsomeantiquatedterminologysuchasthephrase“anaetherwave”todescribelight,buthispointisvalid.Abriefdescriptionofhisconceptwillbegivenhere.Schrodingerlookedatthecollisionasifitwasacontinuousprocess.Inthiscasefourwavesarepresentandcontinuouslyinteracting.Thesefourwavesare1 theelectron’sdeBrogliewavebeforetheinteraction2 theelectron’sdeBrogliewaveaftertheinteraction3 thelightwavebeforetheinteractionand4 thelightwaveaftertheinteraction.SchrodingerfoundthatthetwosuperimposeddeBrogliewavescombinedtomakeawavethathecalleda“waveofelectricaldensity”.Thiscombinedwavehadtheperfectperiodicitytoreflecttheincidentlightbeamandcreateareflectedbeamwiththecorrectfrequencyshiftandscatterangle.Thetwosuperimposedlight waves incident and scattered produce an interference pattern that matches theinterferencepatternproducedbythetwosuperimposeddeBrogliewaves.Schrodingermade an analogy between Compton scattering and light interactingwith soundwaves Brillouinscattering .Soundwavesproduceaperiodicchangeintheindexofrefractionofanacousticmedium.Lightwavespropagatinginanacousticmediumcanreflectoffthesoundwavewhichproducesperiodicchangesintheindexofrefraction. Themaximumreflectionisobtainedifthefollowingequationissatisfied:λ 2ΛsinθWhere: λ lightwavelength, Λ acousticwavelength and θ the angle between the lightpropagationdirectionandaplaneparalleltotheacousticwaves.Thisequationwouldbeexactiftheacousticwavewasstationary.Sincetheacousticwavehasaspeed much less than the speed of light, the condition of a stationary acoustic wave isapproximately met. However, relativistic corrections would be required if the sound wavepropagatedatasignificantfractionofthespeedoflight.TheequationcorrespondstotheBragglaw firstorder andbecomesexactwhentheacousticwavesarestationary.Whentheacousticspeedofsoundistakenintoconsideration,thenitappearsasifthelightwavesarereflectingoffamovingmultilayerdielectricmirror.Thereisafrequencyshiftinthereflectedlightandtheangleofincidencedoesnotequaltheangleofreflectionbecausethemirrorismoving.Schrodinger considered the superposition of the two sets of an electron’s de Brogliewavesbeforeandaftertheinteraction toresultina“waveofelectricaldensity”thatcouldinteractwithlight.HereareSchrodinger’stranslatedwords:

2 Annalen der Physik (4), vol 82


“According to the hypothesis of wave mechanics, which up to now has proventrustworthy,itisnottheψ‐functionitself,butthesquareoftheabsolutevaluethatisgivenaphysicalmeaning,namely,densityofelectricity.Asingleψ‐wavethereforeproducesadensitydistributionwhichisconstantinbothspaceandtime.Ifhowever,wesuperimposetwosuchwaves,…weseethata ‘waveofelectricaldensity’arisesfromthecombination…”NowitisthisdensitywavethattakestheplaceofthesoundwaveofBrillouin’spaper.Ifweassumethatalightwaveisreflectedfromitasfromamovingmirror, subjectto the fulfillmentofBragg's law thenweshall showthatour fourwaves twoψ‐waves and the incident and reflected light waves stand exactly in the Comptonrelationship….AsallthefourwavesareinvariantwithrespecttoLorentztransformation,wecanbring the density wave to rest by means of such a transformation…. Bragg’srelationshipholdsexactlyifλdenotesthewavelengthofthelightwave,Λthatofthedensitywaveandθtheglancingangle.Itcanbeputintheform:2hλ/sinθ h/Λ”

Vector Diagrams: IwillelaborateonSchrodinger’spointthatitispossibletobringthetwointeractingdeBrogliewavesintoastationaryframeofreferencebyaLorentztransformation.NormallyCompton scattering involves an incidentphoton strikinga stationaryelectron.Themomentum transfer produces a recoiling moving electron and reduces the energy of thescatteredphotoncomparedtotheincidentphoton.


Figure10‐12showsavectordiagramthatdepicts thisnormalComptonscatteringcondition.Thisdiagramshowsthemomentumvectorofboththeincidentandthescatteredphotons.ItalsoshowstheelectronsmomentumvectoraftertheComptonscattering.Allofthesearecommonlyincluded in Compton scattering diagrams, but figure 10‐12 includes two additional features.First, the electron’smomentumbefore scattering is designated momentum 0 . Secondly,thereisamomentumvectordesignated“halftheelectron’smomentumafterscattering”.Thisvectorshouldbesuperimposedontheparallelvectorbutithasbeendisplacedslightlyforclarity.Thereasonfortheadditionaldesignationsoftheelectron’smomentumbeforescatteringandhalf theelectron’smomentumafterscattering is that thesedesignationswillhelpexplaintheframeofreferenceusedforFigure10‐13.Infigure10‐13,weadoptaframeofreferencethatisrequired tohave theelectronmovingwith theoppositemomentumas thevectordesignated“half the electron’s momentum after scattering”. If the scattered electron’s velocity is nonrelativistic,thenthemovingframeofreferenceissimplyhalfthescatteredelectron’sspeedandtheoppositevectordirectionasshowninfigure10‐13.Inthisframeofreference,theelectronismovingatvelocity vbeforescatteringandismovingatvelocity–vafterthescattering thesamespeedbutoppositedirection .ThisistheframeofreferencedescribedbySchrodingerasthe Lorentz transformation that “brings thedensitywave to rest”. The superposition of theelectron’s de Broglie waves before and after the interaction results in a stationary butoscillating deBrogliewavepattern.ItisveryeasytoanalyzeComptonscatteringfromthisframeofreference.Thereismomentumtransferbetweenthephotonandtheelectron,butthereisnoenergytransferred.Inthiszeroenergytransferframeofreference,theelectronmomentummovingtowardstheorigin beforescatteringthephoton isthesamemagnitudebutoppositedirectionastheelectronmomentummoving away from the origin after scattering the photon . The reversal in direction is themomentumtransferredtothephoton. Thesuperpositionof thetwosetsoftheelectron’sdeBrogliewavesproducesastationarystandingwavepattern densitywave withperiodicityofΛd ħ/mv. Thisstationarywavepatterneffectivelyreflectsaphotonwithoutanychange infrequency.Also,theangleofincidenceequalstheangleofreflection–justlikereflectionfromastationarymirror.AllComptonscatteringevents involvingan initially stationaryelectroncanbe lookedatasaspecialcaseofthezeroenergytransferComptonscatteringwheretheframeofreferencehasbeen adjusted Lorentz transformation so that the electron is initially stationary. Once weunderstandascatteringeventinthissimplestframeofreference,wecaneasilyswitchbacktothecommonlyusedframeofreferencedepictedinfigure10‐12.Thefrequencyshiftandanglechangeissimplytheresultofreflectingoffamovingmulti‐layerdielectricmirror.


Figure10‐5showsarotarmodelinamovingframeofreference.Thisfiguredepictsaninstantintime.Thelargeblackhorizontalfringesaremovinginthedirectionoftranslationatavelocityofwd c2/v.Thissaysthattheinterferencefringesarealwaysmovingfasterthanthespeedoflight. When we superimpose two rotar models moving at the same speed but in oppositedirections, we again obtain an instantaneous picture similar to figure 10‐5. However, thesuperpositionofoppositemovingwavesresultsinthedeBrogliewavesbecomingstationary.Thiscanbevisualizedasthehighfrequencywaves yellowandbluewaves infigure10‐5beingstandingwaveswhichareoscillatingatafrequencyofapproximately1020Hz.ψ Function: Theenvelopeof thesewaves isawave that isproposed tobeSchrodinger’sψfunction.Squaringthisgivestheprobabilityoffindingtheelectroninareasofgreatestoscillationamplitude. Schrodinger calls theseareasof greatestoscillation “wavesof electricaldensity”.Since thesewavesareproposed tobedipolewaves in spacetimeoscillatingat theelectron’sComptonfrequency,itiseasytoseewhythesquareofthesewavesrepresentstheprobabilityof“finding”anelectron.Schrodingerarguesthatwhenlightinteractswithstationaryψ‐waves deBrogliewaves theyrepresenttheequivalentofadensityvariationthatcanreflectlight.Onceagain,histranslatedwordsare:

“Asingleψ‐wavethereforeproducesadensitydistributionwhichisconstantinbothspaceandtime.Ifhowever,wesuperimposetwosuchwaves,…weseethata‘waveofelectricaldensity’arisesfromthecombination…”NowitisthisdensitywavethattakestheplaceofthesoundwaveofBrillouin’spaper.Ifweassumethatalightwaveisreflectedfromitasfromamovingmirror, subjecttothefulfillmentofBragg'slaw thenweshallshowthatourfourwaves twoψ‐wavesandtheincidentandreflectedlightwaves standexactlyintheComptonrelationship.”


MorewillbesaidaboutthephysicalexplanationoftheΨfunctioninchapter12.Fornowwewillattempt to illustrateSchrodinger’s ideaofComptonscatteringwith thenext two figures.Figure10‐14sets thestageby illustrating thewavepropertiesof lightreflectingoffamirrorfrozenintime .Thebeamofwavesentersfromtheleft,reflectsoffthemirrorandleavestotheright.Theareaofoverlapbetweentheincidentandreflectedbeamsistheareawhereastandingwavepatterniscreated.Thisstandingwavepatternhasstandingwavenullswhichinthisfigurearehorizontalbandsparalleltothemirrorsurfacewherethereisnoelectricfieldoscillation.The antinode bands are also illustrated and these are regions of maximum electric fieldoscillation. For example, if the lightwave is linearly polarizedwith the electric field vectoroscillating perpendicular to the plane of the illustration, then the blue regions might beconsideredregionswheretheelectricfieldmomentarilyispointingtowardsthereaderandtheyellow regions might be considered regions where the electric field vector is momentarilypointingawayfromthereader.Iftimewasallowedtoprogressforward,theseblueandyellowregions in thestandingwavewouldmove from left to right. Thenodeplaneswouldremainunchanged.


Nowfigure10‐15representsthecombinationoffigures10‐14and10‐5.Inthiscaseonlythestandingwaveportionoffigure10‐14isillustrated,sothereadermustimaginethattheincidentbeamiscominginfromtheupperleftandthereflectedbeamisleavingtotheupperright.Alsothelowerportionoffigure10‐15representsthesuperpositionofanelectronbeforeandafterthescattering.Weareusingthezeroenergytransferframeofreference,sothedeBrogliewavepatternwouldappearstationary. TheconditionsthatproduceComptonscatteringresultinaperfectmatchbetweenthespacingof thestandingwaveantinodesof the lightbeamandthestandingwaveantinodesoftheelectron.Itisasifthelightbeamisreflectingoffamultilayerdielectricreflector.Ifwemovedtoadifferentframeofreferencewherewewouldperceivesomeenergytransferbetween the lightand theelectron, then theangleof incidencewouldnot equal theangleofreflectionandthewavelengthofthereflectedbeamwouldbedifferentthanthewavelengthofthe incidentbeam. Intheabovedescription,someartistic licensehasbeentakenbothintheillustrationandthechoiceofwords. Weshouldreallybetalkingaboutthescatterofasinglephotonfromasingleelectron.Thewaveamplitudeshouldnotbeuniformandnumerousothercorrections.


ThisspacetimewaveexplanationofComptonscatteringisproposedtoactuallybebetterthanthe particle based explanation because several quantum mechanical mysteries are alsoexplainedbythespacetimewaveexplanation.Forexample,thespacetimewaveexplanationhasthe wave model of an electron going from its initial velocity to its final velocity withoutaccelerating through all the intermediate velocities. An explanation of Compton scatteringinvolvingthestandardpointparticlemodelofanelectronwouldseemtoimplythattheelectronundergoes acceleration as it transitions through intermediate velocities. This concept ofintermediatevelocitiesisnotconsistentwiththequantummechanicaldescription.Also,Schrodingerindicatedthathisψfunctionhadnophysicalmeaning;itonlygainedphysicalmeaningwhenitwassquaredtogivetheprobabilityoffindingaparticle.Ihavegivenaproposedphysicalmeaningtotheψfunction.Itisthewaveenvelopeshowninfigure10‐7anddepictedinfigure10‐5.Theenvelope ψfunction isundetectablebecauseitisaninterferenceeffectwithnodesandantinodesofdipolewavesinspacetimewithaninterferencepatternthatpropagatesfasterthanthespeedoflight.Onlywhenthereisaninteractionbetweentwosuchenvelopesofwavesdoesthepropagationslowdowntoaspeedlessthanthespeedoflight asdepictedinfigure10‐15 .Squaringthisthengivestheprobabilityof“findingtheparticle”.Plausibility, Not Proof:TosuccessfullycompletethisComptonscatteringanalysis,itwouldbenecessarytoshowthatthissuperpositionoftheelectron’swavesinspacetime twodifferentvelocities producesaperiodicchangeintheproperspeedoflight.Furthermore,itwouldalsobenecessarytocharacterizeaphotonusingwavesinspacetimeandshowthatthecombinationofthesemodelsproducesthecorrectscatterprobability.While Ihaveproposedexplanations that contain theelements required in thisexplanation, Icannot conclusively show that the effects on the speed of light are sufficient to accomplishComptonscattering.Itishopedthatothersmightbeabletocompletethistask.SchrodingerhasshownthatawaveexplanationofComptonscatteringisplausible.Ishowthattherotarmodelin a moving frame of reference is plausibly equivalent to Schrodinger’s ψ‐function waves.Therefore,thiswillbedeclaredasuccessfulplausibilitytesteventhoughitisalongwayfrombeingconclusivelyproven.Double Slit Experiment: Another plausibility test of the rotarmodel is to see if thismodelproducesadiffractionpatterncharacteristicofsendinganelectron orotherparticle throughadoubleslit.Thediffractionpatternproducedbysendingastreamofelectronsthroughadoubleslithaslongbeenofferedasproofthatanelectronexhibitswave‐particleduality.Evenbeforeworkingontheideasexpressedinthisbook,Ialwaysfoundtheimplicationsofthedoubleslitdiffractionexperimenttobeaproblemforthewave‐particleduality“explanation”.Ifanelectronisalsoapointparticle,thenthepointparticlemusthavepassedthroughonlyoneofthetwoslits.Evenifsomewavepropertiesoftheparticleexploredtheotherslit,itseemsasifthediffractionpatternshouldimplyanunequalilluminationofthetwoslits.Alargeinequalityofillumination


should greatly reduce the visibility of the diffraction nulls. Instead, the diffraction patternproduced by electrons implies that the electron possesses onlywaveproperties as it passesthroughbothslitsequallyandsimultaneously. I findit fareasiertoconceptuallyunderstandhow the rotarmodelof anelectron canpossessparticle‐likeproperties thanunderstand thecontradictions of wave‐particle duality. Imagine an electron as a unit of quantized angularmomentumwithaspecificrotationalfrequencyexistinginaseaofvacuumenergy.Withthismodel it ispossibletoseehowthisquantizedangularmomentumcouldpossiblyreassembleitselfontheothersideofadoubleslit.Itwouldhavepassedthroughbothslitsandwouldexhibitthedoubleslitdiffractionpattern.Thefollowingexplanationwillcontinuetouseanelectronasanexample,butthisalsoappliestocompositeparticlessuchasneutronsormolecules. Thereasoningaboutneutronsandothercompositeparticleswillbediscussedlaterinthechapterabouthadrons.Thediffractionpatternproducedbylightofwavelengthλpassingthroughasingleslitofwidthd1producesawell‐knownsingleslitdiffractionpattern.ThesingleslitintensityprofilecanbecalculatedfromtheFraunhoferdiffractionintegral.Ingeneral,thedoubleslitdiffractionpatterncanbethoughtofasasuperpositionofthesingleslitdiffractionpatternonthediffractionpatternfor two coherent narrow λ line sources of light separated by the double slit separationdistanced2.Hereweareonlygoingtodoagreatlysimplifiedversionthatcanillustratesomeinterpretationsofthewavepatternsthatwillbeobtainedinadoubleslitsimulation.Inthissimplification,westartwiththemodelofamovingrotarsuchasshowninfigure10‐5.Symmetricalportionsoftheexternalvolumeofarotararepresumedtopassthroughbothslitssymmetricallyandbecometwonewsourcesofwaves.Forsimplificationtheemissionpatternfromeachslitisspherical.Anactualslitwidthwouldintroduceanadditionalintensitydistributionsuperimposedonthissphericalemissionpattern.Thekeydifferencebetweenthismodelandastandarddoubleslitexperiment with light is that the waves emanating from both slits in this model have thebidirectionalpropagationcharacteristics deBrogliewaves ofamovingrotar.Thismeansthatweareinterfering4setsofwaves 2counterpropagatingwavesfromeachslit .


Figure10‐16showstheresultofthecoherentinteractionofthese4waves. Thetwoslitsareidentifiedas“source1”and“source2”. Theresultant interferencepattern issimilar towhatmightbeexpectedfromlightpassingthroughdoubleslits,buttherearesomekeydifferences.First,thethreedarkhorizontalbandsaretheresultofthedeBrogliewavesandaresimilartothebandsshowninfigure10‐5.Thesebandswouldbemovingatfasterthanthespeedoflightwm c2/v andwouldnotbeinthepatternproducedbylight.Secondly,theblueandyellowwave representationswouldbealternating color dipolepolarity at the electron’sComptonfrequency.Third,theseblueandyellowwaverepresentationswouldbepropagatingawayfromtheslitsattheelectron’spropagationspeed ud v andnotatthespeedoflightaswouldoccurwithlight.Fourth,itisimpossibletodetectthefinewavepatternrepresentedbytheblueandyellow waves. These have displacement amplitudes less than Planck length/time and areundetectableasdiscretewaves.Theyareattheelectron’sComptonfrequencyandthesquareofthetimeaveragedwavesrepresenttheprobabilityof“finding”theelectron.Whenamovingelectronencountersadoubleslit,itnolongerisanisolatedelectron.Themodelmustchangetoreflectthechangedboundaryconditions.Partsofthequantizedwavethatistheelectronpassthroughbothslitsandpartsoftheelectronencounterthematter otherwavesinspacetime thatformstheblockingareas.Whetherornottheelectron rotar reformsontheothersideof thedoubleslit isaprobability.Eventhoughmostof thedipolewave isblockedperhaps 99% , apparently the remaining 1% has a probability of reconstructing the entire


dipolewaveon theothersideof thedoubleslit. If itdoesreform, ithasnewcharacteristicsimposedby the interactionwith thematter waves thatsurrounds thedoubleslitopenings.TherearenewboundaryconditionsthatareexpressedastheradialinterferencepatternshowninFigure10‐16.EnergyDensityofaGravitationalField: Inchapter8wecalculated theenergydensity in thecenterof the rotarmodelof a fundamentalparticle. This is a rapidly rotating timegradient1020Hz whichisequivalenttoarotatinggravitationalfield.Ifarotating“gravfield”atthe

centerofafundamentalparticlehasenergydensity,doesthegravitationalfieldexternaltotheparticle externaltoλc alsohaveenergydensity?Thewave‐amplitudeenergydensityequationU A2ω2Z/c seems to imply thatanoscillatingwave is required for there tobeanyenergydensitybecauseifωc 0thenUq 0.Thisisreasonablebecauseitisnotpossibletoextractenergy from the static strain component of a gravitational field. However, as previouslydiscussed,agravitationalfieldhasbothastaticcomponentwhichwecaneasilydetectandanoscillating component which we cannot detect. In figure 8‐3 the sloping line with smallundulationsismadeupofaDC‐likecomponentthatdoesnotoscillateandanAC‐likecomponentthatdoesoscillate.TheAC‐likecomponentimpliesanenergydensitytoagravitationalfield.ItmaybeimpossibletoeverextractenergyfromtheAC‐likecomponentofgravitythesamewaythatitisimpossibletoextractenergyfromvacuumenergy,butonthequantummechanicallevelagravitational fielddoesappeartohaveenergydensity. Ifso,whataretheequations? Howmuchofablackhole’senergy is in thegravitational fieldexternal to itseventhorizon? Is itpossibletoderivethecurvatureofspacetimeatapointifweknowthegravitationalfieldenergydensityandinteractiveenergydensityatthatpoint?Gravitational Energy Density: The Oscillating Component of Gravity:Mostofthischapterwasspentdescribingthepartoftheexternalvolumeassociatedwiththeoscillatingandnon‐oscillatingcomponentsassociatedwiththeelectricfield.Whiletherehaspreviouslybeensomediscussionoftheoscillatingcomponentofgravity,thereisstillseveralimportantpointsonthissubjectwhichhavenotbeenpreviouslymentioned.Earlierinthischapterthetwoamplitudetermsassociatedwitharotar’sgravityweregivenas:

Ag gravitationalstandingwaveamplitudeoscillatingat2ωc

AG 2 β gravitationalnon‐oscillatingstrainamplitude

For a single particle such as an electron, this oscillating term associated with gravity is anextremely weak effect because the amplitude of this oscillating component isAg Aβ2/ 2 Lp2/r2. Furthermore, this amplitude is squared again in the energy densityequationsoAg2 Lp4/r4 where Lp

4 ≈ 10-140 m4. However,theimplicationisthatagravitationalfielddoeshaveenergydensitysinceithasfrequency,amplitudeandtheimpedanceofspacetime.


Therefore,wewillcalculatetheimpliedenergydensityofagravitationalfieldusingU A2ω2Z/candsettingA Ag Aβ2/ Lp2/r2andZ Zs c3/G

ħ

ħ

Thisequationignoresanumericalconstantnear1TotalEnergyinaGravitationalField:UsuallyIhaveavoidedincludingconstantsnear1sinceevenPlanck lengthandPlanck timemighthavea constantassociatedwith itwhenexactcalculationsarerequired.However,inthecaseoftheenergydensityofagravitationalfieldIcanmakeaneducatedguessaboutthevalueoftheconstant. Einstein’sfieldequationisessentially an equation of energy density equals pressure as shown in chapter 4. Theconstantassociatedwiththatenergydensityis1/8π.AlsotheenergydensityoftheelectricfieldproducedbyaPlanckchargeisUe 1/8π ħc/r4 .IseeananalogybetweenbothoftheseequationsandtheequationfortheenergydensityofagravitationalfieldUg Gm2/r4.Inbothoftheseequationstheconstantis1/8π.Therefore,Iamgoingtorestatethisequationincludingthisconstant.

wheregravitationalaccelerationis

Thiscalculationusedtheassumptionthatweweredealingwithindividualrotars.AllothercalculationsinthisbookthatinvolvingfundamentalparticlesapplyeventoahypotheticalPlanckmassparticlewhichwouldbeablackholewithSchwarzschild radius definedasRs≡Gm/c2)ofλc Lp Rsandω ωp.Nowthequestionis:Canweswitchfromindividualparticles and assume that themass termm in the above equation can apply tomassiveobjectswithmanyparticlessuchasplanets,starsandblackholes?Theunknownis:Howdoes the oscillating component of the gravitational fields produced by many individualparticlesaddtogether? Sinceallothergravitationaleffectsscalewith totalmasswithnoindicationofadifference inthetypeornumberofparticles, Iwillassumethat theaboveequationappliestothetotalmassandproceedwiththeanalysis.Theimplicationisthatgravityisproducingastrainonthehomogeneousvacuumenergythatformsthespacetimefield.Ifgravityhasanoscillatingcomponent,thentheindicationisthatweshouldbeabletocalculatethemagnitudeoftheenergydensityproducedbyamassatknownradiusr.Outofcuriosity,whatistheenergydensityproducedbytheearth’sgravityat the surfaceof the earth? The earth’smass is about6x1024Kgand its radius is about6.38x106m.ThisworksouttoaboutUg 5.8x1010J/m3atthesurface.Convertingthistotheequivalent mass density dividing by c2 this is about 6.4x10‐7 kg/m3. This calculation


includestheconstant1/8π.Totestthisfurther,wewillcalculatethetotalenergyoutsideofaspecificradiusr. Obviously forablackhole, “r”mustbe interpretedascircumferentialradius. Ifwe integrate the above equation to find the total energy ormass equivalent externaltocircumferentialradiusr integrationlimitsrtoinfinity weobtain:

Eext

SetE Ebh energyofaBlackholeandsetr Rs≡Gm/c2,theSchwarzschildradius

Eext

Therefore,weobtaintheresultthat,ignoringnumericalconstantsnear1,abouthalf perhapsall of the energyof a blackhole is contained in the gravitational field external to the eventhorizon3.AfterIindependentlyderivedtheaboveequations,Ifoundthatothershavereachedthesameconclusionabouttheenergydensityofagravitationalfield.4,5However,thisisnotagenerallyacceptedideaamongphysicists.Myapproacharrivedatthisimpliedenergydensityofagravitationalfieldusingthemodelofgravitythathasbothanoscillatingcomponentandanon‐oscillatingcomponent.Thisappearstobeatotallynewconcept.Thefactthatagravitationalfieldhasenergydensityalsoimpliesthatthereshouldbeframedraggingifthemassisrotating.Thepredictedmagnitudeoftheframedraggingwillhavetobedeveloped.Theoscillatingcomponentofgravityisalsoveryimportantincosmology.Chapters13and14describe how the Big Bang and the expansion of the universe can be explained as atransformation of the properties of the spacetime field. A key part of this transformationinvolvestheoscillatingcomponentofgravitytransformingtheobservableenergydensityintheearlyuniverseintothesituationwehavetodaywhereonlyabout1partin10120isobservableenergyintheformoffermionsandbosons.Flat spacetime is the homogeneous energy density of the dipole waves in spacetimepredominately at Planck frequency previously described. Matter experiences “interactiveenergy density” of spacetime Ui ω2/G as previously described. Introducing oscillating

3 Note:Theterm“eventhorizon”ofablackholeisusedbecausethisistheeasiestwaytoexplainaconcept.However,Idoubtthatablackholehasatrue“eventhorizon”wheretherateoftimeistrulystopped.Itispossibletohavetherateoftimesloweddownbyavastamountsuchas1020times,buthavingittrulystoppedrateoftimepresentsproblemsforspacetimeparticleswhichcannotsurvivehavingtherateoftimestopped. 4 http://www.grc.nasa.gov/WWW/k-12/Numbers/Math/Mathematical_Thinking/possible_scalar_terms.htm 5 Lynden-Bell, D. R Astro. Soc. 213, (1985) pp 21-25, http://adsabs.harvard.edu/full/1985MNRAS.213P..21L


standingwavesintothishomogeneousenergydensityofdipolewavesproducesadistortionoftheenergeticvacuumcharacteristics.Wecallthisdistortioncurvedspacetime.InsightintotheCreationofCurvedSpacetime:Weinteractwiththenon‐oscillationcomponentofthegravitationalfield.Howeveritistheoscillatingcomponentofthestandingwaveswhichcreatetheenergydensityofbothanelectricfieldandtheenergydensityofthegravitationalfield.Nowthatwehavetheenergydensitycontainedinarotar’sgravitationalfield,canwegainanewinsightintothecurvatureofspacetimeproducedbyagravitationalfield?Tounderstandthis,recallthattheinteractiveenergydensityofthespacetimefieldatdistancerfromamassisgivenbyUi Fp/r2.Therefore,thegravitationalfieldisintroducingorganizedenergydensityintowhatwaspreviouslyflatspacetimepossessingenergydensitywithoutquantizedangularmomentum.Weknowthatmattercurvesspacetime.NowthatwehavebeenabletoquantifyboththeenergydensityofagravitationalfieldUgandthe“interactiveenergydensityUi”ofthespacetimefield,wewillseeifitispossibletoconnecttheratioUg/Uitothecurvatureofspacetime.Inchapter2wede inedΓ≡dt/dτ 1/ 1–β .Whereβhasbeennamedthe“gravitationalmagnitude”.Forfundamentalparticles theweakgravityapproximation isβ Gm/c2r . Thisdefinitionof thecurvature of spacetime is accurate to better than about 1 part in 1040 for theweak gravityproducedbyknown fundamentalparticles. SinceUi ignoresnumerical constantsnear1, thecalculation of the ratio of Ug/Ui will also ignore the numerical constant associated with Ug.Thereforewewillset:

and

β2

1– thecurvatureofspacetime weakgravityapproximation

Thisisawonderfulresult!Thedistortionproducedbyintroducingagravitationalfield’senergydensity into flat spacetime creates a curvature of spacetime associated with the weak fieldgravity.Ibelievethatamorerigoroustreatmentwillyieldgeneralrelativitysincenonlinearitiesarebeingignoredinthissimplifiedcalculation.I initially found the square in Ug/Ui Gm/c2r 2 β2 a bit surprising. Before Imade thiscalculation,Iwasexpectingtheratiotoequalthegravitationalbeta β Gm/c2r .However,Icannowseethatsincegravityisanonlineareffectthatscaleswithamplitudesquared,thereisalsoasquareeffectthatextendstoUg/Ui.IfindtheconnectionbetweenUg/Uiandthecurvatureofspacetimeissoreasonablethatitconfirmsthreethings.1 ItconfirmsthatgravitationalfieldshaveenergydensityUg kGm2/r4 kg2/G . 2 Itconfirmstheaccuracyof the interactiveenergydensityofspacetimeUi kc2ω2/G kFp/λ2 k ω/ωp 2Up.3 .Itconfirmsthatwehave


foundthekeytounderstandingthemechanismbywhichmatterinteractswiththespacetimefieldtoproducecurvedspacetime.Thecalculationthatshowsthatgravityhasenergydensityalsoimpliesthatgravityisarealforce.Avariationoftheequivalenceprincipleisoftencitedtosupporttheideathatgravityisnotatrueforcebecauseaninertialframeofreferencecanmakeagravitationalfielddisappear.However,aninertialframeofreferenceisreallyanacceleratingframeofreferenceinagravitationalfield.Theopposingargumentinfavorofagravitybeingatrueforceisasfollows:A body in free fall in a gravitational field is just experiencing offsetting forces. The gravitational force is still present but the accelerating frame of reference causes an opposite inertial “pseudo-force” on every particle. The two opposing forces exactly offset each other creating the impression that the gravitational force has been eliminated. However, to draw this conclusion, it is necessary to assume that the “force” of inertia has also been eliminated even though acceleration is taking place. The acceleration also produces an offsetting rate of time gradient and an offsetting spatial effect. The conclusive way to resolve this argument is the proof that gravity possess energy density.


Chapter11

Photons

Photons:Inchapter9weexaminedthesimplifiedcaseofphotonsconfinedtoareflectingcavity.When we reduced the dimensions of the cavity to the minimum size that would supportelectromagnetic radiation of a particular wavelength, we called this condition “maximumconfinement”.Equationsfortheelectricandmagneticfieldweredevelopedforthismaximumconfinementcondition.Inthischapterwewilldevelopamodelofafreelypropagatingphoton.This model will be incomplete, but hopefully this attempt at developing such a model willencourageotherstoimproveonthismodel.It is commonly argued that photons cannot just be waves because the photo‐electric effectdemonstratesthatalltheenergyofaphotonistransferredtoasingleelectroninanabsorbingatom.Thisconcentrationofenergycanejectanelectronfromthesurfaceofthephoto‐electricmaterial. The reasoning is that if a photonwas awave, then thewavewoulddistribute thephoton’senergyevenlyoverasurfaceandnosingleelectronwouldreceivetheenergyrequiredtoejectanelectronfromasurface. Thisreasoningassumesthat thewavesofaphotonaresimilartoeitherasoundwaveorperhapsawaveintheaether.Asoundwaveinairforexampleproducesanoscillatingdisplacementofmanymolecules.Whenasoundwave inairstrikesasolid surface, it is reallymany individualmolecules striking the surface independently. Theenergyofthesoundwaveisdistributedoverthesurface.Similarly,iflightisimaginedasawaveinanomnipresentfluidcalledtheaether,thenwavesintheaetherwouldbeevenlydistributedoverasurfaceandanelectronwouldnotbeejectedfromthesurface.A photon possesses quantized angularmomentum, not quantized energy. It is the quantizedangularmomentumthatgivesaphotonitsparticle‐likeproperties.Recallthataphotonisanħunitofangularmomentumpropagatingasawaveinthespacetimefield.However,thiswaveisalsoquarantinedinsomewaybythesuperfluidpropertiesofthespacetimefield.Thisresultsinaphotonhavingquantizedangularmomentum.Itisimpossibleto distribute quantized angularmomentum tomultiple locations. All the quantized angularmomentum andalltheenergy isdepositedasasingleunitatasinglelocation.Thiseffectgivesaphotonparticle‐likepropertieswhenitinteractswithmatter.However,thisdoesnotrequirethat a photon is an actual particle; it only requires quantization of angular momentum.Furthermore,thephoto‐electriceffectislowenergyComptonscattering.Inchapter10itwasshownthatComptonscatteringhasaplausiblewaveexplanation.Eventhoughthephotoelectricmaterialhasasurfaceworkfunction,thebasicComptonscatterwaveinteractionisthesame.Thequantizedwavecharacteristicsofthephotonandtheelectroncanejecttheelectronfromthesurface.


InComptonscatteringthephotonusuallytransferssomeofitsenergytothescatteredelectronFig.10‐12showstherareexampleofaphotontransferringlinearmomentumbutnotenergy .Evenwhenaphotonisreflectedoffa“moving”mirror,thereflectedphotonisDopplershiftedandthereforehasdifferentenergythanithadbeforeitreflectedoffthemirror.Thepointisthatphotonsdonotpossessquantizedenergy. Whenaphoton isemittedorabsorbed, there isatransferofħquantizedangularmomentum.Energyisalsotransferred,butthetransferofenergyisjustabyproductofthetransferofquantizedangularmomentum.Thischapterwillshowaproposedmodelofaphotonandexplainhowaphotoncanpossessquantized angular momentum. Usually the angular momentum of a photon is an abstractconcept that is dealtwithmathematically. It is easy towrite an equation that incorporatesangular momentum but the hard part is to develop a physical model of a photon whichincorporatesangularmomentum.Whyisangularmomentumquantized?Whatenforcesthisquantization of angular momentum? Answering these questions are another test of theproposedspacetime‐basedmodeloftheuniverse.How Big Is a Photon? Thestandardexplanationforaphoton’spropertiesistoclaimthataphoton exhibits “wave‐particle duality”. Of course, this is not a conceptually understandableexplanation;itismerelyaname.Inadoubleslitexperiment,aphotonseemstohavetheabilitytopassthroughbothslitssimultaneously.Thisimpliesthataphotonhasaphysicalwidth.Aphotonalsoseemstohaveaphysicallengththatisafunctionofthephoton’sspectralwidth.Forexample,arubidiumatomhasaspectrallinecalledtheD1transition.Whenarubidiumatomgoesthroughthistransition,itemitsaphotonoverabout26ns.Thisimpliesthatthisphotonisextendsoveradistanceofabout8meters.Furthermore,thespectrallinewidthofthisrubidiumtransitionhas abandwidth that also implies awavepacketwith thisphysical lengthusingaFouriertransform.Thisisnotjustan8meteruncertaintyinthelocationofthephoton;itisanactual wave train that is 8meters long at a wavelength of about 795 nm. The hyper finetransitionofcesium133thatisusedinatomicclocksemitsatamicrowavefrequencyofabout9.2 109Hz.Thisemissionfrequencyisstabletobetterthanonepartin1013.Thisextremelynarrowbandwidth implies that theemittedphoton is continuouslyemittedoverabout1000secondsandthelengthofthephotonwavetrainisabout3 1011m.Attheoppositeextreme,therecordfortheshortestpulseoflaserlight intermsofapproachingthetheoreticallimit isamere 1.3 cycles per pulse. The current record for the shortest pulse of laser light is about8 10‐17secondsbutthatpulsecontainedseveralcyclesandharmonicsofashortwavelength.Theemissionofaphotonbyanatomisoftendepictedasifitisaninstantaneousevent.However,this is known tobe incorrect1because there is a timedependenceof thewave function in aquantum transition. Experimental measurements have been made of samples undergoingspectroscopictransitions.Theseexperimentsconfirmthattherearenoinstantaneousquantum

1 Macomber, J. D. The Dynamics of Spectroscopic Transition; John Wiley and Sons: New York, 1976.


jumps. Instead, the electric and magnetic properties undergo a smooth and continuoustransitionoccurringovertheemissiontimeperiodwhichusuallycorrespondstothelifetimeoftheexcitedstate.Thereisaverygoodpapertitled“HowaPhotonisCreatedorAbsorbed”2thatisalsoavailableonline3. Thisonlineversionhastwogoodanimationsshowingtheoscillationsofahydrogenatomduringtheemissionofaphoton.Thispapershowsthatthereisanoftenignoredtransitionperiodrequiredfortheemissionorabsorptionofenergyinatransitionbetweenenergylevels.There are numerous experimental observations that confirm that photons are emitted orabsorbedoveratimeperiodcorrespondingtotheinversebandwidth.Quotingfromtheabovearticle:The first experimental measurements of bulk samples undergoing spectroscopic

transitions were obtained from nuclear magnetic resonance observations of thetransitionnutationeffect4andspinechoes5,6usingcoherentradiationproducedbyasingleradio frequencyoscillator. Morerecently, theanalogoustransitionnutationeffect7,8andsocalled“photonechoes”9,10,11havebeenobservedinmolecularspectrausingpulsedcoherent laserradiation. Theseexperimentsconfirmthatthereisno“quantum jumps” in thenon‐stationarystate;rather therearesmooth,continuousperiodicchangesinthemagneticandelectricalpropertiesofthesystemundergoingatransition.

Anelectronbound inanatompossesses lessenergy thanan isolatedelectron. Forexample,13.6 eV of energy is released when an isolated electron combines with a proton to form ahydrogenatominthegroundstate.Thebindingenergycanbeconsideredtobeanegativeformofenergywhichmeansthatbindingreleasesenergyandbreakingabondrequiresenergy.Therotarmodelof fundamentalparticlessaysthatan isolatedelectron isarotatingdipole inthespacetimefieldwitharotationalfrequencyofabout1.24 1020Hz.Whenanelectronandprotoncombinetoformahydrogenatominthegroundstate,aphotonisreleasedwithafrequencyofabout 3.3 1015Hz ~13.6 eV . According to the rotarmodel, the energy releasedwhen ahydrogenatomformscomesprimarilyfromtheelectronlosingenergyandreducingitsComptonfrequencybyabout3.3 1015Hz. Theoscillationsof theelectroncloudshownintheaboveanimation canbe thought of as the interactionbetween the electron in twodifferent energy

2 Henderson, G J. Chem. Educ. 1979, 56, 631-634 3 http://www3.uji.es/~planelle/APUNTS/ESPECTROS/jce/JCEphoto.html 4 Torrey, H. C. Phys. Rev. 76, 1059 (1949). 5 Hahn, E. L. Phys. Rev. 77, 297 (l950). 6 Hahn, E. L. Phys. Rev. 30, 580 (l950). 7 Tang, C. L.; Statz, H. Appl. Phys. Lett. 10, 145 (1967). 8 Hocker, G. B.; Tang, C. L. Phys. Rev. 184, 356 (1969). 9 Kurnit, N. A.; Abella, I. D.; Hartmann, S. R. Phys. Rev. Lett. 13, 567 (1964). 10 Abella, I. D.; Kurnit, N. A.; Hartmann, S. R. Phys. Rev. 141, 391 (1966). 11 Hartmann, S. A. Sci. Amer. 218, 32 (1968).


states twodifferentfrequencies interferingwithitself.Theseoscillationscreatewavesinthespacetimefieldthatremovethisenergyatthefrequencyoftheoscillations. Thesewavesareproposedtobethephoton.In chapter 9 itwas shown that a photonwas not an energy packet traveling THROUGH thespacetimefield,butawavetravelingINthemediumofthespacetimefield.Nowwewilldevelopthemodelofaphotonfurtherbasedonwavespropagatingwithinthevacuumfluctuationsofthespacetimefield.Thesewaveswillbeshowntobedistributedoverasubstantialvolumeofthespacetimefield.Clearlysuchastructurecannotbevisualizedasapointparticle.Itonlyexhibitsparticle‐likepropertiesbecauseitpossessesquantizedangularmomentumcombinedwiththepropertyofunity.Thiscombinationgiveswavesinthespacetimefieldwithquantizedspintheabilitytoactasaunitandtransfertheirenergyandquantizedspintoasinglerotar.Thename“photon”isnotreallyappropriatesincethesuffix“on”wascoinedspecificallytoimplyparticleproperties.However,thename“photon”isflexibleenoughthatitcanadjusttoaquantizedwaveexplanation.Thereforenoattemptwillbemadetoreplacetheword“photon”.Vacuum Energy Versus the Aether:Theaetherwasoncebelievedtobeanomnipresentfluidwithasingleframeofreferencethatservedasthepropagationmediumforlightwaves. Theconceptoftheaetherimpliedthatitshouldbepossibletodetectevidenceoftheearth’smotionrelativetotheaether. Forexample,ifthismodeloftheaetherwascorrect,thereshouldbeadetectableshiftininterferencefringesintheMichelsonMorleyexperiment.Thisandnumerousmorerecentexperimentshaveconfirmedthatthereisnoevidencethatanysuchrelativemotionexists. Withthisexperimentalevidence,theconceptoftheaetherhasbeenabandoned. Thisbackgroundmakesthefollowingseemlikearadicalproposal:A photon is a wave disturbance possessing quantized angular momentum that propagates in the medium of the vacuum fluctuations of the spacetime field.Thisconceptwasfirst introduced inchapter9but it isrepeatedheresinceanydiscussionofphotonsmustbebasedon this concept. This sounds like I ammerely substituting the term“vacuum fluctuations” for aether. However, there is abigdifference. Thepropertiesof thevacuum fluctuations are such that it is impossible to detect any motion relative to thesefluctuations ignoring hypothetical experiments which probe the Planck frequency limits .Recallthatvacuumenergy vacuumfluctuations isaseaofdipolewavesinthespacetimefieldthatlackangularmomentum.Thesewavesarealreadypropagatingatthespeedoflight.Everypartofawavebecomesthesourceofanewwaveletsothespacetimefieldbecomesaseaofdipolewavedistortions thatare rearranging themselvesat the speedof light. Aspreviouslyexplained, these chaoticwaves that lack angularmomentummake the quantummechanicalversion of a vacuum. They are responsible for the appearance of virtual particle pairs, theuncertaintyprinciple,theCasimireffect,theLambshift,etc.


Gravitational waves also propagate in themedium of the spacetime field. It is known thatgravitationalwavespropagateatthespeedoflightinanyframeofreference.IfitwaspossibletodoaMichelson‐Morleyexperimentusinggravitationalwaves,itwouldbeimpossibletodetectmotionrelativetothemediumofthespacetimefield.Thereforethespacetimefieldpossessestheabilitytopropagatewavesatthespeedoflightasseenfromanyframeofreference.Itisalsoimpossibletodetectmotionrelativetothedipolewavesthatareanessentialpartofthespacetimefield.Chapter7discussedthesubjectofspectralenergydensityinvacuumenergy.Thatdiscussion isrepeatedherebecause it takesonnewmeaningwhenappliedtodetectingmotionrelativetovacuumenergy.

Inquantumfieldtheory,spacetimeisvisualizedasconsistingoffields.Everypointinspacetime is treated like a quantized harmonic oscillator. The lowest quantummechanical energy level of each oscillator is E ½ ħω. This concept leads to aspectralenergydensityρ ω dωthatis:

ρ ω dω kħ

dω

Thisspectrumwithitsω3dependenceofspectralenergydensityisuniqueinasmuchasmotionthroughthisspectraldistributiondoesnotproduceadetectableDopplershift. It is a Lorentz invariant random field. Any particular spectral componentundergoesaDopplershift,butothercomponentscompensatesothatallcomponentstakentogetherdonotexhibitaDopplershift. Itshouldalsobenotedthatneithercosmologicalexpansionnorgravityaltersthisspectrum.12

Therefore,vacuumenergyhascompletelydifferentpropertiesthantheaetherwhichisthoughtofasanomnipresentfluidwhichhasaspecificframeofreference.Normallywewouldsaythatitisimpossibletodetectmotionrelativetothevacuumenergy.However,inchapter14wewillexamine the implications of the vacuum fluctuations having amaximum frequency equal toPlanck frequency. Since it is impossible for the spacetime field to propagate awavewith aDopplershiftedfrequencyhigherthanPlanckfrequencyorawavelengthshorterthanPlancklength,itispossibletoimagineextremeframesofreferencewherethelawsofphysicsarenotcovariant.However,sincethesearevastlyoutsideofanyexperimentalcondition,thisexceptionwillbeignoreduntilchapter14.

12 Puthoff, H.E. Phys. Rev. A Volume 40, p.4857, 1989 Errata in Phys. Rev A volume 44, p. 3385, 1991 See also New Scientist, volume 124, p.36, Dec. 2, 1989


Thebosonssuchasphotonshaveangularmomentumofħ.Thesequantizedangularmomentumdisturbancesinthespacetimefield,arenotconfinedtoaspecificlocationliketherotarmodel.Insteadthequantizedangularmomentumofaphotonisdistributedintoanexpandingwavethatwillbedescribedinthischapter.Whenaphotonisabsorbed,thedisturbancewithquantizedangularmomentumwouldcollapseandtransferallthequantizedħangularmomentumtotheabsorbingbody rotars .A Photon Is Not a Dipole Wave in the Spacetime Field:Aphotoncannotbeaquantizeddipolewaveinthespacetimefield.Adipolewavecreatesanoscillatingrateoftimegradient.Arateoftimegradientiscapableofacceleratinganymatter,evenaneutralparticlesuchasaneutron.Topreventaviolationoftheconservationofmomentum,dipolewavesinthespacetimefieldarelimitedtoamaximumdisplacementofspacetimeofPlancklengthandPlancktimeaspreviouslydescribed. If a photon was a dipole wave, a focused laser beam would easily violate thisrestriction.RecallthatrotarshavethisquantummechanicallimitofPlancklengthandPlancktime.Therefore,themaximumenergydensityforadipolewaveinthespacetimefieldisequaltotheenergydensityoftherotarvolumeofarotar Uq .Thisknowledgecanbeconvertedtoamaximumintensity foradipolewaveinthespacetimefieldwithreducedwavelengthλsince Ucforradiationpropagatingatthespeedoflight.Frompreviouscalculationsofrotars,weknowthattheenergydensityintherotarvolumeofarotaris:Uq ħc/λ4.Thisenergydensitythensetsanupperlimittothemaximumintensitythatcouldbeachievedatthefocusofalaserbeamifphotonsweredipolewavesinthespacetimefield.Using Uc,themaximumintensityofelectromagneticradiationthatwouldviolatetheconservationofmomentumifphotonsweredipolewavesinthespacetimefieldis: max ħc2/λ4.Ifphotonsweredipolewavesinthespacetimefield,thenthemaximumintensityallowableforalaserbeamwithawavelengthofabout10‐6m reducedwavelengthλ 1.6 10‐7m wouldbeabout1010w/m2 104w/cm2 . Intensitiesinexcessof1020w/cm2havebeenachievedatthefocusofapulsedlaseratthisapproximatewavelength,soaphotonisdefinitelynotapropagatingdipolewave in the spacetime field. Amodelwill be developed that is a wave in the spacedimensionsofthespacetimefieldwithoutaffectingtherateoftime.GravitationalwavescanalsohavedisplacementamplitudethatexceedsdynamicPlancklengthLpbecausetheyalsodonotcauseadisplacementoftherateoftime. Theyalsoarenotdipolewavesinthespacetimefield. Waves in Vacuum Energy:Sofarwehavetalkedaboutwavesintheabstract.Whatarethewavesofaphotonmadeof?Photonsarenotpropagatingdipolewavesinthespacetimefieldbuttheproposedansweristhattheyareapropagatingpolarizationinthedipolewavesthatformthespacetimefield vacuumenergy .Inchapter9theproposalwasmadethatanelectricfieldisapolarizedstraininthespacetimefield.Thisproducestheunsymmetricaleffectsassociatedwithpropagationofaneutralparticle in thepositiveornegativeelectric fielddirection.Thisresultsintheonewaytimeofflightdistancebetweenpointsbeingslightlydifferentforoppositepolaritydirections.Forexample,ifthereisanelectricfieldpresent,thenthetimerequiredtogo


frompointAtopointBislongerthanthetimerequiredtogofrompointBtopointA.However,theroundtriptimeisthesameasexpectedfromthespeedoflightandnoelectricfield.Thismeans that there isnochange inpropervolumeandnochange in therateof time. AsinglephotonwithreducedwavelengthλinmaximumconfinementhadadifferenceinpathlengthoveradistanceofλofLpandnphotonsinmaximumconfinementproducedapathlengthdifferenceof:√ .SinceitistheoreticallyimpossibletomeasureadistanceaccuratetoPlancklength,thisexplainswhyitisimpossibletomeasurethewavepropertiesofasinglephotonbutitispossibletomeasurethewavepropertiesofmanyphotons forexample,theelectricfieldofaradiowave .Themodelofasinglephotonwouldbeawavepossessingquantizedangularmomentumthatpropagates in vacuum energy. The dipole waves that form vacuum energy cannot possessangularmomentum,soadisturbancecarryingquantizedangularmomentumwouldbe likeapropagatingphasetransitionthatcausesasmallamountofdipolewaves thephoton’senergyandvolume tomomentarilyloseitssuperfluidproperties.Suchawavewouldmomentarilybegiving adistributedangularmomentum to vacuumenergy. Apparently thisdisturbance is atransversewave thatpossesses thepolarizationcharacteristicswenormallyassociatewithaphoton.Suchawaveinvacuumenergywouldpropagateatthespeedoflightforallframesofreference. Furthermore, the impedance of free space Zo associated with electromagneticradiationwas found in chapter9 tobeequal to the impedanceof spacetimeZo/η2 4πZs 4πc3/G.Electron-Positron Annihilation Thought Experiment: Wearegoingtodevelopthephotonmodelfurtherusingathoughtexperiment.Inthisthoughtexperimentwewilllookatthetwoentangledphotonsproducedbyannihilationofanelectron‐positronpair.Thismightseemlikeanexoticwayofproducingapairofphotons,but it actually is the simplest case to examinebecauseunlikeatomicemissionorComptonscattering,noparticlesremainaftertheannihilationtocarryawaymomentum.Weareassumingthatwestartwithanelectron‐positronpairwithantiparallelspin.Thisformof positronium typically has a lifetime of about 10‐10 seconds and usually decays into twoentangledphotonswithantiparallelspins.Wewillassumethisnormaltwophotondecay.Thesetwogammarayphotonshaveoppositespinsandoppositemomentumvectors. However,thespinsandmomentumvectorsareonlydefinedwhenthefirstphotonisdetected.Eachphotonhas511,000eVofenergy,sothefrequencyandwavelengthofeachphotonmatchestheComptonfrequencyandComptonwavelengthoftheannihilatedrotars.Theconventionalpictureofthisannihilationistheemissionoftwophotonswhichhavebothparticleandwaveproperties.Theparticlepropertiesimplyapacketofenergythatcanbefoundsomewherewithintheuncertaintyvolumedefinedbythedecaytimeandspinorientationoftheelectron‐positronpair.Thisconventionalpicturehasthetwoentangledphotonswithoppositemomentum,butthemomentumdirectionsarenotsetuntilthefirstphotonisabsorbed.Atthat


momentthemomentumandpolarizationofthesecondentangledphotonhasbeendetermined.One counter intuitive part of this model is that the information about momentum andpolarizationmustsomehowbecommunicatedtothesurvivingphotonwhenthefirstofthetwoentangledphotonsisabsorbed.Even if the two entangled photons are separated by a distance of one light‐year, they stillsomehowareininstantaneouscommunication.Ifoneofthephotonshappenstointeractwithapolarizerof anyorientationorellipticity, theotherphoton instantlybecomes theorthogonalpolarization.Manylogicalquestionsarisefromthispicture.Howdothetwophotonskeeptrackofeachother?Whattypeofcommunicationsignalissentoutwhenonephotonencountersapolarizer?Howdoesthiscommunicationhappenfasterthanthespeedoflight?Iproposethatthereason that thisexplanation is impossible toconceptuallyunderstand isbecause it is thewrongpictureofaphoton.Electron-Positron Annihilation – The Quantized Wave Model: Wewill now restate thisinteraction using the photon and rotar models that incorporate distributed waves in thespacetimefield.Supposethatweusetherotarmodelofanelectronandapositronwithoppositeantiparallel spins that are initially far apart compared to distance . Both rotars have aComptonangularfrequencyofabout7.76 1020s‐1orafrequencyof1.23 1020Hz.Ifthesetworotarsmovetowardseachother, itmeansthat theywouldbothperceivetheothertobeDopplershiftedandthetwofrequencieswouldnotbeexactlythesame.Sincethesetworotarsaregoingtoeventuallyemittwoentangledphotonsofthesamefrequency,wewillpresumethattheformationofpositroniumincludessometypeofsynchronizationofthesetwofrequencies.Whatwill happenwhenwe bring together an electron and a positron? It appears as if thisinteractiondestabilizesbothrotars. Therotarmodelofanisolatedelectronproposesthatanelectron isstablebecausethere isa typeofresonancebetweentheelectron’srotatingdipolewaveandthesurroundingvacuumenergy.RecallthattheelectronhasaquantumamplitudeofAβ 4.18 10‐23. Thisdimensionlessnumber isalsotheelectron’s frequency,energy,mass,inverserotarradius,etc.inPlanckunits.Thisfrequencyandamplitudeachievesaresonanceinvacuumenergythatcancelsthelossofpowerandproducesconstructiveinterferencewiththerotatingdipolecore.The lifetime of positronium with antiparallel spins has been calculated13 from QED as:τ 2ħ/mec2α5 2/ωcα5 1.25 10‐10s. Thiscalculatedlifetimeagreeswithexperimentallymeasuredlifetime.Theannihilationofpositroniumwithantiparallelspinsusuallyproducestwoentangledgammarayphotons.Thesetwophotonshavethesamefrequency,wavelengthandenergy as the electron and positron in the rest frame. Since the electron‐positron pair hadantiparallelspins,thetwoentangledphotonsalsohaveacombinedspinofzero.

13http://arxiv.org/PS_cache/hep-ph/pdf/0310/0310099v1.pdf


Photon Model of Annihilation: Now the model of this annihilation using waves in thespacetime fieldwill be presented. The electron and positron both have a Compton angularfrequency of 7.76 1020 s‐1. When these two rotars annihilate each other the stabilizationmechanismwithvacuumenergyisdestroyed.Thecancellationwaveformedinvacuumenergynolongerpreventsthedissipationoftheelectron’sandpositron’senergy.Wavesinspacetimeattheelectron/positronComptonfrequencypropagateawayfromthesiteoftheannihilationatthe speedof light. Thesepropagatingwaves are twoentangledphotons that result from theannihilation.Thewavesarepropagatinginthemediumofthevacuumfluctuationsthatareanessential characteristicof thespacetime field. Aspreviouslydetermined, the impedanceandbulk modulus encountered by these waves corresponds to the impedance of spacetimeZs c3/G andthebulkmodulusofspacetimeKs Fp/λ2.Thespeedofthiswavepropagationis equal to c which was previously calculated from the interactive energy density of thespacetimefieldandtheimpedanceofspacetime.

Figure11‐1showsthisannihilationevent.Atthecenterofthisfigure,asmallvolumeofspaceislabeledasthelocationoftheannihilationoftheelectron‐positronpair.Thisfigureshowstheresultsofthisannihilationsometimeaftertheannihilationtakesplace.Thisisacross‐sectionalviewofthewavesinthespacetimefieldthatarethetwoentangledphotons. Thereisnowasphericalshellofwavesinthespacetimefieldaroundtheannihilationvolumewiththeradiusincreasingatthespeedoflight.


Thewavesinthisshellhaveanangularfrequencyof7.76 1020s‐1whichisalsotheelectron’sComptonangularfrequency.Sincethereisnofrequencychangebetweentheelectron/positronandthephotonsemitted,theannihilationeventcanbeconsideredmerelythedestabilizationofthevacuumenergycancelationwavesthatwerekeepingtheelectron’senergyconfined. Thetime required for annihilationwas1.25 10‐10s so this amounts to about1.5 1010 cycleswavelengths formingashellofwaveswithathicknessofabout3.8cm. Figure11‐1showsmultiple concentric circles to convey the idea that the expanding shell contains manywavelengths.Alsotheentangledsphericalshellofwaveshaszeronetspinsincethisexampleassumedthattheelectronandpositronhadantiparallelspin. Entanglement:Supposethatthesphericalshelloftwoentangledphotons propagatinginthevacuumfluctuationsofthespacetimefield expandsintowhatmightbecalled“emptyspace”toaradiusofonelight‐year.Reallythisspaceisfilledwithaseaofvacuumenergyandthewavesareadisturbanceinthisvacuumenergy.Atthispoint,supposethatasmallportionofthewaveshellencountersanabsorbingobjectthatwewillgenericallycallanabsorbingparticle.Itcouldbeanatomorothergroupofrotars.Tomaketheabsorptioninteresting,wewillpresumethattheabsorbingmaterialhasastrongabsorptionpreferenceforclockwisecircularpolarizationatthefrequencyofthetwoentangledphotons.Thisabsorbingmaterialisillustratedinfigure11‐1asapointlabeled“particlepriortoabsorbingoneofthetwoentangledphotons”.Theabsorbingparticle or group of rotars is capable of absorbing quantized angular momentum of ħ.However,thesphericalshellofwavesistwoentangledphotonsthatweregeneratedwhentheelectronandpositron bothspacetimedipoles wereannihilated.Theabsorbingmaterialcannotinteractwithonlyasmallpercentageofthequantizedwave.Thequantizedangularmomentumtransferredmustbeeitherħornothing.Nowitgetsinteresting.Anyinteractionmustbewithacompletephoton quantizedenergyandangularmomentum . In this case, thismeans that the interaction is between the absorbingmaterial and one of the two entangled photons that together made up the entire sphericalwavefront,onelight‐yearinradius.Theinteractioncannotbewithbothphotonsbecausethetwophotonshaveatotalspinofzero.Thereappearstobeaprohibitionagainstenergytransferwithoutanaccompanyingspintransfer.If there is absorption, then the proposed property of unity causes one of the two entangledphotonstocollapse.Alloftheenergy 511,000eV ,alltheangularmomentum ħofclockwisecircular polarization and all themomentum ~ 2.7 10‐22 kgm/s of the single photon isdepositedintotheabsorbingmaterial.Eveniftheabsorptionhappensoverafiniteabsorptiontimethatiscomparabletoafiniteemissiontime forexample,severalnanoseconds ,theentirequantizedwaveenergy,distributedoveronelight‐yearradius,mustcollapseinthisshorttime.


Thedetailsofhowphotonsareproposedtocollapsewillbediscussedlaterwhenwedealwiththeeventualabsorptionofthesecondofthetwoentangledphotons.Thecollapseofthefirstofthe two entangled photons removes from the spherical shell all of the wave characteristicsnecessary to make a circularly polarized photon with clockwise angular momentum. Thisincludesnotonlyspinandenergy,butthecollapsealsoimpartsmomentumof~2.7 10‐22kgm/s.withanaccuratelydefinedmomentumvectorthatwillbediscussedlater.Whatremainsinthesphericalshellofwavesareproposedtobeallthecharacteristicsrequiredtomakeaphotonwiththeorthogonalpolarization counterclockwisespin andtheoppositemomentumvectorwhichisnotquitepreciselydefined.Thisimpliesthatthesecondphotoncanonly impart theoppositemomentumandoppositecircularpolarizationwhen it iseventuallyabsorbed.Thephotonthatwasabsorbedmustbetheinverseofthephotonthatremainssincethetwoentangledphotonsoriginallyformedauniformamplitudeshellofwaves.Therefore,toobtainadescriptionofbothphotonswewillexaminetheremainingphoton.

Single Photon Model:Figure11‐2showstheproposedmodelofthesurvivingphotonaftertheotherentangledphotonpreviouslydiscussedwasabsorbedandcollapsed into theabsorbingparticle.Inotherwords,figure11‐2showsaslightlylatertimethanfigure11‐1.Infigure11‐2theenvelopeofwaveshasexpandedpasttheparticlethatabsorbedtheotherentangledphoton.Thisparticleisshownnearthebottomofthefigure.Comparetheplacementofthisparticletotheplacementshowninfigure11‐1.


Themajorvisibledifferencebetweenfigures11‐1and11‐2isthatthecirclerepresentingtheenvelopeofwavesisnowshownwithdifferentshadingrangingfromblack highestamplitude atthetopofthefigurethroughshadesofgraytowhite lowestamplitude atthebottomofthefigure. Thewaves are still present but thewaves arenot shown in figure11‐2because theemphasisin11‐2istheamplitudeofthesewaves.Theamplitudedistributionforthewavesintheremainingphotonisproposedtobethesameastheamplitudedistributionofthewavelets intheHuygens‐Fresnel‐Kirchhoffprinciple. RecallthattheHuygens‐Fresnelprincipleaccuratelymodelsdiffractionofanopticalwavebyassumingthatallpointsonanadvancingwavefrontbecomethesourceofanewwavecalledawavelet.Anewwavefrontisformedbycoherentlyaddingtogetherthesesecondarywaves,includingtheirphases. This principle was perfected by Gustav Kirchhoff when he added an amplitudedistributiontothewavesthatformedeachnewwaveletthatpreventedbackwardspropagationtowardsthesourceandimprovedtheaccuracy.Previouslythewaveletsweremerelyconsideredto be limited to the forward hemisphere. This arbitrary limitation worked well for mostapplications, but Kirchhoff’s addition perfected the principle for all cases. The amplitudedistributionformulatedbyKirchhoffiscalledtheobliquityfactorK θ .Itcanbeexpressedeitherin Cartesian or spherical coordinates. The spherical coordinate representation is:K θ cos2 θ/2 .


AsinglephotonisproposedtohavethesameamplitudedistributionasthewaveletsrequiredfortheHuygens‐Fresnel‐Kirchhoffprinciple.Theshadingoftheenvelopeofwavesinfigure11‐2has this distribution. However, figure 11‐3 is a graphical representation of the amplitudedistributionofthesurvivingphoton.Theabsorbedphotonwouldhavehadtheinverseofthisamplitude distribution which is the same as inverting figure 11‐3. Adding these twodistributions together produces the uniform distribution of the original entangled pair ofphotons sin2 θ/2 cos2 θ/2 1 .Photon’s Momentum:Wearenowgoingtoreturntofigure11‐2toaddressthequestionofthemomentumofasinglephoton. Recall that the thoughtexperiment thatgeneratedthis figurepresumedthatthetwoentangledphotonsexpandedinavacuumtotheradiusofonelightyearwhenfinallyoneofthetwophotonswasabsorbedbytheparticleshowninfigure11‐2.Theseassumptions mean that the momentum vector for the surviving photon must be very welldefined.Theoriginalannihilationoftheelectron/positronpairhadanuncertaintyvolumethatcanbecalculatedknowingthemassoftheelectron/positronpair ~1.82 10‐30kg andthe

lifetime ~1.25 10‐10s togiveanemissionuncertaintyradiusΔx ħt/2m 6 10‐8m.Alsotheparticlethatabsorbedthefirstphotoncouldhavebeenpartofadetectorthatcouldspecifythelocationoftheabsorptiontoaradiusmuchsmallerthanthe6 10‐8muncertaintyoftheemission. Therefore the uncertainty in the emission dominates and we can ignore theuncertaintyindefiningtheabsorptionlocation.Sincethesetwouncertaintyvolumesareseparatedbyonelightyear ~1016m ,thismeansthatthat the momentum vector uncertainty angle of the single photon in this example is about6 10‐24radians 6 10‐8m/1016m .Thesurvivingphotonmusthavetheoppositemomentum,soatthemomentthefirstphotonisabsorbed,weknowagreatdealabouttheallowedvolumewherethesecondphotoncanpossiblybeabsorbedinthefuture.Thereasonforthisexerciseisthatthemodelofasinglephotonmustbecapableofthismomentumaccuracy.Infigure11‐2theenvelopeofwavesshouldbepicturedasbeingonelightyearinradius.Thisfigurealsoshowsanangularspreaddesignated“momentumuncertaintyangle”whichforthisexampleisabout6 10‐24radians. Howisitpossibleforthiswavestructuretopossessthisnarrowamomentumuncertainty?Justlookingatthefigure,itseemsasifthesurvivingsinglephotoncouldbeabsorbedbyanabsorbingparticlelocatedinalmostanydirectionaroundtheexpandingshellofwaves exceptperhapsthezeroamplitudedirection .Therequirementofawelldefinedmomentumhelpsusdefine themodelandhelps todefine theway thatphotonscollapse.


Collapse of a Single Photon: Figure 11‐4 shows the eventual absorption of the survivingphoton.Theabsorbingbodyisdesignatedanditmustliewithinthevolumelimitedinwidthbythemomentumuncertaintyangle.Alsotheabsorbingbodymustbelocatedwithinthethicknessoftheenvelopeofwavesduringtheabsorption.Thecollapseofthesinglephoton’senergyandangularmomentumisdepictedbythearrowsshowninfigure11‐4.Thesearrowsindicatethatthecollapseproceedsalongthecircumferentialroutedefinedbytheenvelopeofwaves.Thishasagreatdealofappeal.Themomentumtransferredtotheabsorbingbodycanonlyhavea radial vector relative to the emission uncertainty volume. It would be a violation of theconservationofmomentumfortheretobeatangentialvectorcomponentthatislargerthantheuncertainty limit.Thismeansthat theonlyvolumeof thephoton’swavestructurecapableofinteractingwithmatterisrestrictedtothesmallvolumeboundedbythemomentumuncertaintyangle. This is theonly volumewhere the collapse is sufficiently symmetrical toprevent thetransferofsubstantialtransversemomentum.Theremustbeoffsettingtransversemomentumcomponentsoneithersideoftheabsorbingbodysothatthecollapseisbalancedandresultsinthecorrectnetmomentum.Next, we are going to examine whether the photon structure proposed here is capable ofcollapsingtoavolumeassmallaswouldberequiredtosatisfythemomentumuncertaintyanglewhichisabout6 10‐24radians.Ifthesecondphotonwasabsorbedshortlyafterthetimeofthefirst photon absorption, then the shell of waves would be about the same size which was


postulatedtobe1lightyearinradius ~1016minradius . Thewavesaredistributedoveradiameterofabout2lightyears ~2 1016m andthewavelengthproducedbytheannihilationofanelectron/positronpairisλ 2.4 10‐12m.Ifwemerelycalculatethediffractionlimitofthiscombinationofaperturesizeandwavelengthweobtainaλ/Duncertaintyangle divergenceangle of2.4 10‐28radians.Thereforethiswavestructurecaneasilysatisfytherequirementofcollapsingtoavolumeequivalentto6 10‐24radiansuncertainty. Infact,thisphotonmodelshould always collapse to an area about one wavelength in circumference. At this point,quantizedangularmomentumtakesoverandthecollapseintoasingleatomispossible.Popper’s Thought Experiment: Beforeproceedingwith thedescription,wewillpauseandmentionthatthethoughtexperimentdescribedinfigures11‐1and11‐2havesomesimilaritytoPopper’s experiment proposed by Karl Popper and published in 1982. The Copenhageninterpretation of quantum mechanics implies that making an experimental measurementincreasesuncertainty. Popperwrote:“Iwishtosuggestacrucialexperimenttotestwhetherknowledgealoneissufficienttocreate'uncertainty'and,withit,scatter asiscontendedundertheCopenhageninterpretation ,orwhetheritisthephysicalsituationthatisresponsibleforthescatter”. He suggested an experiment incorporating entangled particles, slits and detectorsdescribedelsewhere1415.ThecontentionwasthatdetectingoneoftheentangledparticlesshouldintroducescatterinstantlycommunicatedtotheotherparticleiftheCopenhageninterpretationwascorrect. AnactualexperimentwasperformedbyKimandShia in1999usingentangledphotons16Theconclusionwasthatthesecondofthetwophotonsactuallyhadamuchsmallerscatter smaller uncertainty thanwouldbe expected from theCopenhagen interpretation17.This experimental result supports the model of two entangled photons proposed here. Asproposed in figure 11‐2, the “momentum uncertainty angle” defines the uncertainty of theallowedinteractionvolumeforthesecondphoton.Gainingknowledgeofthelocationofthefirstphotontobeabsorbeddecreasestheuncertaintyaboutthelocationwherethesecondphotonwillbeabsorbed.Limits on Absorption: The portion of the single photon that lies outside themomentumuncertaintyvolumecannotinteractwithmatter.Themodelofphotonssaysthatthesewavescanpassthroughmatter rotars withoutbeingabsorbedoraffected.Inthefigures11‐1to11‐4wecarefully avoided the question of the waves encountering other matter by postulatingpropagationinanemptyvacuum.However,thewavesexternaltothemomentumuncertaintyangleshouldmerelypassthroughmatterwithoutanyinteraction.Recallthatmatterismadeofrotarsthatarejustemptyspacetimethatisveryslightlystrained. roughly1partin1020foranupquark .Ifadisturbanceinvacuumenergyisincapableoftransferringangularmomentum

14 http://en.wikipedia.org/wiki/Popper%27s_experiment 15 Popper, K.R. Quantum theory and the schism in physics, Routledge, 1992, p.27 16 Y.-H. Kim and Y. Shih (1999). "Experimental realization of Popper's experiment: violation of the uncertainty principle?". Found. Phys. 29 (12): 1849–1861. 17P. Sancho (2002). "Popper’s Experiment Revisited". Found. Phys. 32 (5): 789–805


becauseitisoutsideofthemomentumuncertaintyvolume,thenthisportionofaphotonshouldbemoreinertthananeutrinoandshouldeasilypassthroughmatter rotars .Unity Connection:Thecircumferentialcollapseshowninfigure11‐4istheproposedpropertyofunityatwork.Thisproposedpropertypermitsfasterthanthespeedoflightcommunicationandcollapsewithinasinglequantizedwaveinspacetime.Inchapter14aspeculativemechanismwillbeproposedforfasterthanthespeedoflightcommunicationwithinasinglequantizedwave.However, superluminal communication is an experimentally established property ofentanglement and it is quite reasonable that a single photon would also possess this samecapabilityevenifthecompleteexplanationisnotcurrentlyavailable.Thesuperluminalcollapseofaphotonshowninfigure11‐4issymmetricallybalancedanddoesnotinvolvethetransferofanyinformationorexternalmomentum.Themomentumthatistransferredisentirelywithinthesinglephoton.Whilethecollapseisfasterthanthespeedoflight,itisnotinstantaneous.Itprobablyrequiresatimeofatleast2π/ωwhichisaminimumofonecycleofthephoton.Itisalsoclearthatthereisnomysteryhowthephotonmodelcancarryangularmomentum.Theshellofwavesmustbepicturedasa3dimensionsphericalshell. Thewavesthatmakeupacircularlypolarizedphotonwouldhaveaphaseprogressionthatcirculatesthesphericalshellatafrequencyequaltothephoton’sfrequency. Figure10‐9inchapter10showedwavesintheequatorial plane of the external volume of a rotar. It is obvious that waves forming anArchimedesspiralcarryangularmomentum.Itisproposedthatacircularlypolarizedphotonhas a similar Archimedes spiral wave distribution in the equatorial plane. The spatialdistributionofthisshellofwavesmakesiteasytoseehowitispossibleforthisphotonmodeltocarry angular momentum and transfer the angular momentum when the wave structurecollapsesontoanabsorbingbody.Itisnotclearhowtheconventionalmodelofaphotoncarriesortransfersangularmomentum.Photon Emission from a Single Atom:Inthethoughtexperimentinvolvingtheannihilationofanelectron/positronpairtherewasnoremainingmattertoremovemomentumandcomplicatetheanalysis.Wewillnextaddresstheemissionofaphotonbyasingleatom.Ifaprotonandanelectroncombinedtoformahydrogenatom,theenergyofthephotonemittedwouldbeabout13.6eV.Theemissionofthisenergywouldcausethehydrogenatomtorecoilwithavelocityofabout4m/s.Hypothetically,itispossibletodeterminethedirectionofthephoton’smomentumbymonitoringthemotionoftheelectronandprotonpriortoformingthehydrogenatomandbymonitoringtherecoilofthehydrogenatomaftertheemissionofthephoton.Thereisuncertaintyinmakingthesemeasurements,buttheaccuracyindeterminingthedirectionofthephoton’smomentumincreaseswiththeamountoftimebetweenemissionanddetectionoftherecoilingatom.Thephotonhastheoppositemomentumoftherecoilinghydrogenatom.Thefurthertherecoiling hydrogen atom travels before its position is detected, the more accurate that themomentumvectoroftherecoilinghydrogenatomcanbedetermined.


Thephotoniscarryingtheoppositemomentumvectorastherecoilinghydrogenatomtowithinthe limitsof theuncertaintyprinciple. I claim thatnotonlydoesourability tomeasure themomentumvectorof therecoilinghydrogenatom improvewith time,but there is continuedinteractionbetweentheexternalvolumeoftherecoilinghydrogenatomandthephotonevenafterthephotonhasbeenemittedfromtheatomicvolumeofthehydrogenatom.Thehydrogenatomissurroundedbywavesthatarepartoftheexternalvolumesoftherotarsthatformtheatom. As thephotonspherical shell expands, it interactswith theseexternalwavesand thisinteraction fine tunes the momentum uncertainty vector of the expanding photon. Thisinteraction reduces themomentum uncertainty angle of the expanding photon over time tocoincidewith the improved ability tomeasure the recoilmomentumvectorof the hydrogenatom. Compton Scattering Revisited:Ahydrogenatommadeof4rotars 1electronand3quarks istoo complicated to analyze the interaction between the expanding photon and the externalvolumeoftherotarsthatformthehydrogenatom.ThereforewewillswitchbacktoComptonscattering between a single electron and a single photon. This interaction was previouslyexaminedusingtheseriesoffiguresfrom10‐12to10‐15.Figure10‐15isthesuperpositionof4waves. Thewavesatthetopofthisfigurerepresentthephoton’swavesbeforeandafterthescattering.Thewavesatthebottomofthefigurerepresenttheelectron’swavesbeforeandafterthescattering.Themiddleportionofthefigureshowsbothpairsofwavesinteracting.Becauseof thestandingwaves intheelectron’sexternalvolumethis interactioncontinuestooccurlongafterwethinkthatthescatteringeventhashappened.Picturetheelectron rotar ashaving diminishing standing waves in its external volume that extend a long way from thelocationofthescattering.The“news”thattherotarhasundergoneaccelerationpropagatesintothese surrounding standing waves at the speed of light. There is a spherical shell that isexpandingatthespeedoflightwheretheoverlapofthebeforeinteractionandafterinteractionwavesoverlap.Withinthisexpandingshellofoverlappingwaves,thefringepatternshownatthetopoffigure10‐15stillexists,butatgreatlyreducedamplitude.Thisissuperimposedonthephoton’swaveswhicharealsospreadingawayfromthescatteringsiteatthespeedoflight.Theinteractionbetweentheseoverlappingwavescontinuestomakesuccessivelyfineradjustmentstothephoton’smomentum.Thisexplainshowtheuncertaintyofthephoton’smomentumcanimproveovertime,justlikewereasonedbylookingattherecoilofahydrogenatom.Inallofthis,theelectrondoesnotundergoagradualaccelerationasmightbeexpectedforaclassicalparticlethatischangingitsmomentum.Instead,thewavemodelofthescatteredrotarchangesfromthewavepatternoftherotarbeforethescatteringtothewavepatternoftherotarafterthescatteringwithoutundergoingtheintermediatevelocities.Similarly,ahydrogenatomwouldnotgraduallyaccelerateto4m/sasitemitsa13.6eVphoton.The“before”wavepatternfadesasthe“after”wavepatterncomesintoexistence.


From theuncertaintyprincipleweknow thatadecrease in themomentumuncertainty Δp mustbeaccompaniedbyanincreaseinthepositionuncertaintyΔx.Whenwethinkofapointparticlephoton,thenthephysicalinterpretationofΔxisdifferentthanwhenwethinkofasinglephoton as a shell of waves with a large radius. To accommodate this improvement in ourknowledge of the photon’smomentum decreased Δp , it is necessary for the radius of thephotontoincreasewithtime increasedΔx .Thisrequirementfitsperfectlywiththeproposedphotonmodelbecausetheradiusofthephoton’sshellofwavesincreaseswithtime.Recoil from Coherent Emission:Intheaboveexamples,itwasrepeatedlyemphasizedthattheydescribedthepropertiesofasinglephoton.Thecharacteristicsofphotonschangedramaticallywhentheycongregateintocoherentbeams.Wearegoingtoeaseintoadiscussionofabeamoflightmadeofmanyphotonswiththefollowingpreliminarythoughtexperiment.Supposethatwehavearotatingelectricallychargeddipole.Thisimaginedelectricallychargeddipoleismadeofapositivechargeandanegativechargephysicallyseparatedbyashortrod.Forexample,imagineanelectronandapositronseparatedbyarod1mmlongandrotatingaboutaperpendicularaxisatthecenteroftherodatafrequencyof1010Hz.Eventhoughitwouldbevirtuallyimpossibletohavethiselectricaldipolemechanicallyrotateatthisfrequency, inthethought experiment there is no such limitation. We would expect to see the emission ofmicrowaveelectromagneticradiation 1010Hz inaclassicalrotatingdipoleemissionpatternfromthisrotatingdipole.Thispatternhasemissioninalldirections,buttheintensityofemissionistwiceasstrongalongtherotationaxisastheintensityintheequatorialplane.However,nearthe equatorial plane there aremore ster‐radians for emission so all emission directions areimportant. Theclassicalrotationdipoleradiationpatternissymmetricalaroundtheaxisofamechanicallyrotatingdipole.Thispatternisreallytheresultoftheemissionofmanyincoherentphotons.Thissymmetricalemissionpatterndoesnotproducerecoilinanyparticulardirection.Nowsupposethatwehaveatrillionsuchrotatingdipolesdistributedoverasphericalvolumewithradiusabout1000timeslargerthanthemicrowaveemissionwavelength.Thedipolesarethereforedistributedinawaythatindividualdipolesareseparatedfromtheirnearestneighborbymuchlessthanonewavelengthbutthegroupismuchlargerthanawavelength.Iftherotatingdipolesareallrotatingincoherently,theyemitincoherentradiationinalldirections.Sincetheemissionissymmetricallybalanced,thereisnonetrecoildirectionfeltbyindividualrotatingdipoles.However, if all dipoles are rotating coherently same frequency, parallel rotation axis andcontrollable phase the radiation from the group of rotating dipoles can be controlled. Forexample, the microwave radiation can bemade into a diffraction limited beam that can besteeredinanydirection. Thisdirectionalcontroldependsentirelyontheabilitytoadjustthephaseofindividualdipolesinthegroupsothatthemultipleemissionsaddconstructivelyinthedesiredemissiondirection.


Nowlet’sthinkabouttherecoilfeltbyeachmechanicallyrotatingdipolefromtheemissionofradiation.Ifonlyonedipoleismechanicallyrotating,theemissionofmultiplephotons photons is symmetricalandnospecific recoildirection is feltby thesingle rotatingdipole. However,whenthemultiplerotatingdipolesareproperlyphasedtoconstructivelyinterfereinaparticulardirection, theneachrotatingdipolemust feela force intheoppositedirectionastheemittedbeam. We would say that this force is the momentum recoil required for conservation ofmomentum.However,eachrotatingdipoleisjustinteractingwiththelocalEMfieldgeneratedbythecoherentadditionofsphericalwavesgeneratedbyotherrotatingdipoles.Thecollimationanddirectionalityoftheemittedbeamisachievedbythegroupinteraction.Thepointisthattheemissiondirectionandtherecoildirectionaretheresultofthecoherentadditionofproperlyphasedrotatingdipoles.Huygens-Fresnel-Kirchhoff Principle:Atthispointtheanalysisconvertstoaclassicexampleof the Huygens‐Fresnel‐Kirchhoff principle. With this principle, each point on a wavefrontbecomesthesourceofanewwaveletthatemitsintotheamplitudedistributionformulatedbyKirchhoff:cos2 θ/2 .Thisisexactlythesameemissionpatternasaphotonwithitsmomentumvectoralignedwiththebeamvector.Thereforeeachcycleofthecoherentphotonemissionisidenticaltothewaveletthatwouldbeformedatthelocationoftheemittingrotatingdipole.TheHuygens‐Fresnel‐Kirchhoff principle describes amplitude addition and how intensity isproportionaltothesquareofamplitude.Therefore,theindividualphotonjoinsthebeamasifitwasmerely a series of newwavelets. The only difference is that in the Huygens‐Fresnel‐Kirchhoff principle the total energy of the beam remains constant while the emission of acoherentphotonincreasesthetotalamplitudeandenergyofthebeam.Itisashortstepfrommechanicallyrotatingdipolestomanyatomsinanexcitedstateinalasergain medium. The propagation of a laser beam is well described by the Huygens‐Fresnel‐Kirchhoffprinciple.Eachpointonthelaserbeambecomesanewwaveletandthebeamevolvesbycoherentadditionofsuccessivegenerationsofwavelets.Whenalaserbeampassesthrougha laser gain medium, it interacts with atoms in an excited state. The interaction not onlystimulatestheemissionofphotonswiththeproperfrequencyandphase,buttheinteractionalsoimparts the correct recoil to the atoms so that the photons are emitted with the correctmomentumvector. The additionof newphotons to the laserbeam then corresponds to thewaveletadditionoftheHuygens‐Fresnel‐Kirchhoffprinciple.Beam of Light:Howcanthemodelofasinglephoton sphericalshellofwaveswithaKirchhoffamplitudedistribution bereconciledwiththeconceptofawellbehavedbeamoflightthatcanbeeasilyreflectedoffmirrorsandbroughttoafocus?Abeamoflaserlightdoesnotseemtohaveanyof thepropertiesofasphericalshellofwaves justdescribed. However, thephotonmodelforasinglephotonisdifferentthanthephotonmodelformanyinteractingphotons.Manyinteracting photons are better described by the Huygens‐Fresnel‐Kirchhoff principle which


incorporateswavelet addition from theotherphotons. Eachphotondoeshaveamomentumuncertaintyanglesetbytherecoilfeltbytheemittingatom.Thegroupbehaviorislimitedtophotonswhichhaveamomentumuncertaintyanglewhichpermitspropagationinthedirectionoftherestofthebeam.Therefore,thereisalimittotheamountofsteeringthatcanoccurbytherestofthephotons.Ifaphotonwasemittedwithanuncertaintyangleattoosteepofananglerelativetothedirectionoftherestofthebeam,thenthatphotonwillmerelyexitthebeam.Forexample,a1mwHeNelaserbeamcontainsabout3 1015photonspersecond.Multiplephotons,suchasalaserbeam,togetherachievethefamiliarbeamof light. Eventhougheachindividual photonwould propagate into a spherical shell ofwavespreviously described, theinteraction with the other photons in the beam causes a group behavior described by theHuygens‐Fresnel‐Kirchhoffprinciple.Forexample,theintensityscalesasamplitudesquaredandtheonlyamplitudethatcountsistheamplitudewithinthemomentumuncertaintyangleforeachphoton.Theseconditionsachievethefamiliarpropertiesofabeamoflight.Still,thegenerationofwaveletsimpliesthatthephotonsareexploringeverypossiblepathbetweentwopointsasrequiredbythepathintegral.Furthermore,thereisapresenceoutsidethevolumethatappearstobethediameterofthebeam.Eventhoughtheportionsofwavesthatexistoutsidethebeamvolumelacktheabilitytostronglyinteractwithmatter,isthereanyproofthatthese“external”wavesmustexist?Theanswertothisquestionisyes.Lightisaformofenergyandthereforeabeamoflightmustcauseagravitationalfield curvedspacetime inthesurroundingvolume.Itisnotsufficienttocite“curvedspacetime”andclaimtohaveprovedthatlightcausesgravity.Whenweadopttheassumptionthattheuniverseisonlyspacetime,theimplicationisthateverythingisknowableincluding themechanics of light causing curved spacetime. The gravity and electric field offundamentalparticleswasexplainedasbeingcausedbystandingwavesintheexternalvolumeoftheparticle.Wavefrequency,amplitudeandnonlineareffectswerederived.Similarly,abeamoflightcausesgravityinthesurroundingvolume.Howisthisaccomplished?Ifwethinkofabeamoflightasastreamofparticle‐likephotons abundleofquantizedenergythatlacksrestmass ,thenthereisnomechanismtocreategravityinthesurroundingvolume.Alsothereisnomechanismforthelighttoexploreallpossiblepathsbetweentwopointsasrequiredbythe“pathintegral”. However, ifwe think of light as awavedisturbance in spacetimewith eachwavebecomingthesourceofnewwavelets,thenthereisamechanismtoexplainboththegravitationalfield and thephysics behind thepath integral. Aspreviously explained, thesewaves extendbeyondthemomentumuncertaintyvolumewhichdefinesthelimitwherephotonscaninteractwithmatter.Normally Iwoulddoaplausibilitycalculationhere toshowthat theproposedphotonmodelresulted in the correct gravitational field. For example, if we imagined many photons nγphotons inmaximumconfinement,thenthis“confinedenergy”propagatingatthespeedoflightwould exhibit rest mass as described in chapter 1. An analysis of photons in maximum


confinementcouldalsobemadetolooksomethinglikearotar. Itwasalreadydemonstratedhow a rotar with internal energy E produces the correct gravitational field for this energy.However,neitheroftheseareexactlyapplicablebecausethemaximumconfinementvolumeofmanyphotonshas adifferent radius thana single rotarof equal energy. Ananalysisof thissubtletyisincompleteandoffthemainpoint.Itwillmerelybestatedthattheproposedphotonmodeldoesextendintothesurroundingvolume.Furthermore,itappearstohavethecorrectpropertiestocreateboththerequiredgravitationalfieldandtheabilitytoexploreallpossiblepaths.Themathematicalproofofthiswillhavetobelefttoothers.Why Is there No Amplitude Dependence In a Photon’s Energy? Oneof themysteriesofquantummechanics is contained in the photon’s energy equation E ħω. Why does thisequationonlycontainthefrequencytermωwithnoamplitudeterm?Iftwodifferentwaveshavethesamefrequencybutdifferentamplitudes,theyshouldhavedifferentenergy.Canthespacetimebasedmodelofphotonsanswerthiscenturyoldmysteryofphysics?Themaximumdisplacementof thespacetime fieldallowed foradipolewave inspacetime issubject to the previously discussed Planck length/time limitation. While a photon is nottechnicallyadipolewaveinthespacetimefield,itisproducedandabsorbedbydipolewavesinspacetime.AsinglephotoninheritsaconnectiontoPlancklengthandPlancktime.Aspreviouslyshown,multiplephotonscanproduceadistortionofthespacetimefieldthatappearstoexceedthePlancklength/timelimitation.However,theroundtripchangeindistancedoesnotactuallyexceed this limitation. The strain amplitude of a single photon inmaximum confinement isAs Lp/λ. We will calculate the energy of a photon in maximum confinement usingE A2ω2ZV/c.E A2ω2ZV/cset:A As Lp/λ,ω c/λ,Z Zs c3/GandV λ3

E ħ

ħω

Therefore,theamplitudetermdisappearsandweareleftwithE ħω.Itiseasytoseehowthishappens. Aspreviouslyexplained,therearetwotypesofamplitude:displacementamplitudeand strain amplitude. The displacement amplitude for photons inmaximum confinement isalwaysequaltodynamicPlancklengthLp.Thestrainamplitude As Lp/λ isusedintheenergyequation,butthestrainamplitudeincorporatesthedisplacementamplitude. Thefactthatallphotonsproducethesamedisplacementofthespacetimefieldmakesitpossibleforcancelationstoeliminateamplitudefromtheenergyequation.However,inanothersense,thepresenceofħ

isaremnantofamplitudetermbecause: / ħ 3 2⁄ . Everythingcancelsfromtheamplitude squared termexcept forħ. In one senseħ is the amplitude termbecause it is theamplitudeofthequantizedangularmomentum.


Requirements of a Photon Model: A carbon monoxide molecule (CO) is a good source of photons if we are attempting to understand photons. CO has a carbon atom and oxygen atom separated by about 1.1×10-10 m. It is a polar molecule since it has a nonzero permanent electric dipole moment. Experiments have shown that the carbon atom is negatively charged and the oxygen atom is positively charged. This charge distribution is the opposite of what might be expected from chemical valences. The CO molecule can rotate around its center of mass axis perpendicular to its bond length. The CO molecule can only possess integer multiples of ħ angular momentum. (J = 0, 1, 2, 3 etc.) The fundamental rotational frequency is about 115 GHz and higher rotational frequencies are approximately integer multiples of this frequency. The higher frequencies are not exact multiples of 115 GHz because the physical rotation of the molecule slightly stretches the bond length because of centrifugal force. The CO molecule also has vibrational energy levels. This is all mentioned because the rotating polarized molecule will be useful in visualizing the model of a photon. The emission or absorption of a photon is always accompanied by an ħ change in angular momentum of the rotating molecule. All photons, even linearly polarized photons, must possess ħ of angular momentum. Furthermore, this angular momentum that is transferred between the CO molecule and a photon is not a random rotational direction. For example, the emission of a photon, even a linearly polarized photon, must always remove ħ of angular momentum and slow down the rotation of the CO molecule by ħ (one “J” unit). Conventional models of photons do not explain how linearly polarized photons can possess angular momentum. For example, if we take a rotating polarized molecule as the source of photons, then the emission pattern for rotating dipoles has linearly polarized photons emitted in the equatorial plane and circularly polarized photons emitted from both poles. Elliptically polarized waves are emitted in all other directions. All the emitted photons, regardless of polarization, are each removing ħ of angular momentum. This is a requirement to preserve the conservation of energy and momentum. Also, photons transfer ħ of angular momentum when they are absorbed. For circularly polarized photons, the rotational axis is parallel to the propagation direction and it is easy to do experiments that demonstrate that circularly polarized photons carry angular momentum. For linearly polarized photons the angular momentum should have a rotational axis that is perpendicular to both the propagation direction and the plane of the electric field. In other words, the rotational direction should have its axis in the photon’s magnetic plane. It should be possible to do an experiment that demonstrates this. I have some ideas on this point, but a description of possible experiments would be long and beyond the scope of this book. The point of this is that any model of a photon must be able to explain how individual linearly polarized photons must be able to transfer angular momentum with the axis parallel to the magnetic field.


AngularMomentumofaCircularlyPolarizedPhoton:Itispossibletoexperimentallydemonstratethatcircularlypolarizedlightpossessesangularmomentum.Forexample,themineralmicaisopticallybirefringent.Athinsheetofmicaofaparticularthicknesscanhavetheabilitytoreversethedirectionofrotationofcircularlypolarizedlight.Anotherwayofsayingthisisthatmicacanformahalfwaveplate.Ifaverysmallpieceofmicaissuspendedwithlowfriction, forexampleinaliquid thenshiningacircularlypolarizedlaserbeamatthemicachipwillcausethemicatorotate in the liquid. The photon’s angular momentum is being reversed by the mica. Thistransfersangularmomentumfromthephotonstothemicaandcausesthemicatorotate.Nophotons are being absorbed, but the frequency of the transmitted photons is being loweredslightlybytheinteractionwiththerotatingmica.Thisreductioninfrequencyisequaltotwicetherotationalrateofthemica.Thislossofphotonenergyprovidestheenergyrequiredtorotatethemica.The absorption of a circularly polarized photon by an absorbing object imparts both atranslationalmomentumcomponentp E/c mvandanangularmomentumcomponent.TheangularmomentumcomponentcanbeexpressedasL r mvwhichisthecrossproductofthetranslationalmomentumcomponentmvandtheradiusrrelativetotheaxisofrotation.Imaginethatwehaveapolarizeddiatomicmoleculewithtwoatoms,eachwithmassofm/2.Thebondbetweentheatomsapproximatesarigidrodthatmaintainsaconstantseparationdistanceof2rbetweentheatoms.Allmoleculescanonlypossessintegermultiplesofħangularmomentum.Thistranslatesintothemoleculeonlybeingabletorotatearounditscenterofmass distancerfromeachatom atafundamentalangularfrequencyωfandintegermultiplesofthisfrequency.Wearegoingtomakeacalculationofthetranslationalmomentum pt impartedtothemoleculeas it recoils from absorbing a photon of frequency with wavelength ⁄ and

comparethistotherotationalmomentum pr impartedtothemoleculebythephotoncausingthetwoatomstorevolvearoundthecenterofmass.

ħ

ħ

ħ translationalmomentum

ħ rotationalmomentum

Thisisaveryinterestingresult.Forallmolecules, ≫ .Therefore,themomentumtransferred

tocauserotation ismuchlargerthanthemomentumtransferredthatcausestranslation .

Wewilluseacarbonmonoxidemolecule CO toexaminethispoint.Thismoleculeiselectricallypolarized CandOatomsoppositelycharged sowhen it rotates it isa rotatingdipole. Themoleculecanpossessrotationalangularmomentuminintegermultiplesofħ.Forexample,oneħunitisdesignatedJ 1andtwounitsofħisJ 2,etc.AtthefundamentalrotationalfrequencyJ 1 themolecularrotationfrequencyis115GHzandthephotonassociatedwiththetransitionofJ 0toJ 1isaphotonwithfrequencyof115GHz.TheCOmoleculeisarotatingdipoleand


theemission frequencycorresponds the frequencydifferencebetweenrotationalstates. ThephotonemittedgoingfromJ 2toJ 1is230GHz,twicethefrequencyemittedgoingfromJ 1toJ 0.Toillustrateapoint,wewillbeusingthefundamentaltransitionat115GHzwhichcorrespondsto the fundamental angular frequency of ωf 7.2 1011 s‐1 and a reduced wavelength ofλf 4 10‐4m.Ifthemoleculeiscarbon12andoxygen16,thisisnotquiteequalweightatomsasassumedinthecalculation,but it iscloseenoughthatanapproximatecalculationwillstillillustrate thepoint. Even though theelectroncloudspartiallyoverlap, thismoleculehas thecarbonatomnucleusandoxygenatomnucleusseparatedbyaboutr 1.1 10‐10m.Thereforetheradiustothecenterofrotation compensatingforslightlydifferentmasses isr 5 10‐11m.Therefore the ratio of λf/r 8 106. This says that themomentum required to cause theincreaseinrotationvelocityisabout8,000,000timesgreaterthanthemomentumtransferredwhich causes the increase in translation velocity recoil velocity . Transferring ħ of angularmomentumrequiresamuchlargerforce momentumtimesinteractiontime thantransferringthetranslationalmomentum.Wewill check this surprising result to see if it is reasonable. Weknow that themolecule isspinningat115GHzwhenithasabsorbedħofangularmomentum J 1state .Iftheradialdistanceisr 5 10‐11m,then115GHzrotationfrequencyproducesacircumferentialvelocityofabout36m/s.TheCOmoleculehasmassofm 4.65 10‐26kg.Absorbinga115GHzphotonE 7.6 10‐23J producesarecoilvelocity v E/mc ofabout5 10‐6m/s.Therefore,theratiooftherecoilvelocitytotherotationalvelocity 36m/svs.5 10‐6m/s isindeedabout8,000,000.Thisexerciseraisesanimportantquestion.Ifthefullmomentumcarriedbythephotonisonlyabletoachieveavelocityof5 10‐6m/s,thisistotallyinadequatetoachievethetransferofħofangularmomentumandacceleratethetwoatoms CandO toarotationalfrequencyof115GHzrequiringarotationalspeedofabout36m/s.Thisinsightisrevealingsomethingimportantabouthowaphotoninteractswithamoleculeoratom.Toachievetherotationalvelocityofabout36m/sitisnecessarytounderstandwhatforcethephotonusestoapplyabout8,000,000timesmoremomentumthanthephotoniscarrying.Aspreviouslydiscussed,thesetransitionsarenotinstantaneous.Theylastforatimeperiodthatisequaltotheinversebandwidthoftheemissionorabsorption.Therefore,somehowaforceisbeinggeneratedthatisabout8,000,000timesgreaterthantheforceobtainedbydividingthemomentumby the application time. There is one possibility thatwill be examined. The COmoleculeispolarizedmeaningthatthereischargeseparation.WhenarotatingdipolesuchasaCOmoleculegeneratesaphoton,theacceleratedchargegeneratesarotatingelectricfieldwhichchangesfromthenearfieldpatterntothefarfieldpatterwhicheventuallyresultsinpropagatingtransverseelectromagneticwaves.Thepointisthatrotatingelectricfieldsareresponsibleforthegenerationofthephoton.Theemissionofacircularlypolarizedphotonmustgeneratean


electricfieldwhichretardstherotationbecausetheCOmoleculemustloseenergyanddroponeħquantumofangularmomentum forexampleJ 1toJ 0 .It isquitereasonablethatthemechanical process of absorbing the same frequency photon is the reverse of the emissionprocess. Inotherwords, theabsorptionof a circularlypolarizedphotongeneratesa rotatingelectricfield.Itistheelectromagneticcouplingoftherotatingelectricfieldandtheoppositelychargedatomswhichpermitsaforceabout8,000,000timeslargerthanthemomentumforceF p/t . ThiscouplingtransfersħofangularmomentumtoaCOmoleculeandchangestherotationalenergylevelfrom,forexample,J 0toJ 1.ThesuggestionbeingmadeisthattheabsorptionofaphotonisinteractingwiththeCOmoleculeintwodifferentways.TherelativelylargeforcethatisincreasingtherotationalvelocityoftheCOmolecule is the result of the collapsing photon’s electric field causing the rotation of thechargedatomstoincrease.Toaccomplishthisthecollapsingphotonmustgeneratearotatingelectric fieldwith thepropercharacteristicsofphase,orientation,etc.Acalculationhasbeenmade notshownhere whichshowsthatanelectricfieldcomparabletotheelectricfieldpresentin a CO molecule could easily achieve the transfer of ħ of angular momentum even if theabsorptiontimeisassumedtobetheshortestpossibletime–about9 10‐12second,whichis1cycleof a115GHzphoton. Normallythetransferwouldtakeplaceoveramuchlongertimemanycycles .However,thisexercisedoesshowoneadditionaldefectintheconventionalmodelofaphotontransferringenergyfromonemoleculetoanother.Ifspacetimeisanemptyvoidandifaphotonisacompactpacketofenergywithprobabilitywaves,thenhowdoessuchaphotongenerate8,000,000timesmoremomentumintherotationaldirectionthanthelinearmomentumbeingcarriedbythephoton?Thisisnotaviolationoftheconservationofangularmomentumifyoumerelylookatthephotoncarryingħofangularmomentumandthemoleculegainsħofangularmomentumuponabsorbingthephoton.However,theproblemappearswhenyoutrytoexplaintheoriginoftheforceappliedoverthetimerequiredtotransfertheangularmomentumtothemolecule.Afavoritetrickisforaphysicsteachertoexplaintoastudentthatthephotonpossesses“intrinsicangularmomentum”–endofdiscussion.Theimplicationisthatthestudentandallotherhumansarenotintelligentenoughtounderstandthephysicalprocesseswhichproducequantummechanicaleffects.Whenanisolatedatomormoleculeemitsaphoton,thewavelengthofthephotonistypically1,000to100,000timeslargerthantheradiusoftheatomormolecule.Forexample,arubidiumatomhasaradiusofabout2.3x10‐10mandtheD2resonancewavelengthis780nm.Therefore,the emitted wavelength is about 3400 times larger than the atomic radius. We are notparticularlysurprisedthatthissmallanatomcanemitamuchlargerwavelengthphoton.Thepreviously cited article titled “How a Photon is Created or Absorbed” shows that the wavepropertiesof thetwoorbitalsbeatat theemissionfrequencyof thephotonforatimeperiodequaltotheemissiontime.FortheRbemissionexample,theemissiontakesabout26ns.Theabsorption process is the reverse of the emission process. There is not an instantaneouscollapse.Theorbitalsbeatforabout26nsandthephoton’swavescollapseintotheatom.


Similarly,anisolatedCOmoleculehasabeatbetweenthewavepropertiesofthetwodifferentrotational rateswith angular frequencyω 115 109 s‐1. The relatively longwavelengthofλ 0.4mmreversestheemissionprocesscollapsesintoamuchsmallerCOmoleculewithradiusofabout1.4Angstroms.Thepointisthattheenergydensityofspacetimepermitstheprocesstogeneratethenecessarytorquetotransferthenecessaryangularmomentum.Recallinchapter4wecalculatedthebulkmodulusandinteractiveenergydensityofspacetime.Thepropertiesof the spacetime field have no problem generating torque beyond the force implied by thephoton’smomentum.


Chapter12

Bonds,Quarks,GluonsandNeutrinos

Introduction: Thischapterlumpstogetherseveraldifficultsubjectsnotpreviouslycovered.Theseincludebonds,theψfunction,quarks,gluons,theweakforceandneutrinos.Mostoftheseare not clearly understood in mainstream physics. This vagueness prevents plausibilitycalculationstotestthespacetimemodelinvolvingthesesubjects.Forexample,quarksdonotexist in isolation, so their properties are alwayspartly hidden. Even theirmass/energy is amystery.Perhapsthemostshockingconclusionisthatthespacetimebasedmodelinitscurrentstateofdevelopmentdoesnotneedgluons. Aspreviously explained, there isonlyone trulyfundamental force therelativistic force andthis forceisalwaysrepulsive. Thissingleforceappearstobecapableofgeneratingaforcewiththestrengthandbondingcharacteristicsofthestrong force. With thisbondingaccomplishedbyan interactionwith thespacetime field,aregluons required? Color charge is another property currently assigned to gluons, but thepropertiesrequiringcolorchargecanprobablybeexplainedasaninteractionbetweenquarksrotars andthespacetimefield.Theeliminationofgluonsisapreliminaryconclusionthatneedstoberefinedandperhapsmodifieduponfurtheranalysis.Thisisthelastchapterthatdealswithparticlesandforcesbeforeswitchingtocosmology.Thesubjectswiththegreatestunknownsarelumpedintothischapter.Virtual Photons Examined: The standard model considers the electromagnetic force to betransferredbetweenpointparticlesbytheexchangeofvirtualphotons.Thisraisesinterestingquestions.Wheredoesthelossofenergyoccurwhenanelectronisboundtoaproton?Itisnotsufficienttosaythatthereisareductionintheelectricfield.ThevirtualphotonsAREtheelectricfield.Dothevirtualphotonsthatsupposedlybondoppositelychargedparticlespossessnegativeenergy? nosuchthing–onlytheabsenceofpositiveenergy Doesaprotonboundinahydrogenatomhavemoreorlessvirtualphotonssurroundingitcomparedtothesameprotoninisolation?Whatisthewavelengthofavirtualphoton?Thesequestionsareintroducedtoraisesomedoubtsaboutvirtualphotonsandthegenerallyacceptedexplanations.The spacetimemodel of the universe proposes that there are no virtual photonmessengerparticles. Thesearereplacedby fluctuationsof thespacetime fieldwhichexertpressure.Allrotarspossessan“externalvolume”ofstandingwavesandnon‐oscillatingstraininspacetimepreviously discussed. The external volumes of interacting rotars overlap. Two oppositelychargedrotarsinteractinawaythatdecreasestheComptonrotationalfrequencyofeachrotarωcisreduced .Thelocationofthelostenergyiseasytoidentify.Alsotheoverlappingexternalvolumesaffectthepressureexertedonoppositesidesofarotarbyvacuumenergy.Thispressure


differencedistributedovertherotar’sareacausesanetforce.Ifarotarisfreetomove,thereisanetmigrationoftherotationalpathofeachrotartowardstheoppositelychargedrotarwitheachrotation.Weconsiderthisforceandmigrationtobeelectromagneticattraction.Electrons Bound in Atoms:IntheBohrmodelofahydrogenatom,theelectron’s1sorbital thelowestenergylevel isdescribedashavingaradiusofao ħc/Eeα 5.3 10‐11m. Alsotheorbitalangularmomentumof theelectron’s1sorbital isħaccording to theBohrmodel.Thecombinationofthisradiussizeandthisangularmomentumcorrespondstotheelectronhavingvelocityofv αc about137timesslowerthanthespeedoflight .ThedeBrogliewavelengthforanelectronatthisvelocityisλd 2πao.ThismeansthatthedeBrogliewavelengthequalsthecircumferenceof thisBohrorbit. This isanappealingpicture,butaccordingtoquantummechanics,theBohratommodelisanoversimplification.

Figure12‐11showsthegraphandthe3dimensionalplotrepresentingtheψfunctionofthe1sorbitalofanelectroninahydrogenatom.Squaringthisψfunctiongivestheprobabilityoffindingtheelectron.Forthe1satomicorbital,thepeakcorrespondstothelocationoftheprotoninthehydrogenatom. Thecloserweprobetotheproton,thehighertheprobabilityoflocatingtheelectron.Thisplot,obtainedfromtheSchrodingerequation,looksnothinglikewhatmightbeexpectedfromtheBohrmodel.Thereisnoexclusiveorbitwithradiusao.Thereisnonetorbital

1 Mark J. Winter http://winter.group.shef.ac.uk/orbitron/


angularmomentum.Theonlyangularmomentumofthisorbitalisthe½ħangularmomentumoftheelectron.

Figure12‐2issimilartofigure12‐1exceptfigure12‐2showstheΨfunctionandprobabilityoffindinganelectronthatisinthe2porbitalofahydrogenatom.Tounderstandthesefiguresintermsofthespacetimewavemodelofrotars,itisnecessarytoexaminetheΨfunction.Ψ Function: It is an axiom of quantum mechanics that the ψ function has no physicalsignificance;onlythesquareoftheψfunctioncanbephysicallyinterpretedasrepresentingtheprobabilityoffindingaparticle. It isproposedherethatthespacetimewavemodelofrotarsdoesgiveaphysicalmeaningtotheΨfunction.Thisphysicalmeaningiseasiesttoexplainbyreturningtothehypotheticalexampleofa“particleinabox”.Inthisexercise,studentscalculatewhatwouldhappenifaparticlewasplacedinasmallcavitysurroundedbyimpenetrablewallsaninfiniteenergywell . Therearenosuchcavities,butifthereweretheparticle’squantummechanicalpropertieswouldbeexhibited.Theparticlecanonlypossessafewspecificpositiveenergiescorrespondingtoafewspecifickineticenergieswhichproducewavenullsatthewalls.Furthermore, theparticle canneverbe stationarywithin the cavity neverhave zerokineticenergy .


Ifthe“particle”isarotarwiththeexternalvolumewavestructureproposedinthisbook,thenitis conceptuallyunderstandablewhy the “particle”musthaveawavestructure thatproducesnullsatthewalls.Inthisexample,asingleparticlemovingrelativetotheboxproducesasinglestandingwavepatternwithnullsatthewalls.Thisstandingwavepatternconsistsoftwowavefrequenciespropagatinginoppositedirections.Thesetwocounterpropagatingwavesalwayspossessmoreenergythananisolatedparticlethatisnotconfined.The“particleinabox”thoughtexperimentisanidealizedthoughtexperimentsinceinnaturetherearenocavitiessurroundedbyimpenetrablewallscreatinganinfiniteenergywell.Ifsuchacavityexisted,thequantummechanicalpropertiesofaparticle rotar trappedinsidewouldberevealed.Inorderfortherotartomeettheseboundaryconditionsimposedbythewalls,itisnecessary for the rotar to possess slightly more energy than an isolated rotar. Recall thatattraction bonding involves a loss of energy and hypothetical repulsive bonding, such as aparticleinabox,requiresagaininenergy.Inorderfortherotartoachievezeroamplitudeatthetwo100%reflectorsof thebox, it isnecessary fortwospacetimewavefrequenciestobepresentratherthanthesingleComptonfrequencyofanisolatedrotar.


Figure12‐3showstheconventionaldepictionofonepossibleΨfunctionofaparticletrappedina smallboxwith impenetrablewalls. Different resonantmodesarepossible, soa three loberesonanceischosenforeaseofillustration.NotethattheΨfunctionrepresentedinfigure12‐3hasbothpositiveandnegativevalues aboveandbelowthezeroline .QuantummechanicsdoesnotgiveaphysicalmeaningtotheΨfunction.OnlythesquareoftheΨfunctioncanbephysicallyinterpretedastheprobabilityoffindingaparticle.TherotarmodelgoesagainstthisconventionandgivesaphysicalmeaningtotheΨfunction.

The ψ function of a bound rotar is the wave envelope of the waves in spacetime that form the rotar. Figure12‐4showsthespacetimewaveinterpretationoftheΨfunctionofaparticleinabox.Wewillreinterprettheparticleinaboxtoarotartrappedbetweentwohypotheticalbarriersthatare 100% reflectors for waves in spacetime. The box is essentially a repulsive type ofconfinement. Placingarotarinarepulsiveconfinementrequiresthatenergybeaddedtotherotarinexcessoftheenergythatarotarwouldhaveifitwasisolated.Thisaddedenergyshowsupintherotarhavingtwofrequencies onehigherandonelowerthantheComptonfrequency .TogethertheyaveragemorethantheComptonfrequencyofanisolatedrotar.


Placingarotarinsuchacavitychangestherotar’sboundaryconditionscomparedtoanisolatedrotar.Inorderfortherotartomeettheconditionofzeroamplitudeateachofthetwo100%reflectors,therotarmustpossesstwofrequenciesthatarebothpropagatingtotheleftandrightin figure 12‐4. These two frequencies traveling both directions produce the “stationary”standingwavepatternshowninthisfigure.ThispatternlookssimilartothedeBrogliewavespreviouslyshown.However,deBrogliewaveshavethewaveenvelopemovingfasterthanthespeedoflightwhilethewaveenvelopeshowninfigure12‐4isstationaryinthesensethatitsnodesandantinodesdonotmove.Next,wewillreturnbacktothehydrogenatomthathasitselectroninthe1sorbitaldepictedinfigure12‐1. Theelectronandprotonareboundtogetherbyattraction. Thecombinationlost13.6eVenergywhentheywentfrombeinganisolatedelectronandprotontobeinganelectronboundinthe1sorbitalofahydrogenatom.Newfrequencieswereintroducedtotheelectronand the proton’s quarks in this bonding process so that the average frequency of thiscombinationislessthanthesumoftheComptonfrequencyofanisolatedelectronandisolatedproton.Alltheorbitalsofanelectroninahydrogenatomhaveawavestructurethatinvolvestwoormorefrequencies.Thetotaloftheenergyisequaltothecombination’senergyinisolationminus the energy lost to form thehydrogenatom in thedesignated orbital. Orbital angularmomentumalsohasawaveexplanation that involveswavesof slightlydifferent frequenciespropagatinginoppositerotationaldirectionsaroundtheproton.Thisisanalogoustothewaveshavingarotatingframeofreference.WhiletheBohratommodelhasbeenreplacedbythequantummechanicalmodel,thepointisthatthesuperpositionofcounterpropagatingwavesinthespacetimefieldtravelingatthespeedoflightcanachievethedesiredorbitalangularmomentum.Itisproposedthatitisgoingtobepossibletocombinethespacetimewavemodelwiththequantummechanicalatomicmodeltogiveaconceptuallyunderstandablemodelofanatom.Themechanismthateliminatesenergyloss radiationloss fromanatomisunknown,butpresumablyitissimilartothemechanismproposedforstabilizingisolatedrotars.Thisexplanationleavesalotofquestionsunanswered.Perhapsthemostobvious:Istherestillarotatingdipolerotarvolumeburiedsomewherewithintheboundelectron’slobes?Theelectronretains its½ħangularmomentuminadditiontoorbitalangularmomentumpresent inmostatomicorbitals.Further insightsclearlywill involve themarriageofquantummechanicsandimprovedversionsofthespacetimewavetheoryofrotars. Thedifferenceinthespectrumofhydrogenanddeuteriumisduetothedifferenceintheamountofnutationthetwodifferentmassnuclei experience. This seems to indicate that the electron retains a substantial amount ofconcentratedinertiawithintheelectroncloudthatiscausingthenucleustonutate.


Thereasonforbringinguptheparticleinaboxexercisenowistomakeapointaboutboundparticles.Thislessoncreatestheerroneousimpressionthatboundparticlesinnaturecanalsopossessapositivebindingenergy theboundstateismoreenergeticthantheunboundstate .Theparticleinaboxexercisedescribeswhatwouldhappenifaparticleissurroundedbywallsthatarealwaysrepellingtheparticle.Iamproposingthatinnature,particlesarealwaysboundbyattraction;notconfinedbyrepellingwalls.Electronsinanatomareboundbyattractiontothenucleus.Gravitationalattractionbindsaplanettoastar.Theseareallexamplesof“negativebinding energy”. This means that bound state has lower total energy than the unboundcomponentparts. Anotherwayofsaying this is that theboundstate isanenergywell. It isnecessarytoaddenergytobreakthebond.Fromthis,thefollowingstatementcanbemade:Rotars bound by attraction always possess less energy than the same rotar when it is isolated. Attraction binding energy can be thought of as negative energy.Energyisemittedwhenrotarscombine.Thisseemsveryreasonableandnotcontroversial. However, thisgoesagainstthecommonlyacceptedmodelofagluonwhichisdepictedasapositiveenergybindingmechanism.Thesubjectofgluonsandquarkswillfollow,butfirstIwanttosupportthecontentionthatallotherformsofbindinginnatureisbyattractionforminganenergywell lessenergythantheunboundstate .Inchapter#3wedeterminedthataparticlehasmoreinternalenergyinzerogravitythanwhentheparticleisingravity. Eo ΓEg .Gravitationalpotentialenergyisanegativevaluethatisreferencedtozeroatinfinitedistance.SupposeanobserverusingazerogravityclockmonitorstheComptonfrequencyoftherotarastherotarisrestrainedandslowlyloweredtowardsalargemass. Gravitational energy is removed by the restrainingmechanism as the rotar is slowlyloweredintostrongergravity.TheComptonfrequencyωcoftheloweredrotar,measuredbyazerogravityclock,decreasesasgravityincreasesandmoreenergyisremoved.Thisdecreaseinfrequencyisnotnoticeablelocallybecauseofthegravitationaldilationoftime.Locally,aslowclock isused tomonitor a slowCompton frequency. Thepoint is that gravitationalbondingenergyisnegativeenergy.TheComptonfrequencyofagravitationallyboundrotarislowerthanthe same rotarnot gravitationallyboundwhenboth frequencies aremeasuredusinga clockrunningatzerogravityrateoftime.For another example of a bond reducing energy, an electron and aproton release a 13.6 eVphotonwhentheybecomeelectromagneticallyboundtogethertoformahydrogenatomwiththeelectroninthelowestenergylevel. Thisreleasedenergyrepresentsthenegativebindingenergyofthehydrogenatom.Itispossibletogoonestepdeeperinunderstandingthisnegativebindingenergy.AnisolatedelectronhasaComptonfrequencyofabout1.24 1020HzandtheCompton frequencyof aproton sumof all components is about2.27 1023Hz. Whenanelectronandaprotonareboundtogethertoformahydrogenatom,thesumofalltheComptonfrequenciesisabout3.3 1015Hzlessthanthewhentheelectronandprotonwereisolated.This


differenceof3.3 1015Hzisthefrequencyofthe13.6eVphotonreleasedwhenthehydrogenatomformed.Thereforeitispossibletoseethedifferenceinenergywhenanelectronandprotonareboundtoformahydrogenatom. Tobeperfect,thisexampleneedstoalsoaccountforthesmallamountofkineticenergycarriedawaybythehydrogenatomasitrecoilsfromemittingthephoton. Amoreextremeexample is thebondingofanelectrontoauraniumnucleuswhichhasbeenstrippedofallelectrons.Thisbondingissostrongthatthebondingenergyisequivalenttoabout¼ the mass/energy of an electron. The energy lost when this bonding first takes place isremoved by the emission of one or more gamma ray photons. Isolated particles are moreenergeticthanboundparticles.Thisconceptwilllaterbeappliedtoboundquarkswithsomesurprisingimplications.BindingEnergyinChemicalBonds:Next,anexamplefromchemistry.Thereareafewmoleculessuchasozone O3 andacetylene C2H2 whicharecommonlydescribedasbeing“endothermic”.Thismeansthatstartingwithastandardizedsetofconditionsfromchemistry,ittakesaninputofenergy heat toformthemolecule.Thiswouldseemtoimplythatallendothermicmoleculesformpositiveenergybonds.Theimplicationisthatthisisanexamplewheretheboundstateismoreenergeticthantheunboundstate.Forexample,ozonerequiresaninputofenergywhenitisformedfrommolecularoxygenO2understandardizedconditions.However,weareprobingafundamental question about the nature of bonds. Wemust compare the bound state to theunboundstatewhichinthiscasewouldbeindividualatoms.Ittakes1½moleculesofO2toform1moleculeofO3 writtenas:3O2→2O3 .ToseparateoneoftheO2moleculesintotwooxygenatomstakesabout2.6eV/moleculeor249kJ/mole.Thisgivestheerroneousimpressionoftheboundstatecanbemoreenergeticthantheunboundstate.However,ifweformedozonestartingwith3atomicoxygenatoms,thenthisreactionreleasesenergywhenamoleculeofO3isformed.Similarly, the formation of acetylene is an endothermic reaction if the standardized startingcomponents of graphite andmolecular hydrogen H2 are used. However, the formation ofacetyleneisanexothermicchemicalreactionifthestartingcomponentsareatomiccarbonandatomichydrogen.Infact,therearenoendothermicchemicalreactionsifthestartingmaterialisindividual atoms. Therefore, even chemistry supports the contention that bonds in naturereducetheenergyofthecomponentparts.Inthiscasemoleculesalways possess less energy than the component atoms when the atoms are isolated. Binding Energy of Nucleons:Whileitisverydifficulttomakeenergyandforcemeasurementsinsideaprotonorneutron,wecanobtainahintofwhatisgoingoninsideprotonsandneutronsbylookingatthebindingthatoccursbetweennucleons.Whenprotonsandneutronsareboundtogethertoformatomicnuclei,isthereagainorlossofenergy?Theanswerisobvious.Aheliumatom 4He haslessmass/energythan2deuteriumatoms.Thebindingenergyofnucleonsisanegativeformofenergy energyreductioncomparedtothesumoftheunboundcomponents .Thereisadecreaseinmass lossofenergy whenhydrogennucleifusetoformnucleiofheavier


atoms.Atfirstitmightappearthat235Uandotherheavyatomsareanexceptiontothisrule,butthisisincorrect.Thestrongestboundatomicnucleolusis56Fewithabindingenergyof8.79MeVpernucleon.235Uhasabindingenergyof7.79MeVpernucleonandthisiscomparabletothebindingenergypernucleonofcarbonornitrogen.Breaking235Uapartreleasesenergybecausethetwolighternucleiformedhaveagreaterbindingenergypernucleonthan235U. Agreaterbindingenergymeansthatexcessenergymustbereleaseduponformation. Thepointisthateven 235U has lessmass/energy than the energy of the protons and neutrons that form theuraniumnucleus.Itisproposedthatquarksareboundtogethertoformhadronsbynegativeenergy.Thehadronswouldhavelessenergythanthetotalenergyofthecomponentquarksifthequarkswerestableinisolationsothattheirenergyinisolationcouldbeexperimentallymeasured.

Gluons,QuarksandHadronsBackground: Thepreviousdiscussionasserted that innature theboundstatewasalwaysalowerenergycondition that the individual components in theunboundstate boundstate isalwaysanenergywell .Thisconceptisgoingtobeimportantinthefollowingdiscussionaboutquarksandgluons.Theproblemisthatifwefaithfullydevelopamodeloftheuniversestartingwiththeassumptionthattheuniverseisonlyspacetime,weobtainthestrongforcefromthepressureexertedbythespacetimefield.Thereisnoneedtopostulategluons!Sincegluonsareverymuchapartofmodernparticlephysics,thisseemstopresentaproblemforthespacetimemodeloftheuniverse.However,itwillbearguedherethatallthefunctionscurrentlyattributedto gluons color charge, etc. can be converted to functions attributed to the rotarmodel ofquarksexistinginthepressureexertedbythespacetimefield.Previously we rejected virtual photons as the “messenger particles” that carried theelectromagnetic force and rejected gravitons as the “messenger particles” that carried thegravitationalforce.Thenextremainingvirtualmessengerparticleisthegluonandthisisgoingtobequestioned.Thereisnoexperimentalobservationsthatcanbeinterpretedasbeingproofthatvirtualphotonsorgravitonsexistbutthereissomeexperimentsthatseemtoimplythatgluonsexistalthoughtheyhaveneverbeendirectlyobserved. Alsothereisavastamountoftheoretical calculations thataremodelingeffects thatareattributed togluons.Therefore thediscussionaboutgluonsisgoingtobemorenuanced. Theanalysisofquarksandgluonswillbeginwithadiscussionaboutthemass/energyofquarks.


Mass/EnergyofQuarks: Sincequarksdonotexistintheunboundstate,it isnotpossibletosimplycomparetheenergyofanunboundquarktoaboundquark.ThefollowingisaquotefromanarticleonquarkmasseswrittenbyA.V. Manohar and C.T. Sachrajda2

“Quarkmasses therefore cannot bemeasured directly, butmust be determinedindirectly through their influence on hadronic properties. Although one oftenspeakslooselyofquarkmassesasonewouldofthemassoftheelectronormuon,any quantitative statement about the value of a quarkmassmustmake carefulreference to the particular theoretical framework that is used to define it. It isimportant tokeep this schemedependence inmindwhenusing thequarkmassvaluestabulatedinthedatalistings.Historically,thefirstdeterminationsofquarkmasseswereperformedusingquarkmodels.Theresultingmassesonlymakesensein the limited context of a particular quarkmodel, and cannot be related to thequarkmassparametersoftheStandardModel.”

Iwill expandon this thought somewhat. Suppose that there is anexperimentwhich collidesprotonsandanti‐protonstogetherathighenergy.Thecollisionproducesacomplicatedshowerofparticlesthatishardtointerpret.Ifprotonsareassumedtoconsistofpointparticlequarksplusgluonsandvirtualparticlepairs formingandannihilating, thenthe interpretationof theresultswillbedifferentthaniftheprotonsareassumedtobetherotar‐modelquarkspossessingquantizedangularmomentumcollidinginseaofdipolewavesdesignatedasthespacetimefield.Withthisbeingsaid,wewillfirstlookatthewidelyacceptedinterpretationsofthesecomplicatedexperimentalresults.Thedescriptionofaprotondiffersgreatlydependingontheexperiment.Forexample,inlowenergy collisions, a proton seems to be made of 3 quarks with each quark possessingapproximately1/3oftheproton’senergy.Inhighenergycollisionsaprotonseemstohavemanyquarks over10 witheachquarkpossessingenergyofonlyafewMeV.Wewillexamineeachmodel.Atthelowenergylimitaprotonappearstohaveatotalof3quarks‐twoupquarksandonedownquark.Inthelowenergyconditionthese3quarksarereferredtoas“constituentquarks”.Thetwoconstituent“up”quarksofaprotonappeartohaveenergyof336MeVeach3.Thesingledownconstituentquarkappearstohaveenergyof340MeV.Hereisanotherdescriptionoftheconstituentquarks.

Nonrelativisticquarkmodelsuseconstituentquarkmasses,whichareoforder350MeVfortheuanddquarks.Constituentquarkmassesmodeltheeffectsofdynamicalchiralsymmetrybreaking…Constituentmassesareonlydefinedinthecontextofaparticularhadronicmodel."4

2 Quark Masses Updated Jan 2012 by A.V. Manohar (University of California, San Diego) and C.T. Sachrajda (University of Southampton) http://pdg.lbl.gov/2013/reviews/rpp2012-rev-quark-masses.pdf 3 Griffiths, D., (2008) Introduction to Elementary Particles p. 135, Wiley-Vch 4 "Note on Quark Masses":http://pdg.lbl.gov/2010/reviews/rpp2010-rev-quark-masses.pdf


Thisname“constituentquarks”isusedwhendescribingthemodelthatemergesfromlowenergyexperiments.Thename“currentquarks”isusedtorepresentthequarkmodelwhichisobtainedinhighenergycollisionexperiments.Inhighenergycollisionexperimentsthereappearstobemanymore than3quarks andeachof theseappears tohaveenergyof only a fewMeV. Toreconcile this difference the difference between the low energy and high energy collisionexperiments,the3constituentquarksobservedinlowenergycollisionsareoftenvisualizedasbeingcomposedofacorecurrentquarkwithenergyofafewMeVwhichissurroundedbyeitheraclusterofvirtualparticle‐antiparticlepairsorsurroundedbygluons.Ineithercase,thetotalenergyofwhatisinterpretedtobeaclusterisapproximately1/3oftheprotonenergy.According to theproposed spacetimemodel, theproblemcomes in interpretinghigh energycollisions.Intherotarmodelofahighenergycollision,oneormoreofthe3quarksmomentarilyabsorbsfarmoreenergythantherestenergyofthequark.Recallthepreviousdiscussionofthedifficultydeterminingthesizeofanelectronfromenergeticcollisionsoftherotarmodelofanelectron.Thekineticenergyofthecollisionismomentarilyconvertedintotheinternalenergyoftheelectron rotarmodel .Thiscausestheelectron’sradiustomomentarilydecreasesothatitwasalwaysbelowthedetectablelimitoftheexperiment.Thecollisionofprotonsismorecomplicated,butonethingisclear. Theenergeticconditionswhichprevailatthemomentofcollisionarenotthesameasanisolatedproton.Ifthemodelofaquarkisapointparticle,thenthereisnointernalstructureanditisvalidtoassumethatthequarkretainsitspropertieseveninaviolentcollision.However,ifeachquarkisassumedtohavearotarmodel, thenthis is three“fluffy”rotatingdipolewavesexisting in thespacetime field.They each are a rotating dipole wave in spacetime possessing a quantized unit of angularmomentum.Thisrotatingwaveisdistributedoveravolumeofspace.Addingthekineticenergyofthecollisionmomentarilydecreasesthesizeofaquark.Thischangesthebondingconditionsbutretainsaconstantangularmomentum.Therotarmodelofprotonsisinanearlystageofdevelopmentandcannotpredicttheresultswhichwouldbeexpectedinanenergeticcollision.However, it is possible to imply that the complicated results of energetic collisions can bemisinterpretedifpointparticlequarksandgluonmessengerparticlesareerroneouslyassumed.Thequarksobservedinhighenergycollisionsappeartobeverydifferentfromtheconstituentquarksatthelowenergylimit.Thethree“currentquarks”thatformaprotonappeartohaveenergylessthan10MeV.Inthispictureobtainedfromhighenergycollisions,onlyabout1%oftherestmassofaprotonappearstobeintheformthequarks.Inthiscaseabout99%ofthemass/energy of a proton is therefore assigned primarily to the energy of gluonswith somecontributionfromthekineticenergyofquarks.Inotherwords,thegluonsmustpossesspositiveenergyinthismodel.Morewillbesaidaboutgluonslater.


AlternativeExplanation:Theproposedalternativeexplanationofthemassofaquarkisthatupanddownquarkswouldbeintrinsicallyhighenergyrotatingdipolewavesiftheywerestableinisolation.Theyarestronglyboundtogetherbyaninteractionwiththespacetimefieldwhentheyformhadrons.Therearenoisolatedfirstorsecondgenerationquarksbecausetheysimplydonotattainstabilityasisolatedrotars. Thetopquarkwillbediscussedlater .Whenthismodelofaquarkisboundintoahadronitisinalowenergystate,andeepenergywell.Attemptingtoremovequarksfromahadronagainstthestrongforceincreasesthequark’senergy Comptonfrequency towardsthehigherenergystatewhichwouldexist ifaquark rotarmodel couldexistinisolation.Theenergyexertedinattemptingtoremoveaquarkfromahadronrequiressomuchenergythatanewmeson pion isformedbeforeanisolatedquarkisobtained.Whennewmesons are formed, the binding force between former components of the split hadrondecreasestonearzero.Singlefirstorsecondgenerationquarksareneverproduced.However,toillustratetheconcepts,wewillimaginewhatitwouldbelikeifisolatedquarkswereallowed.Itisproposedhere justifiedlater thatifisolatedupanddownquarksexisted,theywould be rotating dipoleswith energy substantially greater than 400MeV. If up and downquarksexistedasisolatedrotars,thentheywouldshedenergywhentheybondtoformaprotonorneutron.Hadronscomeintoexistencealreadyformedsincethisisthelowestenergystateandtheonlyformthathasstability.However,itisinformativetoimaginethestepsthatwouldoccurifahadronwasformedfromisolatedquarks.Energy of Bound Quarks:Theworkingproposalisthatalltheenergyofaprotoniscontainedinitsthreeboundquarkrotars rotatingdipolewavesinspacetime .However,themodeloftheinteractionofquantizedwavesshouldnotbeequatedtotheexpectedinteractionifthequarkswerehardshelledparticles.Furthermore,itisproposedthatthebindingenergyissogreatthattheboundquarkshavemuchlessenergythantheywouldhaveashypotheticalisolatedrotars.Wewill first survey thehadrons that aremadeofonlyupanddownquarks to compare theenergyoftheupanddownquarksundervariousboundconditions.Thenucleons protonsandneutrons havealmostthesameenergy~938MeV. Ifweassumethatall thisenergycanbetracedtotheenergyof3rotatingspacetimedipoles 3quarks thentheimplicationisthatupanddownquarkshaveaboutthesameenergy.Therearedifferentproportionsofupanddownquarks, yetapproximately the same totalenergy. Wewill assume thatboundupanddownquarksinanucleonhasenergyofabout313MeV ~1/3ofthetotal .Therearetwootherfamiliesofhadronsthatconsistofonlyupanddownquarks.Thesearethepimesons pions andthedeltabaryons.ThedeltabaryonshavespinofJ 3/2ratherthanthespinof½forthenucleons.Thereare4deltabaryons.Theseare:Δ uuu ;Δ uud ;Δo udd andΔ‐ ddd .


Theseallhaveaboutthesameenergy 1,232MeV thereforetheupanddownquarksboundinthishadronhaveenergyofabout1/3ofthisvalue:~411MeV.Thepionsconsistofaquarkandananti‐quarksuchasanupquarkandananti‐downquark.Thenetspinofthepionsiszero counterrotatingdipoles .Theenergyofthepionsis139.5MeVforthetwochargedpions π andπ‐ andabout135MeVfortheneutralpion πo .Thismeansthattheupanddownquarksinapionhaveaverageenergyofabout70MeVforthechargedpionsπ andπ‐ andabout68MeVforaneutralpionπo.Thereforewehaveexampleswhereupanddownquarkshaveenergyrangingfrom411MeVto68MeV.ThestandardmodeldealswiththisdifferencebyassumingthatanisolatedupordownquarkhasenergyofonlyafewMeV.Theextraenergyrequiredtoreach68MeV,313MeVor411MeVisassumedtobepredominatelyintheenergyofthegluons.Energy of a Hypothetical Isolated Quark: Theproposed spacetimewavemodelofquarksoffersadifferentanswer. Itsaysthatanisolatedrotarupordownquarkwouldhaveenergysubstantiallygreater than411MeV.Whenquarksarebound intoahadron, thebondsaresostrongthatalargepercentageofthehypotheticalisolatedenergyislost.Itwouldberadiatedawayifitwaspossibletodothisexperiment.Thedifferencebetweenthe68MeV,313MeVor411MeVreflectsdifferentamountsofbindingenergy.Thebindingenergypernucleonin56Ferepresentsapproximately1%ofaproton’senergy.Anelectronboundtoauraniumatom’snucleolusstrippedofallotherelectronshasabindingenergythatisabout25%ofanisolatedelectron’senergy.Inorderforanelectrontoattachtoastrippeduraniumatomnucleolus,ithastoemitoneormoregammarayphotonstoshedatotalofabout25%of theelectron’senergy.Actuallypartof the lostenergycomes fromtheprotons in thenucleus,butthatisoffthesubject.Itisproposedthatthebindingenergyofquarksinahadronismuchgreaterthantheseexamples.Toillustratethisconcept,wewillchooseanenergysubstantiallylargerthan411MeVfortheenergyofahypotheticalunboundupordownquark.Forillustration,wewillusethenumberofabout600MeVfortheenergyofisolatedupanddownquarks.Withthisassumption,anisolatedupordownquarkwouldloseabout1/3ofitsenergywhenitformsadeltabaryon 411MeV .Itwouldloseabout½ofitsenergywhenitformsanucleon 313MeV anditwouldloseabout89%ofitsenergywhenitformsaneutralpimeson 68MeV .Electron vs. Proton Size:Beforeproceedingtoofar,itisdesirabletodoacalculationtoseeiftheideasproposedhereareplausible.Wewillattempttocalculatethesizeofaproton.However,firstitisnecessarytorecognizewhyaprotonhasameasurablesizeandanelectrondoesnot.Aprotonhasameasurablesizebecauseaprotonismadeupof3fundamentalrotars.Thethreequarksof aprotondonot respond to ahigh energy collision as a singlequantizedunit. The


propertyofunityonlyexistswithinasinglequantizedwave rotar .Acollisionwithaprotoninvolves speedof light communicationof forcesbetween the threequarksof aproton. Thismeansthattheprotonexhibitsaphysicalsizeinacollisioneventhoughthissizedoesnotexhibitahardboundary.Whenanelectroncollideswithoneofthethreequarksinaproton, itappearsasifthereisacollision between twopoint particles. The reason is the same as previously explained for acollisionbetweentwoelectrons.Boththeelectronandthequarkarequantizedrotatingdipolesinthespacetimefield.Inadirecthit,theybothconvertthekineticenergyofthecollidingelectronto internalenergyof therotatingdipole.Thishappens faster than thespeedof lightbecausepreserving thequantizedangularmomentumresults in thepreviouslyexplainedpropertyofunity.TheconversionofkineticenergymomentarilyincreasestheComptonfrequencyofeachrotar. In order for angular momentum to be conserved, the rotar radius of each rotarmomentarily decreases. The amount of decrease in size makes each rotar experimentallyindistinguishablefromapointparticlebecause aspreviouslyexplained themomentaryradiusislessthantheresolutionlimitsetbytheuncertaintyprinciple.Theothertwoquarksthatwerepartoftheprotononlylearnaboutthecollisionthroughspeedoflightcommunication.Calculation of Proton Radius:Thepresenceofonlythreequantizedwavesmeansthataprotonstillexhibitssubstantialquantummechanicalpropertiessuchasnodefiniteshapeandthelackofahardedge.Alsotheshapeofaprotondependsonthealignmentofthespinsofthequarks.Ananalysis5ofallmeasurementsoftheprotonradiusconcludesamostprobablechargeradiusforaprotonis:0.877 10‐15meter 0.15 10‐15meterWewilldoaplausibilitycalculationtoseeiftheproposedrotarmodelofaprotongivesroughlythe correct size. This calculationwill assume that theproton ismadeof3 rotars, eachwithenergyof313MeV 5 10‐11J .Wemustdecidehowthesethreerotatingdipolesfittogether.Aretherevoidsorexcessiveoverlaps? It isknownthatprotonsarenotnecessarilysphericaldepending on the alignment of the spins of the up and down quarks. Still, the simplestassumptionisasphericalprotonwithquarksthataresointimatelyboundthattheprotonhasthreetimesthevolumeofanindividualrotarquarkwithradius .Theplausibilitycalculationwillassumethissimplifiedmodel.Wewillfirstcalculatetherotarradius ofarotarwithenergyof5 10‐11Joule 313MeV .Thenwewillincreasethisradiusbyafactorof31/3toobtaintheradiusofaspherewith3timesthevolumeofanindividualrotarquark. ħc/Eisubstitute:Ei 313MeV 5 10‐11J 6.3 10‐16meterradiusofonequarkwithenergyof313MeV 31/3 9 10‐16metercalculatedprotonradius 3quarks

5 http://arxiv.org/abs/hep-ph/0008137v1


Theexperimentallydeterminedradiusofaprotonis:8.77 10‐16 1.5 10‐16meter,thereforethecalculatedradiusof9 10‐16meteragreeswithintheexperimentalerror. Thisdegreeofaccuracyisprobablymorethanwedeserveconsideringthecrudenessofthecalculation.Infact,therotarmodelcouldbecorrectforchargedleptonsandtoosimplifiedtobeappliedtoquarkswhicharetightlyboundwithveryrestrictiveboundaryconditions.Still,thecalculationdoesgiveareasonableanswerandcannotbeignored.Thiscalculationcouldeasilyhavebeenoffbymanyordersofmagnitudeiftherotarmodelwascompletelywrong.It isalsointerestingtonotethatthedensityofthequarksintheabovecalculationisroughly5.9 1017kg/m3.Thedensityofthenucleusofanatomwithmanyboundnucleonsissubjecttosomeinterpretation,butisroughly2.3 1017kg/m3.Therefore,thedensityofanatomicnucleusisroughlyhalfthedensityofindividualquarks. Thisisfardifferentfromthestandardmodelthat depicts quarks as point particles and therefore infinitely more dense than an atomicnucleolus.To summarize, the rotarmodelofquarks implies that if quarks couldexist in isolation, theywouldhavesubstantiallymoreenergythantheyhavewhenboundtogethertoformhadrons.Inthiscase,thequarkswouldloseenergy emitphotons whentheyareboundtogethertoformhadrons. Thisresults inanenergywell, just likeallotherbonds innature. Therotarmodeldependsoftheexistenceofthespacetimefieldtostabilizetherotars.Aspreviouslystated,thespacetime‐basedmodelof theuniversehasonlyone truly fundamental force therelativisticforce andthatforceisalwaysrepulsive.Thebondingofthequarksisultimatelytraceabletopressureexertedbythespacetimefield.Gluon Background:Next,wewillexaminegluons.Accordingtothestandardmodel,gluonsareexchangeparticlesthatcarrythestrongforce thestronginteraction betweenquarks.Gluonscarry the color charge and also participate in the strong interaction in addition to being amediatingexchangeparticle.The8typesofgluonshavecolorcharge.Whilethereare3colorcharges red,greenandblue ,gluonscarrycomplexcombinationsofthesecolorcharges.Forexample,adescriptionofoneofthe8possiblegluonsis: red‐antiblue blue‐antired .Gluonsarealsodescribedasmasslessvectorbosonswithspinof1whichhavetwopolarizationstates.The standard model description of gluons is very complex and quantum electrodynamicscalculationsinvolvinggluonsarealsocomplex.According to theory, bonds between quarks become weaker as the distance decreases.Thereforequarksboundinhadronsareinastateof“asymptoticfreedom”.Whenquarksareintheirunperturbedstate, theycanmigratewithin thehadronas if theyarenot tightlybound.However, the asymptotic freedom state only occurs when quarks maintain the prescribeddistancerelative toeachother. If thedistance is increased, thestrong force increases.As theseparationdistanceis increased,energyisbeingaddedtothehadronandtherestoringforce


increases. The added energy supposedly goes into the energy of the gluons. The quarkssupposedly maintain a constant energy of less than 10 MeV.When the separation distancebetweenquarksreachesroughly10‐15m,enoughenergyhasbeenaddedtothegluonstoformanewmeson quark–antiquarkpair andtherestoringforcedropstonearlyzero.Therearethreeproblemswiththis.

1 This gluon bondmodel does not follow the example of the particles possessing lessenergyintheboundstatethantheyhaveinisolation.Itistruethatwecannotobtainanisolatedquarkforcomparison,butthepointisthatthemodelisthateachquarkretainsaconstantenergyasthequarksarebeingseparated.Theworkbeingdonedoesnotgointoincreasingtheenergyofthequarks.Thisisnotatypicalenergywell.

2 The gluons possess positive energy and are traveling at the speed of light. Since theprotonhasenergydensityofabout3 1035J/m3,ifalmostalltheenergyisattributedtogluons this implies that the gluons should exert pressure of at least 1035N/m2. Thepressure of this high gluon energy density should destroy protons, not bind themtogether.Howdothegluonsachieveattraction?

3 Why does the force start at zero and increasewith distance? This is “explained” byphysicistspostulatingthatasthequarksareseparated,thegluonsform“fluxtubes”andthisconcentrationactuallyincreasestheforceofattractioneventhoughthedistanceisincreasing.Whatisthephysicsbehindthisconcept?Whatsuppliestheforcetoconstrainthesizeofthefluxtubes?Dependingonthevolumeofthefluxtubes,theconfinedgluonsshould exert evenmorepressure than1035N/m2. All of this is so far removed fromanythingelseinnaturethatitspeakstotheproblemscreatedtheconceptofgluons.

The experimental evidence for the existence of gluons is that very energetic collisions ofelectrons and positrons sometimes produce “three jet events” a jet is a narrow shower ofparticles .Theexplanationforthesethreejeteventsisthatatleastoneofthethreejetsresultedfromagluon“hadronize”intonormalparticles colorlesshadrons whichresultsinthenarrowshowerofparticlesknownasajet.TheRelativisticForce:Aspreviouslystated,thespacetimemodelhasonlyonetrulyfundamentalforce:therelativisticforce.Thisforceisgeneratedwhenenergy,travelingatthespeedoflight,isdeflectedinsomeway.Forexample,theabsorption,emissionorreflectionofaphotonresultsinthetransferoflinearmomentum.TherelativisticforceFristhenamegivenwhenmultiplelinearmomentumtransferscanbeconsideredtobeacontinuousforce.TheequationisFr Pr/cwherePrisrelativisticpowerpropagatingatthespeedoflight.Therelativisticforceisalwaysrepulsive.Aspreviouslyexplainedtherelativisticforcecanappeartobeanattractingforcewhenthe spacetime field exerts the relativistic force in a way that brings particles together. Forexample,theelectrostaticorgravitationalattractionisactuallytheresultofthespacetimefieldapplyingunequalpressureonrotarsproducinganetforceofattraction.Thisisanintroductionof how the relativistic force also produceswhatwe perceive to be the strong forcewithoutgluons.


All rotars possess energy densitywhich implies that they have internal pressure. The rotarpressureis:ℙr ωc4ħ/c3 Ei/ .Tostabilizethisinternalpressurethespacetimefieldmustexertanopposingpressure.Ifthispressureisbalancedonallsides,thenthereisnonetforce.However,otherrotarsinthevicinityresultinadistortionofthespacetimefieldandthisinturnresultsinanunbalancedpressure anetforce exertedontherotar.Whenthisnetforceisinthedirectionoftheotherrotar,thenwesaythatthereisaforceofattractionbetweenthetworotars.Themaximumforce Fm thatarotarcanexertwaspreviouslyshowntobeFm ℙr / .CaremustbeusedinapplyingthisequationbecauseEiistheinstantaneousinternalenergyandλc is the instantaneous rotar radius. For example, in a collision, the kinetic energy getstemporarilyaddedtotherotar’sinternalenergyEi.ThereforeattheinstantofthecollisionthevalueofEiincreasesandthevalueofλcdecreases.TheresultisthatthevalueofFmisgreaterinacollisionthanthetheoreticalvalueforarotarnotundergoingacollision.Itispossibletodoaroughcalculationofthevalueofthemaximumforcethatthe3quarksthatmakeupaprotoncanexert.Wewillstartbyassumingthattheproton’senergyof938MeVisevenlydistributedamongits3quarks rotars .Ifeachquarkhasenergyofabout313MeVor5 10‐11J,thenthevalueofthemaximumforceavailabletothethreequarksintheunperturbedstateofasymptoticfreedomisaboutFm 80,000N.Thismaximumforcecalculationdoesnotattempt to identifywhether therotarcanactuallyexert thismaximumforce, it ismerely theupperlimit.Inchapter7 thesection titled“AsymptoticFreedom”gave theexplanationofhowthestrongforcecanbezeroattheasymptoticfreedomseparationdistance.Recallthatatthisseparationdistanceopposing forcesbetween thespacetime fieldand the repulsionof adjacentparticlesoffseteachothersothenetforceonaquarkattheasymptoticfreedomseparationdistanceiszero.However,displacingaquarkineitherdirection closerorfurtherfromtheotherquarks produces a net force which attempts to restore the separation to the asymptotic freedomseparation. The energy required to change the separation distance is stored in the quarkincreasingitsrotationalfrequency.Recallthehypotheticalassumptionthatanupordownquarkinaprotonhasabout313MeVbutitwouldhaveenergyofabout600MeVifitcouldexistasastableisolatedparticle.Thisistheenergywellpreviouslydiscussed.Inthisexample,removingaquarkfromaprotonwouldrequireabout300MeVbutapionformsat135MeV,sothishappensfirstandtherestoringforcefallstonearlyzero.Wecandoaroughcalculationtoseeifthismodelisreasonable.Weknowthattheradiusofaproton isabout8.8 10‐16mandpionsare formedwhenaquark isdisplacedbyroughlythisdistance. It ispossibletodoaroughcalculationtoseehowfaraquarkwouldhavetomoveagainstagainstaconstant80,000Nforcebeforeenoughenergywouldbestoredtomakeapion


withenergyof135MeV Ei 2.2 10‐11J ?Theanswerisabout2.7 10‐16m.Thiscalculationrepresentsanunrealisticlowerlimitbecausetheforcewouldnotgofromzeroattheasymptoticfreedomseparationdistance to the fullmaximum forceof80,000Nabruptly. Instead, therewouldbeagradualincreasestartingatzeroandendingatsomeforceprobablylessthan80,000N.Anothersimplifiedcalculationoftheseparationdistancerequiredtomakeapionwouldstartatzeroforceattheasymptoticfreedomseparationandlinearlyincreasetheforceto80,000Natthetotalenergyrequiredtomakeapion.Inthiscasetheseparationdistancewhereapionisformedwouldberoughly5.5 10‐16m. Eventhisoversimplifiedassumptionclearlygivesananswerintherangeofprotonradius.Again,thiscalculationcouldhavebeenoffbymanyordersofmagnitude.GluonsNotNeededforBonding:Thepointofthisdiscussionistosupportthecontentionthatthespacetime‐based model of the universe can explain the strong force using only the singleuniversal force: the relativistic force. The spacetime‐based model of the universe is notdeveloped sufficiently to make predictions about the properties of specific fundamentalparticles.Inparticular,itisnotpossibletosaywhetheragluon‐likeparticlewithspinof1exists.Themainreasonforpostulatingtheexistenceofagluon amessengerparticlethatconveysthestrongforce hasbeenremoved.However,somethingcauses3jeteventsinenergeticcollisionsofelectronsandpositrons. Quantumchromodynamicsassignsadditional functionstogluonsbesidesbeingamessengerparticleandtheseadditionalfunctionsarestillrequired.Therulesofthecolorforcewhichcurrentlyrequire8gluonsmaystillrequire8bosonsbutwithredefinedwave‐basedcharacteristics.Thereforegluonsarenotneededforbonding,butsomeoftheotherfunctionscurrentlyassignedtogluonswillneedtobereinterpreted.Stability of Hadrons:Supposethatweimaginestartingwiththesuperfluidspacetimefieldthatlacks angularmomentum, thenwe introduce one unit of½ħ quantized angularmomentum.Merely stating the amount of angular momentum does not specify the energy, frequency,amplitude,rotationalsize,etc.thatthisangularmomentummighttake.Thespacetimefieldhascharacteristicsthatdeterminewhatcombinationachieveslongtermstabilitysuchasanelectronandwhatissemi‐stablesuchasamuonortauon.Thereisaninfinitenumberofotherpossiblecombinations of frequency, amplitude, rotational size that do not possess any stabilizingcharacteristicsfromthesurroundingspacetimefield.Thesetotallyunstablecombinationslastforonlyaunitoftimeequalto1/ωcwhereωcisthehypotheticalComptonangularfrequency.Forfrequenciesinthegeneralrangeoftheknownparticles,thesurvivaltimewouldberoughlyintherangeof10‐25to10‐20seconds.Thefewcombinationsofangularmomentum,frequency,amplitude,etc.thatachievestabilityaretherareexception.Thisismentionedasanintroductiontoadiscussionofthestabilityofhadrons.Whiletheleptonsarestableorsemi‐stableasindividualrotars,thequarksarenot.Somehowtwoorthreequarksacting together can find stabilitywhere individual quarks do not. Do the hadrons that findstabilityachievethisstabilitybyexhibitingthestandingwavepropertiesofasingleunit?Itis


possibletoanswerthisquestionbylookingatthediffractionpatternofneutronspassingthrougha crystal. The diffraction pattern produced by a neutron 3 quarks implies a de Brogliewavelengththatischaracteristicoftheneutron’stotalmass λd h/mv ratherthanthemassofthethreeindividualquarkswithapproximatelyonethirdthistotalmass.Apparentlythestabilityconditionrequiredforvacuumenergytostabilizeaneutronresultsinfrequency summation in the external volume. In other words, the 3 quarks present in theneutronlosetheirindividualityattheboundaryoftheneutron.Externally,thevacuumenergystabilizesaneutronbytreatingitasasingleunit.ThestandingwavesgeneratedinthevacuumenergyapparentlyareequivalenttoaComptonfrequencycharacteristicoftheentireenergyoftheneutron. ThedeBrogliewavesgeneratedbyaneutron implyasinglewavelengthof thebidirectional standing waves in the neutron’s external volume. Besides neutrons, largercompositeparticlessuchasalphaparticlesandevenentiremoleculesexhibitdiffractionpatternscharacteristic of the total energy. An improved rotar model should address the issue offrequencysummationfurther.Formation of the First Hadrons: The formation of rotars in the Big Bangwas previouslydescribed as a trial and error process where a few combinations of angular momentum,frequency,andamplitudecondensedoutofthechaoticenergeticwavesinspacetimepresentinthe early stages of the Big Bang. It is now proposed that besides single rotating spacetimedipoles,naturealsofoundafewcombinationsofrotatingdipolesthatcouldachievestability.These also condensed out of the energetic waves in the spacetime field as already formedhadrons.Thesourceoftheangularmomentumrequiredtoformquarks,leptonsandphotonswillbeaddressedinchapters13and14.Itwillbeshownthateventhestartingconditionoftheuniverse Planckspacetime musthavepossessedquantizedangularmomentum.Presumablythefirsthadronsthatcondensedoutoftheenergeticwavesinspacetimecreatedbythe Big Bang were highly energetic hadrons made from generation III and II quarks. Theprobable sequence that eventually arrives at the dominance of protons and neutrons in theuniversetodaywillhavetobedevelopedbyothers.W and Z Bosons:SofarthisanalysishasnotmentionedW ,W‐andZbosons.Thereisclearlyalarge bodyof experimental observations that support these particles in the standardmodel.However, like gluons, it is proposed that these bosons have a wave explanation. W and Zparticleshaveħspincharacteristicwhichisnormallyassociatedwithaboson.HoweverWandZparticlesalsohaverestmasswhichisnormallyassociatedwithafermion spin½ħ .Clearlythisismorecomplicatedthanpreviouslyencounteredwithotherfermionsorbosons.Isthereanyothercasediscussedinthisbookwhereaboson spinħ possessesrestmass?Theansweris:yes.Inthefirstchapterwediscussedthecaseofphotonsconfinedinareflectingbox.Photonshaveħspinyetwhentheyareconfinedinsomewaytheyareforcedtohaveaspecific


frameofreferenceandtheyacquirerestmass.Infact,itwaspointedoutthatanytimeaphotonhasenergyofE pc,thephotonwillhaverestmass.Whenaphotonisconfinedbetweentworeflectors it can be thought of as propagating in both directions simultaneously. Sincemomentumisavector,thetwovectorscancelandp 0.Thisconditiongivesthephotonrestmasseventhoughthephotonisaboson.ThereforeevenaphotonpropagatingthroughglassispropagatingatlessthanthespeedoflightinavacuumandE pc.Thepointofthisexampleisthatbosons,withspinħ,canhaverestmassiftheyinteractwithfermions rotars inawaythatresultsinE pc.Apparentlyspacetimeandthehadronstructuremusthaveatypeofresonancethatoccursat1.22 1026s‐1 Wboson80,38GeV and1.39 1026s‐1 Zboson91.19GeV .ImaginetwoquarksactingsomethinglikemirrorsmomentarilyconfiningaWorZboson.WithComptonwavelengthsof1.54 10‐17mand1.36 10‐17m WandZrespectively thesearethesizerangethatplausiblycouldinteractwiththerotarstructureofhadrons.TheWandZbosonswouldhaveastandingwavestructurewhichwouldlooklikestandingwavesinteractingwiththewavestructureofthetwoquarks.ThiswouldgivetheseWorZbosonsrestmassandtheshort range properties normally associated with the weak force. This is not a completeexplanation,butitdoesshowhowthespacetimemodelofforcesandparticlescanaccommodateWandZbosons.Hopefullyotherswillanalyzethisfurther.TheconceptoftheHiggsfieldandHiggsbosonwasoriginallydevelopedto“explain”howWandZbosonsacquiredrestmass inertia .WhileIfavorthepreviousexplanationofhowWandZbosonsacquireinertia,therestofthescientificcommunitycurrentlybelievesthattheHiggsfieldinteractswithWandZbosonsandthatinteractiongivesthemrestmass.Inthespacetimemodeloftheuniverse,thespacetimefieldistheonlyonetrulyfundamentalfield.Multipleresonanceswithin the spacetime field are responsible for all the various particles. These multipleresonancescanbethoughtofasseparatefields.Inthatsense,itispossibletosaythatthereisaHiggs field the sameway that all other particles can be considered to have their own fieldsresonances .SincethespacetimemodeloftheuniversehasnotbeendevelopedsufficientlytomakepredictionsaboutWandZparticles, it isnotpossibletoconclusivelysayhowWandZbosonsobtaintheir inertia. Thereforeit ishypotheticallypossiblethatthemechanismmightincludeaninteractionwiththeHiggsresonance Higgsfield .However,aspreviouslyexplained,fermionsexhibitrestmassthroughthesamemechanismthatconfinedphotonsexhibitrestmass.FermionsdonotrequireaHiggsfieldtoobtaininertia.


NeutrinosNeutrino Introduction: Neutrinos are the least understood of the “fundamental particles”.There are three types or “flavors” of neutrinos electronneutrinos,muonneutrinos and tauneutrinos . These are three flavors are sometimes designated m1, m2 and m3 respectively.Previouslythestandardmodelconsideredneutrinostobemasslessparticlesbutexperimentsprove thatneutrinos can change fromone type of neutrino to another type of neutrino as aneutrinopropagatesthroughspace.This“neutrinoflavoroscillation”isinterpretedasindicatingthatneutrinosmusthavesomerestmass.Thereasoningisthatanythingtravelingatthespeedoflightcannotexperiencetimeandthereforeneutrinoflavoroscillationimpliesthatneutrinosmust be traveling at less than the speedof light. The flavor oscillation data could not giveabsolutevaluesforneutrinomass,butthedatacouldgiveanindicationofthedifferencebetweenthesquareofthemasses6.Forexample,thisdifferencecanbeexpressedas:Δ 7.6 10 eV2 and Δ 2.4 10 eV2. Now there has been the first experimental results whichpurporttogiveanabsolutevalueofthesumoftherestmassofthethreeflavorsofneutrinos7as0.32 0.08 eV. I will interpret this as the average rest mass since the flavor oscillation issequential. This was not a directmeasurement ofmass. Instead it combined cosmologicalmeasurements and a theory of the effect neutrinomass should have on these cosmologicalmeasurements.Therefore,thismeasurementcouldbewrongifthetheoryiswrong.Still,thismass/energyisreasonableandwillbeusedinfurtherdiscussionofneutrinosbothhereandinchapter13.Forexample,usingavalueof0.32eVforthemassofthemuonneutrinoandusingthepreviouslymentionedvaluesof thedifference in the squareof themassesweobtain thefollowing:Anelectronneutrinoshouldbeabout1.2 10‐4eVlessthanamuonneutrinoandatauneutrinoshouldbeabout3.7 10‐3eVlargerthanamuonneutrino.Thereforethisestimateimpliesthatthetauneutrinoisabout1%moreenergeticthantheelectronandmuonneutrinos.Thespacetimewavemodeloftheuniversecanaccommodateneutrinosthathaveasmallrestmass.Thestartingassumption theuniverseisonlyspacetime issorestrictivethatitgreatlynarrowsthepossibilitiesforphysicalmodels.Allparticlesthatexhibitrestmassareproposedtogenerallyhavetherotarmodel.Besidesthedifferenceinmass,thereisobviouslyabigdifferenceinpropertiesbetweenanelectronandanelectronneutrino.Thisimpliesthatthereshouldbeatangibledifferenceinthemodelsofthesetwospacetimeparticles.Thisdifferenceisunknownatthepresenttime.However,sincethemechanismforgivinganeutrinorestmass,energyandangularmomentumis thesameasanelectron,wewillstartbyusingthesamegeneralrotarmodelforanisolated,stationaryneutrinoasforanisolated,stationaryelectron.

6 K. Nakamura et al. (2010). "Review of Particle Physics". Journal of Physics G 37: 1. 7Battye, R. A., Moss, A., Evidence for Massive Neutrinos from Cosmic Microwave Background and Lensing Observations, Phys. Rev. Lett. 112, 051303 (2014)


Modeling a Neutrino:Ifweassumethatthethreeneutrinoshaveenergyintherangeof0.32eV,thisimpliesthatanisolatedneutrinoinitsrestframehasaComptonangularfrequencyofωc 5 1014s‐1andarotarradiusofabout6 10‐7m.Thislargesizeandlowfrequencyforrotarswouldseemtopresentaproblemfortherotarmodelthatcanbeillustratedwithanexample.Amuondecaysintoanelectron,anelectronantineutrinoandamuonneutrino.Themuonhasarotarradiusofabout1.9 10‐15mandaComptonangularfrequencyofabout1.6 1023s‐1.Ifneutrinoshaverestmass,howisitpossibleforthedecayofamuontoproduceneutrinosthatareabout108timeslargerradiusthanthemuon? ~10‐7mcomparedto~10‐15m This apparent incompatibility occurs because we are erroneously comparing the size andfrequencyofisolatedrotarsinarestframewhenweshouldbelookingatthesecharacteristicswhenrotarsareintheverycloseproximitytootherrotarsatthemomentofdecay.Recallthatwhenanelectroncollideswithaprotonoranotherelectron, there isamomentwhenall thekineticenergyoftheelectronisconvertedtointernalenergyoftheelectron.Thismomentarilyincreasestheelectron’sComptonfrequencyandmomentarilycontractsitsrotarradius. A50GeVelectroncollisioncausestheelectrontomomentarilydecreaseitsrotarradiusbyafactorofabout100,000andincreaseitsfrequencybythesamefactor.Itisproposedthataneutrinocreatedinaparticledecay amuondecayforexample isinitiallycreatedinthehighenergy,compressedconditioncharacteristicofacollision.Thisextraenergyisconvertedtotheultra‐relativisticvelocitiesofthethreedecayproductsproducedbythemuondecay. Thiscanbeseen fromthe followingexample:Supposethatwe imaginereversingthedecayprocess. Thedecayproducts consistingof anelectron, anelectronantineutrinoandamuon neutrino would reverse directions and produce a collision that forms a muon. Thiscollisionwouldbehighlyrelativisticandmomentarilyreturnthethreedecayproductstotheirenergetic,compressedstatepresentwhenthemuoninitiallydecayed. Infact, thesumofthefrequenciesofthethreedecayproductsinthecompressedstate beforeseparation wouldequalthemuon’sComptonfrequency.Inthisexplanation,theneutrinoswouldnotdevelopthelargesizeandlowerfrequencyuntiltheyseparatefromtheotherrotarsandeachisviewedinitsrestframe.Almost all the quantized angular momentum in the universe is contained in neutrinos andphotons.Alltheotherleptonsandquarkstogethercontainlessthanonepartin108comparedtoquantizedangularmomentumofphotonsandneutrinos.Thismakesneutrinosanimportantconsideration when examining the evolution of the universe. Therefore, neutrinos will bediscussedagaininchapters13and14oncosmology.


Chapter13

CosmologyI–TheSpacetimeTransformationModel

The Big Bang theory is one of only a few theories in science that enjoys virtually universalacceptance.Beforeabout1929the“SteadyStatetheory”wasfavoredbutsincethentherehavebeennumerousexperimentalobservationsthatsupporttheBigBangtheoryanddisprovetheSteadyStatetheory.IamgoingtoproposeanalternativetothestandardBigBangmodel.Thismightbeconsideredbysomeastheequivalentofscientificheresy.However,wewillfaithfullyfollowourstartingassumption theuniverseisonlyspacetime toitslogicalconclusion.Asanintroductiontothecosmologicalimplicationsofourstartingassumption,wewillusealineoutofthechildren’sstoryofAliceinWonderland.InthisstoryAliceasksthequestion:“IstheroomgettingbiggeroramIgettingsmaller?”Scientistsdonotasktheequivalentquestionabouttheuniverse. Theyalwaysassumethattheuniverseisgettingbiggerwhiletheirmetersticksandclocksstaythesame. However, thisassumptionleadstonumerousmysteries. Howis itpossiblethatdistantgalaxiesarerecedingat fasterthanthespeedof light?Isnewspacetimecontinuously being created? Ismysterious dark energy required to explain the acceleratingexpansionoftheuniverse?Allofcosmologylooksdifferentwhenweassumethattheuniverseisonlyspacetimeandthepropertiesofspacetimearechanging.BeforeCopernicus,itwasassumedthattheearthwasthecenteroftheuniverse.Nowweknowthatwedonotliveinaspecialplaceintheuniverse,butweatleastassumethatbothoursizeandourrateof timeareconstant. Iamproposing that the final indignity tomankind is thatspacetime isgoing througha transformation that contractsallphysicalobjects, including theearthandourbodies.Furthermore,therateoftimeisalsoslowingdown.Itisproposedthatthespacetimefieldisundergoingatransformationwhichisresponsibleforwhatweperceivetobetheexpansionofthevolumeoftheuniverse.Supposethatweimaginegoingbackwardsintimefromtoday.Thedistancebetweengalaxieswoulddecreaseand the cosmicmicrowavebackground CMB photonswould increase theirenergy.Todayonlyasmallpercentageoftheobservableenergyintheuniverseisintheformofelectromagnetic EM radiation.However,goingbackwardsintimewewouldreachatimewhenthedominantformofobservableenergyintheuniversewasEMradiation.Whathashappenedtothisenergythatoncedominatedtheuniverse?Individualphotonsarenotdisappearing.Evenasphotonsloseenergytheyretainalltheirangularmomentum.Electronsandprotonsappeartobe immune from this energy loss, butwe have to questionwhether our standard of energyremainsconstantovertime.


Whenweassumethattheuniverseisonlyspacetimebothtodayandthroughoutitshistory,weget an entirely different perspective on cosmology. There still is something that can bedesignatedthe“BigBang”butthereisnosingularitywithinfiniteenergydensity. Infact,theenergydensityoftheuniverseneverexceedsPlanckenergydensity∿10113J/m3.Allofthiswillbeexplained,butwewillstartbyreviewingthecurrentcosmologicalprinciples.The Cosmic Microwave Background (CMB):TheprecisemeasurementoftheCMBhashadaprofoundimpactoncosmology.ThePlanckSpacecrafthasmappedthefullskydistributionoftheCMBtoaresolutionbetterthan0.1°.Theageoftheuniversehasbeendeterminedtobe13.8billion years 0.4%. Also the universe has been determined to not have any large scalecurvature.Itis“flat”toameasurementaccuracyofabout0.1%.1,2TheCMBwasalsomappedbytheWilkinsonMicrowaveAnisotropyProbe WMAP whichhadlowerresolution,butsomeofthedataismoreaccessibleandstillquiteaccurate.Therefore,numbersquotedherewillbeamixtureofthesetwosources.AnalysisofthedatafromWMAPhasalsodeterminedthatabout380,000yearsaftertheBigBang,theenergyintheuniversewas10%neutrinos,15%photons,12%ordinarymatter baryonicmatter , and 63% dark matter. For comparison the percentages quoted for today are:4.9%ordinarymatter,26.8%darkmatterand68.3%darkenergy.The15%energythatwasphotonenergyat380,000yearshastodaybeenredshiftedbyafactorofabout1080.ThishasreducedthecurrentenergycontentoftheseCMBphotonstoanalmostinsignificantpercentage ~0.014% ofthetotalenergyintheuniverse.Similarly,neutrinosandother relativistic particles have also lost momentum kinetic energy due to the redshift.Therefore,over20%ofthetotalobservableenergypresentintheuniverseatanageof380,000has disappeared. Ifwe extrapolate back further, the universewas radiation dominated andalmost all the observable energy present in the early universe has disappeared observableenergyexcludesvacuumenergy . Thequestionaboutwhathashappenedto thisobservableenergywillbediscussedlaterItcanbeshownthattheenergythatwaspresentinordinarymatteranddarkmatterat380,000years is still approximately present in the universe today. The percentages look differentbecause of the addition of dark energy and the subtraction of the energy in photons andneutrinos.Darkenergyhasneverbeenobserved,but it isrequiredbythecurrentlyacceptedcosmologicalmodel.Theneedfordarkenergywillalsobediscussedlater.ThemeasurementoftheCMBhasalsoshownthattheuniversedoeshaveapreferredframeofreference.Fromanygivenlocation,thecosmicmicrowavebackgroundonlyappearsisotropic

1 http://lambda.gsfc.nasa.gov/product/map/dr4/pub_papers/sevenyear/cosmology/wmap_7yr_cosmology.pdf 2 Astrophysical Journal Supplement 180: 225-245 February 2009


fromaparticularframeofreferencecalledthe“CMBrestframe”.Forexample,thesunappearstobemovingatabout369km/sinthedirectionoftheVirgoconstellationrelativetotheCMBasseen from the sun’s location. This relativemotionproducesaDoppler shift in theCMB thatshowsupasaredshiftintheCMBinonedirectionandablueshiftintheCMBintheoppositedirection dipoleanisotropy .ThisanisotropyissubtractedfromCMBpicturestoproducethespeckled but uniform CMB pictures commonly exhibited showing only small temperaturevariations. Each location in the universe has a unique frame of reference that is stationaryrelativetotheCMB.Hubble Parameter: The expansion rate of the universe will be referred to as the “Hubbleparameter ”.Itisnotatrueconstantsinceitchangesovertime.Theexpansionoftheuniversequoted by astrophysicists usually is given in units of kilometers/second/mega‐parseckm/s/Mpc .Thismightbeconvenientunitsforastrophysicists,butsincewearestandardizingonSIunits,wewillconvertthisintounitsofm/s/m.Theconversionis:km/s/Mpc 3.24 10‐20m/s/m. TheHubbleparametermeasuredby thePlanck spacecraft hasbeen combinedwithothermeasurementstoyieldcompositemeasurementsrangingfromabout67to68km/s/Mpc.Thisbookwilluse 67.8km/s/Mpc 2.2 10‐18m/s/m.Toputthisinperspective,twopointsatrestrelativetothelocalCMBrestframeandseparatedby4.5 1017m ∿49lightyears wouldbemovingawayfromeachotherat1m/s.TheMilkyWaygalaxyandevengalaxyclustersaregravitationallyboundanddonotexpandwiththeexpansionofspacetime.Galaxiesseparatedbymorethanabout300millionlightyearsexhibitthisexpansion.TheHubbleparameterhasunitsofinverseseconds.IftheuniverseexpandedlinearlyfromtheBigBangtotoday,then1/ wouldequaltheageoftheuniverseinseconds.Usingthemeasuredvalue,1/ 4.5 1017seconds 14.4billionyearscomparedtothe13.8billionyearsageoftheuniversemeasuredotherwaysbythePlanckspacecraft.Comoving Coordinates:TheperspectiveofauniqueCMBrestframeateachlocationhasledtotheconceptthattheuniversecanbemodeledashavinga“comovingcoordinatesystem”.ThisisacoordinategridthatexpandswiththeHubbleexpansionoftheuniverse.Eachpointonthisgrid is in theCBM rest frame for that location. This is also called the “comoving frame”. Acomoving observer is the only observer that will see the universe including the CMB asisotropic.Galaxiesarenearlyinthecomovingframe,soanyvelocitytheyhaverelativetothecomovingframeistheir“peculiarvelocity”.Atanygiveninstant,allpointsintheCMBrestframeareexperiencingthesamerateoftimeiflocal gravitational disturbances are ignored. Anotherwayof saying this is that on the scalewheretheuniverseishomogeneous about300millionlightyears ,allpointsinthecomovingframeareexperiencingthecomovingcoordinate’scosmologicaltime.


Comovingdistanceχistheproperdistancebetweentwopoints bothinthecomovingframe atthepresentinstantofcomovingtime.Whilecomovingdistancecorrespondstoproperdistanceatthepresentinstant,comovingdistanceisimaginedtoremainatafixedvalueofχovertime.InthepastorfuturetheproperdistancebetweenthesetwopointschangeswiththeHubbleflowbutthecomovingdistanceisafixeddesignationonanexpandingcoordinatesystem.Thecosmic“scalefactor a ”isafunctionoftimeandusuallydesignatedas:a t .Thisscalefactorquantifiesthe relative expansionof theuniversebetween twomoments in time. The term lt is properdistance between the two points at a different time different ages of the universe . Therelationshipbetweenthesetermsis:χ lτa t Thecurrentscaleofthecomovingcoordinatesystemisdesignated“ao”andusuallysettoequalone ao 1 .The Λ-CDM Cosmological Model: Λ‐CDMisanabbreviationforLambda‐ColdDarkMatter.ThisiscurrentlyconsideredtobethestandardBigBangmodel.In1929whenEdwinHubblediscoveredtheredshiftofgalaxies hecalledthemnebula ,theinitial interpretationwasthatthesewereDopplershiftsduetorelativemotionofgalaxiesexpandingintoapreexistingvoid.Hubbleproposedthattherewasalinearrelationshipbetweenvelocityanddistance.Thecurrentinterpretationisthattheredshiftisduetoacosmologicalexpansionoftheuniversewherespaceitself isbeingcontinuouslycreatedeverywhere. OnedifferencebetweenHubble’sconcept apreexisting void and cosmological expansion is that cosmological expansion is currentlybelievedtobeabletoproduceseparationvelocitiesthatexceedthespeedoflight.Observationsstrongly support the general relativistic interpretation of a cosmological expansion over thespecialrelativityinterpretationofmassexpandingintoapreexistingvoid.AdetaileddescriptionoftheΛ‐CDMmodelwillnotbegivenhere.However,thereareseveralphysical interpretationsandpredictionsof thismodel thatwill bedescribed toestablish thecurrent perspective of cosmologists. The favored Λ‐CDM model is usually designated as:ΩM 0.31andΩΛ 0.69withtheHubbleparameter 67.8km/s/Mpc 2.2 10‐18m/s/m inMKSunits

The matter density parameter ΩM 0.31 represents that all forms of matter makes upapproximately31%ofthecriticaldensityoftheuniverseandthecosmologicalconstantdensityparameterΩΛ 0.69indicatesthatdarkenergymakesupabout69%ofthecriticaldensityoftheuniverse.DarkenergyisthehypotheticalenergythatexpandstheuniverseandisassociatedwithEinstein’scosmologicalconstantΛ.


Figure 13‐1 is taken from a very good article titled “Expanding Confusion: CommonMisconceptionsofCosmologicalHorizonsandSuperluminalExpansionoftheUniverse”3Figure13‐1istwospacetimediagramsthatplottimeversusproperdistanceDonthetoppanelandtimeversuscomovingdistanceonthebottompanel.Thepanelsaredrawnfromtheperspectiveofanobserverlocatedattheintersectionofthe“now’line 13.8billionyearsaftertheBigBang andthedistance 0line. Thelowerpanelhasverticalworldlinesforobjectscurrentlywithvariousredshifts z 0,1,3etc. becausethecomovingdistanceχdoesnotchangeovertime.Theproperdistancechangesovertimebutthecomovingdistanceisacoordinatedistancethatexpandswiththecomovingcoordinatesystem.Thetwopanelsofferdifferentperspectivesandthefollowingdescriptionappliestobothpanels.Thedashedlinelabeled“particlehorizon”crossesthe13.8billionyear“now”lineatabout46billionlightyears.Itistobeunderstoodthateventhoughasignalfromthemostdistantsourcecurrentlyreachingus,calledtheparticlehorizon,hasbeentravelingatthespeedoflightfor13.8billionyears,thecurrentdistancetothatsourceislargerthan13.8billionlightyears.Thisisbecause the cosmic expansion of the universe has continued to increase the volume of theuniverseandthedistancebetweenpointsafterthespeedoflightsignalhaspassedanylocation.Nextwewill lookat the line labeled“Hubblesphere”. Currently theboundaryof theHubblesphereis13.8billionlightyearsfromus.Thisisthedistancewherespaceitselfissupposedly

3 T. M. Davis; C. H. Lineweaver; Publications of the Astronomical Society of Australia 2004, 21, 97-109.


recedingfromusatavelocityequaltothespeedoflight conceptexaminedlater .AccordingtotheΛ‐CDMmodel,thespacebeyondtheHubblesphereisrecedingfromusfasterthanthespeedoflight.Thisgivesrisetoan“eventhorizon”.The line labeled“eventhorizon”representsthe furthestdistancethatwecanreceivecurrentinformation.Forexample,accordingtotheΛ‐CDMmodel,galaxiesthatwecurrentlyobserveashavingaredshiftofz 1.8arecurrentlycrossingoureventhorizon.Thismeansthatlightbeingemittedbythesegalaxiestodaywillneverreachusbecauseoftheacceleratingexpansionoftheuniverse.Allgalaxieswithredshiftsgreaterthan1.8havealreadycrossedoureventhorizonandwilldisappearfromviewsometimewithinthenext14billionyears largerredshiftsdisappearsooner .According to the Λ‐CDMmodel eventually the Hubble sphere and the event horizonwill beapproximatelythesamedistance.Inthedistantfuturetheuniversewillhaveaneventhorizonataconstantproperdistanceofabout17billionlightyears.Galaxieswillcontinuetodisappearfromviewastheyarecarriedbycosmologicalexpansionbeyondthisdistance.Eventuallyonlythegravitationallyboundgalaxieswillremainvisible.Iftheconceptofacceleratingexpansionof the universe is carried to its logical conclusion, we would end with the “Big Rip” whereexpandingspacetimeeventually tearsapartourgalaxy, thenoursolarsystemandeventuallyevenatoms.ProblemswiththeBigBangSingularityandInflation:Themostwidelyacceptedmodelforthestartoftheuniverseisa“singularity”withnovolumeandinfiniteenergydensity.Thatstartingconditionrequiresthatthelawsofphysics“breakdown”.Italsocreatesaproblemforexplaininghowtodaytheenergydensityoftheuniverseishomogeneousonthelargescalelarger.Startingwithasingularityresults in inhomogeneousdistribution,evenon the largescale. Therefore,physicistspostulateaperiodof “inflation”whentheuniverseexpandedmuch faster thanthespeedoflight.Thishypothesiswouldresultintheportionoftheuniversethatwecanobservebeinghomogeneous.However,itisgenerallyagreedthattheuniverseextendsfarbeyondourcurrentparticlehorizon.Somecosmologists even suggest that the “extendeduniverse” is infinite. Wedonotneed todecidethatquestioninordertodothefollowingthoughtexperiment. Supposethatwecouldobtainamapoftheuniverseextendingfarbeyondourparticlehorizon.Wouldthemaprevealthattheuniverseisinhomogeneousontheextendedscale?Inflationsupposedlydidnotreallyeliminate the inhomogeneties causedby starting froma singularity. Itmerely expanded thevolumesomuchthatanysingleobserverdoesnotseeobviouslargescaleinhomogeneties.Theimplication is that today there are parts of the extended universe that have substantiallydifferentdensitythanthedensityofourobservableportionoftheuniverse.


It has been established byWMAP space probe that the portion of the universe thatwe canobservehasflatspacetimetowithinthe0.4%marginoferror.Doweliveinaspecialpartoftheuniversethathappenstohaveachievedthisspecialcondition?Surelyifthereareunobservablepartsof theextendeduniverse thathavesubstantiallydifferentdensity, then theywouldnothaveflatspacetime.Animportantcosmologicalprincipleisthatwedonotliveinaspecialpartoftheuniverse.Thisshouldapplyeventotheextendeduniversebeyondourparticlehorizon.Theimplicationisthat13.8billionyearsaftertheBigBang,allpartsoftheextendeduniverseshould have the same density. Inflation does not explain how the extended universe can behomogeneous.Spacetime Transformation Model: We are about to start the description of an alternativecosmologicalmodelthatwillbereferredtoasthe“spacetimetransformationmodel”.Thismodelwillbemorefullyexplainedinthenextchapter,butkeypartsofthemodelwillbepresentedinthis chapter. Initially, these key parts will be explained using the familiar concepts andterminology of the current Big Bang model. When explaining the spacetime transformationmodel,theinitiationofthetransformation thebeginningoftime willbereferredtoasthe“BigBang”. This is a highly descriptive term that is being adapted to express the start of atransformationofoneformofspacetimetoanother. Thiswasaccompaniedbyatremendousincreaseinpropervolume,sotheterm“BigBang”isstillapplicable.Iftheuniverseisonlyspacetime,howdoesthespacetimefieldcreatenewvolume?Today,alotofattentionisbeingpaidtotheapparentaccelerationoftheexpansionoftheuniverse.Whilethisisanimportantquestion,itisnotpossibletoanswerthisquestionuntilwefirstunderstandhow any new proper volume is created in the universe. If the expansion of the universe isimaginedasmass/energyexpandingintoapreexistingvoid,thenmomentumwouldcontinuethisexpansionandtherewouldbenomystery.However,theΛ‐CDMmodelofcosmologysaysthattheuniverseisundergoingacosmologicalexpansion.Thisexpansionistheresultofnewvolume continuously being created everywhere in the universe. This new volume isindistinguishablefrompreviouslyexistingvolume.Therefore,thenewvolumemustalsocontainvacuumenergywithenergydensityofmorethan10112J/m3.Supposethatweattempttoputthis in perspective. TheHubble parameter is 2.2 10‐18m/s/m. Thismeans that eachsecond,eachcubicmeterofspacetimeisexpandingby 2.2 10‐18 3m3 10‐53m3.Thissmallvolume seems insignificant until it is realized that the energy density of this new volume isapproximatelyPlanckenergydensityUp 10113J/m3.Therefore,10‐53m3requiresabout1060Jofvacuumenergy. Themassof theMilkyWayisabout1042kgequivalenttoanannihilationenergyofabout2 1059J.Therefore,eachcubicmeterintheuniverseseemstobeaddingtheequivalenttheMilkyWay’senergyeachsecond.Thisseemsridiculous,butitappearsthatthereareonlythreechoicestoexplainthisdiscrepancy.Either 1 vacuum energy is canceled by some other equally vast offsetting effect; 2 a vastamountofnewvacuumenergyisbeingcontinuouslyaddedtotheuniversetomaintainaspecific


vacuumenergy density or 3 a fixed amount of vacuumenergy is being distributed over anincreasingvolumewhichvastlydecreasestheenergydensityofvacuumenergyastheuniverseexpandsfromtheBigBangtoday. Allofthesealternativesseemunappealing.Thespacetimetransformationmodel of the universe will propose a fourth explanation which involves thetransformationofthepropertiesofspacetime.Theadditionofnewvolumetotheuniverseisnotsomesubtleeffectthatistakingplaceinremotepartsoftheuniversebetweengalaxies.Itispossibletoillustratethescaleofcosmicexpansionusingourownsolarsystemasanexample.SupposethatweconsiderasphericalvolumewiththeradiusofNeptune'sorbit radius 4.5 1012m .Thissolarsystemsizesphericalvolumewouldcontainallthemajorplanetsandhaveavolumeofabout3.8 1038m3.CalculatingfromtheHubbleparameter 2.2 10‐18s‐1 ,thisvolumewouldincreasebyabout1021m3eachsecondifthesphericalvolumeexpandedproportionaltotheHubbleexpansionoftheuniverse.Toputthisexpansioninperspective,thissolarsystemsizevolumeisaddingtheequivalentofaboutearth’svolume 1.08 1021m3 everysecond.WedonotnoticeanychangeinthevolumeofthesolarsystemortheMilkyWaygalaxybecausetheseobjectsaregravitationallybound.Theadditionofnewvolumebecomesobviouswhendistance properlighttraveltime ismeasuredbetweengalaxies thatarenotgravitationallyboundtogether. Onlyafterwehaveaplausibleexplanationforthecreationofthisnewvolumeeverywhereintheuniverse,canweseriouslyaddressthequestionaboutthenonlinearityinthiscreationprocess acceleratingexpansion .The new volume is also spacetime so a rephrasing of the question is: How does spacetimerearrange itself to creatingnewpropervolume? If theuniverse isonly spacetime, and if theuniverseappearstobeexpanding,thenthepropertiesofspacetimemustbechangingwithtime.Somethingmustbechangingbecausetodayittakesalongertimeforlighttotravelbetweentwogalaxiesthanitdidabillionyearsago assumingeachgalaxyisatrestrelativetotheCMB .Proper Volume:Whatexamplesdowehaveofspacetimeadjustingitselfinawaythatchangesproper volume? In the example of the Shapiro experiment, the sun’s gravity changed thespacetimebetweentheearthandthesuninawaythatincreasedtheproperradialdistancebyabout7.5kmcomparedtotheradialdistancethatwouldbeexpectedfromEuclidiangeometry.Thisincreaseinradialdistanceincreasedthevolumewithinaspherethatis1astronomicalunit1AU inradiusbyΔV 3.46 1026m3 previouslycalculated .Thisnon‐Euclidianincreaseinproper volume is more than 300,000 times the earth’s volume. This volume increase wasaccompaniedwithadecreaseintherateoftime.Also,therotarmodelhastwolobeswherethepropertiesofspacetimeareslightlydistorted.Theslowtimelobehasarateoftimethatloses1unitofPlancktimeinatimeperiodof1/ωc.ThatslowtimelobehasavolumethatislargerthanwhatwouldbeexpectedfromEuclidiangeometry.The fast time lobehas lesspropervolumethanwouldbeexpected fromEuclidiangeometry.


Againitappearsas if thereisaninverserelationshipbetweenpropervolumeandtherateoftime.Neartheendofchapter2itwasshownthatthestandardsolutiontotheSchwarzschildequationhasa4dimensionalvolumethatisindependentofthegravitationalgammaГbecausethechangeinthetimedimensionoffsetsthechangeintheradialspatialdimension.Inalltheseexamples,thereisaninterconnectionbetweentherateoftime,thecoordinatespeedoflightandpropervolume.Wewillexplorethefollowingidea:New proper volume is created when the rate of time decreases. Itisproposedthatspacetimeisabletoexchangetheabsoluterateoftimeforpropervolume.Thisconceptisimpliedinthefollowingequation:

ГnoteinverserelationshipbetweendτanddLR

WhiledLR/dRappliestotheradialdirectionintheSchwarzschildcoordinatesystem,theaboveequationalsocanbeinterpretedasshowinganinverserelationshipbetweentherateofpropertimeandpropervolumeevenwhenappliedtotheentireuniverse.Weseethepropervolumeoftheuniverseincreasing,butprovingthatthisiscoupledwithadecreasingrateoftimeismoredifficult.Anabsoluteproof that therateof time isslowing in theuniversewouldrequire thecomparisontoahypotheticalcoordinateclockoutsideof theuniverse. This isan impossibleexperiment,soinsteadwewillstartwithseveralthoughtexperiments.Cavity Thought Experiments:Firstwewillimagineasphericalshellwithmassmandinternalradiusr.Wewillplacethissphericalshellinspacewhereitisinaninertialframeofreference.If the spherical shell hasmass, then on the outside surface of this spherical shellwewouldexperience a gravitational acceleration accompanied by the standard time dilation andgravitational effect on volume. Inside this spherical shell, there would be no gravitationalaccelerationandwecouldconsiderthisflatspacetime.However,theescapevelocityfromtheshellisgreaterifwestartfrominsidetheshellcomparedtostartingfromtheoutsidesurface.Even though there is flat spacetime, inside the spherical shellwe still have the gravitationaleffectsthatareafunctionofescapevelocityandscalewithΓ.Thispointwasalsomadeinchapter2 with examples using a cavity at the center of the earth and the Andromeda galaxy. Thegravitationaleffectontherateoftime,thecoordinatevelocityof lightandonpropervolumeremainsevenwhenthereisnogravitationalacceleration.However,onlythegravitationaleffecton the rate of time is easy to experimentally see so wewill temporally concentrate on thegravitationaleffectontherateoftime.Background Gravitational Gamma Г:PreviouslywedefinedthegravitationalgammaГandacloselyrelatedconcept,thegravitationalmagnitudeβ.Theseweredefinedasfollows:


Г≡ 1/ 1 / –

Г gravitationalgamma;ve escapevelocity

β≡1– 1– 1– 1–Гβ gravitationalmagnitude

Г 1 βandβ weakgravityapproximations

Inside a cavity uniformly surrounded by mass m, there is a gravitational effect, but the

gravitationalgammadefinitionГ needstobereinterpreted.Thisequationpresumes

anisolatedmassinanotherwiseemptyuniverse.Theinteriorofasphericalshelldoesnotmeetthis assumption. However, the definition of Г that incorporates escape velocity ve is easily

definable:Г 1/ 1 / . Forexample,acavityatthecenteroftheearthwouldhaveanescapevelocity about22%greater than theescapevelocity starting from theearth’s surfaceassumingauniformdensityapproximationandignoringairfriction .Theotheralternativedefinitionforthegravitationalgamma stationaryframe alsoworksforacavitysurroundedbymass:Г dt/dτ.Inthiscase,wedefineacoordinateclockfarfromthesourceofgravitywitharateoftimedtandaclockinthecavitywithrateoftimedτ.Actually,thedefinitionГ dt/dτispreferablebecauseitisalsoapplicabletotheentireuniverse.Thereasonfordiscussingcavitiessurroundedbymassisthatourlocationintheuniversehassimilarities to a cavity surrounded by mass. In fact, the universe is something like beingsurroundedbyashellthatisincreasinginmasseverysecond.Whatwoulditbeliketobeinsideashellthatisincreasingitsmass? assuminginertialframeofreference Therateoftimewouldbeslowingdownandthecoordinatespeedoflightwouldbeslowingdown,butwewouldnotbeable to directly detect these changes unlesswe had communication to an outside standard.However, it is proposed thatwewould notice something strangewas happening. A carefulexperimentwouldrevealthat lightwasundergoingaslightredshift. Thewavelengthof lightpropagatinginsidethecavitywouldincreasewithpropagationtime.Aproofofthiscontentionwillbegiveninthenextchaptertoexplaintheredshiftobservedintheuniverse.We often make the assumption that a location in space far from the earth and sun can beconsidered to be a location with zero gravity. It is true that there might be negligiblegravitationalacceleration,butwearestillsurroundedbyallthemass/energywithintheparticlehorizonforourlocationintheuniverse.WeknowfromobservationsoftheCosmicMicrowaveBackground Planckspaceprobe thattheuniverseisflatspacetimetowithina0.1%marginoferror.Thissuggeststhattheuniverseextendsinfinitelybeyondourparticlehorizon.However,only the mass/energy within our particle horizon can have any influence on us. Themass/energywithinourparticlehorizonexertsabackgroundgravitationaleffectwhichmust


havegreatlysloweddownboththerateoftimeandthecoordinatespeedoflightcomparedtoahypotheticaluniversewithoutthisinfluence.Wewilluseanewnametodiscussthissubject.Background Gravitational Gamma of the Universe Гu:Thename“backgroundgravitationalgammaoftheuniverse”andthesymbolГuwillbeusedtorepresenttheconceptthattheuniversepossessesagravitationaleffectthatproducesnoaccelerationbutaffectstherateoftime,propervolume, thecoordinatespeedof lightandmanyotherphysicalproperties. Withbackgroundgammathereisdistributedmass/energyinalldirectionssowedonotmeettheSchwarzschildassumptionofasinglemassinanotherwiseemptyuniverse.However,thedefinitionofГbasedontheeffectontherateoftimeisthesame:Гu dt/dτuwheredτuistherateoftimeinaframeofreferencethatisstationaryrelativetothecosmicmicrowavebackground CMB andfarfromlocalizedsourcesofgravity.InthiscasedtisthecoordinaterateoftimewhichiseithertherateoftimeinahypotheticaluniversewithoutgravitationalinfluencewhereГu 1ortherateoftimeattheinstanttheBigBangstarted whenГu 1 .Thiswillbeexplainedlater.Toillustratetheconceptofabackgroundgravitationalgamma,anexamplewillbegivenusingtheMilkyWaygalaxy.Thisgalaxyhasavisibleradiusofabout50,000lightyears,butdarkmatterisbelievedtoextendtoaradiusofatleast150,000lightyears.Thesun’sorbitisabout26,000lightyearsinradiusaroundthegalacticcenter.Thegalacticgravitationalaccelerationfeltbythesunisdueentirelytomatterthatisinsidethesun’sorbit.MatteranddarkmatterintheMilkyWaygalaxythatliesexternaltothesun’sorbitdoesnotproduceanygravitationalaccelerationontheSun.However,thisexternalmatterdoesstillproduceagravitationaleffectinthesensethat itaffects therateof time, thecoordinatevelocityof light, thegravitationalpotential, theescapevelocityfromthegalaxy,thestandardofenergy,etc.Inotherwords,thisexternalmatterisproducingasignificantbackgroundГatthesun’sorbitaldistance.It is interestingtoestimatethebackgroundГof theMilkyWaygalaxycausedbythegalaxy’smassexternaltothesun’s26,000lightyearorbit.Thetotalmassofthegalaxy,includingdarkmatter,isestimatedatabout1.2 1042kg.Themassinsidethesun’sorbitrequiredtoproducetherequiredgravitationalaccelerationonthesunisonlyabout15%ofthistotalmassof thegalaxy.ThereforethemassoftheMilkyWaygalaxyexternaltothesun’sorbitisabout1042kg.AlmostallofthismassisdarkmatterthatissphericallydistributedaroundtheMilkyWaygalaxy.Forthiscalculationwewillestimatethatallthemassexternaltothesun’sorbitcanbesimulatedby a spherical shell 110,000 light years in radius with mass of 1042 kg. Substituting intoβ Gm/c2rweobtainβ 7 10‐7 orГ 1 7 10‐7 .Theearth’sgravitationalmagnitudeattheearth’ssurfaceisβ 7 10‐10 orГ 1 7 10‐10 .WhileneithertheearthnortheMilkyWaygalaxyhavealargeeffectontheabsoluterateoftime,theMilkyWay’suniformbackgroundГ slows down the rate of time about 1,000 times more than the earth’s own gravity7 10‐7/7 10‐10 .


ExtendingtheconcepttothebackgroundgammatotheuniverseГu,allthemass/energywithintheparticlehorizoniscreatingaverylargevalueofГu.IndividualstarsandgalaxiesrepresentonlyarelativelysmallperturbationofcurvedspacetimeintheflatandverylargebackgroundГuoftheuniverse. Blackholeswillbediscussedlater. Forexample,iftheuniversehad99%ofthem/Rratiorequiredtoformablackhole,thebackgroundgravitationalgammaoftheuniversewould be: Гu 10. Laterwewill attempt to calculate the actual current value of Гu of theuniverse. General relativity calculations normally ignores the background gamma of theuniversewhichisthesameasassumingthatthisbackgroundisГu 1.Thisisacceptableforallcalculations involvingdiscretemassor evengalactic clusters. However,whenattempting toexplaintheexpansionoftheuniverse,thiscannotbeignored.Besides Гu, there is a related concept that is the background gravitationalmagnitude of theuniverse:βu.TherelationshipbetweenГuandβuisthesameastherelationshipbetweenГandβ. SomeequationsaresimplerwhenexpressedintermsofgravitationalmagnitudeβratherthangravitationalgammaГ.TheexactconversionbetweenГandβis:β 1–1/Г 1–dτ/dt.Theserelationshipsalsoholdforβu.Thegravitationalmagnitudeβhasascalethatrangesfrom0to1whilethegravitationalГhasascalethatrangesfrom1toinfinityIsotropic and Homogeneous Universe:Allpointsonthecomovingcoordinatesystem CMBrestframe perceivetheuniversetobeisotropicandhomogeneous.ThisincludestherateoftimewhichisthesameeverywhereintheCMBrestframe.Sincepointsonthecomovingcoordinatesareexpandingawayfromeachother,whatdoesitmeanforthesepointstohavethesamerateoftimeatagiveninstant?Thiscanbeillustratedwithathoughtexperiment.Supposewehave3 spaceships located far from sources of gravity in intergalactic space. Each of the threespaceships isexactlystationaryrelative to theCMBfor its location.Thethreespaceshipsarewidelyseparatedformingastraightlinewiththemiddlespaceshipexactlyhalfwaybetweenthetwooutsidespaceships.Supposethetwooutsidespaceshipssentoutastablemicrowavesignalatthe9.19gigahertzcorrespondingtothefrequencyoftheircesiumatomicclocks.Whenthetwo signals reach themiddle spaceship, the frequency would be slightly lower redshifted comparedtotheatomicclockonthemiddlespaceshipduetocosmicexpansion.However,theimportantpointisthatboththefrequenciesfromthetwooutsidespaceshipswouldbeexactlythesame.Themiddlespaceshipobservesthatboththeoutsidespaceshipswereexperiencingthesamerateoftimeattheinstantthesignalsleftthespaceships.Withrelativitytherecanbeconfusionaboutdifferentdefinitionsof the term“simultaneous”. Therefore, theconceptofa“midpointobserver”specifiesonewayofdefiningsimultaneous.Thetwooutsidespaceshipsaresimultaneouslyexperiencingthesamerateoftimeaccordingtothemidpointobserver.Thismeansthatallpointsonthecomovinggridareexperiencingthesamerateoftimeaccordingtomidpointobservers. Therefore, this rateof time isusedas coordinate rateof time in theRobertson‐Walkermetric.Nowsupposethatallthreespaceshipsarestationaryrelativetoeachother.Anextrapolationofthethoughtexperimentshowsthattheevenifthethreespaceships


aremovingrelativetotheCMBbutstationaryrelativetoeachother,themidpointobserverwillstillseethattheoutsidespaceshipsareexperiencingthesamerateoftime.Thisconceptcanbeextendedtoanytwopointsintheuniverseinthesameframeofreference.On the scale where the universe is homogeneous, any two points in the same frame of reference experience the same rate of time according to a midpoint observer.No Large Scale Gravitational Acceleration:Thisthoughtexperimentwillbeexplainedusingtherotarmodelofanelectronandtheexplanationforgravitypreviouslydeveloped.Supposethatweplaceanelectron rotar inintergalacticspaceandignoreanylocallygeneratedforces.We are only interested in examining gravity on the scale larger than300million light yearswheretheuniverseishomogeneous.Wearegoingtotestthelargescalegravityoftheuniverseusingtherotarmodelofanelectron.Recallthatarotarexperiencesgravitationalaccelerationwhenthereisarateoftimegradientacrosstherotar.Thiscausesthevacuumenergytoexertanunbalancedpressureonopposite sidesof therotar. Thisproducesanet force that is thegravitationalforce.Theaccelerationofgravity g waspreviouslyshowntobe:

g c2 –c2 g gravitationalacceleration

Itwaspreviouslyshownthatthedistancedesignatedinthedenominator dr isproperlength,notcoordinatelength circumferentialradius fromgeneralrelativity.Ifthereisnogradientintherateoftime,thend dτ/dt /dr 0andg 0.Thereisnogravitationalaccelerationwhenthereisnogradientintherateoftime.Twopointsontheoppositesidesofthequantumvolumeofarotarcanbeconsideredtobetwopointsinthesameframeofreference.Inahomogeneousandisotropicuniverse,thesamerateoftimeispresentonoppositesidesoftherotar accordingtothemidpointobserver .Thereisnogravitationalaccelerationonthehomogeneousscaleof300millionlightyearsorlarger.Allthismightseemobvious,butitimpliesthattheuniverseisnotstrugglingtoexpandagainstagravitationalforcethatisattemptingtocollapsetheuniverse.Thegravityoflocalizedobjectslikestarsandgalaxiescancurvespacetimeonalocalscale,butthere is not a rate of time gradient in the universe on the scale where the universe ishomogeneous.If there is no large scale rate of time gradient in the universe, then there is no tendency for there to be gravitational deceleration or acceleration at this scale.Itmightbearguedthatdarkenergyiscurrentlyprovidingsomethinglikeanti‐gravity.Todefendthispositionitwouldhavetobearguedthatgravityiscurrentlyproducingarateoftimegradientthatisattemptingtocollapsetheuniversebutdarkenergyisproducinganoppositerateoftimegradient.Thecombinationoffseteachotherandeliminatesanylargescalerateoftimegradienttherebyeliminatinglargescalegravitationalacceleration.However,eventhisargumenthasan


obvious flaw. Dark energy only became a significant fraction of the energy in the universeroughly7billionyearsago.Itwouldhavetobearguedthatpriortothistimetherewasalargescalerateoftimegradientintheuniverse.ThisiscountertotheassumptionscontainedintheRobertson‐Walkermetric.Itisnotpossibletohaveanisotropicuniverseifthereisalargescalerateoftimegradient.Infact,iftherewasalargescalerateoftimegradientpresentwhentheCMB radiation was emitted, evidence of this rate of time gradient would be preserved asanisotropy intheCMBunlesswehappentobe locatedat theexactcenterof therateof timegradient.Rate of Time Gradient in a Dust Cloud:Itisanaxiomofgeneralrelativitythatadistributedcloudofdust nopressure shouldexperienceauniformgravitationalcollapse.Ifthecloudisdividedintosphericalvolumesofsuccessivelylargerradii,thenaparticulardustparticle thetestparticle inthecloudexperiencesgravitationalaccelerationonlyfromthegravityofdustparticleswithinthesmallerradiispheres.Massthatliesoutsidethesphericalvolumecontainingthetestparticledoesnotcontributetothegravitationalacceleration.Enlargingtheradiusoftheimaginarysphericalvolumeincreasestheenclosedmass.Theneteffectisthatthecollapseofthedustparticleshappenseverywhere.However, gravitational acceleration does not happen in the abstract. To have gravitationalacceleration,thecloudofdustparticlesmusthaveadefinablerateoftimegradient.Ittakesarateof timegradient of about1.11 10‐17 seconds/second/meter toproduce accelerationof1m/s2.Itwouldbepossibletodrawathreedimensionalmapshowingtheequivalentofisobarlineswithinthecloudexceptdepictingregionsofconstantrateoftime.Adustcloudoragalaxydoeshaverateoftimecontoursthatcanbemapped. However,theuniversehas andalwayshad thesamerateoftimeeverywhereonthelargescaleaddressedbythecomovingcoordinatesystem.Isitpossibletoproposeamodeloftheuniversethatdoesnothavearateoftimegradientonthescaleofthecomovingcoordinatesystem?Furthermore,doessuchamodelfollowlogicallyfromstartingtheuniverseasPlanckspacetime?Thesequestionswillbeexaminedbystartingwithathoughtexperiment.Dust Particle Cloud Thought Experiment: Supposethatthepreviouslymentionedcloudofdustparticlesisuniformlydistributedinspaceoveravolumeaboutthesizeoftheearth.Also,supposethatitispossibletoturnoffthegravityofallthedustparticlesinthecloud.Thisisanunrealisticassumptionforacloudofdustbutitwillbeshownthatitisnotunrealisticforthebeginningoftheuniverse. Thereforeattempttofollowthishypotheticalthoughtexperiment.Afterturningoffthegravityofeachdustparticle,thegravitationalmagnitudewithinthecloudwouldquicklydroptoβ 0whichisequivalenttoГ 1.Theproperrateoftimeeverywherewithinthecloudwouldbeequaltothecoordinaterateoftime,thereforewewouldstartwithdτ dtorГ dt/dτ 1.Thereobviouslywouldbenorateoftimegradient.


Next,thegravityofalltheparticlesissimultaneouslyturnedon.Atspeedoflightpropagation,therewouldbeaperiodoftimewheretheproperrateoftimedτwouldbeslowingdownwhilethegravitationalinfluenceofsuccessivelymoredistantparticlesisbecomingestablished.Forexample,ittakesabout20millisecondsforthegravityofaparticleatthecenterofthisearthsizecloudtoaffecttherateoftimeofparticlesneartheouter“surface”ofthesphericalcloudandviceversa.Itwouldtakeabout40msforthemostdistantparticlesonoppositesidesofthecloudtomake gravitational contact. Therefore, it takes about 40 ms for the mature gravitationalacceleration maturerateoftimegradient withinthecloudtobecomefullyestablished.SincewearedefiningthegravitationalgammaasГ dt/dτ,wecouldcharacterizetheestablishmentofgravityataparticularlocationinthecloudbyreferringtothevalueofГasafunctionoftimeafterthegravitywasturnedon.Supposethatwelookjustatthefirstfewmillisecondsafterturningonthegravity.Alsowewillexamineseveralpoints testpoints withinthecloudthatarecloseenoughtothecenterofthecloudthatthereisnotenoughtimetoestablishgravitationalcommunicationwiththeboundarycondition that occurs at the outer surface of the cloud. In the first few milliseconds thegravitationalГisincreasingexactlythesamewayateachofthesetestpoints.Eventhoughtherateofpropertimeisslowing Гisincreasing ,thereisnogradientintherateofpropertime.This isbecauseall thetestpoints andallotherpoints far fromthesurface haveundergoneexactlythesameamountofgravitationalinteractionwiththeirsurroundingparticles.Thelackofarateoftimegradientmeansthattherewouldbenogravitationalaccelerationattemptingtocollapse thecloudduring the first fewmillisecondsaftergravity is turnedon in this thoughtexperiment.Thereisnotendencytowardsgravitationalcollapseonlyduringthenonequilibriumcondition of an increasing value of the background Г. Within the dust cloud there is flatspacetimewhileГisincreasing.Thisflatspacetimedoesnotrequirea“criticaldensity”.Thelowdensityof thedustcloud isvastly less thanmeeting theconditionsofa “criticaldensity”,yetduringthenonequilibriumphaseofincreasingГthedustclouddoesachieveflatspacetime nogravitational acceleration . In fact, any homogeneous density can achieve flat spacetimeprovidedthatthenonequilibriumconditionofanincreasingbackgroundГissomehowachieved.Immature Gravity: This thought experiment describes a condition that will be called“immaturegravity”orthenonequilibriumconditionwherethegravitationalinfluenceofdistantmass/energy is in theprocessofbeingestablished. This is tobe contrasted to the “maturegravity”conditionwheretherehasbeensufficienttimetoestablishthegravitationalgradientsresultingfromtheexistenceofdistantboundaryconditions distantchangesindensity .Inthisexample,afterabout40mstherearedifferencesintherateoftimethroughoutthecloudbecauseofdifferentdistancestothecloudboundary.Therateoftimegradientproducesgravitationalacceleration thatwas previouslymissing. This is themature gravity conditionwe normallyassumewhenwetalkabouttheuniformgravitationalcollapsethatweexpectfromacloudofparticles.Theseconceptscanbeillustratedusingthefollowingfigures.


Figure13‐2showsavolumedeepwithinthedustcloud.Supposethatthisisasnap‐shotabout1msafterthegravitywas“turnedon”. Thecirclelabeled“particlehorizonforpointO”wouldhave a radius of about 300,000mwhich represents the speed of light gravitational contactestablishedfrompoint“O”after1ms.Thishorizonisactuallyanexpandingsphereandthefigureisacrosssectionalrepresentationofamomentintime.Thereisalsoanotherpointdesignated“X”thatisrelativelyclosetopoint comparedtotheparticlehorizon.Forexample,supposepoint“X”isseparatedby30,000mfrompoint“O”.Thisseparationdistanceisabout10%ofthedistancetotheparticlehorizonafter1ms. However, itshouldbenotedthatpoint“X”hasadifferentexpandingparticlehorizondesignatedbythedoublelinecircle.After1msthisparticlehorizonisalsoasphereabout300,000minradius,butcenteredonpoint“X”.

Figure13‐2alsoshowsa“testvolume”.Thisisaanimaginaryspherethatiscenteredonpoint“O”andhasaradiusequaltothedistancebetweenpoint“O”andpoint“X” about30,000m .Therefore,after1mstherehasbeenenoughtimeforthecenterofthis30,000mradiussphericalvolumetoestablishgravitationalcontactwithalltheotherdustparticleswithinthetestvolume0.2msisrequiredforoppositeedgestoestablishgravitationalcontact .Nowwearegoingtoexaminethegravitationalforcesondustparticleswithinthetestvolumeifthetestvolumeisisolatedasshowninthelowerrighthandcorneroffigure13‐2.Thisisolatedillustrationshowsbothpoints “O” and “X”. If this small test volume is isolatedas shown, thenafter0.2msallparticles in the test volume have sufficient time to establish gravitational contact with theboundaryofthetestvolume.Ifthetestvolumeisisolatedasshown,the“mature”rateoftime


gradientwouldbeestablishedwithinthetestvolumeafter0.2ms.Inthiscase,particleswithinthistestvolumewouldstarttoundergoagravitationalcollapseandadustparticleatpoint“X”experiencesagravitationalforcetowardspoint“O”asshownbythearrow.

Figure13‐3showsadifferentperspectivecenteredonpoint“X”.Thereforetheparticlehorizoninfigure13‐3isaspherecenteredonpoint“X”.Nowwearegoingtoexaminetheforcesonpoint“X”.Inthepreviousfigure,weconcludedthatadustparticleatpoint“X”hadaforcetowardspoint“O”after0.2mswhenthetestvolumewasisolated.Fromthisnewperspectiveshowninfigure13‐3,supposethatweweretophysicallyremovethetestvolumeasshowntotheright.Now1msaftergravityhasbeen“turnedon”thedustparticleatpoint“X”hasunbalancedforcefromthecomparablevolumetoitsleftthatisdesignatedas“thesymmetricalvolumethatoffsetsthetestvolume”.Thelengthanddirectionofthisarrowrepresentsthegravitationalforceonadustparticle.Thisforcearrowispointingintothe“symmetricalvolume”andexactlyoffsetsthelengthanddirectionoftheforcearrowpointingtowardspoint“O”inthetestvolume.Withthetestvolumeremoved,point“X”wouldexperienceagravitational force in thedirectionof thearrowpointingtothecenterofthe“symmetricalvolume”.However,ifthetestvolumeispresentandpoint“X”issurroundedbyahomogeneousdistributionofparticles,thenthereisnonetforceon a dust particle located at point “X”. There is also no rate of time gradient and no netgravitationalacceleration.


The point of figures 13‐2 and 13‐3 is to illustrate that the immature gravity increasingbackground Г creates a unique condition where there is no rate of time gradient and notendencyforgravitationalcollapse.Bothpoint“O”andpoint“X”hadtheirownparticlehorizonsthatwerepartlyoverlapping,butalsoeachpointhadahorizonthatwasuniquelycenteredonthem. This is the conditionwhere all the gravitational forces are balanced and there is notendency towardsa gravitational collapse. Anotherwayof saying this is that the conditionsshowncreatearateoftimethatisthesameeverywhereasdefinedbyamidpointobserveratanygiveninstant.Itisironicthattheconditionwherethereisnorateoftimegradient midpointobserverperspective onlyhappensinimmaturegravitywheretherateoftimeiscontinuouslydecreasingrelativetoanoutsideconstantrateoftime.Observable Effects of an Increasing Г:TheslowingoftherateoftimewhenГisincreasingwouldnotbeobviouswithinthecloudwhileitishappeningbecauseitwouldnotbepossibletocomparetheproperrateoftimewithcoordinaterateoftimeusingspeedoflightcommunication.However, itwill be shown that the condition of a homogeneous increase in the backgroundgravitationalgammawouldtheoreticallyproduce1 anincreaseinthepropervolumeofthedustcloudand2 wouldproduceaslightredshiftinradiationcomingfromotherpartsofthedustcloud.Theproofforthisstatementwillbegiveninthenextchapteronceaneasiermodelforanalysishasbeenintroduced.Implications of an Increasing Гu in the Universe:Atthestartofthischapter,thestatementwasmadethatbeforewecanattempttounderstandtheapparentaccelerationintheexpansionoftheuniverse,itisfirstnecessarytounderstandwhythereisanyexpansionintheuniverse.Itisproposedthatthereasonfortheapparentexpansionoftheuniverseisthatthebackgroundgravitational gamma Гu of the universe is presently increasing. Furthermore, Гuwas alwaysincreasingoverthelifetimeoftheuniversebuttherateof increasehasbeendifferentduringdifferentepochs.Thepurposeofthepreviousthoughtexperimentistointroducetheconceptofimmaturegravityusingamodelthatiseasiertounderstandthantheentireuniverse.However,acloudofduststartingwith thegravity turnedoff isnotaperfectanalogy for theuniverse. Thedustcloudillustrates how it is possible for there to be no rate of time gradient midpoint observerperspective aslongasthebackgroundГisincreasing,buttheexampledescribesaconditionwherethereisonlyaminutechangeinГ.Scaledoesmatter.ThechangeinГuexperiencedbytheuniverseinitsfirstmomentsaftergravityis“turnedon” thestartoftheBigBang comparedtothedustcloudoverasimilartimeperiodisadifferenceofmorethanafactorof1030.Thisenormousdifferenceintroducesmanyotherdifferencesbetweentheuniverseandthecloudthatwillbedescribed.Tomyknowledge,noonehasmathematicallyanalyzedtheimplicationsoflivinginauniversewherethebackgroundГuisincreasingasafunctionoftheageoftheuniverse.TheSchwarzschild


solutiontoEinstein’s fieldequationassumedasinglesourceofstaticgravity inanotherwiseempty universe. The solution describes the effect on spacetime that surrounded this staticsource of gravity. The Friedman equation assumes a uniform distribution of matter in theuniverse. However,even theFriedmansolutionassumesamaturegravitationaldistribution.ThestandardBigBangmodelalsoimpliesamaturegravitationaldistribution.Itisproposedthatthisisanerroneousassumptionthatultimatelyleadstotheneedtoinventdarkenergy.NotetotheReader: Wewilldivertfromthedustcloudexampleandimmaturegravityforashort time to introduce additional properties of spacetime. Wewill then combine all theseconceptstodevelopamodeloftheexpandinguniverse.Energy Density of Planck Spacetime:Untilnow,thisbookhasignorednumericalfactorsnear1.Fromobservationsandcalculations,theenergydensityoftheuniverseasafunctionoftimeiswellknown.Inorderforatheoreticalmodeltobecredible,itisnecessarytobeabletomatchobservations. Therefore, it is necessary to introduce a missing numerical factor into thedefinitionofPlanckenergydensitytodeterminetheenergydensityofPlanckspacetime.ItisnecessarytocarefullydefinetheenergydensityofPlanckspacetimebecausethisisproposedtobethestartingenergydensityoftheuniverseandthisvalueaffectstheevolutionoftheuniverse.PlanckenergydensityisdefinedasUp c7/ħG2 4.6 10113J/m3.ThisdefinitiondescribestheenergydensitythatwouldoccurifPlanckenergyEpwascontainedinacubewithdimensionsofPlanck lengthon a side Up Ep/Lp3 . Thisdefinition lacksdimensionlessnumerical factors.EventhoughtheenergydensityofPlanckspacetimeiscloselyrelatedtoPlanckenergydensity,theydifferbyadimensionlessconstant.Forcosmology,amorereasonabledefinitionofenergydensitywouldbebasedonthevolumeofaspherethatisPlancklengthinradiusratherthanacubethatisPlancklengthonaside.Thedifferencebetweenthesetwovolumesis 4/3 π.Zero Point Energy: There is one other dimensionless numerical factor that must also beincluded.Itisproposedthatvacuumenergyisequivalenttozeropointenergy.Zeropointenergyisthelowestenergythataquantummechanicalsystemmayhave. Theenergyofthegroundstateofaquantumharmonicoscillatoris:E ½ħωzeropointenergyItisproposedthatPlanckspacetimewasthehighestzeropointenergydensitypossible. Thedipolewaves in spacetime can be visualized as quantum harmonic oscillators. The highestpossiblezeropointenergydensitycanbevisualizedasaquantumoscillatorthatisoscillatingatthehighestpossiblefrequencyinthesmallestpossiblevolume.Thehighestpossiblefrequency

isPlanckangularfrequency:ωp c /ħG.Thezeropointenergyofthisoscillatoris:

E ½ħωp ½ ħc /G ½Ep 9.78 108J.


Therefore,thehighestenergydensityofthespacetimefieldistohavethisenergy ½Ep inthevolumeofaspherethatisPlancklengthinradius.ThisenergydensityofPlanckspacetimewillbeidentifiedbythesymbolUps.

Ups≡½ ħ

k’Upwhere:k’≡

Ups 5.53 10112J/m3Ups≡energydensityofPlanckspacetimeThe correction factor3/8πwill alsobeused frequentlyandwill be identifiedby the symbolk’ 3/8π.AspherePlancklengthinradiuswhichhassphericalPlanckenergydensitycontainsenergyof½Ep 9.78 108J.However,wewilloftenroundthisofftoaboutabillionJoules 109J .Theconstantk’ 3/8πistheconversionfactorbetweenPlanckspacetimeenergydensityUpsandPlanckenergydensity Up Ups/k’ .Theseconceptsareexpressedusingenergydensity U ratherthanmassdensity ρ becausemassisameasurementofinertia.Wavesinspacetimearenaturallyexpressedinunitsofenergydensityratherthanunitsofmassdensitywhichimpliesinertia.Difference between Planck Spacetime and Vacuum Energy:PlanckspacetimeatthestartoftheBigBangpossessedthesameproperenergydensityasvacuumenergytoday. Therefore,whatisthedifference?Oneofthekeydifferencesishiddeninthequalificationof“properenergydensity”.PlanckspacetimehadabackgroundgammaoftheuniverseofГu 1whiletodaythevalueofГuisanextremelylargenumberthatwillbecalculatedlater.ThislargevalueofГuaffectseverythingincludingtherateoftimeandourstandardofaunitofenergy.EventhoughoneJouletodayappearstobethesameasoneJouleabillionyearsagooroneJouleattheBigBang,thesearecomparisonsofpropervaluesthatdonottakeintoaccounttheeffectofadifferentvalueofГu.OnanabsolutescaleofenergythatcompensatesforthechangeinГu,oneJouletodayisfarlessenergythanoneJoulewhenГu 1atthestartoftheBigBang explainedbelow .Anotherdifferenceisthatvacuumenergyhasnoquantizedangularmomentum nospin whileatthestartoftheBigBang100%oftheenergyinPlanckspacetimepossessedquantizedangularmomentum.Thelackofquantizedangularmomentuminvacuumenergymeansthatthisisaperfect superfluid thatdoesnot interactwithourobservableuniverseexcept throughsubtlequantummechanicalinteractions.Therearenofermions halfintegerspin orbosons integerspin in“pure”vacuumenergy.Wewouldsaythatvacuumenergyhasatemperatureofabsolutezero.Eventhoughthereisenergydensity,vacuumenergyisincapableofgivingkineticenergytemperature toparticles.Ontheotherhand,inPlanckspacetimeatthestartoftheBigBangeachofthe½Epunitsofenergy zeropointenergy possessedħofangularmomentum.Thisresultsinthehighestpossibleinteractionwithotherquantizedunitsofenergy.ThetemperatureofPlanckspacetimehadthehighestpossibletemperatureforzeropointenergywhichisequalto½Plancktemperature ½ p 7 1031°K .


Proposed Alternative Model of the Beginning of the Universe: The alternative modelproposedherestartstheuniversenotasasingularitybutasPlanckspacetimewithvolumeandfiniteenergydensity.RecallthatPlanckspacetimehasasphericalcorrectionfactorandazeropoint energy correction factor that total 3/8π 0.12 compared to Planck energy density.However, Planck energy density eliminates numerical factors by assuming a cubic volume.Therefore,Planckspacetimeisthehighestzeropointenergydensityachievable. What istheimpliedrateoftime ortheimpliedcoordinatespeedoflight requiredtoachievethehighestpossibleenergydensityonanabsolutescale?Thisquestioncanalsobestatedasfollows:WhatvalueofthegravitationalgammaisrequiredtoachievePlanckspacetime?Thehighestpossibleenergydensity on an absolute scale that takes into consideration the valueof Г can only beachievedifГ 1.IfГ 1thentherateoftimeisslowerthancoordinaterateoftimeandtheunitofenergyislessthanthecoordinateunitofenergy.Thetheoreticallargestenergydensityonanabsolutescalecanonlybeachievedatthetheoreticalfastestrateoftimewhichisalsothefastestcoordinatespeedoflight.Therefore,Planckspacetimehaswhatmight seem likeacontradiction. Thehighestpossibleenergydensitystartswitharateoftimethatwewouldexpecttofindinahypotheticalemptyuniverse noslowingoftimecausedbygravitationalfields .Ifwearetrulyatthebeginningoftime,thentherewasnopriortimeforgravitytobecomeestablished.ThereisnocontradictionforГ 1ifthiswasthestartingcondition.The source of Planck spacetime is an open question. Perhaps spacetimemerely came intoexistenceasPlanckspacetimeorperhapsthisisaphasechangeinarepeatingcycle.Ineithercase,weassumethatPlanckspacetimestartswitharateoftimecommensuratewithГu 1.ThisassumptionmeansthatatthestartoftheBigBangtherateofpropertimeequaledtherateofcoordinatetime dτu dtatthebeginning .Extending the Dust Cloud Example:Previouslywehadthethoughtexperimentinvolvingacloud of dust distributed in space. In this thought experiment we imagined eliminating allgravitationaleffectsfromthiscloudofparticles gravityturnedoff .ThismeansthatwestartedwithabackgroundgravitationalgammaofГ 1.Thenwe“turnedon”gravityeverywhere.Thestarting condition for the entire universe is proposed to be similar in some respects to thisthoughtexperiment.Planckspacetimehadanenergydensityroughly10100timeslargerthantheenergydensityofthedustcloudbutPlanckspacetimewasnotasingularity.Planckspacetimehadanextremelyuniformenergydensitybecausetheenergydensitywasdeterminedbythelimitationsofspacetimeitself.Therefore,itisproposedthattheuniversestartedwiththefiniteenergydensityofPlanckspacetime.Thisquantifiableconditionwasthestartingconditionfortheevolutionoftheuniverse.


Therewasnopreexisting gravitypresent in this volume, sowhen time started thewaves inspacetimebegantodistortbecauseofthenonlinearityofspacetime.ThebackgroundГuoftheuniversebegantorapidlyincreasefromthestartingconditionofГu 1ofPlanckspacetime.Thisissimilartogravitybeing“turnedon”inthethoughtexperiment,exceptthechangesbroughtaboutbyanincreasingГuwerevastlylargerthanoccurredinthethoughtexperiment.Ratherthananalmostimperceptibleeffectonpropervolumeinthethoughtexperiment,therewasatremendousincreaseinthepropervolumeoftheuniverseasГurapidlyincreased.Therateoftimeintheuniverseslowedandthecoordinatespeedoflightslowed.Allofthisproducedtheimmaturegravitycondition.Thisimmaturegravitylacksgravitationalaccelerationonthelargescaleofhomogeneityaspreviouslydescribedinthedustcloudexample.Comparison to the Big Bang Model:Wearenowbeginningtoresolvethedifferencebetweentwomodelsoftheuniverse.TheBigBangmodeldoesnotattempttoexplaintheprocesswhichresults incosmicexpansion.Matter isnotexpandingintoapreexistingvoid. Instead,theBigBangmodelhasavastexpansioninthevolumeoftheuniversebutnoexplanationisgivenastohow new volume is created. The proposed model of the universe that starts with Planckspacetime will be called the “spacetime transformation model”. This model attributes theexpansioninpropervolumetoachangeinthepropertiesofspacetimethatcanbequantifiedasacontinuousincreaseinГu.Mostimportant,thereisauniformincreaseinГueverywherewhichimpliesthatthereisnolargescalegravitationalgradientattemptingtocollapsetheuniverse.ThiscontrastswiththeBigBangmodeloftheuniversewhichhasgravityattemptingtoslowdowntheexpansionoftheearlyuniverse. ThefollowingquotebyP.J.E.Peeblesinthebook“PrinciplesofPhysicalCosmology”describesthecurrentreasoning.

“Newton’s ironspheretheoremsays that theNewtoniangravitationalaccelerationinside a hollow spherical mass vanishes. The relativistic generalization is thatspacetime is flat in a hole centered inside a spherically symmetric distribution ofmatter.Nowweareinapositiontofindtherelationbetweenthemassdensity,ρ,andthelocalrateofexpansionorcontractionofthematerial.Weareconsideringaspatiallyhomogeneousandisotropicmassdistribution.Supposethematterwithinthespacewithinasphereofradiusr isremovedandset tooneside. Thenspacetimeis flatwithinthesphere. Nowreplacethematter. Ifr issmallenough,wehaveplacedasmall amount of material into flat spacetime. Therefore, we can use Newtonianmechanicstodescribethegravitationalaccelerationofthematerial.”

Thisgoeson to show that the sphericalvolumewithhomogeneousdensityhasgravitationalaccelerationthatattemptstocontractthedistributedmass.Whenthisreasoningisappliedtotheentireuniverseitnaturallyresultsingravityopposingtheexpansionoftheuniverse.Eveniftheuniverseisundergoingacceleratedexpansionbecauseofdarkenergy,theBigBangmodel


hasgravityattemptingtocontracttheuniverse.Thissoundsreasonableandithasbeenacceptedbygenerationsofphysicists.However,itisproposedtobewrong.IfweassumethattheuniversehasimmaturegravityandanincreasingГu,thenthefallacyintheabovereasoningisthatremovingasphericalvolumeofhomogeneousdensitymaterialfromtheuniversewouldnot leaveavoidwithflatspacetime. Instead,theremainingvoidwouldhavenegativecurvaturewhichhasgravitationalaccelerationawayfromthecenterofthevoid.Figure13‐3showedthatpoint“X”ontheedgeofthevoidhasgravitationalaccelerationawayfrompoint“O”atthecenterofthevoid.Point“X”infigure13‐3isnotunique.Allotherpointswithinthevoidalsohavegravitationalaccelerationawayfrompoint“O”atthecenterofthehypotheticalvoid.Thisnegativecurvatureeffectcanbeunderstoodsinceeachpointinthevoidcanbethoughtofasexisting inasphericalparticlehorizoncenteredonthem. Therefore theseotherpointswithinthevoidaresimilartopoint“X”andalsohaveagravitationalaccelerationvectorpointingaway from the central point “O”. These vectors exactly offset the gravitational accelerationtowardspoint“O”causedbythetestvolume.Replacingthetestvolumewouldresultinoffsettingvectorsthateliminateanygravitationalacceleration.Theimmaturegravitydescribedherehasnotendencytowardsgravitationalcollapseoftheuniverseonthecurrentscaleofhomogeneity~300millionlightyears .Thepreviousparagraphmightbemisunderstoodifquicklyread,sohereisanotherattempt.Onthelargescaleof300millionlightyears,thereisflatspacetime.However,onthesmallerscaleof 1million light years there canbepositively curved spacetimenear a galaxyornegativelycurved spacetime in a volume of spacewithout anymatter. Supposematterwas uniformlydistributed throughout the universe so that every cubic meter of space possessed uniformenergydensityofabout10‐9J/m3.Also,theuniverseextendsperhapsinfinitely,butdefinitelyfarbeyondoutpresentparticlehorizon.Thentheuniversewouldhaveflatspacetimeevenonthesmallscale.Thematterineachsphericalvolume1meterinradiuswouldbeproducingagravitational fieldwhichwouldtendtocollapsethemattertothecenterof thesphere if thatspherical volumewas isolated like the “test volume” in figure 13‐3. However, in the actualuniversetheimmaturegravityconditionisproducinganoffsettingnegativecurvature,sothecombinationresultsinflatspacetime.Low Density Regions of the Universe:Onthescalesmallerthan300millionlightyearstheuniverseiscurrentlynothomogeneous.Therearevolumescontaininggalacticclusterswhichhave above average density and volumes of space between the galactic clusterswith belowaveragedensity.Thesevolumeswithbelowaveragedensityshouldexhibitnegativecurvatureanalogoustothevoidpreviouslydiscussed.Alowdensityregionintheuniversewouldhaveasimilarity to theconceptofanelectronic “hole” ina semiconductor electronsandholes . Avolumeofasemiconductorthatismissinganelectronistheequivalentofapositiveelectricalcharge.Similarly,alowdensityregionintheuniverse missingmasscomparedtotheaverage exhibits the repulsive properties as if themissingmass had anti‐gravity properties had an


inverserateoftimegradient .Thiseffectonlyhappensiftheuniversehastheimmaturegravitycharacteristic.Iflowdensityregionsoftheuniverseexhibitsomethinglikeanti‐gravity,thiswouldsubstantiallyspeedtherateofformationofstarsandgalaxiesintheearlyuniverse.Anymatterwithinthelowdensityregionsoftheearlyuniversewouldtendtobeexpelled.Furthermore,theanti‐gravityaccelerationwouldextendintotheedgesofsurroundinghigherdensityregions.Thiswouldtendtodrive thesehighdensityregions toevengreaterdensity. Theaveragedensityof theearlyuniversewasmuchgreaterthantoday.Thereforetherepulsivegravitationalaccelerationoflowdensity regions of the early universewould bemuch greater than today. Modeling galaxyformationintheearlyuniverseshouldtakethisproposedeffectintoconsideration.Scaling Factor Based on Planck Spacetime au:Thecomovingcoordinatesystemuses“ao”torepresentthepresentscalingfactorofthecoordinatesystemandthescalingisusuallybasedontheconventionofconsideringthecurrentscaleoftheuniverseasao 1.Thisisconvenientsinceamodeloftheuniversethatstartswithasingularitydoesnothaveanyabsolutescalingfactorfrom the start of the Big Bang. However, the concept that the universe started as PlanckspacetimemeansthatthespacetimetransformationmodeldoeshaveadefinableinitialsizeandrateoftimeatthestartoftheBigBang.Itisnaturalthatwewouldchoosethisabsolutestartingconditionasthebasisofourscalingfactor.Thereforewewillusethesymbol“au”torepresentthescalingfactorrelativetothescaleoftheuniversewhenitwasPlanckspacetimeatthestartoftheBigBang.Therefore the spacetime transformation model sets au ≡ 1 for Planck spacetime when theuniversewas1unitofPlancktimeoldandГu 1.Accordingtothisproposal,atthestartoftheBigBangPlanckspacetimehadenergydensityofaboutUps 5.53 10112J/m3and100%ofthisenergywas“observable”.AnotherwayofsayingthisisthatalltheenergyinPlanckspacetimehadquantizedangularmomentum.Todayalmostalltheenergydensityintheuniverseisintheformofvacuumenergywhichlacksanyquantizedangularmomentum.Thereforeonlyroughly1 part in 10122 today has quantized angular momentum and is “observable” to us and outinstruments.Weonlyinteractwiththevastenergydensityinvacuumenergythroughquantummechanicaleffects.Thesenumbersthenbecomethebasisforgaugingthescaleoftheuniverseatanyothertimeinthehistoryoftheuniverse.Cosmic Expansion from Гu:WhathappenstothecomovingcoordinatesystemwhenthereisanincreaseinthebackgroundgravitationalgammaoftheuniverseГu?Inchapter2wedidathought experiment involving two concentric shells. The changes that occur whenmass isintroduced inside the inner shell are: 1 the rate of proper time dτu slowed compared tocoordinatetimedtand2 thepropervolumebetweenthetwoshells increased. We live inauniversewhichisproposedtohavestartedwithabackgroundgravitationalgammaГu 1andthisvaluethenstartedtoincreaseatthebeginningoftime attheBigBang .Wealsoknowthat


the proper volume of the universe rapidly increased initially and the proper volume of theuniversecontinuestoincreasetoday theso‐calledcosmicexpansion .Assumingthattheuniverseisonlyspacetime,thereappearstobeonlyonewaythattheuniversecan rearrange itself to increase its proper volume. I am proposing that the reason for theobservedcosmicexpansion is that thebackgroundgravitationalgammaГuof theuniverse iscontinuously increasing towards Гu ∞. This is a transformation in the characteristics ofspacetime. When the background gravitational gamma of the universe Гu increases, propervolume increases, the rate of time decreases and the coordinate speed of light decreasesproportional to1/Гu2. Itwill be shown later that themodelbeingdeveloped results innewmass/energycontinuouslycomingintoviewatourparticlehorizon. Theproposal isthattherate of time continues to get slower, butwe are unaware of this change becausewe cannotdirectlycompareourrateoftimetoeithertherateoftimeyesterdayortherateoftimeinahypothetical static universe that is not undergoing any change. This obviously is a differentmodelthantheΛ‐CDMmodelcurrentlyconsideredtobethecosmologicalstandard.The proposal is that the expansion of the proper volume of the universe is caused by the background gravitational gamma of the universe Гu increasing.We can see if this proposal is compatiblewith theRobertson‐Walkermetric hereafterR‐Wmetric . The highly successful R‐W metric has been called the standard model of moderncosmology. It assumes an isotropic and homogeneous universe and a spherical coordinatesystemthatexpandswiththe“Hubbleflow”sothatthecoordinatesystemremainsintheCMBrestframeeverywhere.ThestandardwayofwritingtheR‐Wmetricis:

dS2 c2dt2–a2 t –

r2 dθ2 sin2θdϕ2

wherea t isthescalefactoroftheuniverseandrisphysicaldistance.Theconventionistousethecurrentscalefactoroftheuniverseasao 1.Thereforea t isusuallyscaledintothefutureorpastfromthecurrentscalingfactorand“r”issetequaltothecurrentdistancebetweentwopointswhichcanbeconsideredaunitoflength.However,itisalsopossibletochoosesomeotherstandardfor“a”and“r”ifthereisacompellingreason.Inourcase,wehaveaverycompellingreasontochooseadifferentstandardfor“a”and“r”becauseunliketheΛ‐CDMmodel,wehaveanexactscalefortheuniverseatthebeginningofthebeginningoftime atthestartoftheBigBang .Thereforethisscalefactorwhichchangeswithtimebutwasau 1whentheuniversewasPlanckspacetimewillbedesignatedau t .AlsothephysicallengthbetweentwopointswilluseasitsstandardthelengthwhentheuniversewasPlanckspacetime Гu 1 . Thislengthstandard coordinatelength willbedesignated . Therefore,rewritingtheR‐Wmetricwiththesechangeswehave:


dS2 c2dt2–au2 t –

2dΩ2 R‐WmetricwheredΩ2 dθ2 sin2θdϕ2

The spatial curvature term is k and in the general case “k” can take values of 1, 0 or – 1.However, analysis of the CMB using experimental data from WMAP has established thatspacetimeonthelargescaleisflattobetterthan0.4%experimentalaccuracy.Therefore,wearegoingtoassumeaflatuniverseandsetk 0. Wecandeterminethespatialmetric fora flatuniversebysettingdt 0andfollowaradialraybysettingΩ 0.

dS2 c2dt2–au2 t –

2dΩ2 setk 0,dt 0,Ω 0anddS cdτ dL

dL2 au2 t d 2

au t

Theproposedphysicalinterpretationofthisisthatthereasonthatthescalefactorincreasesisbecauseproperlengthcontractsrelativetocoordinatelength whentheuniversehadГu 1orcomparedtoaunitof lengthinahypotheticalemptyuniverse. IfaunitofproperlengthLissmallerthanaunitofcoordinatelength ,thenprogressingintheradialdirectiontherateofchangeinproperlengthdLwillbegreaterthantherateofchangeind .Thisratioisequaltothescalefactorau t .Sincethespacetimefieldishomogeneousandisotropiconthelargescale,this applies to all spatial directions since the choice of a “radial” direction is arbitrary. TherelationshiptothegravitationalbackgroundoftheuniverseГuisproposedtobe:

SeventhStartingAssumption:Гu t au t

Itisnotpossibletoconclusivelyprovethisequation.Therefore,thisequationisanassumption.However,itispossibletosupportitsvaliditybyshowingthatitiscompatiblewithobservationsandgives a compellingmodelof theuniverse. Thepropervolumeof theuniverse is indeedexpanding. This equation is a key step in explaining why the universe is undergoing whatappearstobeacosmicexpansion.TheBigBangmodeldoesnotgofarenoughtoexplaintheunderlyingphysicsbehindthepropervolumeexpansionoftheuniverse.ThisequationallowsustoestimatethevalueofГuoftheuniversetodaybecauseweknowtheinitialscaleofPlanckspacetimeandwecanestimatetheexpansionthathasoccurredsincetheuniversehadtheenergydensityofPlanckspacetime.WewillproceedbyassumingГu autoobtainotherinsights.Iftheanswersobtainedbytheuseofthisequationarereasonable,thiscanalsoserveasacheckontheequation.TheappealoftheconceptthattheuniverseisundergoinganincreaseinГuisthat:1 itexplainstheapparentcosmicexpansionoftheuniverse2 itgivesthecorrectredshifttolightfromdistant


galaxies provenlater ,3 itgivesauniversewithauniformrateoftimeonthecomovinggridand4 iteliminatesthemysteryofexpandingspacesupposedlycarryinggalaxiesawayfromusatfasterthanthespeedoflight5 itisoneofseveralstepsthateliminatetheneedfordarkenergy6 itsolvesthemysteryofhowtheenergydensityofvacuumfluctuations zeropointenergy canremainconstantwhenthepropervolumeoftheuniverseexpands.Coordinate Rate of Time dt:Asexplainedpreviously,thedefinitionofthecoordinaterateoftime designateddt usedbythespacetimetransformationmodelistherateoftimewhenГu 1.ThisonlyoccurredinouruniverseduringthefirstPlanckunitoftimeintheageoftheuniverse.Therefore,thecoordinateclockcanbeimaginedasaclockthatcontinuestorunattherateoftimethatwaspresentatthebeginningoftime.Thisisalsotherateoftimethatwouldoccurinahypothetical empty universe where always Гu 1. After the first unit of Planck time, thebackgroundГuof theuniversebegan to increase Гu 1 and therateofproper time in theuniversedτudecreasedrelativetothecoordinaterateoftime.Thegravitationalgammaoftheuniversecanalsobedefinedbythefollowingtemporalrelationship:Гu dt/dτuCoordinate and Hybrid Speed of Light:Thesymbol“dt”intheR‐Wmetricrepresentstherateoftimeinthecomovingcoordinatesystem.ThisisactuallytheproperrateoftimeintheCMBrestframeoftheuniverse.Weneedtosubstituteadifferentsymbolbecausewewanttouse“dt”asthecoordinaterateoftimewhenГu 1.TherateoftimeusedintheR‐Wmetricisactuallytheproperrateoftimeintheuniverse therateoftimeonthecosmicclock .ThereforeintheR‐Wmetricwewillmakethefollowingsubstitutiondt→dτu.Alsomakingthesubstitutionofauand previouslydiscussed,wehave:

dS2 c2dτu2–au2 t –

2dΩ2modifiedR‐Wmetric

Combiningthesesubstitutionswithsubstitutionspreviouslyexplainedwehave:Set:dS 0,dΩ 0,k 0,au t Гuc2dτu2 au2 t d 2

c au t setau t Гu t and ≡C

C Г

hybridspeedoflight coordinatelengthandproperrateoftime

Next we will determine the coordinate speed of light d /dt which references thecoordinaterateoftimedtandtherateofchangeofcoordinatelengthd bothinahypotheticalemptyuniversewhereГu 1.ThedefinitionofdtusedfortheremainderofthisbookistherateoftimeinahypotheticalГu 1universe.Therefore:dt≡Гu t dτu.


c Гu t setdτu Г

c Гu2 t set

Г

coordinatespeedoflight

Therearetimeswhenitisnecessarytousethecoordinatespeedoflight butusuallythehybridspeedoflightCisadequateandsimpler.Whenpossible,itismoreconvenientandintuitivetousetheproper comoving rateoftime.Ifwearedeterminingthetimerequiredforlighttotraveladistancebetween two stationarypoints,weusuallywant theanswerexpressed inunitsofpropertimethatwouldberecordedonthecosmicclock. ThedifferencebetweenthehybridspeedoflightC d /dτuandtheproperspeedoflightc dL/dτuisthedifferencebetweenaunitofcoordinatelength andaunitofproperlengthL.Thereasonthatittakesmoretimeforlighttotravelbetweentwodistantgalaxiestodaythanitdidabillionyearsagoisthatthehybridspeed of light is less today than itwas a billion years ago. The distance between these twogalaxiesisconstantwhenmeasuredinunitsofcoordinatelength .Aslowinghybridspeedoflightimpliesthataunitofproperlength, suchas1metertoday iscontractingrelativetoaunitofcoordinatelength suchas1meterwhenГu 1 .Reconciliation With Proper Length Being Constant:Clearlythephysicalinterpretationthatis emerging is that aunit of proper length contractswhenГu increases. In chapter3 itwasproposed that itwasanacceptablebasisofacoordinatesystemtoconsideraunitofproperlength,suchasameter,tobethesameeverywhereintheuniverse.AmeterinalocationwithalargegravitationalГwasconsideredtobethesameasameterinalocationfarfromanysourceofgravitywhichwewerepreviouslycalling“zerogravity”.Howisitpossibletoreconciletheviewthatproper length iscontractingasГu increaseswith theviewinchapter3 thatproperlengthisconstanteverywhereinthecurrentuniverse?Theansweristhatthe“normalized”coordinatesystemofchapter3usedtheequationLo Lg.ThisdoesnotimplythatLoandLgareconstantovertime.Itmerelysaysthatthiscoordinatesystemassumesthatinastationaryframeofreference,aunitoflengthinazerogravitylocationLo is the same as a unit of length in a locationwith gravity Lg at a given instant in timemidpointobserverperspective .BothoftheselengthscanbesimultaneouslycontractingastheuniverseagesandyetmaintainLo Lg.Morewillbesaidaboutthisinchapter14.Planck Sphere:Wewillnowdefinetermsnecessarytoquantifyandanalyzetheevolutionoftheuniverse.PlanckspacetimehasenergydensityequivalenttoaspherethatisPlancklengthinradius containing themaximumzeropointenergypermittedby thepropertiesof spacetime.Thismaximumenergyisequalto½Planckenergy ½Ep 9.78 108J .Mostimportant,Planck


spacetime has Гu 1 and all the energy possesses quantized angular momentum. Theimplicationsoftheseconditionswillbeexplainedlater.WewillcallthebasicquantizedunitofPlanckspacetimea“Plancksphere”. BesideshavingaradiusofPlancklengthitalsohasquantizedangularmomentumofħ perhaps½ħ .Thechoicebetweenthesetwowillhavetobeworkedoutbyothers,butħismorereasonableandwewilluseħinthefollowingdiscussion.TheconceptofaPlancksphereisintroducedprimarilybecausethisisaconvenientreferencetohelpusvisualizetheapparentexpansionoftheuniverseinunitsofproperlength.Wecandescribewhathappenstotheradius,energyandangularmomentumthatareinitiallyinthePlancksphere.InthiswaytheevolutionofthePlanckspherebecomesaneasyreferenceforwhathappenstoallthespacetimeintheuniverse.Angular Momentum in Planck Spacetime:Besideshavingenergydensity,Planckspacetimealsomusthaveadistributeddensityofquantizedangularmomentum. OurcurrentuniversepossessesquantizedangularmomentumprimarilyintheformofCMBphotons.CurrentlytheCMBhasthephotondensityofa2.725°Kblackbodycavitywhichisabout4 108photons/m3.Theangularmomentum ħ possessedbythisvastnumberofphotonsexceedsbyafactorofabout 108 the internal angular momentum possessed by baryonic matter in the universeignoringdarkmatter . Ifwe simply imagine reversing time, all theCMBphotonswouldbecompressedandblueshifteduntileachphotonwasreducedtoaspherethatisPlancklengthinradius.Thisideacanbecheckedwithaplausibilitycalculationthatwillbemadelater.Starting the Universe from Planck Spacetime:Theremainderofthischapterwilllookattheevolutionoftheuniverse.ThiscombinestheconceptsofanincreasingГuandPlanckspacetimewiththeBigBangmodelofexpandingspacetime.ThiscombinationallowsustointroducenewideasinthefamiliarcontextoftheBigBangmodel.InthenextchapterwewilldroptheBigBangmodel and switch completely to the proposed spacetime transformationmodel that ismorecompatiblewiththepremisethattheuniverseis andalwayswas onlyspacetime.WearesettingthestageforthestartoftheBigBang.InthisthoughtexperimentwearegoingtowitnesstheBigBangfromthevantagepointofaninfinitelysmallpointthatisatthecenterofaPlancksphereatthestartoftheBigBang.FromthisvantagepointwearegoingtopayparticularattentiontowhathappenstotheapproximatelyonebillionJoules ½Ep thatistheenergyofthePlanckspherethatsurroundsourinfinitelysmallvantagepoint.AllofthisenergyisinitiallywithinadistanceofPlancklengthofourlocation.Wewillimaginethatwecanwatchtheevolutiontheuniverse watchtheBigBang fromourinfinitelysmallvantagepointwithinPlanckspacetime. WearealsogoingtoequipourselveswithtwoclocksthatwillstartrunningthemomentthattimestartstoprogressforwardatthestartoftheBigBang.BothoftheseclocksaredigitalclocksthatruninquantizedincrementsofPlancktime ~5.4 10‐44secondsteps .Oneoftheseclockswillbecalledthe“cosmicclock”


thatmeasurestheproperageoftheuniverseinthecomovingframeofreference.Timeonthecosmicclock,expressedindimensionlessPlanckunitsoftime,willbedesignatedwiththesymbolτu boldandunderlineddesignatesPlanckunits anddefinedas:τu≡τu/tp.Theotherclockwillbecalledthe“Г 1clock”.ThisisahypotheticalclockthatrunsattherateoftimeofanemptyuniversewherethebackgroundgravitationalГuisalwaysequalto1.TimeontheГ 1clockalsoisinPlanckunitsoftimeanddesignatedastu.SincetheuniversestartedwithГu 1,theГ 1clockcanalsobethoughtofasaclockthatcontinuedrunningattherateoftimethatwaspresentatthebeginningoftheBigBang.Thisclockiskeepingcoordinatetimewhichisthe“dt”termintheequation:Гu dt/dτu.The Big Bang Starts:Nowforthebigevent:TimestartsandtheBigBanghasbegun.Actually,theobservable,properenergydensitydefiningPlanckspacetimeonlyexistsinthisformduringthefirstunitofPlancktime.Afterthisbrieftimetheobservableenergydensity energywithspin rapidly drops to less than Ups 5.53 10112 J/m3 and Гu rapidly increases Гu 1 .However,thetotalenergydensityoftheuniverseremainsconstantwhenweaddtogetherthevacuumenergydensity nospin plustheobservableenergydensity withspin asexplainedlater.Itisnotpossibletotalkaboutwhathappens“before”timestartstoprogress whentu 0 because this implies that the rate of time has stopped and there is no motion of waves inspacetime.Wewilladoptthepositionthattherewasnotime“before”tu 1.Inorderforthespacetimefieldtobeenergetic,theremustbemovingdipolewavesinspacetime theremustbedynamicspacetime .Thereisnoenergyinstaticspacetime.IfthespinineachPlancksphereisħ,thenthisistheequivalentofonephotonpersphere.InitiallytheenergyofthissinglephotonperPlanckspherewouldbeequaltothemaximumzeropointenergywhichis½Ep.AnotherwayofsayingthisisthatalltheenergyofPlanckspacetimehadquantizedangularmomentumandwas“observable”energy.Todayonlyabout1partin10122ofthetotalenergyintheuniversehasquantizedangularmomentumandis“observable”.WearegoingtoassumethattimestartssimultaneouslyeverywherewithinthePlanckspacetime.Thisobviouslyrequireseitherfasterthanspeedoflightcommunicationorsomeotherwayofsynchronizingthestartoftime.Theonlyjustificationforthisassumptionisthatobservationstoday indicateasimultaneous initiationof theBigBang. Perhaps thiscanberationalizedbyassuming that in this highest possible energy density, the entire universe was like a singleparticlethatexhibitedtheinstantaneouscommunicationofentanglementorunity.Wewillnotdwell on this point. This assumption seems more understandable than starting with asingularity,butultimatelythesimultaneousinitiationoftheBigBang simultaneousstartingoftime isjustastartingassumption.


AttheinstantthattheBigBangstarts,ourimaginedvantagepointiscompletelyisolated.TherehasbeennotimeforanycommunicationwithanyofthesurroundingenergyinPlanckspacetimeand therehasbeennononlineardistortionof thewaves in spacetime that constitutePlanckspacetime. Thismeans thatat thestartof theBigBang there isnopreexistinggravitationaleffects–nopreexistinggravitationalaccelerationandforthe firstunitofPlancktime,Гu 1everywhereintheuniverse.Therefore,forthefirstunitofPlancktime,thecosmicclockandtheГ 1clockbothrunatthesamerateoftime dτu dt .Thisstartingconditionissimilartothedustcloudthoughtexperimentwherewestartedwithgravity“turnedoff”.Increasing Value of Гu:ThefirsttickofPlancktimewaseasybecauseithappenswithГu 1.Our“particlehorizon”wasequaltoPlancklengthwhichistheboundaryofthePlancksphere.Thequestionis:WhathappenswithsubsequentticksasourparticlehorizonexpandsbeyondtheboundaryofourPlancksphere?SpeedoflightcontactisbeingmadewithadjacentwavesinspacetimejustbeyondtheboundaryofourPlancksphere.ThenonlinearityofspacetimebeginstoaffectthesewavescausinganincreaseinthebackgroundgravitationalgammafromГu 1toГu 1.Therateoftimeslowsandpropervolumeincreases,butthereisnotimegradient.Allpointsareexperiencingthesamerateoftimesothereisnotendencyforgravitationalattractionorgravitationalcollapse.Thisisthestartoftheimmaturegravityconditionthatcontinuestoday.AuniformlyincreasingГueliminatesthetimegradientnecessaryforgravitationalattraction.For the first unit of Planck time the cosmic clock and the Г 1 clock were synchronized.However,all subsequent ticksexhibit the fact that thecosmicclockhasarateof time that isslowingrelativetotheГ 1clock.AlsotheproperdistancetotheboundaryofthePlancksphereincreases.BothoftheseeffectsaretheresultofГuincreasingwhichinturnistheresultofthenonlineareffectsofspacetimeaccumulating.ThewavesinspacetimeresponsiblefortheenergydensityofPlanckspacetimehavebeguntodistort.Unlikethegravityproducedbyanisolatedmass,insidetheearlyuniversethestressedspacetimeisahomogeneousdistributionofenergythatextendsbeyondourparticlehorizon.ThiswavedistortionwouldproduceanincreaseinthebackgroundgravitationalgammaГusimilartotheincreaseinthebackgroundГinthedustcloudthoughtexperiment.However,becausetheuniversestartedwiththemaximumpossiblequantizedenergydensity,theeffectsonvolumeandtherateoftimearevastlylargerthaninthedustcloudthoughtexperiment.Spacetime Evolves into Different Types of Energy: Thedistortion freeoscillating spacetimePlanckspacetime thatwaspresentduringthefirsttickofthecosmicclockbeginstoexhibitdistortionforallsubsequentticks.Thenonlinearityofspacetimecausesthewavesinspacetimetodistort. Thedistortedwavescanbe throughofasbeingsplit into threecomponentparts.Thesethreecomponentsare:


1 Thedistortionfreecomponentoftheoriginalwavesinspacetimethatretaintheoriginalquantized angular momentum spin . This is proposed to become the source ofeverythingintheuniversethatwecansensetoday allfermionsandbosons .

2 Thenon‐oscillatingcomponent theAβ2componentofchapter6 thatresultswhenthewavesinspacetimearedistortedbythenonlinearityofspacetime.ThisisresponsibleforgravitationaleffectsandalsoresponsibleforthebackgroundГuoftheuniverse.

3 Thehighfrequencyoscillatingcomponentofgravity theAβcos2ωtcomponentdescribedin chapter 6 proposed to be responsible for vacuum energy. This component lacksangularmomentumandbecomes thevacuum fluctuations componentof theuniversetoday.

Therefore startingwith Planck spacetimemeans that any nonlinear distortion caused by aninteractionofwavesinspacetime gravitationalcontactwithsurroundingenergy mustincreaseГu.However,asГuincreasesthisalsoincreasesproperdistancebetweenstationarypointsanddecreasestherateoftime.Newenergyisalwaysaddingitsgravitationalinfluenceastheparticlehorizonexpandsandnewenergycomesintoview.AsГuincreases,theproperdistancetotheparticlehorizonincreasesevenfasterthanthespeedoflightbecauseourunitoflengthisalsoshrinkingrelativetocoordinatelength .ThevalueofГuincreasestowardsinfinity.Avoiding Gravitational Collapse: What prevents Planck spacetime from collapsing into asingularity? In fact,whenτu 1eachPlancksphereappears tocontain theexactamountofenergyrequiredtobecomeablackholewithaSchwarzschildradiusequaltoPlancklength.Eventodaytheproperenergydensityofspacetimeisapproximatelythesameastheenergydensityof Planck spacetime. According to general relativity, energy in any form produces gravity.Therefore, general relativity considers the energydensity of the spacetime field requiredbyquantummechanics tobea ridiculously largenumber ~10120 times larger than the criticalenergydensityoftheuniverse .Whatpreventstheenergydensityofthespacetimefieldfromcollapsingintoablackhole?Apartialanswerwillbegivenhereandadditionaldetailswillbegiveninthenextchapter.Thinkoftheenergydensityofthespacetimefieldasbeingthenecessaryingredienttogivethespacetimefieldpropertiessuchasaspeedoflight,impedance,agravitationalconstant,elasticity,etc.Thereforeitisanoversimplificationtosaythatenergyinanyformcreatesgravity.Themorecorrect statement would be that any form of energy that possesses quantized angularmomentum creates gravity. In otherwords, fermions and bosons create gravity. The dipolewavesinspacetimethatformvacuumenergylackangularmomentumandarepartofthefabricofspacetime.Theyhavesuperfluidpropertiesandareashomogeneousasquantummechanicsallows.Theyarealsothenecessaryingredientrequiredtosupporttheexistenceoffermionsandbosons.


When rotars with their quantized angular momentum are introduced into an otherwisehomogeneousvolumeofthespacetimefield,thisisaddinganadditionalenergydensitytothepreviouslyhomogeneousenergydensityofthespacetimefield.Forexample,iftherewasarotarwithPlanckenergy,thenthisrotarwouldhaveaquantumradiusequaltoPlancklength.Theenergydensityofsucharotarwouldbeequaltotheenergydensityofanequivalentvolumeofthespacetimefield.Therefore,introducingsucharotarintoapreviouslyhomogeneousvolumeofthespacetimefieldwoulddoubletheenergydensityofthatvolumeofthespacetimefield.Thisdoublingreachesthetheoreticallimitofthepropertiesofthespacetimefieldandresultsinthecreationofablackhole.Therearenofundamentalparticles rotars withPlanckmass/energy. Theconditionswhichcreateastellarsizeblackholeorasupermassiveblackholedonotmatchthetotalenergydensityof thespacetime field ~10113 J/m3 . However, theydomatchthe interactiveenergydensityUipreviouslydiscussedinchapter4.Recallfromchapter4thefollowing:

TheinteractiveenergydensityUi Fp/λ2isaverylargeenergydensity. Howdoesthisenergydensitycomparetotheenergydensityofablackhole symbolUbh ? We will designate the black hole’s energy as Ebh and its definedSchwarzschildradiusasRs≡Gm/c2.Ignoringnumericalfactorsnear1wehave:

Ubh

Therefore,sinceUi Fp/λ2andUbh Fp/Rs2itcanbeseenthatifλ RsthenUi Ubh.Addingadditionalenergytoaspecificvolumebyintroducingmatterincreasesthetotalenergydensityandproducesthegravitationaleffectsdescribedbygeneralrelativity.Thelargestforcethatthevacuumenergy/pressurecanexertisPlanckforce Fp c4/G .Withtherotarmodelitwasstatedthattheenergydensityofarotarimpliedaspecificpressure.Thispressureneededtobecontainedbyanoffsettingpressureexertedbyvacuumenergy/pressure. Likewise, thematterofalargerbodysuchasastarhasacollectiveenergydensitythatmustbestabilizedbyoffsettingpressurefromvacuumenergy/pressure.Thepressureonastarcanbeconsideredastwoopposingforcesexertedonoppositehemispheresofthestar.ThemaximumforcethatthespacetimefieldcanexertisPlanckforce.Thereforetheconditionswhichformablackholeoccurwhenthespacetimefieldisaskedtoexertthismaximumforce.Theenergydensityofablackhole ignoringnumericalfactorsnear1 isUbh Fp/Rs2.Thepointisthattheenergydensityofspacetimedoesnotformablackhole.Instead,ablackholeformswhenadditionalenergydensitymatter inintroducedtoavolumeandequalstheinteractiveenergydensitywhereλ RsintheequationUi Fp/λ2.


WhentheuniversestartedatthebeginningoftheBigBang τu 1 ,100%oftheenergyofPlanckspacetimepossessedquantizedangularmomentum.TherewasnovacuumenergyduringthefirstunitofPlancktime.Therefore,thetotalenergydensityofaPlanckspheredidnotformablackholeandtheparticlehorizonwasequaltoPlancklength.ThepossibilityofformingblackholesonlyoccurredlaterwhensomeoftheenergyofPlanckspacetimewasconvertedtovacuumenergy.Thenitbecamepossibletooverloadaparticularvolumeofthespacetimefieldwiththecombinationofvacuumenergyandenergywithquantizedangularmomentum multiplerotarsand/orphotons .Inchapter14wewillseehowitispossiblefortheproperenergydensityofthespacetimefieldtoremainconstantwhilethepropervolumeoftheuniverseexpands.Three Epochs in the Evolution of the Universe:Therateofexpansionoftheuniverse rateofchange of Гu is currently believed to have gone through 3 different phases. The followingcalculationsusethecurrentestimatesfortheratesofexpansionoftheuniverseandthecurrentestimates of age of the universe when transitions occur. As improved estimates becomeavailable,theyshouldbesubstitutedforthefollowingpreliminaryestimates.ThefirstphaseistheradiationdominatedepochandiscurrentlybelievedtohavestartedwiththeBigBangandendedroughly70,000yearsaftertheBigBang.Duringthisepoch,thescalingfactoroftheuniversegrewas:au t τu

1/2.Nextcamethematterdominatedepochfromroughly70,000yearsaftertheBigBangtoabout5billionyearsaftertheBigBang.Thematterdominatedepochisbelievedtohaveascalingfactorthatgrewwithaslopecharacteristicofτu

2/3.Howeveritisnotaccuratetosaythatau t τu

2/3becauseduringtheradiationdominatedepochtheslopewasdifferent. Therefore,wewill say thatduring thematterdominatedphase the change inscalingfactorisΔau t Δτu

2/3.Fromabout5billionyearsuntilthepresentweareinwhatisreferredtoasthe“darkenergy”dominatedepoch.Duringthisphaseafirstapproximationofthechangescalingfactorcanbedescribedas:Δau t Δτu. However,thechangeinscalingfactorappears to be more complex than this because the change in the scale factor is probablyaccelerating.TheΛ‐CDMmodelofcosmologyhasparametersthatcanbeadjustedtopermitittomatchobservations.Fortunately,thekeypointsoftheproposedspacetimetransformationmodelcanbestatedwithoutrequiringapreciseequationforthescalefactorinthe“darkenergy”dominatedepoch.Radiation Dominated Epoch: ThestandardBigBangtheory withoutinflation hasthescalefactor of the universe increase proportional to the square root of time during the radiationdominatedepochau t τu

1/2.Eventhemodeloftheuniversethatincludesinflationstartsoffwiththescalefactorincreasingproportionaltoτu

1/2betweenthestartoftheBigBangandabout10‐37secondafterthestart.Theinflationmodelthenhasabriefperiodwhereavastexponentialexpansiontakesplace.ThisisthenfollowedbyareturntoexpansionproportionaltoΔτu

1/2.The spacetime wave model proposed here does not need the hypothetical inflationaryexponentialexpansiontomaketheuniversehomogeneous,isotropicandspatiallyflat.Allofthis


automatically follows from starting the universe at the highest possible energy density thatspacetimecansupport‐Planckspacetime.Thisenergydensityisashomogeneousasquantummechanics allows. These quantum mechanical fluctuations are traceable to the Plancklength/time limitation of wave amplitude in the model of the universe based on waves inspacetime.Thesequantumfluctuationsprovidethesmallamountofinhomogeneityrequiredtoseedtheeventualgravitationalformationofstarsandgalaxies.Thesymbolsandequationsthatwewillbeusingtocharacterizetheradiationdominatedepochare:

au t Гu τu1/2 τ /t radiationdominatedepochrelationships

The Universe at τu = 9:Wewillillustratesomeimportantconceptsabouttheearlystagesoftheexpansionoftheuniversewithanexample.PreviouslyanimaginaryvantagepointatthecenterofaPlanckspherewasdescribed.Supposethatwereturntothatinfinitelysmallvantagepointandlookatthechangesthatoccur9unitsofPlancktime τu 9 afterthestartoftheBigBang.Fromτu

1/2 Гuweobtainthatatτu 9thebackgroundgravitationalgammaoftheuniverseis:Гu 3. Thishasaprofoundeffectonspacetime. Thescale factorof theuniverserelative toPlanck spacetime au has tripled.Therefore, thePlanck sphere that startedwitha radiusofPlancklengthlpnowhasaproperradius3timesaslarge r 3lp . However,thecoordinateradiusalwaysassumesГu 1,thereforethecoordinateradiusofthePlancksphere measuredinunitsof remainsconstantat1unitofPlancklength 1 TheHubblespherewithradiusrharoundtheoriginpoint,hasaproperradiusofrh cτusowhenτu 9thenrh 9lp.TheHubblespheredefinesthedistancewhereobjectsseemtoberecedingattheproperspeedoflight.Theproperdistancetotheparticlehorizon designatedrph ataninstantintimeislargerthantheHubblesphereatthatsameinstantintime.Thisisbecausetheinstantaneousproperdistancetotheparticlehorizonincludesdistancethathasexpandedafteraspeedof lightsignalhaspassedanypoint. DuringtheradiationdominatedepochitcanbeshownthattheproperdistancetotheparticlehorizonisalwaystwicetheradiusoftheHubblesphere.Atageoftheuniverseτu,theradiusoftheparticlehorizonisrph 2cτu 18lpwhenτu 9orτu 4.85 10‐43 s. Today thevalueofГu ismuch larger than3 calculated later .Therefore the proper dimensions quotedwhen τu 9 and Гu 3would be different todaybecauseГuismuchlarger.However,thecoordinatedimensions measuredinunitsof alwaysareconstantbecausetheydonotscalewithГu.Any signalobtained from theparticlehorizonhas infinite redshift but a signal emitted frominsidetheparticlehorizonhaslessofaredshift.Thegravityofanapparentlyrecedingmassislessthanthegravityofastationarymassatthesamedistance.Therefore,theredshiftatvariousdistancesalsoaffectsgravity. It isthecombinationofthegravityofalltheenergyinsidetheparticlehorizon that together is increasing thebackgroundgammaatourobservationpoint.Sinceallotherobservationpointsareexperiencingasimilargravitationaleffect,thebackground


gravitational gammaof the universe is increasing homogeneously because Planck spacetimestartedoffatthemaximumhomogeneitypermittedbyquantummechanics.Wewillnowreturntoourexampleoftheuniversewhenitwas9unitsofPlancktimeold τu 9 .Atthisinstantwehad:Гu τu

1/2 dt/dτu 3.TherateofpropertimeshownonthecosmicclockhasslowedtoathirdtherateoftimeshownontheГ 1clock.AlsothehybridspeedoflightC d /dτuhasslowedtoathirdtheproperspeedoflightexpressedbytheuniversalconstantc dL/dτubecauseofthedifferencebetweend anddL.Therefore:

C Г c/τu

1/2 assumesradiationdominatedepoch

ThePlancksphereinitiallycontainedabout109JoulesduringthefirstunitofPlancktimewhentheuniversewasPlanckspacetime. Allthisenergyinitiallypossessedħofquantizedangularmomentum.Thereforetheseinitialwaveswerephotonsinmaximumconfinement.However,energeticphotonscanbeconvertedtoparticles.Forexample,ithasbeenexperimentallyproventhattwo511,000eVphotonscanbeconvertedtoanelectron/positronpair.Whenτu 9andГu 3thephoton‐likewaveshavebeenredshiftedbyafactorof3.Therefore,thepropervolumeof thePlanck spherehas increasedbya factorof27and the “observableenergy” possessingquantizedangularmomentum inthePlanckspherehasdecreasedbecauseofredshiftbyafactorof3toabout0.33billionJoules.Wheredidthedifferenceinenergygo?Lost Energy Becomes Vacuum Energy:Whenphysicsstudentshearaboutthecosmicredshiftdecreasingtheenergyofphotons,theyoftenaskwheretheenergywent.Answersofteninvolveadiscussionofstretchedwavelengths,andeventuallyimplythatconservationofenergydoesnot apply to the cosmic expansion of the universe. The proposed spacetime transformationmodelgivesadifferentanswer.Whenthepropervolumeoftheuniverseexpands,aphotonlosesenergybutretains100%ofitsproperangularmomentum.Thereforethelostenergypossessesnoangularmomentum.Itisproposedthatalltheobservableenergylostbyphotonsisbeingconvertedtovacuumenergyzeropointenergy thatisinasuperfluidstatethatcannotpossessangularmomentum.Infact,itisnotonlyphotonsthatarelosingenergytocosmicexpansion.Neutrinosandotherrelativisticparticles movingrelativetothelocalCMBrestframe arealsolosingsubstantialamountsofobservablekineticenergy.Thislostenergyisalsobeingconvertedtovacuumenergy.Inthenextchapterwewillattempttocalculatetheratioofvacuumenergydensitytoobservableenergydensityintheuniversetoday.Radiation-Matter Equality Transition: The radiation dominated epoch ended when theenergydensityinradiationfelltoeventuallyequaltheenergydensityofmatterintheuniverse.The proper energy ofmatter such as an electron does not decreasewhen Гu increases so


eventuallyradiationenergydensity falls toequal theaverageenergydensityofmatter. Thistransitioniscalledtheradiation/matterequalityandoccurredabout70,000yearsaftertheBigBang.BeforetheWMAPprobeaccuratelymeasuredtheCMB,thistransitionwasthoughttohaveoccurredearlier,perhaps10,000yearsaftertheBigBang.However,analysisoftheWMAPdatahasdeterminedthattheenergycontentoftheuniverseat380,000yearsaftertheBigBangwas15%radiation,10%neutrinos,12%baryonicmatterand63%darkmatter.Sincebothradiationandneutrinoenergydecreaseinenergyatvirtuallythesamerate assumingnonewsources ,the breakdown at 380,000 years can be generalized as 25% radiation‐like energy and 75%energyinmatter. Thesenumbersimplytheradiation/matterequalityoccurredabout70,000yearsaftertheBigBang τu 4.09 1055Planckunitsoftime .Itispossibletocalculatethevalueofthebackgroundgravitationalgammaattheradiation/matterequalitytransition.Thisbackgroundgammaisimportantforlatercalculationsandwillbedesignatedas:Гeq.Гu τu

1/2setτu 4.09 1055 70,000years Гeq 6.4 1027Гeq Гuattheradiation/matterequalitytransitionTherefore,about70,000yearsaftertheBigBangtheabsolutescalefactoroftheuniversewas:au Гeq 6.4 1027.ThismeansthatthePlancksphereincreasedinradiusfromPlancklength~10‐35m toabout10‐7m obtainedfromlp 6.4 1027 10‐7m .Furthermore,theredshiftthatoccurredduringtheradiationdominatedepochdecreasedtheproperenergyofradiationbyafactorequaltoГeq.Thismeansthatthe“observable”energyinthePlanckspheredecreasedby a factor equal to Гeq from about a billion Joules to about 1.53 10‐19 J. All themissingobservableenergy i.e.energypossessingangularmomentum wasconvertedtovacuumenergy.ThereforethetotalenergyinthePlancksphere observableenergyplusvacuumenergy,bothmeasuredinunitsofcoordinateenergy remainedconstantatabout9.78 108Joulesattheendoftheradiationdominatedepoch discussedlater .Thiscanbeconfusingsincetherearetwodifferentstandards forenergy thesameway that thereare twodifferentstandardsof lengthproper and coordinate . The conclusion is that the energy density of the universe has notchangedfromtheBigBangtotodayifvacuumenergyisincludedintheenergydensity.Whathaschangedisthatthe“observable”energydensityoftheuniverse excludesvacuumenergy hasdecreasedbyafactorofabout10122.After theradiationdominatedepochended, thenextphasewas thematterdominatedepochwhichlastedfromabout70,000yearstoabout5billionyearsaftertheBigBang.ThescalefactorandГu duringthisepochscalesproportionaltoΔτu

2/3.Thecurrentepochisusuallyreferredtoasthedarkenergydominatedepochandextendsfromabout5billionyearstothepresent.ThescalefactorandГuduringthisepochmightbelinearlyincreasingproportionaltoΔτuoritmightbeincreasingfasterthanlinearlyinwhichcaseitwouldbeanexponentialincrease.Thisisthefamousacceleratingexpansionoftheuniverseandtheexactrepresentationofthisphasewillbelefttotheastrophysicists.


Гuo Calculation from Expansion:OurobjectivehereistodeterminethecurrentvalueofthebackgroundgravitationalgammafortheuniversedesignatedГuo.WehavecalculatedthevalueГeq 6.4 1027 for the radiation dominated epoch. The remaining part of the calculationrequiredtodetermineГuoisgreatlyhelpedbythefactthattheredshiftfromtheradiation/matterequalitytothepresenthasbeenexperimentallymeasuredbyWMAP. Thefollowingquoteisfromthe5YearWMAPreport4;“TheequalityredshiftzeqisoneofthefundamentalobservablesthatonecanextractfromtheCMBpowerspectrum5”.Theterm“equalityredshiftzeq”isthevalueoftheredshiftsincetheradiation/mattertransition equality tothepresent.ThisredshifthasbeenmeasuredbyWMAPandfoundtobe:zeq 3253 88.Thereforethisnumberincludesallof the redshift that occurred during both thematter dominated epoch and the dark energydominatedepochuntilnow.Thissinglenumberincludesevenacceleratedexpansion.Thereforesincetheredshiftzeqequalsthechangeinscalingfactorsinceequality,wecansimplymultiplythis timesthepreviouslycalculatedvalueofГupriortoradiation/matterequalitytoobtainthetotalvalueofthebackgroundgravitationalГuoftheuniversetoday.WemakeuseofthecontentionthatГu authereforethecurrentvaluesareГuo auo.Гuo Гeqzeq 6.4 1027 3253Гuo 2.1 1031ГuocalculatedfromcosmologicalexpansionThis calculationofГuoandauo simplyextended thestartingassumption theuniverse isonlyspacetime .ThismeansthatwebegintheuniversewithPlanckspacetimeatoneunitofPlancktimeold τu 1 rather thanstartingwithasingularityat τu 0. Thecalculatedexpansionincorporated the radiation dominated epoch, thematter dominated epoch and the so called“darkmatter”epochwhentherewasacceleratedexpansion.Thecalculationdidnotincorporatean “inflation” phase. Inflation is an ad hoc assumption necessitated when the universe ispresumedtostartfromasingularity.Tokeep trackof theexpansion since theuniversewasτu 1,wewill be referencingasourstandardofcomparisonthepropertiesofaPlanckspherewhenτu 1.Forexample,theproperradiusofthePlanckspherewasPlancklengthlpatthattimeandΓu 1.ThecalculatedvalueofГuo 2.1 1031saysthattheproperradiusofthePlanckspheretodayhasincreasedbyafactorofabout2.1 1031times.Therefore,thepropervolumeofthePlanckspherehasincreasedbyafactorofabout1094times 2.1 1031cubed .Гuo Calculation from Planck Temperature: One advantage of the proposed spacetimetransformationmodeloftheuniverseisthatitdescribesthestartingconditionoftheuniversein

4 Five-Year Wilkinson Microwave Anisotropy Probe ∗ Observations: Cosmological Interpretation, E. Komatsu1 et.al. The Astrophysical Journal Supplement Series, 180:330–376, 2009 February


awaythatwecanmakebothpredictionsandplausibilitycalculationsthatcheckthemodel.Thecurrent characteristics of the universe have been experimentally measured. It is only thephysical interpretationof thesemeasurements that isbeingcalled intoquestion. Wecandosimplecalculationsextrapolatingfromtheproposedstartingconditionstoseeifthemodel isreasonable.ThefirstoftheseplausibilitycalculationsdeterminesthevalueofГuobycomparingthetemperatureofPlanckspacetimetothecurrenttemperatureoftheCMB.TheratioofthesetwotemperaturesshouldbeequaltoГuo.Thereasoningisthatthetemperatureoftheuniversechanged inverselywith the scaling factorwhichmeans that the change in temperature alsoscaledinverselywiththechangeinГu.Toexpressthisrelationshipwewillusethefollowingnewsymbolsps temperatureofPlanckspacetime: ps ½Ep/kB 7.08 1031°Kuo currenttemperatureoftheuniverse CMBtemperature uo 2.725°K

Гuo currentvalueofthebackgroundgravitationalgammaoftheuniverse

Гuo 7.08 1031°K/2.725°K 2.6 1031Гuofromratiooftemperatures

Thisisafantasticresult.FromtheratiooftemperaturesweobtainГuo 2.6 1031comparedtoГuo 2.1 1031fromtheredshiftcalculation.Thissupportstheassumptionthatthestartingcondition for the Big Bangwas Planck spacetime and not a singularity or any other energydensityinexcessofPlanckenergydensity.Гuo Calculation from Total Energy Density:Thereisanotherwaythatwecancalculatethevalue of Гuo using the energy density of Planck spacetime and comparing this to the totalobservableenergydensityoftheuniversetoday. Theenergydensityoftheuniversetodayiscommonlythoughttobeabout8.46 10‐10J/m3 equivalentto9.4 10‐27kg/m3 .However,thisisacalculatednumberbasedontheconceptthattheuniversemustmaintainacriticaldensity.Only about 27.9% of this energy density or 2.36 10‐10 J/m3 is “observable” mass/energyconsisting of fermions and bosons including dark matter . Dark matter is considered“observable”becausethegravitationaleffectsofdarkmatterareclearlyobservable. Onlytheobservable 2.36 10‐10 J/m3 is traceable toPlanck spacetime. The remaining approximately72.1%ofthe“critical”energydensity 8.46 10‐10J/m3 iscurrentlyattributedtodarkenergy.Almostallthisdarkenergyhassupposedlybeenaddedtotheuniversesincetheuniversewasabout 5 billion years old. It did not originate with Planck spacetime and therefore will beexcluded from this calculation. Therefore, the current energy density of the universe thatexcludesdarkenergyisabout2.36 10‐10J/m3. Thiswillbecalledthe“currentlyobservableenergydensityoftheuniverse”anddesignatedwiththesymbolUobs.ThereisanotherwayofcalculatingthevalueofГuousingonlyUps,Uobsandzeq.ThereasoningisthattheuniversestartedwithPlanckspacetimewithenergydensityofUps 5.53 10112J/m3andtodaytheobservableenergydensityisUobs 2.36 10‐10J/m3.Therearetworeasonsfor


thischangeinenergydensity.First,thevolumeofeachPlancksphereincreasedbyafactorofГuo3becausetheradiusofthePlancksphereincreasedbyafactorofГuo.Secondly,thecosmicredshiftreducedtheenergyinthePlancksphereduringtheradiationdominatedepoch.ThisepochstartedwithГu 1andendedroughlyattheradiation/matterequalitywithabackgroundgamma designated as Гeq. We do not experimentally know the value of Гeq, but we doexperimentally know the redshift that has occurred since the radiation/matter equality. Aspreviously stated, this redshift value has been measured by WMAP and found to bezeq 3253 88.ThevalueofГuoisГuo ГeqzeqthereforeГeq Гuo/zeq.TheinitialenergydensityofPlanckspacetimewasreducedbothbecauseofexpansion afactorofГuo3 andbecauseofcosmicredshift afactorofГeq .WecannowcalculatethevalueofГuousingthisknowledge.Ups/Uobs Гuo3 ГeqsetГeq Гuo/zeqUps/Uobs Гuo4/zeqГuo Upszeq/Uobs 1/4setUps 5.53 10112J/m3Uobs 2.36 10‐10J/m3zeq 3253Гuo 2.95 1031ГuocalculatedfromUps,UobsandzeqThis is another successful plausibility calculation that supports themodel that the universestartedasPlanckspacetimebecause thisvalueofГuogenerallyagreeswith theprevious twovaluesofГuo.Calculation from Energy Density of the CMB:TheCMBcurrentlyhasenergydensityequalto theenergydensityof2.725°Kblackbodyradiation:UCMB 4.2 10‐14 J/m3. Throughtheentirelifetimeoftheuniverse,theenergydensityinradiationhasscaledproportionalto1/a4.Thereforeifweonlytracktheenergydensityofradiationwehaveanextremelysimplifiedmodeloftheuniverse.Thisassumptionignoresallthefermionsandotherbosonswhichhaveabout5,000timesgreaterenergydensitythantheenergydensityofphotonsintheCMB.Therefore,thisgreatlysimplifiedassumptionrepresentsaroughupperlimitestimateofГuo. WeassumethattheuniversestartedasPlanckspacetimewithenergydensityofUps 5.53 10112J/m3andthrough expansion achieved the current energy density of just the UCMB. The cosmologicalredshiftplusincreasedvolumeresultedina1/a4scalingofenergydensity.Гuo Ups/UCMB 1/4 5.53 10112/4.2 10‐14 1/4 3.4 1031IfinditsurprisingthatthissimplifiedestimatethatincludesonlyradiationachievesarelativelyclosevalueforГuo.Thereasonisthatthegreatestexpansionfactoroccurredwhentheuniversewasradiationdominatedthereforemostoftheenergylossinfactscaledas1/a4.ThisГuovaluewouldmorecloselyapproachthepreviousthreevaluesifweincludedtherelativisticenergyofparticles in the early universe which are similar to radiation because they also exhibit arelativistic“redshift”.


Calculation from Spin: There isone lastcalculationthat I findamazing. It isbasedontheassumptionthatquantizedspinshouldbeapproximatelyconserved.Theterm“quantizedspin”isbestdefinedwithanexample.Twogammarayphotons,eachwithħspin,caninteracttoformanelectronandapositron.Thesetwofermionseachhavespinof½ħ.Theconceptofquantizedspinignoresspindirectionandmakesnodistinctionbetweenħspinand½ħspin.Todeterminethe totalnumberofquantizedspinunitswemerelyadd together thenumberofbosonsandfermions ħand½ħunitsofspinaretreatedthesame .WeknowthedensityofquantizedspinunitsinPlanckspacetimeandweknowthedensityofphotons and fermions in theuniverse today except for darkmatter . The spacetimebasedmodeloftheuniversedoesnotrequiretheexchangeofvirtualparticlestotransmitforcessonoquantizedspinisallottedtovirtualparticles.However,theassumptionthatthereislargescalepreservationofquantizedspinisquestionable.Forexample,twophotonscanbeabsorbedandreemitted as a single photon. However, with black body radiation there is large scalepreservationofthetotalnumberofphotonsbecausethenumerousabsorptionsandreemissionsaverageout.Forthiscalculationtobeaccuratewewillmakethequestionableassumptionthatquantizedspinhasasimilaraveraging.Iftheanswerisreasonable,thenthiswillsupporttheaccuracyofthisassumption.PlanckspacetimehadħspinineachPlanckspherewithinitialvolumeof 4π/3 lp3 1.77 10‐104m3 or a quantized spin density of 5.65 10103 spin units/m3. Todaymatter dominates theuniversebutthequantizedspinunitscontainedinobservablematterissmallcomparedtothequantizedspin in theCMB. Ordinarymatter is4.6%of the “criticaldensity”of theuniverse.Therefore the density of ordinary matter has a density of about 4.3 10‐28 kg/m3. This isequivalenttoabout¼hydrogenatompercubicmeter.Sinceahydrogenatomhas3quarksandoneelectron,¼hydrogenatompercubicmeterisequivalenttoabout1quantizedspinunitpercubicmeter.Recallthatthespacetimebasedmodelhasnovirtualphotons,gravitonsorgluons.Iftherereallyaregravitons, etc. then this calculation shouldbewrongbymanyordersofmagnitude. Thequantized spin in the photons of the CMB dominate baryonic matter because the 2.725°KblackbodyCMBphotondensityis4.2 108photons/m3comparedtoabout1fermion/m3forordinarymatter.Thephotondensityofstarlightisalsoconsideredinsignificantbecauseithasbeenestimated6thatthedensityofstarlightphotonsislessthan1%ofthephotondensityoftheCMB.Wewillalsotemporarilyignorethespininneutrinosanddarkmatter.Therefore,withtheseexceptionsthequantizedspinoftheCMBdominatestheuniverse.IfeachPlanckspherecontainedoneunitofquantizedspininPlanckspacetimeandiftherewaspreservationofquantizedspin,thenwewouldexpectthattheexpandedPlanckspherewouldstillonaveragecontainoneunitofquantizedspin.Thebestestimateofthecurrentproperradius

6 Paul Davies, 1982 The Accidental Universe, Cambridge University Press ISBN 0-521-24212-6


ofthisPlancksphereis:lpГuo lp 2.6 1031 4.2 10‐4moravolumeofabout3.1 10‐10m3.ThisestimateusesthevalueofГuoobtainedfromtheratiooftemperatures Гuo 2.6 1031 .Thisnumber represents themiddleof the rangeofvaluesand isobtained from the simplestcalculation.Therefore,assumingspinpreservationwewouldexpectthatthecurrentdensityofquantizedspinunitswouldbe3.2 109/m3.ItappearsasiftheCMBphotondensityof4.2 108photons/m3 is low by a factor of about 7.6. However this is actually very good agreementconsideringthatthevolumehasexpandedbyafactorofabout1094from:Гuo3 2.6 1031 3 1094.Infact,comparedto1094,thissmallfactorcouldbeexplainedbynotincludinganumericalfactornear1suchas2π.Dark Matter Calculation:Eventhoughweshouldbesatisfiedwithanagreementthatisquiteaccurateconsideringthatitcoversarangeof1094,westillhavenotaccountedforneutrinosanddarkmatter.Supposethatweassumethatstatisticallythereisvirtuallyprefectpreservationofquantizedspin.Thisassumptiongivesustheopportunitytoestimatetheenergyofadarkmatterparticle rotar . Using 4.2 108 photons/m3 and an expanded Planck sphere volume of3.1 10‐10m3/spherewehave:4.2 108photons/m3 3.1 10‐10m3/sphere 0.13photons/sphereAssumingspinpreservationandГuo 2.6 1031,thereshouldbe1spinunitperspherethereforewearemissing0.87spinunits/sphere. Theaveragedensityofdarkmatterintheuniverseisabout2.2 10‐27kg/m3oranenergydensityof1.9 10‐10J/m3.ThecurrentvolumeofaPlancksphereis3.1 10‐10m3,thereforecombiningthesewehave:1.9 10‐10J/m3 3.1 10‐10m3/spinunit 1/0.87 6.8 10‐20J/spinunitTheimpliedenergyperdarkmatterfundamentalparticleis6.8 10‐20J 0.4eV.Perhapsthisisacoincidence,butoutofthemanyordersofmagnitudeinvolvedinthiscalculation,theanswerofabout0.4eVisintheenergyrangecurrentlyattributedtoneutrinosanditisfarremovedfromthe 1010eVenergyrangeassumedforthehypotheticalWIMPdarkmatterparticlemodel.Forexample,aphotometricredshiftsurvey7hassetanupperlimitof0.28eV/c2onthesumofthemassesofthethreetypesofneutrinosknowntoexist.AnanalysisofthePlanckspaceprobedatahassetanupperlimit 0.23eV/c2 onthismasssum8.Allowingforthespeculativenatureofthisresult,thissurprisingresultwarrantsanewexaminationofdarkmattercandidates.Can Neutrinos Be Dark Matter?Neutrinoshavebeendiscountedaspossibleexplanationsofdarkmatter because they are considered to be “hot darkmatter” propagating at velocity in

7 Thomas, S.A.; (2010). "Upper Bound of 0.28 eV on Neutrino Masses from the Largest Photometric Redshift Survey". Physical Review Letters 105 (3): 031301.arXiv:0911.5291 8 Planck Collaboration, "Planck 2013 results. XVI. Cosmological parameters". Astronomy & Astrophysics 1303: 5076.arXiv:1303.5076


excessof0.1c.Theyhavesuchlowrestmassthatevenatatemperatureofabout1°K,theywouldhaveavelocitymuchgreaterthantheescapevelocityofagalaxy.Forexample,theMilkyWaygalaxyhasanescapevelocityofroughly600,000km/s.Also,intheearlyuniverseahighdensityofultra‐relativisticneutrinoswouldexcessivelyhomogenizetheCMBdistribution. However,the leading candidate for dark matter a WIMP also has problems. For example, thehypothetical fundamental particle of a WIMP ~1010 to 1012 eV must not exhibit anyelectromagnetic or strong interaction. Therefore these hypothetical particles would notexperience collisions with baryonic matter except through the unlikely event of a weakinteraction.Mostimportant,theWIMPscannotlosekineticenergybyEMradiationtoformastablegravitationallyboundorbit.ThinkabouttherequiredpropertiesofaWIMPparticle.Itmusthavesuchahighprobabilityofbeingformedthatintheearlyuniverse,thatover80%oftheenergythatwasdestinedtobecomeafermionwouldhavehadtogointotheformationofthesehighlyenergeticparticles.Inallotherrespects, the lowest energyparticles are themost stable andhave thehighestprobabilityofsurvivingtotoday.IfaWIMPisdarkmatter,thenthistrendshouldbereversed.Ahighenergyislandofstabilitywouldhavetoexistandover80%oftherestmassintheuniversewouldhavetocurrentlyexistasWIMPs.Nowlookat theeasewithwhichneutrinosareproduced. Theyare formedwiththebirthofalmosteverynewparticle.Theyarethelowestenergyfermionknownandtheyareknowntohave long term stability. Also, the lowmass/energy of neutrinosmore easily forms amassdistributionaroundagalaxythatischaracteristicofthedarkmatterspherical“halo”providedthattheycansomehowbeslowedtofitthedefinitionofcolddarkmatter.However,aplausiblemethodforthemtolosekineticenergyandcooltolessthanabout0.01°Kmustbesuggested.Neutrinosat2.7°Kwouldgreatlyexceedtheescapevelocityfromgalaxies.However,neutrinosatatemperatureintherangeof0.01°Kwouldhavetherequiredproperties.Neutrinoshavethefollowingadvantages:1 Theyareknowntoexist,2 Theyareknowntobeproducedinlargenumbersintheearlyuniverseand3 Theyareknowntoonlyinteractwithordinarymatterby theweak force. Using theknownpropertiesofneutrinos, thedensityofneutrinoshasbeenestimated9atabout3/11oftheCMBphotondensityperspecies.Thereforethe 3 species of neutrinos would have a total density of about 82% of the photon density~ 3.4 108/m3 for 3 species . This is not quite the density required to satisfy themissingquantizedspin requires~2.8 109/m3 ,butitiswithinanorderofmagnitude.Eventhoughthisisnotenough,theknownformationmechanismsmakethemthemostabundantfermionintheuniverse.A rotarmodel of a neutrinowith internal energy of 5 10‐20 J about 0.3 eV would have aquantumradiusof ħc/Ei 6.3 10‐7moraquantumvolumeofabout10‐18m3.Theenergy

9 J Rich; Fundamentals of Cosmology (second edition 2009); Springer, Heidelberg, New York


densityofa0.3eVrotarwouldbeabout1024timessmallerthantheenergydensityofanelectron.Theenormousdifference inenergydensityofneutrinoscompared toanyother fundamentalparticlesuggests thepossibility thatperhapsneutrinosalsopossessothercharacteristicsnotobservedwithotherfundamentalparticles.Neutrinoflavoroscillationhasbeenwelldocumentedexperimentally.However,itisdifficulttoexplainhowneutrinoscanchangetheirrestmassiftheyarevisualizedaspropagatingthroughanemptyvacuum.Theusualexplanationinvokestheneutrinosexistingsimultaneouslyinthreestatesthatdifferslightlyinmass/energy.ThesethreestateswouldhavethreeslightlydifferentdeBrogliefrequencieswhentheyaremovingrelativetoanobserver.Thethreefrequenciescanthenconstructivelyanddestructivelyinterferewitheachotherproducingwhatisachangeinneutrino flavor mass . However, this explanation has problems. It is difficult to devise anexplanationwhichallowsforachange inrestmasswhilemaintainingbothaconservationofenergyandaconservationofmomentum.I would like to propose a new consideration which is best explained by an example. The so called “MSW effect” occurs when neutrinos pass through matter such as a star and achieve a resonance with electrons. This is analogous to introducing a different index of refraction for each of the 3 neutrino flavors and results in an increase in flavor oscillation frequency. The mystery of neutrinos changing their mass ceases if we postulate an interaction with other particles because conservation laws hold if other particles are introduced.

Suppose that we assume that there is some mechanism by which neutrinos can be cooled to a temperature less than 0.01 degrees K. Then neutrinos would exhibit the properties of dark matter. At the earth’s distance from the galactic center, the energy density of dark matter is about 3x1014 eV/m3. If we assume neutrinos to have energy of about 0.3 eV, then this works out to about 1015 neutrinos/m3 at the earth. With this density, the experiments conducted on earth are not measuring the flavor oscillation rate in a total vacuum. Our best vacuum would still contain about 1015 neutrinos/m3. The interesting point is that an interaction between neutrinos would produce acceleration and deceleration of fermions with rest mass. This would generate gravitational waves and cooling of neutrinos. Perhaps this is an over simplification, but a flavor oscillation which involves an interaction between neutrinos with different energy resulting in a slight mass change would involve acceleration and deceleration of matter. This should produce a small gravitational wave. If this happens, it would represent a loss of kinetic energy relative to the cosmic microwave background (CMB) rest frame of reference. Neutrinos which originated at the Big Bang could possibly be cooled by gravitational wave emission to a temperature far below the temperature of the CMB photons. If this additional cooling occurs to an extent that neutrino temperature is lowered to less than about 0.01 degree K, then neutrinos would qualify as being cold dark matter. Summary of the Гuo Calculations: We took a deviation in the discussion of darkmatter.However, nowwewill return to the earlier discussion of the Гuo calculation. We have now


calculatedthevalueofГuoseveraldifferentwaysandtheyallgiveaboutthesameanswer.Theclosenessoftheresultsobtainedwithverydifferentapproachessupportsthemodelbeingused.WewillusethevalueГuo 2.6 1031asrepresentativeoftherangeofvaluescalculatedhereforfuturecalculations.Usingthisvalue,todayeachPlanckspherehasexpandedtoaproperradiusoflp Гuo 0.42mm.However,measuredincoordinateunitswhichpresumeГu 1,thePlanckspherestillhasaradiusof 1.Italsoappearstostillhaveonaverage1unitofquantizedspin.TheanalysisofthespinperPlanckspherewillbemadeinthenextchapter.

Graph Showing the Evolution of Гu and Observable Energy: Figure 13‐4 shows twosuperimposedgraphsthatusethesametimelinebutdifferentYaxisscales.NotefirstthattheXaxisisalogscaleoftheageoftheuniverseinseconds.Thetimescaleextendsfrom5 10‐44swhich is1unitofPlanck time to1020secondswhich isanageofabout3 trillionyears. Thepresentageoftheuniverseisabout4.3 1017seconds ~13.8 109years andisdesignatedwithanarrowandaverticaldashedline.ThegraphlinedesignatedГuisthevalueofthebackgroundgravitationalgammaoftheuniverseГu .ThisgraphusestherightscalewhichisalogscaleextendingfromГu 1toaboutГu 1032.It canbeseen from thisgraph thatГu 1at a timeof τu 5 10‐44 secondsandendswithГu 2.6 1031atthepresent τu 4.3 1017s .Notethatthereisaslightchangeintheslopeof


thislineasitcrossestheverticallinedesignatedR‐Mtransition.Thisisthetransitionbetweenaradiationdominateduniverseandamatterdominateduniversethatoccurredabout70,000yearsafterthebeginningoftheuniverse theBigBang . Thereshouldalsobeanotherslopechangeatthetransitionbetweenthematterdominatedepochandthedarkenergydominatedepoch.However,thisslopechangeissosmallthatitdoesnotshowuponthislog‐loggraph.Theothergraph designated “Joules” uses the left scaleand is theobservableenergy in thePlancksphere alsoalogscale .AspherePlancklength lp inradiusstartedwithabout1billionJouleswhentheuniversewasPlanckspacetimeforthefirstunitofPlancktime.OvertheageoftheuniverseasГuincreased,theproperradiusofthissphereincreasedandequaledГulp.TheproperenergyofthisimaginaryPlanckspheredecreasedduringtheradiationdominatedepochby a factor of 6.4 1027 from about 109 J to about 1.5 10‐19 J and then the proper energygenerallyhasremainedconstantsincetheendoftheradiationdominatedepoch.Therefore,notetheflatlinefromanageofabout70,000yearstothepresent.However,theenergyonanabsolutescalewhereГu 1continuestodecreaseasГuincreases.Forexample,thedecreasingrateoftimemakesitappearthattheComptonfrequencyoffundamentalparticlesisconstant.However,ifwetimedtheComptonfrequencyofanelectronusingtheГ 1clock,wewouldfindthatonthisabsolutetimescale,eventheComptonfrequencyofanelectronwouldbeslowingdowneachsecond.Itisnotnoticeablebecauseourcosmicclockisslowingdownatthesamerate.


Chapter14

CosmologyII–TheBigPicture

Inchapter3wedeterminedthechangesinenergy,force,voltage,etc.thatarerequiredtokeepthelawsofphysicsconstantindifferentgravitationalpotentialswherethereisadifferenceintherateoftime.The“normalized”coordinatesystemofchapter3usedtheequationLo Lg.ThisdoesnotimplythatLoandLgareconstantovertime.Itisnowproposedthatbothoftheseunitsof lengtharesimultaneouslycontractingastheuniverseages. ThissimultaneouscontractionmaintainsLo LgintheCMBrestframeaccordingtoamidpointobserver.Nowweareattemptingtounderstandtheevolutionoftheuniverse.TodothisIproposethatitismostconvenienttouseacoordinatesystembasedonthepropertiesofPlanckspacetimewhenГu 1. Eventhoughtheuniversehasalwayshadflatspacetime,therehasbeenacontinuousincrease in Гu since the Big Bang. As previously explained, the current value of Гu is aboutΓuo 2.6 1031andthisnumbercontinuestoincrease.Thisaffectsmanythingsincludingourrate of time, our length standard and our energy standard. The best referencewe have toquantifythesechangesistouseacoordinatesystembasedontheconditionsthatexistedwhenГu 1.AnotherwayofsayingthisisthatweshouldreferencetheconditionsthatexistedatthestartoftheBigBangwhenwehadPlanckspacetime.RelativetothespatialandtemporalcoordinatesystemthatexistedattheBigBangwhenГu 1,therehasbeenadecreaseintherateoftimeandadecreaseinastandardizedunitoflengthsuchasonemeter.Aswillbeexplained,thiscombinationkeepsthelawsofphysicsunchanged.Thishassomesimilaritiestothegravitationaleffectspreviouslydiscussedinchapter3. However,withtheuniversethebackgroundgravitationalgamma Гu iscontinuouslyincreasing.Oneofthefewindicationsthatanythingischangingintheuniverseisthatlightthatwasemittedalongtimeagofromdistantsourceshasundergoneobviouschangesinwavelengthandintensity.Spacetime Transformation Model:AmodeloftheuniversebasedonacontinuouslyincreasingГurepresentsanalternativetotheBigBangmodel.WhatweperceiveasanincreaseinthescaleoftheuniverseisactuallyduetoanincreaseinthebackgroundГuoftheuniversechangingthespatialandtemporaldimensionsofspacetime.Thisisachangeinthepropertiesofspacetimethathasaneffectoneverythingintheuniverse.Theradiusofanatomortherotarradiusofarotarwoulddecreaserelativetocoordinatelength .AnincreaseofГuresultsinthefollowing:1 thehybridspeedof lightof theuniversedecreases;2 proper lengthcontracts relative tocoordinatelength;3 therateofpropertimedecreasesrelativetotherateofcoordinatetime;4 thetotalenergydensityoftheuniverseremainsthesame totalenergyincludesvacuumenergy .


Perhaps most surprising of these is that the spacetime transformation model says that thecoordinate energy density of the universe has remained constant since the beginning of theuniverse sincetheBigBang .Thecoordinateenergydensityutilizescoordinatelength andcoordinaterateoftimedttoquantifycoordinateenergydensity.Theobservableenergydensityoftheuniverse measuredinproperunitsofenergy hasdecreasedbyafactorofroughly10120sincethebeginningoftime sincetheBigBang . However, includingthewavesinspacetimeresponsibleforvacuumenergy,itwillbeshownthatthespacetimetransformationmodeloftheuniverseseesnochangeinthecoordinateenergydensityoftheuniverse.Thisalsoeliminatesthefamous10120discrepancybetweenthe“critical”energydensityoftheuniversederivedfromgeneralrelativityandsupportedbyobservationcomparedtothecalculatedenergydensityoftheuniversederivedfromquantummechanicsandquantumchromodynamics.Thisspacetimetransformationmodelmightseemlikeanunnecessarycontrarianviewthatisfundamentallyequivalenttodepictingtheuniverseasexpanding.However,itwillbeshownthatthismodelisnotequivalenttotheBigBangmodel.ThisproposedmodelgivesthesameredshiftandthesameincreaseinpropervolumeastheBigBangmodel,butthespacetimetransformationmodel offers different predictions about the future of the universe. Probably the mostcontroversialdifferenceisthatthespacetimetransformationmodelpurportstoeliminatetheneedfordarkenergyandacosmologicalconstant.Observable Universe from Planck Spacetime: Aspreviously stated in chapter13, PlanckspacetimehadsphericalPlanckenergydensity.Mostimportantly,PlanckspacetimehadГu 1and all the dipole waves in spacetime had ½ Planck energy about 109 J and ħ angularmomentum.Thismeansthat100%oftheenergyinPlanckspacetimewas“observable” hadquantizedspin .ThevalueofГu 1alsomeansthattherateoftimewasthehighestpossibleandthepropervolumeoftheuniversewasthesmallestpossible.Гu 1alsoimpliesthataunitof energy such as one Joulewas the highest possible valuewhenmeasured on the absoluteenergyscalewhichusescoordinaterateoftimeandcoordinatelength.Incomparison,itwillbeshownthatoneJouletodayisavastlylowerenergyontheabsoluteenergyscalebecausetodayΓuo 2.6 1031.Also,thetransformationofspacetimethathastakenplacesincetheBigBanghas resulted in a decrease in the percentage of the energy in the universe that possessesquantizedangularmomentum.Today,onlyabout1partin10122oftheenergyintheuniverseis“observable” possessesquantizedangularmomentum All the energy required to form our current universe including vacuum energy would becontainedinasphereofPlanckspacetimeabout15 10‐6meters ~15microns inradius.Thisradiusiscalculatedbyreducingthecurrentdistancetoourparticlehorizon ~46billionlightyearsor4 1026m byafactorofГuo 2.6 1031. OurcurrentuniversehasГuo 2.6 1031whichgreatlyreducesbothourcurrentstandardofenergyandthefractionoftheenergyintheuniversethatis“observableenergy”


Itispresumedthattheuniversecurrentlyextendsfarbeyondourcurrentparticlehorizon.ThismeansthattheoriginalvolumeofPlanckspacetimewasfarbiggerthanthe15micronradiussphericalvolumerequired to formeverything includingvacuumenergy withinourcurrentparticlehorizon.ThisoriginalvolumeofPlanckspacetimemightnothavebeeninfinite,butitispresumedtobeeffectivelyinfinitebecausethereisnodetectabledifferenceasfarasamodeloftheuniverseisconcerned.Itisalsopresumedthattherearegalaxies,darkmatter,etc.beyondourparticlehorizonthathaveasimilarappearanceanddensitytoourobservableuniverse.The spacetime transformationmodel of theuniversehas a fixed not expanding coordinatesystem.Forexample,twodistantgalaxiesareconsideredtobeseparatedbyaconstantdistancewhenmeasured inunitsof coordinate length . The coordinategridusedby the spacetimetransformationalmodelhassimilaritiestothecoordinategridusedbytheΛ‐CDMmodel.BothgridscorrespondtotheCMBrestframeatalllocationsintheuniverse.HoweverthedifferenceisthattheΛ‐CDMmodelhasagridthatexpandswiththepropervolumeoftheuniverseandthespacetime transformation model has a grid that remains stationary. In the spacetimetransformationmodel,theexpansioninthepropervolumeoftheuniverseisaccommodatedbythechangeinΓuovertheageoftheuniverse.Hubble Parameter and Shrinking Meter Sticks:Today,astronomersdonotrealizethattheirmetersticksarecontractingduetochangesinspacetime.Theterm“meterstick”representsanymeansoflengthmeasurement.Astrophysicistscalculatethatthedistancetodistantgalaxiesisincreasing.However,thisdistanceismeasuredusingcontractingmetersticks contractingunitsoflength .ThedistantgalaxiesthatarestationaryonthestaticcoordinatesystemappeartoberecedingatavelocitygivenbytheHubbleparameter.In astronomical terminology, the Hubble parameter is often expressed as about 70.8km/s/MpcwhereMpcismegaparsec,aunitoflengthusedinastronomyequaltoabout

3.09 1022meters. Converting theHubble parameter to SI unitswe have 2.29 10‐18m/s/m.Thesecondsusedherearetoday’sproperseconds.ThecommoninterpretationoftheHubble parameter is that this is the current expansion rate of the universe. The spacetimetransformationmodelinterpretstheHubbleparameterdifferently:

Г

Гor

Г

Гnotethedotsrepresentingtimederivative

Thedotsareshorthandfortimederivatives.Therefore,theHubbleparameter equalstherateofchangeofГudividedbythecurrentbackgroundvalueГuo.AlsoauoisthecurrentscalingfactoroftheuniversewhichisequaltoГuo.Ameterstick 1meterlong iscontractingatavelocityofabout2.29 10‐18meters/secondwhencompared to a hypothetical meter stick that is not contracting a meter stick with fixed


coordinatelengthinunitsof .Asexplainedinchapter13,theHubblesphereisa13.7 109lightyear 1.3 1026m radius imaginary spherical shellwheregalaxiesandspace itselfarecalculatedtoberecedingawayfromusataboutthespeedoflight.However,itisproposedthatweareusingacontractingunitoflength,suchasacontractingmeterstick,asreferenceforthiscalculation.TheproperdistancebetweenusandagalaxyattheedgeoftheHubblesphereisindeed increasingby3 108m/s,but that isbecausewearemeasuring thedistanceusingacontractingmeterstick.Ourmeterstickisshrinkingattherateof2.29 10‐18meters/secondsoweobtainanincreaseinproperdistanceof3 108m/s obtainedfrom2.29 10‐18m/s/m 1.3 1026meters .Thisisnotthesameassayingthatthegalaxyisphysicallyrecedingfromusat the speed of light. The spacetime transformationmodel says that the calculated speed iserroneousbecauseitisobtainedwhenwemeasureafixeddistanceusingshrinkingmetersticks.Thereasonableexplanationisthatspacetimeisundergoingatransformationthatchangesthecoordinatespeedof lightwhilekeeping the lawsofphysicsunchanged includinga constantproper speed of light . This is similar to the covariance of the laws of physics in differentgravitationalpotentialsasdiscussedinchapter3.However,withtheuniverseГuincreaseswithtimebutthelawsofphysicsremainunchanged.Oneobservableeffectisthatittakeslongerforlighttotravelbetweengalaxiesastheuniverseages.Sincewemeasurenochangeintheproperspeedoflightweinterpretthisasindicatinganexpansion.However,thealternativeexplanationproposed here is that the change occurring in the properties of spacetime produces adimensional contraction. In chapter 3we sawhow the rate of time can change at differentelevationsofagravitationalfieldwithoutbeingdetectablelocally. Gravityalsowasshowntoaffect proper volumewhich implies a difference in the unit of length. The universe is alsoproducinganomnidirectionalgravitationaleffectthatiscontinuouslyincreasing.Oneresultofthisiswhatweperceivetobethecosmologicalincreaseinthevolumeoftheuniverse.Itwasshowninchapter3thatgravityaffectsmanyoftheunitsofphysicsinawaythatkeepsthelawsofphysicsunchanged.Itisproposedthatsomethingsimilarishappeningwiththeentireuniverseexceptthatthereisanimportantdifference.WiththeuniverseatanyinstantthevalueofГuisuniformlyincreasingeverywhere.Thisistheoppositeofthegravityassumedinchapter3 which was static and had a gravitational gradient. The continuous increase in Гu causeschangesinvariousunitsofphysics energy,force,voltage.etc. whichtogetherpreservethelawsofphysics.Onlywhenwelookatdistantgalaxiesdoweobtainahintthatchangeovertimeisoccurring.No Cosmic Event Horizon: TheΛ‐CDMmodel considers theacceleratingexpansionof theuniversetohaveacosmiceventhorizon.Thisisdefinedasthelargestcomovingdistancefromwhich lightemittednowcaneverreach theobserver in the future1 According to theΛ‐CDMmodel,galaxiesthatweobserveashavingaredshiftgreaterthanZ 1.8arecurrentlybeyond

1 Bergström, L.; Goobar, A.: "Cosmology and Particle Physics", WILEY (1999), p. 65.ISBN 0-471-97041-7


ourcosmiceventhorizon.Lightthatiscurrentlybeingemittedbythesegalaxieswillsupposedlyneverreachusbecausecosmicexpansionofspaceisaddingvolumeatsuchafastratethatthedistanceincreaseexceedsthespeedoflight.EventheexpansionofourHubblespherecannotovercometheacceleratingexpansionoftheuniverse.PhotonsbeingemittednowbygalaxieswithZ 1.8willbesweptawayfromusbytheacceleratingexpansionoftheuniverse.Theonlyreasonthatwecanseethosegalaxiestodayisthatweareseeingthelightemittedfromalongtime ago before they crossed our event horizon. Once again, that is the Λ‐CDM modelinterpretation.ThespacetimetransformationmodeloftheuniversemakesapredictionthatisdifferentthantheΛ‐CDMmodel.TheproperdistancebetweenusandtheZ 1.8galaxiesisindeedincreasingfasterthanthespeedoflight.However,thisisbecauseofthecurrentrateofincreaseinГuiscausing our meter sticks to shrink. The galaxies are actually stationary on the proposedcoordinategrid. Theprediction is that lightcurrentlybeingemitted fromthosegalaxieswilleventuallyreachusbutataslowerhybridspeedoflightthantoday.Eventhoughtheuniverseappearstohaveacceleratingexpansion,thespacetimetransformationmodelsaysthatthereisno event horizon at a distance corresponding to Z 1.8 or at any other distance in theforeseeablefuture.Wecanobtainabetterinsightintothepropertiesofthehybridspeedoflightwithanumericalexample.SinceC≡d /dτu c/Гu,thereforethecurrentvalueofthehybridspeedoflightis:C c/Гuo 2.6 10 10‐23m/s

Thisisthecurrenthybridspeedoflightwheretheunitsm/sarecoordinatemeters dividedbypropercomovingseconds noteuseofitalicincoordinateunits .Thehybridspeedoflightisdeceleratingeverysecondatacurrentrateof:C 10‐23m/s 2.3 10‐18s‐1 2.3 10‐41m/s2decelerationofCFinally,thecurrentrateoftime dτuo isabout2.6 1031timesslowerthanthecoordinaterateoftime dt whichassumesГu 1.Constant Coordinate Energy Density:Theenergydensityofthis15 10‐6msphericalvolumeisequaltosphericalPlanckenergydensity:Ups 5.5 10112J/m3.Atthebeginningoftime theBigBang thisenergywasintheformofdipolewavesinspacetimewiththeuniquepropertiesofPlanckspacetimepreviouslyenumerated. Todaythecharacteristicsofthedipolewavesinspacetimethatformbothvacuumenergyandtheobservablemass/energyinouruniversehavechanged their characteristics compared to Planck spacetime. Almost all the energy in theuniversetodayisintheformofvacuumenergy–dipolewavesinspacetimethatdonotpossessangular momentum. Only an extremely small part is in the form of observable energy that


possesses angular momentum. However, it will be shown that not only the proper energydensity includingvacuumenergy butalsothecoordinateenergydensityoftheuniversetodayisstillthesameasPlanckspacetime. Therehavebeenchangesrelatingtothedistributionofquantizedangularmomentum,therateoftime,properlength,etc.butthetotalenergydensityhasnotchangedevenwhenmeasuredusingcoordinateenergydensity thatassumesГu 1.Therefore,itispossibletoadoptacoordinatesystembasedonthesecoordinatevaluesthatdoesnotexpandovertime.Thisisthestationarycoordinatesystemofthespacetimetransformationmodel.ItmightseemthattheBigBangmodelisultimatelyequivalenttothespacetimetransformationmodelwithitsstationarycoordinatesystem.However,thisisnotacaseofsimplecoordinatetransformation.Forexample,thetwomodelsmakedifferentpredictionsabouttheexistenceofaneventhorizonaspreviouslynoted.Also,theBigBangmodelcannotaccommodatethefactthatnewvolumebeingaddedtotheuniversemustalsopossessthevacuumenergywithenergydensity exceeding 10112 J/m3.Where did this additional energy come from? The spacetimetransformation model can accommodate this requirement as will be explained later in thischapter.

Illustration of Slowing Hybrid Speed of Light: Figure 14‐1 illustrates the concept of astationarycoordinatesystemwithaslowinghybridspeedoflight.Thisfigureusescoordinatelength,thereforethedistanceincoordinatelengthunitsbetweenPointAandthefurthestcurvedsurface 16billionyears isroughly10‐5m.Point“A”canbeimaginedasinitiallyaPlancksphere


withinPlanckspacetimeatthebeginningoftime.ThisspherecontainedaboutabillionJoules,sowhentimebegantoprogressthegravitationalinfluenceofthisenergybegantopropagateawayfrompointAattheproperspeedoflightc.However,therateofpropagationasmeasuredusingthehybridspeedoflightdecreasesasГuincreases.After4billionyearsthegravitationalinfluencehadreached thepropagatingparticlehorizondesignated4billionyears. The term“propagatingparticlehorizon”isusedheretodesignatetheexpandingsphereofinfluenceofapointofmass/energy Similarly,thepropagatingparticlehorizonsfor8,12and16billionyearsareshown.Thepurposeof this figure is to illustrate theslowingrateofpropagationas indicatedby thedecreasingdistanceseparatingthecurvedsurfacesastimeprogressesandГuincreases.Thereisnotendencyforthisprogressiontobesweptbackwardsbycosmicexpansion. Therateofprogresswillcontinuetodecrease,butthereisnoeventhorizonwheretheprogressisstopped.ItisspeculationwhetherГueverreachessuchalargevaluethataquantummechanicaltransitionoccurs.Thespacetimetransformationmodeloftheuniversepredictsthatfortheforeseeablefuture,wewillcontinuetoseenew,moredistantgalaxiesappearinthesky.Thegalaxiesthatwecurrentlyseewillgetdimmer lessphotonspersecondperm2 butalsoparadoxicallybelessredshiftedthantoday.Redshift:ThespacetimetransformationmodelisalsothebestmodeltoseewhyanincreasingbackgroundГuproducesaredshiftonthelightthatweseefromadistantgalaxy.Thepresenceof a redshift in cosmology is counter intuitive when it is realized that the spacetimetransformationmodelclaimsthattherateoftimewasfasterwhenthelightwasemitted dτem thanwhen the light is observed Гem Гobs anddτobs dτem . For example, Schwarzschildassumedasinglestationarymassinanemptyuniverse.Thisassumptionpresumeda“maturegravity” condition no time dependence . Under these conditions, light propagating from alocationfarfromthemass smallgravitationalГ toalocationnearthemass largegravitationalГ ,undergoesa“gravitationalblueshift”.Thiswaspreviouslydiscussedandshownthatadistantobserverusingasinglerateoftimeperceivesnochangeintheenergyofthephoton.Thelocallyobservedapparentincreaseinenergyisduetotheslowrateoftimeingravity largeГ .When the backgroundГu of the universe increases uniformly everywhere, this is completelydifferentthanaphotonpropagatingfromalocationwithasmallvalueofГtoalocationwithalargervalueofГ.ItwillbeshownbelowthatanincreaseinthebackgroundГuoftheuniverseproducesaredshiftwhichincludesanincreaseinproperwavelengthandadecreaseinproperfrequencyandadecreaseinproperenergy.Coordinate Wavelength Constant: When the background Гu of the universe is increasinghomogeneouslythroughouttheuniverse,thismeansthatthehybridspeedoflightisdecreasinghomogeneouslyas:C c/Гu.Lightinflightjustslowsdownhomogeneouslyeverywhere.Thishomogeneousslowingmaintainsthesamecoordinatewavelengthforalightwave.Theentire


wavejustslowsdownwithoutchangingitssizewhenmeasuredusingcoordinatelength.Ifthelightinflightisconstantwavelengthwhenmeasuredinunitsofcoordinatelength,whatresultwillweobtainwhenwemeasurethewavelengthinunitsofproperlengthusingacontractingmeterstick?Wewillobtaintheresultthatthelightisincreasingitswavelengthrelativetothecontractingmeterstick.Inotherwords,wewouldseearedshift anincreaseinwavelength .Redshift – Wavelength Analysis:Toanalyzethiswewillassumethatlightisemittedinlocation#1atanageoftheuniverset1andabackgroundgravitationalgammaГ1. Theemissionisatcoordinate wavelength λ which can be converted to proper wavelength λ1 at the time ofemission. At a later time ageof theuniverse t2 andbackgroundgammaГ2 the coordinatewavelengthisstillthesame buttheproperwavelengthisλ2relativetothecontractedmeterstick. Allwehavetodoiscompareλ1toλ2whichmeansthatweneedtoconvertλwhichisalways in units of coordinate length intowavelength λ expressed in proper length at twodifferent times. The coordinatewavelength λ doesnot change; it only slowsdowndue to achangeinthehybridspeedoflightCwhenГuincreases C c/Гu .Thereforewemustconvertbetweencoordinatelengthandproperlengthattwodifferentvaluesofbackgroundgamma:Г1andГ2.Fromchapter13weknowthattheconversionofunitsofcoordinatelength tounitsofproperlengthisL Гu .Whenweexpressthisconversionintermsofwavelengthsymbolswehaveλ Гuλ.Thissaysthatagivenwaveappearstohaveabiggerwavelength moreunitsofproperlength whenitismeasuredwiththecontractedmeterstickusedforproperlengththanwhen it ismeasuredwith the coordinate scalemeter stick that is not contracted. Since λ isindependentofthebackgroundГuwehave:λ wavelengthoflightwhenmeasuredinunitsofcoordinatelength.λ1andλ2 wavelengthoflight properwavelength attimet1andt2wheret2 t1Г1andГ2 backgroundГuoftheuniverseattimet1 andt2wheret2 t1aem cosmologicalscalefactoratemission aem attime t1aobs cosmologicalscalefactoratobservation aobs attime t2

1Г1andλ

Гconversionofλtoλ1andλ2

Г

ГsinceГ2 Г1thereforeλ2 λ1 wavelengthsinunitsofproperlength

SinceГuisincreasingwithtime,thereforeГ2 Г1andλ2 λ1.Thisallsaysthatλ2isredshiftedlongerwavelength comparedtoλ1.Theamountoftheredshiftis:λ2/λ1 Г2/Г1whichcanalsobe expressed in terms of the ratio cosmological scaling factors at emission aem a1 andobservation aobs a2 orintermsofredshift1 Z.

Г

Г 1 Z


Since λ1 is the properwavelength at emission time t1 and λ2 is the proper wavelength atobservation timet2 ,thereforetherelationshipcanbewrittenas:

1 Z

This answer, obtained from the spacetime transformation model agrees with the answerobtainedfromtheBigBangmodelthatassumescosmicexpansionoftheuniverse.Redshift – Frequency Analysis:Therefore,ithasbeenshownthatlookingjustatwavelengththere is thecorrectredshiftwhenwepresumethat theredshift iscausedbyachange in thebackgroundГuratherthananexpansionoftheuniverse.Itispossibletoworkthissameproblemlookingatthefrequencyoftheradiationratherthanatthewavelength.Inthiscasewewouldexpectalowerfrequencyatalatertimewhenweexpressfrequencyrelativetopropertime.Wewillstartoffbyworkingthisproblemusingcoordinatevalues.Asbefore,thereisnochangeinwavelengthexpressedintermsofcoordinatelengthbetweentheemissionandobservation.Thenewsymbolsare:C1andC2 hybridvelocityoflightattimest1andt2respectivelyλ wavelengthexpressedinunitsofcoordinatelength–thisdoesnotchangeatt1andt2υ1andυ2 properfrequencyoflightattimest1andt2respectivelyproperfrequencyhasaredshiftsinceυ1 υ2andГu2 Гu1Sinceυ1canbeconsideredthefrequencywhenthelightwasemitted υ1 υem andυ2canbeconsideredthe frequencywhen the lightwasobserved υ2 υobs , therefore the following isanotherwayofstatingtheseresults:

Thereforethespacetimetransformationmodelgivesthesameredshift properwavelengthandproper frequency as the Big Bang model. However, this analysis leaves one questionunanswered.Iftherateoftimewasfasterinthepastthanitistoday,whydon’tweobserveablue shift on light from distant galaxies? The answer to this question is not obvious in theprevious analysis because that analysis used the “hybrid speed of light C d /dτu. Thisdefinitionincorporatestheproperrateoftimeintheuniverse dτu whichhidesthequestionabout the blue shift. This question can only be answered if we compare proper values tocoordinatevalues.Thiscomparisonrequiresthatwereworktheproblemusingcoordinaterateoftime dt ,coordinatespeedoflight andcoordinatefrequency υc .


Tobegin,wewillreturntotheexamplepreviouslystatedandexaminelightemittedatlocation#1atanageoftheuniversetu1whichhadabackgroundgravitationalgammaГu1.Thislightislaterobservedatlocation2withtheageoftheuniversetu2andbackgroundgammaГu2.Again,the wavelength of the light measured in units of coordinate length is λ. This coordinatewavelengthdoesnotchange;thelightmerelyslowsasthecoordinatespeedoflightdecreases.Thecoordinatespeedoflightatagesoftheuniversetu1andtu2willbedesignatedas 1and 2respectively.

1 Г

and 2 Г

Wewillnowstatethefrequencyofthislightusingtherateofcoordinatetimedtasourstandard.Recallthatdt Гudτu.Frequencyobtainedusingthecoordinatetimestandardwillbedesignatedνc.Theparticularcoordinatefrequencyproducedbywavelengthλ willbedesignatedν1cwhenthebackgroundgammaisГu1andν2cwhenthebackgroundgammaisГu2.Therefore:

λ set 1Гand 2

Г

Г Г

Г

Гratioofcoordinatefrequencies

Thissaysthatusingcoordinatefrequencyresultsinaredshiftproportionaltothesquareoftheratioofgammas.Thismeansthatthecorrectionduetotheslowingrateoftimedoesnotproduceanobservableblueshift,butinsteadthisblueshiftisusedtoreduceacoordinateredshiftthatisproportionalto Гu1/Гu2 2toaproperredshiftproportionaltojust Гu1/Гu2 .Thisisshownby

makingthesubstitution Гand

Гtoobtaintheratiooffrequenciesexpressedin

properfrequenciesυ1andυ2.

Г

Гratioofproperfrequencies

Rotar’s Frequency:SincephotonsloseproperfrequencyasГuincreases,whydoestheproperComptonfrequencyofrotarsremainconstant?Toputthisquestioninperspective,weshouldfirstacknowledgethatonthecoordinatetimescalethatassumesГu 1theComptonfrequencyofarotardoesdecrease.Therefore,allfundamentalrotarsarecontinuouslyslowingdownonanabsolutetimescale.Wedonotnoticethisslowingbecauseourcosmicclockisalsoslowing.


Thereforeweareusingacontinuouslyslowingclocktotimethe frequencyofacontinuouslyslowingrotarsuchasanelectron. This isamovingstandard,buteachsecondanelectron islosingabout282HzifwemeasuredtheComptonfrequencyusingtherateoftimethatexistedintheprevioussecond ωc /2π 282Hz/s .Therefore,onanabsolutetimescale,whydoesaphoton’sfrequencydecreaseproportionaltoГu1/Гu2 2 while a rotar’s frequency scales with just Гu1/Гu2 ? The answer is that thepropagationrateoftherotar’sdipolewavedecreasesproportionalto Гu1/Гu2 2butthisispartlyoffsetbyadecreaseintherotar’scircumference measuredontheabsolutescaleof . Thereducedcircumferencedistanceandreducedrotarradius bothmeasuredontheabsolutescaleof means that a dipole wave with quantized angularmomentum propagates around theshortened circumference measured in units of . The decrease in a rotar’s circumferenceresultsinarotar’sfrequencyscalingwith Гu1/Гu2 whichmatchestherateoftimedecreaseonanabsolutescale.Therefore,therotar’sComptonfrequencyappearstobeconstantwhilethefrequency of a photon decreases when measured using proper rate of time. The standardexplanationofthecosmicredshiftbasedonexpansionoftheuniverseisproposedtobewrong.Wavelengths are not being stretched. There are no point particles. Particles with finitedimensions are also affected by the transformation of spacetime. However the effects onparticles are not noticeable because there are offsetting effects on the rate of time, on thestandardofenergyandonotherunitsofphysics. Maintaining the Vacuum Energy Density:Throughoutthisbooktherehavebeennumerousreferencestotheenergydensityofthespacetimefieldbeingontheorderof10113J/m3 timesanunknown numerical constant near 1 . The problem is that throughout the history of theexpandinguniverse,theproperenergydensityofvacuumenergywouldhavetoremainconstant.Theexpandingvolumewouldappeartorequireamechanismtocontinuouslyaddatremendousamount of new energy to the expanding universe. For example, the Hubble parameter of 2.3 10‐18 m/s/m indicates that each cubic meter in the universe is expanding and

increasing its volume by ~ 10‐53 m3/s. If the vacuum energy density required to fill thisadditional volume isUvac 10112 J/m3, then theadditional volumegeneratedbyEACHcubicmeter in theuniverserequiresadditional1059 Joules/second. Toput this inperspective, theE mc2totalobservableenergyoftheMilkyWaygalaxyisalsoabout1059Joules.Thereforeitwouldappearthateachcubicmeterofvolumeintheuniversemustbesuppliedwithabout1059Joulesofenergyeachsecond.Therefore, the problem of supplying the universe with dark energy each second is trivialcomparedtotheproblemofsupplyingtheuniversewithnewvacuumenergyeachsecond.Thisallseemstobeimpossible,somostphysicistspresumethattheremustbesomemechanismthatcancels out almost all the implied vacuum energy density in the universe. However, thishypothetical cancelation mechanism has several problems. 1 Zero point energy, vacuumfluctuations,andtheuncertaintyprinciplemustremainandtheseallimplythatvacuumenergy


hasnotbeencanceled.2 Thecancelingmechanismmustbecarefultoleavetheonepartin10122thatconstitutesourobservableuniverse.3 Theeffectcapableofcanceling10113J/m3mustbeequallyaslarge.4 Nomechanismhasbeensuggestedthatiscapableofcausingthisenormouscancelation.Thespacetimetransformationmodelsaysthatthereisnocancelingmechanism.Theenormousvacuumenergyispresentinspacetime.Itisdipolewaveenergythatlacksangularmomentumandthereforeonlyinteractswithourobservableuniverse fermionsandbosons throughsubtlequantummechanicalmechanisms.Thisvacuumenergyisthemostperfectsuperfluidpossible.Itformsthesingleuniversalfieldthatisresponsibleforallotherfields.Italsogivesthefollowingpropertiestospacetime:Zs,c,εo,µo,ħ,G,lp,tp,etc.NewTransformationsofUnits:Recallthatinchapter3wemadeatableoftransformationsoftheunitsofphysicsshowingthedifferencebetweena“zerogravity”locationandalocationwithgravity. In chapter 3we designated the “zero gravity” location as having Г 1. Now it isnecessarytorealizethatthisdesignationincorporatedasimplification.WewereignoringanychangeinthebackgroundgravitationalgammaoftheuniverseΓu.Anotherwayofsayingthisisthatwedefineda“zerogravity location”ashavingΓ 1. Nowthatwearetalkingabouttheevolutionof theuniverse it isnecessary tobemoreprecise. The length transformationwaspreviouslyexpressedas:Lo Lg.However,toputthisinthebiggerperspectivethatincorporatesΓuand ,wecannowsay:Lo Lg /ΓuThisequationsaysthatwhatwewerepreviouslycallingLoandLgwerebothchangingrelativetoourabsolute lengthstandard thatwaspresentat thestartof theuniversewhenΓu 1.When we are dealing with the universe and time scales where the effects of a changingbackground Гu are significant, then it is no longer possible to adopt proper length as thecoordinate unit of length. A different coordinate length transformation is required tocharacterizetherelationshipbetweentheunitsofphysicswhentheyarecomparedatthesamelocationbutatsubstantiallydifferentagesoftheuniverse.ThevalueofГuincreaseswiththeageoftheuniverse.Inchapter3wewereabletoobtainall theothertransformationsusingdimensionalanalysisonce we had the transformations for length, time and mass. Previously these threetransformationswereexpressedas:Lo LgunitoflengthtransformationTo Tg/ГunitoftimetransformationMo Mg/Гunitofmasstransformation


InthesetransformationsthesymbolsLo,ToandMorepresentedcoordinate zerogravity unitsof length, timeandmass respectively. Now it isnecessary toadoptnewcoordinateunits torepresentaunitofcoordinatelength,coordinatetimeandcoordinatemassatthestartoftheBigBangwhen theuniversewasoneunitofPlanck timeold τu 1 andhadГu 1. Wehavepreviouslybeenusing torepresentoneunitofcoordinatelengthinauniversewhereГu 1.However,nowitisnecessarytoaddasubscript“1”tothisdesignationtoconformtoapatternwhereallunitsofphysicsneedtobespecifiedwhenГu 1.Forexample,E1,Q1andU1willbeused to specify a unit of energy, charge and energy density respectivelywhen Гu 1. Thesymbols 1and 1willbeusedtospecifyaunitofmassandtimerespectivelyatthestartoftheBigBangwhenΓu 1.Inchapter3thesubscript“g”wasusedtospecifyalocationingravity.Theanalogousconditionwhendealingwiththeevolutionoftheuniverseistospecifytheunitofphysicswhenitfeelstheeffectofabackgroundgravitationalgammathatisgreaterthan1 Γu 1 .Thisconditionwillbespecifiedbythesubscript“u”.Forexample,aunitoflength,timeandmasswhenΓu 1willbedesignatedasLu,TuandMurespectively.Fortheevolutionoftheuniversethetimeandmasstransformations are similar to those in chapter 3 but with new symbols. Only the lengthtransformation equation is not analogous to Lo Lg from chapter 3. The new lengthtransformationequationneedstospecifythefactthatwearenowrecognizingthechangeinaunitoflengththatscaleswithΓu.Thereforewehave:1 ГuLuunitoflengthtransformation1 Tu/Гuunitoftimetransformation1 Mu/Гuunitofmasstransformation

Inutilizingthemasstransformation 1 Mu/Гu,itisimportanttorecalltheassumptionstatedinchapter3thatthesamerateof timemustbeusedtoquantifyboth 1 andMu. Mass isameasurementofinertia,whichinturninvolvesforceandacceleration.Alloftheseimplytheuseofarateof time. Mass isnotsynonymouswithmatter. Inchapter3weoftenassumedthatcoordinatetimewouldbeused.However,inthecurrentuniversewithГuo 2.6 1031,therateofcoordinatetimeontheГ 1clockisabout2.6 1031timesfasterthantherateoftimeonthecosmicclock,soitmightnotbeconvenienttousecoordinaterateoftime.Allthatisimportantisthatwerememberthatthetransformationofunitsrequiresthatweusethesamerateoftimetoexpressboth 1andMuorotherunitsofphysicsatdifferentagesoftheuniversewithdifferentvaluesofГu.Because of the change in the length transformation, it is necessary to recalculate the othertransformationsusingthedimensionalanalysisproceduresestablishedinchapter3.Usingtheabovetransformationsforunitsoflength,timeandmassweobtain:


Impedance of Spacetime Zs :Zs1→ 1/ 1 Г

→Zsu

Zs1 Zsuimpedanceofspacetimetransformation

Energy E:E1→1 12/ 1 ГГ

Г

→Г3Eu

E1 Гu3Euunitsofenergytransformation

Energy Density U:U1→1/ 1 12 /Г

Г /Г →Uu

U1 Uuunitsofenergydensitytransformation

Coordinate Speed of Light : 1→ 1/ 1 Г

/Г →Г2 u

c 1 Γu2 ucoordinatespeedoflighttransformationThesearethemostimportanttransformationsandsomeofthemwillbeusedtodeterminethecurrentvacuumenergydensityandanalyzethe10120mystery.First,theimpedanceofspacetimeshouldbeunaffectedbyachangeinГu.ThefactthatthetransformationgaveZs1 Zsushowsthat the length, time and mass transformations are correct. This acts as a check on thetransformationprocess.Aboveweassumedthemasstransformationwasthesameaschapter3.In truth, this was not a foregone conclusion. However, the impedance of spacetime shouldremainconstant.Assumingthelengthandtimetransformations,thereisonlyonepossiblemasstransformationthatachievesaconstantimpedancetransformation.Energy and Energy Density Transformations: Next, the units of energy transformationE1 Гu3Eu will be illustrated with an example. Suppose that there was an electron in ahypotheticaluniversewithГ 1.TheenergyoftheelectronintheГu 1universewouldbe8.19 10‐14Joulesmeasuredlocallywhichisthesameenergywewouldmeasurefortheelectroninourcurrentuniverse.However,themeasurementintheГu 1universeusedalocalclockthatisrunning2.6 1031timesfasterthanthecosmicclockinourcurrentuniverse.Furthermore,ameterintheГu 1universeis2.6 1031timeslargerthanameterinourcurrentuniverse.Bothofthesefactorscombinetomake1JouleintheГu 1universeequivalenttoГuo3 1.8 1094joules inourcurrentuniverse. Therefore,even thoughbothelectronshave thesameenergymeasured locally, different standards of energy are being used. When we correct for thisdifference,theelectronintheГu 1universehasГuo3 1.8 1094moreenergy.Using therotarmodel, suppose thatwewanted tocompare theenergydensityof theГu 1electronandanelectronintheuniversetoday.Thetransformationofunitsoflengthis 1 ГuLu.


ThissaysthatameterintheГ 1universewouldbeabout2.6 1031timeslongerthanameterstickinourcurrentuniversebecausewearelivinginauniversewithГu 2.6x1031.Therefore,therotarradiusoftheelectronintheГ 1universewouldbe2.6 1031timesbiggerandtherotarvolumeofthatelectronwouldbeГuo3 1.8 1094timesgreaterthantherotarvolumeofanelectroninouruniverse.Theresultisthatbothelectronswouldhavethesameenergydensitybecause the1.8 1094 difference in the electron’s energy is offset by the factorof 1.8 1094differenceinthesizesoftherotarvolumes.ThetransformationofenergydensityisshownaboveandresultsinU1 Uu.This illustrates how the proper energy density of the universe (including vacuum energy) remains constant even when the universe experiences a vast increase in Гu.This is a fantastic result because it is a key component in solving themystery of the 10122differencebetweenvacuumenergydensityandcurrentlyobservedenergydensity.Whentheuniverse was Planck spacetime, it had energy density of 5.53 10112 J/m3. The spacetimetransformationmodelof theuniverseviews the currentuniverseas the samesize andsameenergydensityasPlanckspacetime.Therefore,thetransformationU1 Uusaysthattheproperenergydensityof theuniverseequals the tremendously largeenergydensityof theuniverseobtainedwhentheenergydensityisexpressedincoordinateunits.Itisnotnecessarytoaddenergytotheuniversetokeeptheenergydensityofvacuumenergyconstant.Instead,natureusestwodifferentstandardsforaunitofproperenergy inadditiontodifferentstandardsoflength,force,therateoftime,etc. Thisdifferenceinenergystandardsexactlyoffsetthechangein proper volume thereby maintaining a constant energy density. The total proper energydensityoftheuniverse includingvacuumenergy hasremainedconstantat5.53 10112J/m3sincethebeginningoftime sincetheBigBang .Todayalmostallofthisenergyoftheuniverseisintheformofvacuumenergy.Additional Transformations: If we carry these transformations further, we obtain a fewcounter‐intuitiveresults. Forexample,thetransformationsofcharge Q andmomentum p are:Q1 ГuQuunitofchargetransformationp1 ГupuunitofmomentumtransformationAtfirstthesetransformationsseemtobesayingthatneithercharge Q normomentum p isconservedwhentheuniverseagesandГu increases. However, thesearethetransformationsrequiredtopreservecharge,momentumandthelawsofphysicswhenmeasuredlocally propermeasurement andassumingaCMBrestframewhichhasthedistancebetweenpointsincreasewiththeHubbleflow.Themomentumtransformation p1 Гupu willbeusedtoillustratethispoint.


Wewillstartwithathoughtexperiment.SupposethatthereisahydrogenatominanexcitedstatethatisatrestrelativetotheCMBandalsoatrestattheoriginofacoordinatesystem.Thehydrogen atom emits a photon in the Y direction and the photon’smomentum causes thehydrogenatomto recoil in the–Ydirectioncarrying theoppositemomentum. Asshown inchapter5,themomentumimpartedtotheatombytheemissionofaphotonresultsintheatomhavingadeBrogliewavelengththatequalsthewavelengthoftheemittedphoton.IfweviewthisfromarigidframeofreferencethatdoesnotexpandwiththeHubbleflow,thenthereisnolossofmomentumovertime.However,ifweviewboththerecoilingatomandthepropagatingphoton from a coordinate system that expands with the Hubble flow, then relative to thiscoordinatesystemthereisalossofmomentum.BoththephotonandthedeBrogliewavesoftheatomundergoaredshift losemomentum relativetoacoordinatesystemthatexpandswiththeHubbleflow.Thecoordinatesystemusedbythespacetimetransformationmodelisrigidbutthe effect of an increasing Γu produces effects similar to adopting an expanding coordinatesystem. Therefore the equationp1 Гu pu ismerely expressing thisdifference inperceivedmomentumbetweenthetwocoordinatesystems.Similarly,thechargetransformationQ1 ГuQukeepstheproperlawsofphysicsunchangedinbothanexpandingcoordinatesystemandinthespacetimetransformationcoordinatesystemasГuincreases.10120 Calculation:Nowwearegoingtocalculatethecurrentratioofvacuumenergydensitytoobservableenergydensity.APlancksphereoriginallycontainedaboutabillionJoulesmeasuredusingthecoordinateenergystandardofenergybecausetheuniversestartedasPlanckspacetimewithГu 1.ThePlanckspherestartedwithradiusofPlancklengthandtodaythepropervalueof this radiushas increasedbya factorofГuo 2.6 1031 to0.42mmradiusoravolumeof3.1 10‐10m3.The109JofcoordinateenergywhenГu 1hashadanapparentincreasesothatcurrentlythismuchenergywouldappeartohaveincreasedbyafactorofГuo3.TheobjectiveofthefollowingcalculationistofindthecurrentvacuumenergydensityUvac.109J Гuo3 1.8 10103Jconversionofcoordinateenergytoproperenergy4π/3 lp3Гuo3 3.18 10‐10m3currentpropervolumeofPlancksphere1.8 10103J/3.18 10‐10m3 5.5 10112J/m3 UvacIgnoringvacuumenergy,thecurrentcriticalenergydensityoftheuniversedependsonthevalueof theHubbleparameterused. Using 70.8km/s/Mpc the critical energydensityof theuniverseUcritisabout8.5 10‐10J/m3ifweincludehypotheticaldarkenergy.Ifweexcludedarkenergywhich represents about 72.1% of the total energy density, thenwe have observableenergydensityUobsofabout2.36 10‐10J/m3.Uvac/Ucrit 5.5 10112J/m3/8.5 10‐10J/m3 6.5 10121ratioincludingdarkenergyUvac/Uobs 5.5 10112J/m3/2.4 10‐10J/m3 2.3 10122ratioexcludingdarkenergy


Either of these numbers qualifies as the famous 10120 discrepancy between the theoreticalenergy density of the universe and the observed energy density. Here is how we achievesphericalPlanckenergydensityusingoneofthe5wave‐amplitudeequations U H2ω2Z/c .Usingtheproperrateoftimeonthecosmicclock,thefrequencyappearstobePlanckangularfrequencyωp.Furthermore,strainamplitudeisadimensionlessnumberthatdoesnotchangewithГu.ThereforewewillinsertH 1.Finallywemustinserttheconstantk’toconvertfromcubictosphericalwiththefactorof½associatedwithzeropointenergy.U H2ωp2Zs/csetωp 1.855 1043s‐1;Zs 4.038 1035kg/s;H 1andaddk’ 3/8πUps k’H2ωp2Zs/c 3/8π 12 1.855 1043 2 4.04 1035 /3 108Ups 5.53 10112J/m3Ups Uvac UobsThespacetimetransformationmodeloftheuniverseproposesthatovertheageoftheuniversetherehasbeennochangeinthetotalenergydensityoftheuniverse.TodayvirtuallyalloftheenergydensityoftheuniverseisintheformofvacuumenergyUvacwhichlacksquantizedangularmomentum.However,atthestartoftheBigBangalltheenergydensityofPlanckspacetimeUpswas observable energy density Uobs because all the energy possessed quantized angularmomentum.Overtimethetransformationofspacetimehasresultedinadramaticdecreaseintheobservableenergydensityoftheuniverseandanequalincreaseinvacuumenergydensityof the universe. Today Uvac 10122Uobs but the total energy density has not changed:Uvac Uobs Ups.TodayweperceivethemaximumfrequencyofthewavesthatformvacuumenergytobeequaltoPlanckangular frequency. However, this isaproper frequencythathasbeenslowedbyafactorofГuo 2.6 1031comparedtothecoordinatefrequencythatoccurredwhentheuniversewasPlanckspacetime.Howisitpossiblefortoday’svacuumenergytopossessvirtuallythesameenergydensityasPlanckspacetimeifthecurrentmaximumfrequencyofthedipolewavesisafactorofabout2.6 1031timesslowerthanthedipolewavesthatformedPlanckspacetime?Theanswertothisquestionisanalogoustotheanswergivenpreviouslyinthesectiontitled“EnergyandEnergyDensityTransformations”.ThereitwasshownhowtheenergydensityofanelectronremainsconstantevenwhenthereisabigincreaseinГu.Theenergyscalesproportionalto1/Гu3butthevolumealsoscaleswith1/Гu3sotheenergydensityoftheelectronremainsconstant..This holds true for any dipole wave in spacetime that has a specific frequency and strainamplitude.ThehighestfrequencydipolewaveshaveaproperfrequencyequaltoPlanckangularfrequencyωpandapropervolumethatisPlancklengthinradius.However,thisvolumeis1/Гu3times smaller than it was in Planck spacetime. The wavelets that form vacuum energy arecontinuously forming new wavelets as previously explained. These wavelets adapt to thechangingscaleoflength.


Illustrations Showing the Effect of Гu on Waves:Next,wewanttoseewhathappenstothewavesinspacetimethatformvacuumenergywhenthereisanincreaseofГu.Themysterytobeexplainedishowthewavestructureofvacuumenergychangestoresultinanincreaseinpropervolumeastheuniverseages.Infigure14‐2wehavetwosinewavesdesignatedwave#1andwave#2. These are crude representationsof thedipolewaves in spacetime responsible forvacuumenergy.Sincethenonlinearityisparticularlystrongintheearlypartoftheevolutionoftheuniverse,insteadimaginetheseasrepresentingvacuumenergyatmorerecenttimes.Infact,wave#2canbethoughtofasrepresentingvacuumenergytodaywithГuo 2.6 1031andwave#1 representing vacuum energy when the comoving grid was 1/3 its current size which isequivalent to Гu 1031 . Therefore, it is important to remember that there is a factor of 3differencebetweenthevalueofГuforwave#1comparedtothebackgroundgammapresentforwave#2.Thisiswrittenas3Гu1 Гu2.Bothofthesewaveswouldbeexactlythesameiftheyweredrawnusingproperunitsoflength.ThedisplacementamplitudeofbothwavesisdynamicPlancklengthwhenthedisplacementofspacetimeisexpressedinunitsofproperlength.However,figure14‐2usescoordinatelengthforboththe andYaxis.Therefore,thespatialdisplacementamplitudeofwave#1is3timeslargerthanthespatialdisplacementamplitudeofwave#2becauseofthefactorof3differencebetweenГu1andГu2.Thedisplacementamplitude Yaxis issetsothatwave#1hasamplitudeof 1. Thismakes the coordinate amplitude ofwave #2 equal to 0.333. If the displacement


amplitudewasexpressedusingtheabsolutecoordinatescalewherePlancklengthequals1whenГu 1,thenwave#1wouldhaveadisplacementamplitudeof10‐31.Thisisbecausewave#1ispresumedtoexist inauniversewithabackgroundvalueofГu 1031. This largevalueofГucontractsproperlengthcomparedtoaunitofcoordinatelength 1.Themaximumslopeofasinewaveoccurswhenthesinewavecrossesthezeroline.Thearrowshowsoneofmanypointswherethetwowaveshavethe“samemaximumslope”.Thisslopeisadimensionlessnumberthatisthestrainamplitudeofthesinewave.Thepointofthisfigureistoshowthatthemaximumslopeisthesameeventhoughthewaveshaveadifferentscale.Thestrainproducedbywavesinspacetimeisproportionaltothemaximumslope.ThereforebothwaveshaveastrainamplitudeofH 1.Naturally,theslopewouldalsobethesameifthewaveswere drawn using proper length because both waves would then be exactly the same indisplacement, wavelength and maximum slope. The waves that form vacuum energy canmaintainthesamestrainamplitudeevenwhenГuisincreasing.Thefrequency,measuredlocally,remainsthesamesotheproperenergydensityalsoremainsconstantwhenГuincreases.ThepointisthatthestrainamplitudeisalwaysH 1forallvaluesofΓu.Thisisakeycomponentinmaintaining the total energydensity of theuniverse at 10113 J/m3 throughout the ageof theuniverse.Itisinterestingtonotewhatthesewaveswouldlooklikeiftheywereplottedinthetemporaldomain rather than the spatial domain. The Y axiswould be labeled “Coordinate TemporalDisplacementAmplitude”andthe axiswouldbelabeled“CoordinateTime”.Thefigurewouldphysicallylookthesameasfigure14‐2exceptthatthelabelsforwave#1and#2wouldbe


reversed.Wave#2 largervalueofГu wouldhavethelargertemporaldisplacementamplitudewhenmeasuredincoordinateunitsoftime.ThiscomparisonhelpstoillustratehowachangeinГuexchangesthetemporalpropertiesofspacetimeforthespatialpropertiesofspacetime.Figures14‐3 showsa3‐dimensionalplotofwave#1 in figure14‐2and figure14‐4 showsa3‐dimensionalplotof3layersofwave#2infigure14‐2 originaldefinitionsofГu .Thesetwofigures are oversimplified. Thewave structure should bemore chaotic and unsymmetrical.Imaginethewavesinfigure14‐4asoscillatingat1/3thefrequencyofthewavesinfigure14‐3.Thegridpatterninfigure14‐4isonly1/3thecoordinatelengthsoeachgridcubehasonly1/27thecoordinatevolumeofthegridcubeinfigure14‐3.However,eachgridcubealsoonlycontains1/27thecoordinateenergyasthegridcubeinfigure14‐3,sotheenergydensityisthesamenomatterwhether it is assessedusing theproper standardof energydensityor the coordinatestandardofenergydensity.Quantummechanicshasbeentellingusthatthevacuumenergydensityshouldbeconstantevenas the universe ages and the proper volume increases. Now it is possible to see that thespacetimebasedmodeloftheuniverseshowsthatthisispossible.Infact,inorderforthelawsofphysicstoremainconstant,itisnecessarythatthevacuumenergydensityremainsconstant.Ifthevacuumenergydensitydecreasedasthepropervolumeoftheuniverseexpanded,thenthehighfrequencyvirtualparticlepairswouldeventuallybelostandthiswouldbedetectable.Does Dark Energy Exist? Darkenergyissupposedlyahomogeneousformofenergythatformsasthevolumeoftheuniverseexpands.Everythingabouthypotheticaldarkenergyconflictswiththeconceptspresented in thisbook. There isnosingleexplanation fordarkenergy,but thesimplestexplanationgivenfortheexistenceofdarkenergythatscaleswithvolumeisthatdarkenergyis“thecostofhavingspace”.Eachtimecosmicexpansionsomehowcreatesanadditionalcubicmeterofspacetime;thisvolumeissupposedlyleftwithanenergydeficitofabout6 10‐10Joulesof“negativeenergy”thatisconsideredtobedarkenergy.Intheearlyuniverse,whentheenergydensityofmatterandphotonswashigher,6 10‐10 J/m3was insignificant. However,todaythepropervolumeoftheuniversehasincreased.Thedensityofmatterandlighthasfallensothattodaydarkmattersupposedlymakesupabout73%oftheenergydensityoftheuniverse.Inthisconcept,gravityisattemptingtocollapsetheuniverse,butdarkenergyopposesgravityandcausesanacceleratedexpansionoftheuniverse.Theexactmechanismusedtoaccomplishthisacceleratingexpansionisvague.Ifitistheoppositeofgravity,thenthiscreatesaproblemforthemodeloftheuniverse.Recallthatitwaspreviouslyshownthatgravitationalaccelerationrequiresagradientintherateoftime.Therefore,anythingthatcausesananti‐gravityrepulsionmustaccomplishthisbyarateoftimegradientthatopposesgravity. However,theobservedredshiftofgalaxieswouldrequirealargescalegradientintherateoftime.Theproblemisthatalargescaletimegradientintheuniverseisincompatiblewiththeconceptthattheuniverseishomogeneousbothspatiallyandtemporallyonthelargescale.


TheΛ‐CDMmodeldoesnotrespecttheconditionsthatmustbemettocreateacubicmeterof“new”space.Creatingevenacubicmeterofnewspacerequiresalotofconditionstobemet.This new spacemust have the impedance of spacetime Zs c3/G and the interactive bulkmodulusofspacetimeKs Fp/λ2.Thenewspacemustbefilledwithzeropointenergyatenergydensity of 10113 J/m3. Therefore, each new cubic meter requires more energy than theannihilation energy of entire observable universe. As before, the problem is in the physicalinterpretationofobservationsandequations.Iftheproperdistancebetweengalaxiesincreases,thiscanbeinterpreteddifferentways.Themodelproposedhereisactuallythesimplestbecauseitdoesnotdemandanynewphysicsornewenergytobeaddedtotheuniverse. Itdoesnotrequiremysteriousdarkenergy.Thereisnodirectexperimentalevidencethatdarkenergyexists.Darkenergyisatheoreticalconceptispostulatedtoexplaintheapparentaccelerationoftheexpansionoftheuniverseandalso toexplain that theenergydensityof theuniversehas fallenbelowthe“criticaldensity”.Baryonic matter, dark matter and radiation only achieve about 28% of the energy densitycalculatedtobenecessarytoachieveflatspacetime.However,thiscalculationdependsontheaccuracyofthemodeloftheuniversebeingused.Theconceptof“criticaldensity”oftheuniverseassumesthattheuniversepossessesagravitationalgradient.Thisdoesnotexistintheconditionpreviously described as “immature gravity”. It does not make any difference whether theimmaturegravityoccursinthelowgravitationalГofthedustcloudthoughtexperimentorthehighgravitationalГuoftheuniverse.TheimportantpointisthatimmaturegravityproducesanincreasinggravitationalГuandauniforminstantaneousrateoftimeintheCMBrestframe.Ifthereisnolargescalerateoftimegradientfromthemidpointobserverperspective,thenthereis no large scale gravitational acceleration and nothing that demands an explanation thatincorporatesanti‐gravity.Theconceptofcriticaldensityoftheuniverseassumesthatthereisagravitationalaccelerationthatisattemptingtocollapsetheuniverse.Iftheuniverseispicturedasthehomogeneousandstatic distribution of galaxies with proper volume increase because of the spacetimetransformationoftherateoftimeandofproperlength,thentheuniverseisnotstrugglingtoexpandagainstgravity. There isno such thingasa criticaldensity. Thedust cloud thoughtexperimentdidnotmeettheconditionsof“criticaldensity”andyettherewasnogravitationalaccelerationinthefirstfewmillisecondsaftergravitywas“turnedon”.As long as theuniversehasnodetectableboundary no edge , themature gravity conditioncannotbeestablished.Ittakesadensitychangeataboundarytoestablisharateoftimegradientand gravitational acceleration. The proposed model of the universe started with Planckspacetimethathadauniformrateoftime. Atspeedof lightcommunication,westillhavenodetectableboundary.Therateoftimehassloweddownbuttherestillisnolargescalerateoftime gradient. Gravitational acceleration and curved spacetime both require a rate of time


gradient. Therefore, the universe has never possessed large scale curved spacetime orgravitationalacceleration.Newmass/energywillcontinuetoappearontheparticlehorizonoftheobservableuniverse.ThebackgroundГuoftheuniversewillcontinuetoincreasetowardsinfinityandtherewillbenorateoftimegradientonthescaleofuniversalhomogeneityunlessone day we become aware of a large scale density discrepancy that is the equivalent of aboundaryconditionthatgivesan“edge”totheuniverse.Dark Energy Not Needed:Whatisbeingproposedisthatthespacetimetransformationmodeldoesnotrequiretheinventionofdarkenergytoprovidethemissingcriticaldensityanddoesnot need any mysterious force with anti‐gravity properties that is causing the apparentexpansionoftheuniversetoaccelerate.Whenviewedfromtheproposedcoordinaterateoftimeandcoordinateunitof length, there isnoexpansionof theuniverse. Nowork isbeingdoneagainstgravity.Theimmaturegravityconditionpreviouslydiscussedeliminatesthetendencyfortheuniversetohaveagravitationalcontraction.Thecoordinatevolumeoftheuniversehasnever changed and the coordinate energy density including vacuum energy has remainedconstantatthelargescaleof300,000lightyears.Atasmallerscalematterhasformedstarsandgalaxies which distort the homogeneous energy density of vacuum energy. We call thisdistortion “curved spacetime”. Our perception of the volume of the universe indicatescontinuousexpansion.However,thisistheresultofacontinuousincreaseinthebackgroundГuof the universe. What we perceive as acceleration of the expansion is due instead to anaccelerationintherateofchangeofdГu/dτu.AllthefactorsthatdeterminedГu/dτu therateofchangeofГuinpropertime arenotknown.Thiswouldbe a function the ageof theuniverse, but it probably also includesother factorsrelating to thecompositionand theobservableenergydensityof theuniverse. Forexample,whentheuniversewasradiationdominated,asubstantialamountoftheobservableenergywasbeingconvertedtovacuumenergy.ThisprocessresultedindГu/dτubeingproportionaltoτ1/2.Duringthematterdominatedepochtheelectromagneticradiationwasasmallpercentageoftheobservable energy of the universe and dГu/dτuwas proportional to τu2/3. Todaywe have anincreaseinpropervolumethatisacceleration.IfthisisviewedasanaccelerationintherateofchangeofdГu/dτu,thenamysterystillexists,butitdoesnotdemandtheinventionofdarkenergyforanexplanation. ThesolutionistobefoundinthepropertiesofspacetimethatcreatetheaccelerationofdГu/dτuforthecurrentconditionoftheuniverse.Offsetting the Rate of Change of Гu:ReturningtotheincreaseinГu,howfastwouldanobjectneedtoberaisedintheearth’sgravitationalfieldinorderforthedecreaseintheearth’sГtooffset the increase inГuof theuniverse? Inotherwords,what rateof increase inelevationachieves dГ/dt /Г 2.29 10‐18s‐1intheearth’sgravitationalaccelerationof9.8m/s2?

g c2 c2Г

Гset

Г

Гweakgravityapproximation


Г

Г set

Г

Г andg 9.8m/s2

.021m/s verticalvelocity

Thereforeanelevationvelocityofabout2.1cm/sorabout75metersperhour intheearth’sgravityoffsetsthetemporaleffectsofanincreaseintheГuoftheuniverse.Obviouslythisisonlyatemporaryreprievemadepossiblebecauseanobjectintheearth’sgravitystartsoffatalowerenergystate largertotalГ thanthesameobjectifitwasisolatedonthecomovingcoordinatesystem.Still,thisexamplegivesaphysicalfeelfortherateofchangethatiscurrentlytakingplaceintheuniverse.AllphysicalobjectsarelosingenergyeachsecondwhenmeasuredwithanabsoluteenergyscalethatdoesnotdecreaseasΓuincreases.Forexample,thesuniscurrentlyradiatingabout4 1026wattsofelectromagneticradiationbutthesunislosingabout1000timesthisenergypersecondastheenergyinthesun’srotar’sisbeingconvertedtovacuumenergy.ThisisanundetectableeffectusingtheproperenergystandardwhichdoesnotacknowledgetheeffectofanincreasingΓuoneverythingintheuniverse.Thereisanotherinterestingwayoflookingatthechangingrateoftimeastheuniverseages.Anelectronhastwodifferentratesoftimeinitstwolobesasexplainedinchapter5.Theseratesoftimedifferbyαβ 4.18 10‐23.HowmanysecondsdoesittakeforΓutochangebyafactorof4.18 10‐23?Inotherwords,whatdifferenceintheageoftheuniverseproducesarateoftimedifferenceequaltotherateoftimedifferenceinanelectron?Aβ/ 4.18 10‐23/2.29 10‐18s‐1 1.8 10‐5secondTime’s Arrow:Theequationsofphysicsseemtobereversibleintime.Exceptforentropy,itappearsasifitshouldbepossibletogobackwardsintime.However,ifthebackgroundГu oftheuniverseisincreasingcontinuouslyandallmatterisconvertingenergyintovacuumenergy,thenitisnotpossibletogobackwardsintime.Yesterdayalltherotarsandphotonsintheuniversehadmoreenergythantheypossesstoday measuredonthescaleofcoordinateenergy .Also,thelowerbackgroundГuofyesterdayalsoaffectsmanyotherthingssuchastheunitsofforce,velocity,voltage,etc.Eventhoughthelawsofphysicsarethesametodayandyesterday,allthecomponents that makeup the universe are different. The universe is undergoing atransformationandthismakesTime’sarrowonlypointonedirection–tothefuture.


Black Holes:Thefollowingdiscussionofblackholesismorespeculativethantherestofthisbook.Thereforethefollowingshouldbeconsideredjustafewpreliminarythoughtsaboutblackholes.Doblackholeshaveadifferentstructureinaspacetimebaseduniversethantheywouldhaveiftheuniverseispopulatedbypointparticles?Sofarthegeneralrelativityanalysisofblackholeshasindirectlyassumedthestandardmodelofparticles.Withthepointparticleassumption,ablackholehasanaccretiondisk,aneventhorizon,avolumeinsidetheeventhorizonandfinallyasingularityatthecenter.Thissingularitysupposedlyhasinfiniteenergydensity thesameaspoint particles . The volume inside the event horizon supposedly has modulation of thepropertiesofspacetimethatwouldrequireinexcessof100%depthofmodulationofspacetime.Clearly these conditions cannot be achieved by the spacetime based model of the universeproposedhere.Theeventhorizonofablackholesupposedlyhasarateoftimethatisstoppedandacoordinatespeedoflightequaltozero.Itisquestionablewhetheracompletestoppageoftherateoftimeandstoppingthepropagationoflightcanbeachievedbythewave‐basedmodelofhadronsandbosonsproposedhere.Ifyourmodelofafundamentalparticleisapointparticlewithnophysicalsizeandnostructure,thensuchaparticlewouldbeabletosurvivetheplungepasttheeventhorizonofablackhole.However,ifweassumetherotarmodelofmatter,thenapreliminaryanalysisseemstoindicateadifferentanswer.Aspreviouslyexplained,arotarisjustaslightdistortionofspacetimethathasaspecificfrequency,rotarradius,anddisplacementamplitude.Itseemsasifaspacetimebasedexplanationoftheuniversecannotformatrueblackholeeventhorizon.Thisisbecausesuchaneventhorizonwouldeliminatethewavesinspacetimerequiredforitsformation.Ifamasscollapsedtoadegreethattherateoftimeissloweddownbyanenormousamountsuchas1020 or more compared to the comoving rate of time , then externally this would beindistinguishablefromaconventionalblackhole.Inthisscenario,afterablackholeforms,alladditionalmass/energythatfallstowardstheblackholeaddstotheorbitingaccretiondiskandneverreachesaneventhorizon.Thespacetimewavepropertiesofrotarsandphotonswouldhave tobe taken into consideration inorder toproperlycharacterize theaccretiondisk thatneverquitereachesaneventhorizon. Ifhadronsandbosonsneverquitereachatrueeventhorizon,thenthiswouldexplainhowitispossibleforinformationabouttheblackhole’scharge,magneticfield,massandrotationaldirectioncanbecommunicatedtotherestoftheuniverseoutsideoftheblackhole.Likeanygravitationalcapture,massmustshedsomeenergyinordertobecapturedbyapseudoblackhole. This sheading of energy is doneby the emission of radiation andby the energyemittedbythepolarjetsassociatedwithblackholes.Theenergythatiscapturedcanchangeitsformbutitsgravitationaleffectremainsconstant.Recalltheexamplepreviouslygivenofaplanetinahighlyellipticalorbitaroundastar.Thetotalgravitationaleffectofthecombinationofthestarandtheplanetisconstanteventhoughtheenergyintheplanetchangesform.Similarly,a


photonfallingintoapseudoblackholewouldappeartobeblueshiftedifthephotoncouldbeobservedlocallyinaregionwithahighgravitationalgammaГ.ArotarwouldgainkineticenergytooffsetthelossofinternalenergyassociatedwithahighГ.Inneithercasedoestheenergypassaneventhorizonwherecontactwiththeoutsideuniversewouldbelost.Themodelofspacetimecurrentlyacceptedisthattheeffectsofcurvedspacetimecansomehowtranscendaneventhorizon.Wecanobviouslyaccuratelymeasurethemass,spinandchargeofablackhole.Theseareexamplesofcommunicationthatappearstobecomingfrominsideaneventhorizon.Thespacetimebasedmodelofgravityrequireswavesinspacetimetoproduceanonlinearinteractioninspacetime.Whentheenergydensityofmatterandradiationapproachesthe energy density thatwould require 100%modulation of the spacetime field thenwe areapproachingtheconditionsofablackhole.However,thespacetimebasedmodelneveractuallyreaches100%modulationofspacetime.Timeneverquitestopscomparedtocoordinatetimeand length never quite contracts to zero compared to coordinate length. The singularityassociatedwiththeconventionalblackholerequiresenergydensityinexcessofPlanckenergydensity. This is usually “explained” by saying that “the laws of physics break down”. Thespacetimebasedmodeloftheuniverseneverrequiresthatthelawsofphysicsbreakdown.Ivisualizethevolumenearthecenterofawavebasedblackholetobeprimarilyphotonsthathavebeenhighlyblueshiftedrelativetothelocalrateoftime.Matterfallingintotheaccretiondiskwillundergohighlyenergeticcollisions. Whilenewparticleswouldbeformed,repeatedcollisionsanddecompositionswouldeventuallyresultinahighpercentageoftheenergybeingphotons. Therefore, photon densitywould increasewith depth. Therewould be no eventhorizon,butenergyintheaccretiondiskwouldcausethegravitationalgammaГtoapproachinfinity.Forexample,supposethatthisenergydensityachievesarateoftimethatis1020timesslower than the surrounding volume of the universe. Thiswould look like a black hole, butinformation about charge,mass and rotational direction could still be communicated to thesurroundingspace.The Spacetime Transformation Model Versus The Inflationary Model: Inchapter13weperformed several calculations to find the value of Гuo. However, the same data can berearrangedtosupportthecontentionthattheproposedspacetimetransformationmodeloftheuniverseiscorrectandthattherewasnotaninflationaryphase.Hereisthereasoning.WhenwecalculatethechangeinscalefactorstartingfromoneunitofPlancktime ~5 10‐44s andendingwith13.7billionyears,weobtainedscalingfactorsof2.1,2.6,2.95andanupperlimitof3.4 all 1031 .Thesenumbersareapproximatelythesameyettheywereobtainedfromdiversesources such as the currently observable energy density of the universe, the observed CMBtemperatureandtheCMBphotonenergydensity.Nowwewillreversethethoughtprocessandextrapolatebackintimetowhentheuniversewas5 10‐44 seconds old 1 unit of Planck time . Startingwith the currently observable energy


density,CMBtemperatureandCMBphotonenergydensityoftoday’suniversewealwaysarriveat the properties of Planck spacetime using an average value of Γuo 2.6 1031. Thisextrapolationmakestheassumptionthattherewasnoinflationaryphaseintheexpansionofthepropervolumeoftheuniverse.However, suppose thatwe include inflation in thisbackwardsextrapolation. Betweenabout10‐35secondsand10‐32secondswehavetodeviatefromtheradiationdominatedconditionthatscaleswithτu1/2and insert the inflationaryexponentialscaling factor. This inflationfactor isunknown,but it is usually considered tobe in excessof 1025. At an ageof 5 10‐44 second,includinginflationimpliesthattheenergyperphotonexceedsPlanckenergybyafactorofatleast1025.SimilarlytheimpliedtemperatureexceedsPlancktemperaturebymorethan1025andtheimpliedenergydensityexceedsPlanckenergydensityalsobyasimilarfactor. Theseareimpossibilitiesaccordingtotheknownlawsofphysics.Thereforephysicistscasuallydisregardthisbysayingthatthelawsofphysicsmust“breakdown”undertheseconditions.Eventheideathattheinflationaryexpansiongreatlyexceededthespeedoflightrequiresabreakdownofthelawsofphysics.Thereisnoexperimentalproofthatitispossibleforthelawsofphysicsto“breakdown”.Thisismerelyatermusedwhenaparticulartheorygivesanimpossibleansweraccordingtotheknownlawsofphysics.Instead,whenatheoryrequiresabreakdowninthelawsofphysics,thisshouldbeastrongindicationthatthetheoryiswrong.Thebeautyofassumingthattheuniverseisonlyspacetimeisthatthereshouldbenocaseswherethetheoryneedstoreverttosayingthatthelawsofphysicsmustbreakdowninordertoexplainaparticularimpliedresult.Cosmicinflationisanadhocsolutionrequiredbyamodeloftheuniversethathaspointparticlesandforcescarriedbytheexchangeofvirtualparticles.Iftheuniverseisonlyspacetime,thenitwasonlyspacetime thecompositequantummechanicalandrelativisticspacetimemodel evenat thebeginningof theBigBang. Extrapolatingbackwards fromtodayresults in the “Planckspacetime”homogeneousstate.Thisisthehighestobservableenergydensitythespacetimefieldcansupport.Thelawsofphysicsneverbreakdown.Forexample,therearenosingularitiesinthisspacetimebasedmodeloftheuniverse.Allthestepsareconceptuallyunderstandableandaccessible to physicists today. Planck spacetime is as homogeneous as quantummechanicsallows,sothereisnoneedforinflationtoexpandspacetimetoachievelocalhomogeneity.Unity and Entanglement Revisited: It was previously proposed that quantized waves inspacetimesuchasrotarsandphotonscanhaveinternalcommunicationfasterthanthespeedoflight.Thispropertyalsoextendstocommunicationbetweentwoentangledphotonsorrotars.Noinformationcanbeimposedonthisinternalcommunicationsothereisnoviolationoftheprohibition against faster than light communication. Still there is a question about howspacetimeaccomplishesthefasterthanspeedoflightinternalcommunication.Onepossibilityisthatthisinternalcommunicationmightbetakingplaceatthespeedoflightcharacteristicof


Planckspacetime.Thisspeedwouldbeabout2.6 1031timesfasterthantheproperspeedoflight. At this speed, internal communicationwithinaphotondistributedoverone lightyearwouldonlytakeabout10‐24secondandaCMBphotongenerated380,000yearsaftertheBigBangwouldcollapsewithinabout10‐14seconds.ThemicrowavephotonsthatmakeuptheCMBhaveapeakfrequencyofabout1.6 1011Hz.Thereforeacollapsewithatimedelayofabout10‐14 s would meet the conditions of being virtually instantaneous. Perhaps the internalcommunicationisactuallyinstantaneous,butcommunicationat2.6 1031timesfasterthancisindistinguishablefrombeinginstantaneous.Are All Frames of Reference Really Equivalent?Abasicassumptionofrelativityisthatallframes of reference are equivalent. The CMB rest frame is clearly the preferred frame forcosmologicalpurposes,butthelawsofphysicsarepresumedtoworkequallywellinallframesof reference. Experimental observations have not detected any preference for frames ofreference,butdoesthismeanthatultra‐relativisticframesrelativetotheCMBrestframeareequivalent?Recallthatinchapters4and11thesubjectofthespectralenergydensityofzeropointenergy quantizedharmonicoscillators wasdiscussed.Itwasstated:

“Thisspectrumwithitsω3dependenceofspectralenergydensityisuniqueinasmuchasmotionthroughthisspectraldistributiondoesnotproduceadetectableDoppler shift. It is a Lorentz invariant random field. Any particular spectralcomponentundergoesaDopplershift,butothercomponentscompensatesothatallcomponentstakentogetherdonotexhibitaDopplershift.”

Thereisoneproblemwiththisconcept.VacuumfluctuationshaveacutofffrequencyequaltoPlanckfrequencyωp. IfthiscutofffrequencyissymmetricalwhenviewedfromtheCMBrestframe,thentheremustbeanultra‐relativisticframeofreference relativetotheCMB wheretheasymmetrybecomesobvious.Anexamplewillhelptodefinethisquestion.Wecancurrentlyaccelerateanelectrontoenergyof50GeV.ThisisarelativisticLorentzfactorofγ 105relativetotheCMBrestframe.However,aframeofreferencewithγ 105doesnotcomeclosetotestingthequestionsrelatedtothelimitsofextremeultrarelativisticframesofreference.Imagineanelectronwithanultra‐relativisticspeedwithγ 2.4 1022asseenfromtheCMBrestframe.ThisistheLorentzfactorwheretheelectron’sdeBrogliewavelengthλdwouldbeshorterthanPlancklength λd λc/γapproximationvalidwhenγ 1 .Thisisveryclosetothespeedoflightbutitdoesnotequalthespeedoflight.Therefore,itishypotheticallyapermittedframeofreferenceforanelectron.However, in the CMB rest frame the electronwould have a de Brogliewavelength less thanPlancklengthanddeBrogliefrequencyexceedingPlanckfrequency.Accordingtothepremiseofthisbook,spacetimeisnotcapableofproducingthiswavelengthandfrequency.Also,intheelectron’sframeofreferencetherewouldbeanextremeredshiftinonedirectionofthedipolewaves in spacetime required to stabilize the energy density pressure of the electron. This


redshiftwouldpreventthevacuumenergyfromexertingthepressurerequiredtostabilizeanelectroninthisframeofreference.Iftheuniverseisonlyspacetime,thenthisframeofreferenceisnotpermittedforanelectron.InstabilitywouldappearasanelectronapproachedthePlancklength/frequencylimitasseenfromtheCMBrestframe.Theelectronwouldexhibitpropertiesinthisultra‐relativisticframeofreferencethattheelectrondoesnotpossessintheCMBrestframe. Other particles would exhibit unusual properties and instabilities at differentultra‐relativisticframesrelativetotheCMBrestframe.Foranotherexample,inchapter9wedeterminedthatphotonsarequantizedwavespropagatinginthemediumofthespacetimefield.ThisimpliesthataphotonisnotpermittedinanyframeofreferencewhichwouldappeartohaveawavelengthshorterthanPlancklengthintheCMBrestframe.Thereisonlyonespacetimefieldforallframesofreference.Thisfieldisnotcapableofpropagatingwavesshorterthantheshortestwavelengthwaveinthefield.Thecurrentrecordfor thehighestenergyphotoneverobserved isa12TeVgammaray ∿2 10‐6 J whichhaswavelengthofabout10‐19m.Thisenergyphotonispermittedinourframeofreference,butitwouldnotbepermittedinanyframeofreferencewhichexceededaboutγ 1016relativetotheCMB. Thereason is that thisphotonwouldhaveawavelength less thanPlanck lengthwhenviewedintheCMBrestframeandtheenergywouldexceedPlanckenergy.Thisimpliesthatthelawsofphysicschangeintheseextremeframesofreference.String theory is based on three starting assumptions which are expressed as mathematicalequations.Theseare1 Lorentzinvariance, thelawsofphysicsarethesameinalluniformlymoving frames of reference 2 analyticity a smoothness criteria for the scattering of highenergyparticlesafteracollision and3 unitarity allprobabilitiesalwaysadduptoone .Thespeedoflightisthesameinalluniformlymovingframesofreference,butthelawsofphysicsarenot. Therefore the contention is that Lorentz invariance is not a valid assumption for alluniformlymoving framesof reference.String theory incorporates thiserroneousassumptionandalltheconclusionsbasedonthisassumptionarequestionable.Lorentz Was Correct:EventhoughtodayweoftenassumethatEinsteindidnotbelieveintheaether,heactuallycontinuedtorefertotheaetheror“physicalspace”untilhisdeath2.Hemerelydeclaredthattheaethermusthaverelativisticpropertieswithnopreferredframeofreference.Lorentzalsobelievedintheaether,buthiscalculations Lorentztransformations assumedthatthe aether had a preferred frame of reference. Lorentz transformations did not confer anddetectable special properties to this preferred frame of reference since the transformationsmadethelawsofphysicsthesamein“all”framesofreference.However,EinsteinandLorentzdisagreed about whether the aether had some special reference framewhich served as thestandardforallotherreferenceframes.

2 L. Kostro, Einstein and the Ether, (2,000) Apeiron, Montreal, Canada


Thespacetimefieldcanbethoughtofasatypeofaether.Sinceallparticles,forcesandfieldsaremadefromthisspacetimefield,thereisaconceptuallyunderstandablereasonwhyexperimentsusingcurrentlytechnologycannotdetectmotionrelativetothespacetimefield.Theparticlesand forces merely undergo the transformations required to achieve Lorentz invariance.However,Lorentzwascorrect.Thespacetimefielddoeshaveapreferredframeofreference.Itis theCMBrest frame. Noexperiment capableofbeingperformedusingcurrent technologywouldbeabletoexperimentallyprovethisbecauseitwouldrequireacceleratingaparticletoaspeedwhereitsdeBrogliewavelengthapproachesPlancklength.Forexample,anelectroncouldbeacceleratedtoaspecialrelativitygammaofγ 2.4x1022iftheexperimentisdoneintheCMBrestframe.Theelectronwouldthenhaveλc Lp.Theelectron’sinternalpressurewouldbeequaltoPlanckpressureandtheabilityofthespacetimefieldtostabilizetheinternalpressureoftheelectronwouldbeatitslimit. Allotherframesofreferencewouldnotbeabletoachievethisvalueofγbecausetheelectronwouldbecomeunstableata lowervalueof γ.Therefore, thespacetimefielddoeshaveapreferredframeofreference,butitiscurrentlyundetectable.The Fate of the Universe:Thecurrentlyacceptedmodeloftheuniversehasmysteriousdarkenergybecomingmoredominantandacceleratingtheexpansionoftheuniverseuntilwelosesightofdistantgalaxies. In themostextremeextensionof thisprocess,aBigRipeventuallyoccurs when the expansion becomes so extreme that gravitationally bound objects such asgalaxiesandstarsaredispersedbytheexpansionofspace.Finallyevenatomsarerippedapartandtheuniversediesassubatomicparticlesareeventuallyconvertedtophotons.Themodelproposedherehasnotbeendevelopedsufficientlytohaveaclearpredictionabouttheeventualfateoftheuniverse.However,aspreviouslyexplained,thenearterm afewbillionyears has distant galaxies getting dimmer but also the currently observed redshift of anyparticular distant galaxy will decrease. This counter intuitive prediction is actually acontinuationoftheprocessthathasoccurredthroughoutthehistoryoftheuniverse.Overthelongertermthespacetimetransformationmodeloftheuniverseoffersanintriguingpossibility. The total energy density of the universe observable energy vacuum energy remainsthesameoverthelifetimeoftheuniverse.Presentlyobservableenergy includingdarkmatter representsonlyabout1partin10122ofthetotalenergyintheuniverse.Aspreviouslyexplained,weonlycanobserveand interactwithwaves inspacetime thatpossessquantizedangularmomentum fermionsandbosons .Furthermore,thefractionofthetotalenergythatpossesses angular momentum 10‐122 is dropping daily and the rate of change of dГu/dτuappearstobeaccelerating.Iffundamentalparticleseventuallydecayintophotonsinthefardistantfutureoftheuniverse,thenan intriguingpossibilityexists. When thequantizedangularmomentumof thephotonsbecomeshomogeneouslydistributedthroughouttheuniverse,thenthisconditionofspacetimebeginstolooklikePlanckspacetime.Theenergydensityisthesameandtheaveragedistribution


ofthequantizedangularmomentumisthesame.ThemajordifferenceisthatPlanckspacetimehasГu 1andthisfinalstateoftheuniversehasГuapproachinginfinity.PerhapstheenergyinthesephotonsbecomessuchasmallfractionofthevacuumenergydensityoftheuniversethatquantummechanicsallowsthebackgroundgammaoftheuniversetoroundofftoГu ∞.Therateoftimewouldstopandthehybridspeedoflightwouldstop.Thisisadiscontinuity thatwould allow a rebirth of the universe. All that has to happen is that thebackgroundgammaoftheuniversehastochangefromГu ∞toГu 1.Nocollapseisrequiredbecausetheuniverseisalreadyattherequiredenergydensity.Also,therequiredquantizedspinunitswouldbepreservedandevenlydistributed.AllthathastochangeistherateoftimeandthespatialcharacteristicsmustrevertbacktotheГu 1condition.ThiswouldproducePlanckspacetimeandtheuniversewouldstartanewcyclewithanew“BigBang”.________________________________________________ClosingThoughts: Einstein’sgreatest contributions to sciencehappenedwhenhecombinedinsightfulnewassumptionswithmathematicalanalysis.Laterinhislifehetendedtowardsmoreadvancedmathematicalanalysisofthesameoldassumptionsandhiscontributionstosciencediminished.Iseeananalogytoallofphysics.Thegreatestadvancesinphysicsoccurredwheninsightfulnewassumptionswerefirstintroduced.Thisintroductionofnewideaswasfollowedby“goldenages”ofphysics.However,overtimethepaceofadvancementslowedwhenthesameassumptionswerejustmathematicallyanalyzedinmoredetail.Theobjectiveofthisbookistoproposeaseriesofnewideasbasedonthesimpleststartingassumption:Theuniverseisonlyspacetime. Whileworkingontheideascontainedinthisbook,thereweretimesthatIquestionedwhetherIshould be undertaking this large project. Was there really a conceptually understandablesolutiontoaparticularproblem?WhyshouldIbetheoneattemptingtofindthissolution?Atthose times I thought about and received encouragement from the following quote by JohnArchibaldWheeler.Predictinganewrevolutioninphysics,hesaid:“Andwhenitcomes,willwenotsaytoeachother,Oh,howbeautifulandsimpleitallis!Howcouldweeverhavemisseditsolong?”–JohnArchibaldWheeler


Chapter15

EquationsandDefinitionsProperties of a Single Rotar

Aβ Tpωc

2

ħ Aβ rotarstrainamplitude

ħ

ħ rotarradius Comptonradius

ωc ħ Aβωpωc Comptonangularfrequency

Ei mc2 ωcħ ħ Fm AβEpEi internalenergy

m ħ

ħ Tp2ωcZs Aβmpm rotarmass

Pc Eiωc ωc2ħ ħ

ħ

ħ Aβ2PpPc circulatingpower

Fm ħ ħ ħ Aβ2FpFm maximumforceatdistance

Uq ħ

Aβ4UpUq rotarvolumeenergydensity

βq ħ

Aβ2βq gravitationalmagnitudeatrotarradius

ħ distance r fromarotarexpressedasthenumberof units

Ae √

√ Ae electromagneticstandingwavestrainamplitude oscillating

A √ √ A electromagneticnon‐oscillatingstrainamplitude

Ag 2

2Ag gravitationalstandingwavestrainamplitudeoscillatingat2ωc

AG β 2 AG gravitationalnon‐oscillatingstrainamplitude

Af Af hypotheticalfundamentalamplitudebeforecancelation

5 Wave-Amplitude Equations kA2ω2Z intensity w/m2 U kA2ω2Z/c ℙU energydensity J/m3 U /c andU ℙE kA2ω2ZV/cE energy J E V/c P kA2ω2Z P power J/s P F kA2ω2Z /cF force N F /c


PlanckUnitsLp PlancklengthLp Tpc ħG/c 1.616 10‐35mmp Planckmassmp ħc/G2.176 10‐8kg

Tp PlancktimeTp Lp/c ħG/c 5.391 10‐44sqp Planckchargeqp e/√ 4πε ħc1.876 10‐18Coulomb

Ep PlanckenergyEp mpc2 ħc /G1.956 109Jωp Planckangularfrequencyωp 1/Tp c /ħG1.855 1043s‐1Fp PlanckforceFp Ep/Lp c4/G1.210 1044NPp PlanckpowerPp Ep/Tp c5/G3.628 1052wpp Planckmomentumpp Ep/c ħc /G6.525kgm/sUp PlanckenergydensityUp Ep/Lp3 c7/ħG24.636 10113J/m3ℙp Planckpressureℙp Fp/Lp2 c7/ħG24.636 10113N/m2 Up p Plancktemperature p Ep/kB ħ / 1.417 1032°K

Ap PlanckaccelerationAp c/Tp /ħ 5.575 1051m/s2ρp Planckdensityρp mp/Lp3 c5/ħG25.155 1096kg/m3p Planckelectricalpotential p Ep/qp c /4πε G 1.043 1027V

p Planckelectricfield p Fp/qp 4 ħ⁄ 6.450 1061V/m

p Planckmagneticfield p Zs/qp 4 ħ ⁄ 2.152 1053Tesla

p Planckcurrent p qp/Tp 4 ⁄ 3.480 1018ampZp PlanckimpedanceZp ħ/qp2 1/4πεoc29.98ΩGravitational Г and β (excludes cosmology equalities)

Г≡ –Г gravitationalgamma

Г 1 1 β weakgravityapproximations

β≡1– 1 1– 22 1–1/Гβ gravitationalmagnitude

β weakgravityapproximation

βq Hβ2 ħβq gravitationalmagnitudeinarotarat

Properties of Spacetime

Zs 4.038 1035kg/sZs impedanceofspacetime

Ks Ks bulkmodulusofspacetime K≡ ℙ

⁄

Ui Ui interactiveenergydensityofspacetime

Amax TpωAmax maximumdisplacementamplitudeofadipolewaveinspacetime


Charge Conversion Constant

η≡ √

8.617 10‐18m/Coulombη chargeconversionconstant

Characteristics of an ElectronAβ 4.1851 10‐23Aβ strainamplitude 3.8616 10‐13m rotarradius

ωc 7.7634 1020s‐1ωc Comptonangularfrequencyυc 1.2356 1020Hzυc ComptonfrequencyEi 8.1871 10‐14JEi internalenergyme 9.1094 10‐31kgme electron’smasse 1.6022 10‐19Coulombe elementarychargePc 6.3560 107wPc circulatingpowerFm 0.21201NFm maximumforceatdistanceof Rs 6.7635 10‐58Rs classicalSchwarzschildradiusU Ei/ 1.4218 1024J/m3U energydensity cubic–ignoresconstant U 3/4π Ei/ 3.3942 1023J/m3U energydensity spherical q 9.7404 106m/s2 g gravaccelerationatcenterofrotarvolume

gq 4.0764 10‐16m/s2gq gravitationalaccelerationat e eνc 19.796amps e equivalentcirculatingcurrente 3.22 107Tesla e internalmagneticfield

Some Useful Dimensional Analysis Conversions DimensionalAnalysisSymbols:U→M/LT2U energydensityM massG→L3/MT2G gravitationalconstantL lengthħ→ML2/Tħ PlanckconstantT timeZs→M/TZs impedanceofspacetimeQ chargeZo→ML2/TQ2Zo impedanceoffreespaceE→ML2/T2E energyF→ML/T2F forceεo→T2Q2/ML3εo permittivityμo→ML/Q2μo permeability→ML/T2Q electricfield→Q/LT “H”magneticfield ampere/meter →M/TQ “B”magneticfield Tesla

Ω→ML2/TQ2Ω resistance→ML2/T2Q electricalpotential voltagerelativetoneutral

Normalized Transformations (assumes coordinate rate of time and proper length is coordinate length)Lo LgunitoflengthTo Tg/Гunitoftime

Mo Mg/ГunitofmassQo Qgcharge coulombs


Θo ΘgtemperatureCo ΓCgnormalizedspeedoflightdR dL/ΓcircumferentialradiusEo ΓEgenergyvo ΓvgvelocityFo ΓFgforcePo Γ2PgpowerGo Γ3GggravitationalconstantUo ΓUgenergydensityℙo Γℙgpressureρo ρg/Гdensityωo Γωgfrequencyko ΓkgBoltzmann’sconstant

σo Γ2σgStefan‐Boltzmannconstanto Γ gelectricalcurrento Γ gelectricalpotential

εoo εog/Гpermittivityofvacuumµoo µog/Гpermeabilityofvacuumpo pgmomentumαo αgfinestructureconstantΩo Ωgelectricalresistanceo gmagneticfluxdensity

Zoo ZogimpedanceoffreespaceZso Zsgimpedanceofspacetime

Transformation of Planck Units into Spacetime Units PlanckUnitsStandardConversionSpacetimeConversionPlancklengthLp ħ ⁄ Lp cTpPlanckmassmp ħc/Gmp ZsTpPlanckfrequencyωp c /ħGωp 1/TpPlanckimpedanceZp 1/4πεocZp ZsPlanckchargeqp 4πε ħcqp cTpPlanckenergyEp ħc /GEp c2TpZsPlanckforceFp c4/GFp cZsPlanckpowerPp c5/GPp c2ZsPlanckenergydensityUp c7/ħG2Up Zs/cTp2Transformation of Standard Units into Spacetime Units

η≡ √

8.617 10‐18m/Coulombη chargeconversionconstant

NameSpacetime ConversionElementarycharge e e √ Lp/η √ cTp/ηPlanckcharge qp qp Lp/ηCoulombforceconstant 1/4πεo 1/4πεo cZs/η2Permeabilityoffreespace μo μo/4π η2Zs/cElectricfieldofEMradiation γ γ Hω HωZsηMagneticfieldofEMradiation γ γ Hω ⁄ Hω/ηImpedanceoffreespace Zo Zo η24πZsPlanckimpedance Zp Zp η2ZsPlanckconstant ħ ħ c2Tp2Zs Lp2ZsGravitationalconstant G G c3/Zs

The Universe is Only Spacetime ©2012 [email protected] 15-5

Useful Cosmic Information 2.29 10‐18m/s/m 70.8km/s/Mpc Hubbleparameter

1parsec pc 3.086 1016m 3.262lightyears1lightyear ly 9.4607 1015m 6.3241 104AUSolarmass1.989 1030kgSolarradius6.96 108mEarthmass5.974 1024kgEarthradius6.378 106m equator and6.357 106m polar Earth:averageaccelerationofgravity9.807m/s2atequator:9.78m/s2Earth–Sun meanradialdistance 1.496 1011m 1AUMilky Way galaxy; mass: (1.2 to 3) ×1042 kg; radius: ~50,000 ly; rotation rate ~ 200,000 years Sun distance to galactic center ~ 27,000 ly CMB ~ 2.735 °K, ~ 4 108photons/m3, CMB energy density 4.2 10‐14J/m3 Sun’s motion relative to the CMB ≈ 369km/s Planckspacetime:Ups 5.53 10112J/m3,quantizedenergyunits:½Ep 9.78 108J Cosmology

Гu t au t

Гu backgroundgammaoftheuniverse

Гu t

Г

coordinatespeedoflightintheuniverse

C≡ c/ГuC hybridspeedoflight

Г

Гor

Г

Г Hubbleparameter

Г

Г 1 z

Гuo3 Гeq

energydensityratio‐Planckspacetime/observablespacetime

Ups≡ħ

5.53 10112J/m3Ups Planckspacetimeenergydensity


Symbol Definitions Roman alphabet :A waveamplitude area acceleration

Af fundamentalamplitudeofaspacetimewavepriortoanycancellationAf Lp/rAβ strainamplitudeintherotarvolumeofarotarAβe waveamplituderequiredforarotar’selectromagneticcharacteristicsat Aβg amplitudeofthenonlinearwaveatdistance Aβg Aβ2 Aβw speculativeamplitudeoftheweakforceAe amplitudeofthewaveresponsibleforelectricfieldofchargeeAe √ Lp /r2

A electromagneticnon‐oscillatingstrainamplitudeA √ √ Aeq electricfieldstandingwaveamplitudeatdistance Aeq √ Lp/ Agw amplitudeofagravitationalwave Agw 2ΔL/L kGω2 ε/c4r Amax maximumstrainamplitudepermittedforadipolewave

o crosssectionalareainbulkmoduluscalculationao cosmologicalcomovingscalefactoratthepresenttimeap cosmologicalscalefactorofPlanckspacetime whentu 1 au cosmologicalscalefactoroftheuniverserelativetoPlanckspacetimeaem cosmologicalscalefactoratemissionaobs cosmologicalscalefactoratobservationg rotar’sgravaccelerationatthecenteroftherotarvolume “B”magneticfield;magneticfluxdensity;magneticinductione electron’sinternalmagneticfield e 3.22 107Teslaoand g normalizedmagneticfluxdensity

c speedoflight 3 108m/s Co normalizedspeedoflightinzerogravityCo cCg normalizedspeedoflightingravityCo ГCgCr speedoflightintheradialdirectionrelativetoCoCt speedoflightinthetangentialdirectionrelativetoCoC hybridcoordinatespeedoflightC d /dτuu cosmologicalcoordinatespeedoflight u d /dt c/Γu21 cosmologicalcoordinatespeedoflightforΓu 1T coordinatespeedoflightinthetangentialdirection SchwarzschildmetricdR 0 R coordinatespeedoflightintheradialdirection SchwarzschildmetricdΩ 0

dm dipolemomente elementaryelectricalcharge e 1.6 10‐19coulomb E energyE energyinPlanckunits:E E/Ep electricfieldγ electricfieldstrengthinEMradiation

Ei internalenergyforaparticle:Ei mc2 ωcħEp PlanckenergyEp ħc /G 1.956 109JEE energyintheelectricfieldexternalto“r”forchargeeEE αħc/2r


Eel elasticpotentialenergyinthecontextofbulkmodulusEg energyofthemassingravitybutmeasuredusingthezerogravitystandardofenergyEg normalizedenergyofanobjectingravitybutmeasuredusingzerogravitystandardsEk kineticenergyofmassfallingfromzerogravitytodistancerfrommassMEo energyofamassinzerogravitymeasuredusingthezerogravitystandardofenergyEy Young’smodulusEy stress/strain FLo/AoΔLEu CosmologicalunitofenergyforΓu 1E1 CosmologicalunitofenergyforΓu 1F forceFe electromagneticforce‐assumeschargeeparticlesFe e2/4πεor2Fe electromagneticforceindimensionlessPlanckunits‐assumeschargeeparticlesFE electromagneticforce‐assumesPlanckchargeparticlesFe qp2/4πεor2FE electromagneticforceindimensionlessPlanckunits‐assumesPlanckchargeparticlesFg gravitationalforceFg GmM /r2Fg gravitationalforceinPlanckunits:Fg Fg/FpFg normalizedforceingravitybutusingzerogravitystandards notesymbolduplication Fo normalizedforceinzerogravityFm maximumforcepossibleatadistanceof m2c3/ħFs thestrongforceatdistance Fp PlanckforceFp c4/GFp 1.210 1044NFr relativisticforceFr P/cFw weakforceatdistance g accelerationofgravityG gravitationalconstantGoandGg normalizedgravitationalconstantusingzerogravitystandardsgoo,g11,g22,g33 generalrelativitymatrixcoefficients “H”magneticfield;magneticfieldstrength;magneticfieldintensityγ magneticfieldstrengthinEMradiation Hubbleparametercurrently 2.29 10‐18m/s/m

h Planckconstanth 6.626 10‐34Jsħ reducedPlanckconstant:hbar ħ h/2π 1.055 10‐34Js electricalcurrente electron’sequivalentcirculatingcurrentIe eνc 19.796amps intensity momentofinertiak dimensionlessconstants k1,k2,etc. k’ 3/8π aconstantusedincosmology Kb bulkmoduluskB Boltzmannconstant 1.38 10‐23J/m3

KB bulkmodulusKp Planckbulkmodulus c7/ħG2Ks bulkmodulusofspacetime:Ks Fp/λ2 Fp ω/c 2L dimensionalanalysissymbolrepresentinglengthLg normalizedlengthdLg cdτLo normalizedlengthdLo cdt


Lo inthecontextofYoung’smodulus,Loistheoriginallengthbeforestress angularmomentumoand g normalizedangularmomentumLr&Lt properlengthintheradialortangentialdirectionrespectivelylp Plancklength ħG/c 1.616 10‐35m astaticunitoflength Lp dynamicPlancklength waveamplitudeoflp Lu cosmologicalunitoflengthforΓu 1m massm massindimensionlessPlanckunits:m m/mpmp Planckmassmp ħc/G 2.176 10‐8kgme massofanelectron 9.1094 10‐31kgmr pseudorestmasswhenindexofrefractionn 1M dimensionalanalysissymbolrepresentingmassMoandMg normalizedunitofmassMu cosmologicalunitofmassforΓu 1

1 cosmologicalunitofmassforΓu 1N anintegernumber thedistancebetweentworotarsexpressedasamultipleof r/

nk theindexofrefractionwhichincludestheopticalKerreffectcontributionno theindexofrefractionatzerointensityp momentumpp Planckmomentumpp mpc ħc /G 6.525kgm/sP powerPc aParticle’scirculatingpower:Pc Eiωc

Pc circulatingpowerinPlanckunits:Pc Pc/Ppℙ pressureℙp Planckpressureℙp c7/ħG2 4.636 10113N/m2 Up Pp PlanckpowerPp c5/G 3.63 1052wℙq pressuregeneratedbyarotarℙq ωc4ħ/c3 Ei/ Uqq electricalchargeQo,&Qg dimensionalanalysisunitsofchargeusedinvarioustransformationsQ1 cosmologicalunitofchargeforΓu 1Qu cosmologicalunitofchargeforΓu 1qp Planckcharge 4 ħ e/√ 11.7e 1.876 10‐18Coulombr radialdistance properlength R circumferentialradiusfromgeneralrelativity circumference/2π aunitofcoordinatelengthpertainingtocosmologyd dLu/Гu1 cosmologicalunitoflengthforΓu 1

rh radiusoftheHubblesphererph radiusoftheparticlehorizonRs classicalSchwarzschildradius:Rs Gm/c2rs relativisticSchwarzschildradiusrs 2Gm/c2≡rotarradiusofarotar ≡ħ/mc λc reducedComptonradius rotarradiusinPlanckunits: /Lp

t eithertimeorcoordinatetime dependsoncontext


tc timeindicatedonthecoordinateclockT dimensionalanalysissymbolrepresentingtimeTg normalizedunitoftimeingravityTo normalizedunitoftimeinzerogravityTu cosmologicalunitoftimeforΓu 11 cosmologicalunitoftimeforΓu 1

tp Plancktimetp ħG/c 5.391 10‐44sTp dynamicPlancktime awaveamplitudewithdimensionofPlanktime temperaturep Plancktemperature Ep/kB 1.4168 1032°K

ud deBrogliewavegroupvelocityud vU energydensityUel energydensityofelasticpotentialenergy bulkmodulus Ui interactiveenergydensityencounteredbyawaveinspacetimeUi c2ω2/GUo energydensityincoordinateunits assumesГ 1 Ug energydensityinalocationwithgravity Г 1 Uu energydensityintheuniversewhenГu 1Up PlanckenergydensityUp c7/ħG2 4.636 10113J/m3Ups energydensityofPlanckspacetimeUps 3/8π c7/ħG2 5.53 10112J/m3

Uq energydensityofarotarUq Ei/ ωc4ħ/c3 ℙqUu cosmologicalunitofenergydensityforΓu 1U1 cosmologicalunitofenergydensityforΓu 1v velocityve escapevelocityve 2Gm/r 1/2V VolumeVr rotarvolumeVr cubic Vr 4π/3 spherical Electricalpotential Voltage electricalpotential inPlanckunits: / p

p Planckelectricalpotential voltage p c/4πε Gwd deBrogliewavephasevelocitywd c2/vwm velocityofthemodulationwaveenvelope movingresonator wm c2/v maximumdisplacementproducedbydipolewaveinspacetimez cosmologicalredshiftzeq cosmologicalredshiftsincetheradiation/matterequalitytransitionГeq Гuattheradiation/matterequalitytransitionZ impedanceZs impedanceofspacetimeZs c3/G 4.04 1035kg/sZs1 cosmologicalimpedanceofspacetimeforΓu 1Zsu cosmologicalimpedanceofspacetimeforΓu 1Ze electromagneticimpedanceoffreespaceZe 377ΩZa acousticimpedanceZa ρca density speedofsound Zoo normalizedimpedanceoffreespace zerogravity Zog normalizedimpedanceoffreespace ingravity Zp Planckimpedance electromagnetic Zp 1/4πεoc 29.98Ω


GreekSymbolsα finestructureconstantα e2/4πεoħc 1/137.036β gravitationalmagnitudeβ≡1– 1–2Gm/c2R 1/2 1–1/Г 1–dτ/dt gm/c2rβq gravitationalmagnitudeatdistance βq Hβ2 Gm2/ħcβu backgroundgravitationalmagnitudeoftheuniverseβu 1–1/Гu 1–dτu/dtГ gravitationalgammaГ 1‐2Gm/rc2 ‐1/2Гq gravitationalgammaatdistanceof Гq–1 Gm2/ħc Гu backgroundgravitationalgammaoftheuniverseГu dt/dτu au/apГuo thecurrentvalueofГuwhere:Гuo 2.6 1031Гobs ГuatthetimeanobservationofaphotonismadeГem ГuatthetimeofemissionofaphotonГeq Гuattheradiation/matterequalitytransitionγ specialrelativitygammaγ 1–v2/c2 ‐1/2ε asymmetryofanobject‐uniformspherehasε 0,twoequalpointmasseshaveε 1εo permittivityofvacuumξ spinaxisprobabilityξa acousticamplitude particledisplacement η chargeconversionconstantη≡Lp/Qp 8.617 10‐18meters/CoulombΘoandΘg normalizedtemperatureθ anglesymbolλ wavelengthλ reducedwavelength:lambdabar λ λ/2π c/ωλd DeBrogliewavelengthλd h/mvλdd wavelengthofconfinedphotoninmovingframeofreference relativisticcontraction λm modulationenvelopewavelength;λm λoc/vλc Comptonwavelengthλc h/mcλc reducedComptonwavelengthλc ħ/mc λc/2πλo originalwavelengthorwavelengthinzerogravityλg wavelengthingravity blueshifted λem&λobs wavelengthatemissionandobservationrespectivelyλγ wavelengthofconfinedlight chapter1 wavelengthoflightwhenmeasuredinunitsofcoordinatelength.Λ cosmologicalconstantμB Bohrmagnetron eħ/2me 9.274 10‐24J/Teslaμo permeabilityofvacuumμo 4π×10−7mkg/C2 1.257 10‐6mkg/C2ν frequencyνc Comptonfrequencyνc mc2/hυobs&υem cosmologicalobservedfrequency&emittedfrequencyρ matterdensityρp Planckdensityρp c5/ħG2


σ Stefan‐Boltzmannconstant:σ 5.670373×10−8 W m−2 K−4

τ propertime–timeintervalonalocalclockτd timeindicatedonthedipoleclockτobs cosmological properageoftheuniverseatthetimeanobservationismadeτem cosmological properageoftheuniverseatthetimeofemissionofaphotonτu cosmological properageoftheuniverseatarbitrarytime cosmictime τuo cosmological currentproperageoftheuniverse cosmictime τu cosmological ageoftheuniverseinnondimensionalPlanckunitsτu τu/tp emissionlifetime

χ distanceincomovingcoordinatesΨ psifunctionω angularfrequencyωc ComptonangularfrequencyofaParticle ωc mc2/ħ ωc ComptonangularfrequencyinPlanckunits:ωc ωc/ωpωp Planckangularfrequencyωp c /ħG 1.855 1043s‐1Ω asolidangleinasphericalcoordinatesystem dΩ2 dθ2 sin2θdΦ2 Ω symbolrepresentingelectricalresistanceΩM matterdensityparameterΩΛ cosmologicalconstantdensityparameter

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