Wandering Towards a Goal: How can mindless mathematical ... · Wandering Towards a Goal: How can...
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Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry 1 The short answer is that mindless mathematical laws cannot, and therefore that they do not, give rise to aims and intentions. How is it that human history can lead to an essay contest on the above theme? And what does that choice of theme tell us about ourselves today? It tells us that we are intellectually still young indeed—that we still do not know what the right questions are, though (even so) we have made many wonderful strides toward eventual answers. For please do remember that it was only about 50,000 years ago, in East Africa, that we humans even developed spoken language. That happened with the appearance of the FOXP2 gene, in one small group of humans, giving them (well, giving us— for the rest of the million or so people then on Earth left no descendants) the incredible gift of speech: the ability to symbolically represent both things and ideas. The ideas were there already, inchoate—for example, people over the ages looking at Venus in the western sky after sunset! I was a guest, long ago, in Western Samoa, and I said to my host, "Tapui T'ear." I had learned that from a friend who had visited Western Samoa years before. My host turned, and he looked into the western sky: Venus! What is Venus? Marcus Aurelius thought that the stars were gods—so, perhaps, did pre-FOXP2 humanity. We humans are not very smart, for how can it be that our most brilliant ancient civilizations—those of India, China, and some others—failed to find the magic key to our present intellectual explosion: that key being mathematics (yes, mindless mathematical laws) mapping astoundingly to our perceptions of what we call the universe? How could it be that, instead of being the fruit of those ancient and most glorious civilizations, the truly magical and liberating connection appeared and took fire only amidst the chaos of horrible primitive medieval Europe? Newton modestly (and correctly) said that he stood on the shoulders of giants: yes indeed of Galileo, but Newton really was referring to the tag-team of Tycho Brahe and Johannes Kepler: the fantastic conjunction of Tycho, a builder of giant celestial observing devices— and, more, the user of same to pursue Mars relentlessly—working with a human mathematical machine, Kepler, a man of incalculable ability and persistence. Kepler worked directly with Tycho, and after Tycho's death used Tycho's data to pursue Kepler's own (and quite wrong) ideas about the mathematical structure of the solar system. Kepler died without understanding what his true accomplishment was—it took Newton to extract it from Kepler's work and thereby open up the physics of today: the advent of our wonderful but mindless mathematical laws. Above, I criticized the theme of this contest. But now let me withdraw that criticism, simply because of the marvelous opening phrase of our theme: "wandering towards a goal." The realization of ignorance is the beginning of wisdom. I have mentioned how Kepler, supreme genius, wandered incorrectly, yet achieved greatness; well, Newton, too, wandered incorrectly, but he also, like Kepler, achieved success despite the falsity of his core beliefs—for Newton was flabbergasted and repelled by the success of his own law of
Transcript of Wandering Towards a Goal: How can mindless mathematical ... · Wandering Towards a Goal: How can...
Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry
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Theshortansweristhatmindlessmathematicallawscannot,andthereforethattheydonot,giverisetoaimsandintentions. Howisitthathumanhistorycanleadtoanessaycontestontheabovetheme?Andwhatdoesthatchoiceofthemetellusaboutourselvestoday? Ittellsusthatweareintellectuallystillyoungindeed—thatwestilldonotknowwhattherightquestionsare,though(evenso)wehavemademanywonderfulstridestowardeventualanswers.Forpleasedorememberthatitwasonlyabout50,000yearsago,inEastAfrica,thatwehumansevendevelopedspokenlanguage.ThathappenedwiththeappearanceoftheFOXP2gene,inonesmallgroupofhumans,givingthem(well,givingus—fortherestofthemillionorsopeoplethenonEarthleftnodescendants)theincrediblegiftofspeech:theabilitytosymbolicallyrepresentboththingsandideas. Theideasweretherealready,inchoate—forexample,peopleovertheageslookingatVenusinthewesternskyaftersunset!Iwasaguest,longago,inWesternSamoa,andIsaidtomyhost,"TapuiT'ear."IhadlearnedthatfromafriendwhohadvisitedWesternSamoayearsbefore.Myhostturned,andhelookedintothewesternsky:Venus!WhatisVenus? MarcusAureliusthoughtthatthestarsweregods—so,perhaps,didpre-FOXP2humanity. Wehumansarenotverysmart,forhowcanitbethatourmostbrilliantancientcivilizations—thoseofIndia,China,andsomeothers—failedtofindthemagickeytoourpresentintellectualexplosion:thatkeybeingmathematics(yes,mindlessmathematicallaws)mappingastoundinglytoourperceptionsofwhatwecalltheuniverse?Howcoulditbethat,insteadofbeingthefruitofthoseancientandmostgloriouscivilizations,thetrulymagicalandliberatingconnectionappearedandtookfireonlyamidstthechaosofhorribleprimitivemedievalEurope? Newtonmodestly(andcorrectly)saidthathestoodontheshouldersofgiants:yesindeedofGalileo,butNewtonreallywasreferringtothetag-teamofTychoBraheandJohannesKepler:thefantasticconjunctionofTycho,abuilderofgiantcelestialobservingdevices—and,more,theuserofsametopursueMarsrelentlessly—workingwithahumanmathematicalmachine,Kepler,amanofincalculableabilityandpersistence.KeplerworkeddirectlywithTycho,andafterTycho'sdeathusedTycho'sdatatopursueKepler'sown(andquitewrong)ideasaboutthemathematicalstructureofthesolarsystem.Keplerdiedwithoutunderstandingwhathistrueaccomplishmentwas—ittookNewtontoextractitfromKepler'sworkandtherebyopenupthephysicsoftoday:theadventofourwonderfulbutmindlessmathematicallaws.Above,Icriticizedthethemeofthiscontest.Butnowletmewithdrawthatcriticism,simplybecauseofthemarvelousopeningphraseofourtheme:"wanderingtowardsagoal."Therealizationofignoranceisthebeginningofwisdom.IhavementionedhowKepler,supremegenius,wanderedincorrectly,yetachievedgreatness;well,Newton,too,wanderedincorrectly,buthealso,likeKepler,achievedsuccessdespitethefalsityofhiscorebeliefs—forNewtonwasflabbergastedandrepelledbythesuccessofhisownlawof
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gravitation,whichwasnothingbutatinyformula,entirelylackinginthemechanismsthatNewtonsought. TheproblemthatNewtonhad,andthattoomanyofustodaystillhave,istheincorrectnotionthatmathematicsisjustatoolthatisusefulforexploringaphysicaluniverse.Butwhat"physical"meansisnotdefined—itstilltodayhasnomeaningfuldefinition.Acommonwordthatisusedis"real."Well,thereisno"real." Thatistheheartofmyessay.Theworldisnotreal—itreallyisnot.Theworldismental,notphysical.Physicalhasnomeaning.ArthurStanleyEddingtonknewthat,andSirJamesJeansknewthat.In1990IpublishedmyessayinNature,"Thementaluniverse,"anattempttoreviveEddingtonandJeans'pioneeringproposedintellectualbasisforustoproceedmoresuccessfullywithourinvestigationofthatwhichwecalltheuniverse. Whatdoesitmatterthattheworldisnotreal?Itmattersbecauseuntilonegraspsthe,well,reality,ofthatidea,onecannotmosteffectivelyworktowardsthegoal:ifyoudon'tknowtherules,youwillnotplaythegameverywell.Ofcoursemostpeopledonotknowtherules,yettheyneverthelessdoachievesomesuccesses.Thatisbecausemostotherpeoplehavethesameblindnessandthusgoalongwithwhatisinfactanaïveapproachtotheuniverse.Wewouldbemuchbetteroffshouldweadopta,well,again,realistic,approachtoourstudyoftheuniverseandofourselves.Weourselves,unliketheuniverse,arereal. Let'slookatasequenceof"casestudies"ofmytheme,whichIexpectwillconvertyou—startingwiththePythagoreantheorem,whichisthemostfundamentalkeyofall:
Noticethatthisproofofthatmostfamoustheoremdoesnotinvolveanymathematicsatall:wesimplyseethetruthofthetheorem,asapropertyofthatwhichwecalltheexternalworld.Myaimandintentionwiththisfigurewillgiverisetomathematicallaws—firstofall,tothe"planeordinary"Pythagoreantheorem,𝑑𝑠! = 𝑑𝑥! + 𝑑𝑦!. Thatequationisamentalhumanconstruct—itwasdevisedinItalyduringthefourteenthcenturytorepresentthealready-known,fromnature,Pythagoreantheorem.
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Fastforwardnowtothenineteenthcentury,andtoWilliamRowanHamilton,thatsupremegeniusofphysicsandofmathematics,theinventorofquaternions(nowvectors),bywhichinventionHamiltonshowedthatthegreatGausswaswrongwhenGaussclaimed(in1830)that"wemustconfessinallhumilitythat,whilenumberisaproductofourmindalone,spacehasarealitybeyondthemindwhoseruleswecannotcompletelyprescribe."Well,Hamiltonwasindeedagenius—yesindeed;butgeniusdoeshaveitslimits,asweshallnowsee! ForHamiltontriedtoextendthePythagoreantheoremtoencompasstimeasafourthdimension:andour"supreme-genius-of-physics-and-mathematics"Hamiltonfailedutterlyinhisattempt—ahugeembarrassmentinretrospect,butonethatilluminatesthecoreweaknessofourhumanminds.Understandingthatilluminationofourweaknessisofimmensevalueintryingtoremovetheblindnesstodayexemplifiedinourcontesttheme. HowdoIknowthatHamiltontriedandfailed?Iknowitbecause(blesshim)Hamiltonwrotethefollowingpoemrevealinghisabjectfailure:
THE TETRACTYS by William Rowan Hamilton
Of high Mathesis, with her charm severe, Of line and number, was our theme; and we
Sought to behold her unborn progeny, And thrones reserved in Truth’s celestial sphere: While views, before attained, became more clear; And how the One of Time, of Space the Three,
Might, in the Chain of Symbol, girdled be: And when my eager and reverted ear
Caught some faint echoes of an ancient strain, Some shadowy outlines of old thoughts sublime,
Gently he smiled to see, revived again, In later age, and occidental clime, A dimly traced Pythagorean lore,
A westward floating, mystic dream of FOUR.IputthisforwardnottoshameHamilton'sghostbuttoencourageeveryreaderofthisessay:thegreatHamilton(andhewasindeedgreat)wassubjecttoanutterlynaïveprejudice:Hamiltoncouldnotbringhimself,inplaceof
ℎ! = 𝑎! + 𝑏! + 𝑐! + 𝑡! towrite,andtoexploreℎ! = 𝑎! + 𝑏! + 𝑐! − 𝑡!Thelatter,ofcourse,isEinstein's1905TheoryofSpecialRelativity,complete.HadHamiltonexploredthat(inretrospectobvious)possibility,thenwithHamilton'smathematicalabilitythereisnoquestionthathewouldhaveanticipatedEinstein.Andindeed,EinsteinhimselfwasjustasblindaswasHamilton—itwasonlyEinstein'steacher,
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HermannMinkowski,whofinallyrealizedexactlywhatEinstein'stwooriginal(andpatheticallynaïve)"postulates"actuallyimpliedabouttheinnernatureoftime. Carefulnow,dearreader:don'tmaketheterriblemistakeofacceptingtheresultbutmissingthepoint:wemustanalyzetheresult,indetail,tograspitshugeimplications—notfortheuniverse,butforourownminds. Geometryisahumaninvention:possiblyoriginallycreatedsimplytore-establishlandboundariesaftertheannualfloodingoftheNile?Thatearlymathematicswas,muchlater,expandedbyItalianmathematicianstoencompassalgebraicequations.Andnow,post-Einstein,abruptly,throughthemindofMinkowskiin1908,revealedtoallowhumanaccesstothesecretofthenatureoftimeitself,throughMinkowski'sℎ! = 𝑎! + 𝑏! + 𝑐! − 𝑡! Thatsingle,simple,equationcontainsthemostimportantphysicswehaveeverdiscovered.Wearelookingformeaninginthisessay,notjusttechnicalanswerstoquestions!Soletmepersist:ifweinsist,asweusuallydo,ontinourequationbeingexpressedinseconds,thenwemustmultiplytbysomenumberctoconvertittometers,soastohaveourequationbeconsistent.Butwhatisthenumericalvalueoftherequiredunits-conversionfactorc? Byexperiment(aclockflownathighspeedaroundtheworld,andthetimetaken,asmeasuredaboardthemovingplane,comparedwiththe(supposedly)sametimetaken,butasmeasuredwithanidenticalclockthathadbeenleftbehindonthetarmac)thenumericalvalueofcwasdeterminedtobeveryclosetoc=299,792,458meterspersecond—recognizedofcourseastheknownspeedoflight. So,today,weusuallychoosetowrite
h2=a2+b2+c2-(ct)2
or,ifwealsodecidetowrite,fortheseparationinspace-aloneoftwoevents(say,oftwosnapsofmyfingers,asIwaveanarmabout)
d2=a2+b2+c2thenweareabletomorecompactlywrite
h2=d2-c2t2
Thattiny“Pythagorean”equationis—complete—Einstein’stheoryofrelativity:allelse,includingE=mc2andtheatomicbomb,canbededucedfromit(aftermultiplyingthroughbym2)usingnothingbutthemagic—andIusethatwordadvisedly—ofourhuman-inventedalgebra.(Forthereader’sstudentsIprovidemyproofintheAPPENDIX.) Butwait—algebraisindeedahumaninvention—andindeed,isarecenthumaninvention. Whyshouldalgebra,whichwasinventedbyus,produceapredictionoftheatomicbomb?Yet,algebradoes!Wehavenoideawhy—butitisso.
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Our“Pythagorean”equationnoweasilyshowsusthat"photonsoflight"arebuta,dareIsay,“whitelie”ofthephysicists(thoughstillausefultoolforstudentsandforengineers):forif,insteadofourtwoeventsbeingtwosnapsofourfingersatdifferenttimesandlocations,wechooseforourtwoevents:
Event#1,aphotonoflightiscreatedonthesunandheadstowardEarth;and,
Event#2,8minuteslater,thatphotonarrivesatEarth,andisabsorbedinaretina theninthatcasedisthedistancefromthesuntotheEarth,andtisthetimetakenbythephotoningoingfromthesuntoyoureye. Now,ofcoursedistanceisequaltovelocitytimestime:d=vt.Substitutingthattinynewequationintoourredrelativityequationgives
h2=v2t2-c2t2
wherehistheseparationinspace-and-time(thatis,inouruniverse)ofthetwoevents:thebirthofthephotononthesun,andtheextinctionofthatsamephoton,ingeneratinganelectricalsignalintheeyeofsomepersononEarth. Wehavereachedthecrux:thespeedvofaphotonoflightis,byexperiment,cmeterspersecond.Settingv=cshowsthattheseparationinouruniversebetweenthetwoeventsofaphoton’sbirth,andofthatsamephoton’sextinction,is
c2t2-c2t2=h2=0
Soinspacetime—thatis,inouruniverse—thereisnoseparationatallbetweenthetwoevents:thusoursupposedphotoncanhavenoexistence.BetweenitsemissiononthesunanditsreceptiononEarth,thephotonismostdefinitelynot"outthereandonitsway."
“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.” (Albert Einstein, 1954) Thatisabigpartofwhywegettheodditiesofthewell-knownquantum-mechanicaldouble-slitexperiment:theresimplyarenophotons—thereareonlyresults:whicharethemeasurementsthemselves. Butwait,there'smore! Supposetwoobservers(onemovingwithvelocityv1andtheothermovingwithvelocityv2)witnessthosesametwoevents:byhypothesishisinvariant,so
v12t12-c2t12=h2=v22t22-c2t22orv12t12-c2t12=h2=0-c2t22ifthesecondobserverhasv2=zeroorv12t12-c2t12=-c2t22
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Looksharplyatthatlastequation! Thesquareofanynumberisalwayspositive.Sobecauseoftheminussign,therighthandsideoftheequationisnecessarilyanegativenumber.Butifv1isfasterthanlight(thatis,ifv1isgreaterthanc)thelefthandsideofthatequationwouldbeapositivenumber,whichcannotbeequaltoanegativenumber.Soourequationassertsthatvelocitiesgreaterthanthatoflightsimplycannotoccur:ifanyonesaysthattheycan,thatwouldbeanalogoustosomeone—absurdly—announcingthattheycangonorthofthenorthpole. Ourequationproducedtheatomicbomb—ourequationisknowntobetrue:soweconcludethatvelocitiesfasterthanlightarenotjustnotpossible,theyareinconceivable—the4-dimensionalgeometryofspacetimesimplydoesnotpermitit—theresimplyisnofaster. Solightisdiscoveredtogoasfastasitispossibletogo.Asdogravitonsandgluons. Ifphotonsdonotexist(andwehavejustseenthattheydonot)neitherinfactdoevenelectronsoranyotherparticles(andthishasalsobeenconfirmedbyexperimentaltests).Sotheuniversedoesnotexist,exceptinourmindsasexperiences.Yes,yourmindisreal,andyes,theobservationsarereal:buttheobservationsarenotobservationsofanything. Goodphysicistsareperfectlywellawareofthis,butitisasmind-bogglingtothemasitistoanyone,andthereisnoclearconclusionastowhatallofthisreallymeans:arewebutdreamsinthemindofGod?Whoknows?Sophysicistsgenerallysticktoonly—atleastinpublic—discussingafakeuniverseofthings—forwhymakeyourselfsoundlikeafool,evenifyouknowthatyouarenot?Tomakeprogressintheirphysics,allphysicistsknowthattheymuststicktotheirequations,becausetheyhavelearnedthatonlybymeansofthoseequationsdotheygetanswersthatturnouttobecorrectwhencheckedbyexperiment. IfthePythagoreantheoremisworkedoutnotinaflatspacetimebutinacurvedspacetime,itsalgebraicformchanges,ofcourse.Andifweapplythesamemagicthatwelearnedtoadoptfortime(simplychangingtheone+signtoa-sign)wequicklygetEinstein’stheoryofGeneralRelativity. Curvedspacetime—asjustslightlypatchedupbyEinsteinsoastoensureconservationofenergy—wasfoundin1915toyieldNewton’sLawofUniversalGravitationasitsfirstapproximation(butonlyifafactorof8πwasstuckin—for,still,nounderstoodreason)and,wonderfully,Einstein'snewtheorypredictedmanysince-confirmedeffects(includingthemostgloriousrecentlydetectedgravitationalwaves)neitherpredictedbynorexplicablewithNewton’soriginalgravitation:another,andhuge,successforAlbertEinstein. Relativity—eveninitsgeneralform—isasimpletheory.Itisperhapsverystrange,butitiscertainlytrue:true—bytheacidtestofmanyexperiments.Andithelpsmakecleartousthattheuniverseisamentalmystery.
Thereisabiggermysterythanrelativityorevenquantummechanics:themysteryofthehumanmind—ofitsstrengthsandofitsfoibles.ThehumanracehasproducedsuchasRamanujan,instinctivelysuperblymathematical.Thatsuggeststhatevolutioncouldresult
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ininstinctivemathematicalbrillianceinall,induecourse.Andthankfullythehumanraceneednotpayanypriceforthat:thebrilliantphysicistArthurStanleyEddington,forexample,wasbothmathematicallysupremelygifted,andyetwasalsoasuperblyrounded,albeitimperfect,humanbeing. ButevenEddingtonwasunabletothinkupthatmagicminussign—theminussignthatcantodayeasilyrevealrelativitycorrectlyeventoelementaryschoolstudents.Butforthatmatter,aswehaveseen,neitherdidEinsteinhimself!Instead,Einstein,in1905,putforwardtwoclumsypostulates,postulatesthatwerenotmathematical,butratherweresentencesinGerman.Yes,thosesentencesdidwork,despitetheirclumsiness,butthreeyearslater,Einstein’sformerprofessor,HermannMinkowski,finallydidcomeupwiththePythagoreanminussignthatabruptlyrevealedEinstein’srelativitytheorytobemostbeautifulandnatural,ratherthanfreakishandarbitrary. Sowhataboutquantummechanics? Inmyexperience,thevastmajorityofwell-educatedpeoplewhoareacquaintedwithquantummechanicsbelieveittobedeeplymysterious.Indeed,Irecallarevealingtalk,beforeascientificaudience,byRogerPenrose.Itispowerfultoreceivetheideasofluminariesinperson,ratherthanjustfromtheirwritings,becauseinperson,theysaythingsthattheydonot,incoldblood,write. Rogerwasemotionalonthesubjectofquantummechanics! Now,thisessayofminesummarizesmypersonaljourneyoversomedecades. TeachingphysicshadgraduallyrevealedtomewhatIhavesummarizedabove,andso,happily,itfinallydawnedonmethatquantummechanicstoo,mustsurelybenatural,andmustbepurelymentalinitsorigins. So,onesummeratLosAlamos,IsatdownandIsaidtomyself,supposeIjustask,“whatisanobservation?”Anyobservation,ofanything?Theanswerclearlywas,thatwhencloselyexamined,anyobservationis“anumber.”Thatwasthelaunchingpointformypaper“Quantummechanicsmadetransparent.”AsIcomposedmypaper,quantummechanicsappeared,tomygreatpleasure,automatically,simplyfromconsideringnumbers,Noether’stheorem,andnothingelse.Quantummechanicswasnatural,andquantummechanicswasinevitable! Subsequent(andmuchmoresophisticated)abinitioderivationsofquantummechanicshavecometoexist:Shapiro(2008). Soletmefinallysumup:wanderingTowardsaGoal:Howcanmindlessmathematicallawsgiverisetoaimsandintention:badquestion!Lawscannotgiverisetoanythingatall:lawsarequantitativeabstractmathematicalconclusionsthatcanonlybedrawnfromobservationsandthedisciplinedpowerofourminds.Absentobservations,mindscouldonlyhypothesizeconceivablelaws.Absentobservation,mindhasnothingtosearchforanypossiblelaws.Sotheentirepremiseofthiscontestishopelesslymuddled.Butwhyisthat?
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Itisclearlybecauseofyearningonthepartofthosehumanspirits—thoseminds—whose“aimsandintentions”gaverisetothecontestinthefirstplace! Goodforthosespirits!Theyshallberewarded: Icouldproposeafollow-upessaycontest:Howcanaimsandintentiongiverisetomathematicallaws?Iwouldwinthatcontest—indeed,I’llhappilyshowhowI'dwinitrightthisminute:itisclearthataimsandintentions—consciousspirits—thatdonotgiverisetomathematicallawswillbe‘aimsandintentions’—thatis,minds—thatonlywanderinchoatein—literal—darkness.Onlyaimsandintentionsthatgiverisetomathematicallawswillfindthemselvesinapositionwheretheycan,forexample,launchintellectualcontests.Keplerwasthereforewell-motivatedindeedinlookingforsymmetries—hewassimplylookingforthewrongsymmetriesinthewrongplaces.ItwasEmmyNoetherwhofoundforusthenecessarysymmetries—thosesymmetriesthatsnatchcoherenceoutofchaosandthattherebyallowthehumanmindandthatalso,necessarily,requirethattherebeavastandemptyuniverse—withnopurposeotherthantosimplyprovidethestageonwhichwehumansdelightfullypranceanddance.References:Henry,RichardConn,"Quantummechanicsmadetransparent,"1990,AmericanJournalof
Physics,58,1087,1990Henry,RichardConn,"Thementaluniverse,"2005,Nature,436,29,2005Henry,RichardConn,citedin:https://en.wikipedia.org/wiki/Kretschmann_scalarHenry,RichardConn,additionalhelpforyou:http://henry.pha.jhu.edu/trivial.pdfShapiro,Moshe,"Derivationoftherelativistic'proper-time'quantumevolutionequations
fromcanonicalinvariance,2008,J.PhysicsA:Mathematical&Theoretical,41,175303
(Alltheweblinkswereverifiedon2017March07Tuesday:whichwasmy77thbirthday!)
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Appendix: to show that E = mc2 Manyreadersofthisessay,Iexpect,willbeteachers,whomightappreciateatransparentderivationofthatmostfamousformulaE=mc2fromtheequallyfamoustheoremofPythagoras(sotrivially—butsoconsequentially—extendedbyMinkowski'simprovingonAlbertEinstein)—
h2=d2-c2t2wherehistheseparation(inspacebyd;intimebyt)ofanytwoevents,suchasthespaceship(redheart)leavingthesun,andarrivingatEarth,ofmyessay.[OfcourseIignoredgravitycompletely(Ihopeyounoticedthat)—Ichosethatgeographyjusttoestablishaneasilyvisualizedandacceptedgreatdistance.]Hereisthattrip:d=dxisthedistancefromthesuntoEarth,t=dtthetimethetriptakes,andtheprimedcoordinatesystemisthatofyourmovingsisterredheart,theunprimedcoordinatesystembeingthatofEarth.
𝑑ℎ! = 𝑑𝑥! − 𝑐!𝑑𝑡! = 𝑑𝑥!! − 𝑐!𝑑𝑡!"Thatequalityistheonlyhypothesisofspecialrelativity.Thehypothesisisthatdhisinvariant:thatdhisthesameforyouonEarth,asitisforyourmovingsister,redheart.Sotheclaimisthat(slightlyreorganizingourequation)
𝑐!𝑑𝑡!" − 𝑑𝑥!! = 𝑐!𝑑𝑡! − 𝑑𝑥!Ourtwochoseneventsareredheartsnappingherfingersassheleavesthesun,andredheartsnappingherfingersonceagainasshearrivesatEarth.WeonEarthhavestrategicallylocatedtwocarefullysynchronizedclocks,oneonEarth,theotheroperatedbyagraduatestudentlocatedatthesun.dt′isthedurationofherjourneyasmeasuredbyherwristwatch,whiledtisthesametimeintervalasmeasuredbyus,stationary.Snaptest:studythetwodiagramsagain,andtellmethenumericalvalueofdx′.
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Right!dx′=0becauseitisthedistancebetweenthetwofinger-snapsasmeasured,inherownreferenceframe,byyoursister.Shecarriesthatreferenceframewithherasshemoves.Thetwoeventstakeplaceattheidenticallocationinherreferenceframe.Nowwelaunchonabunchofalgebra.Itisnothingbutalgebra:noadditionalphysics,ofanykind.(ButhowdoesMotherNatureknowalgebra?Noonehasanyidea,butknowitMotherNaturedoes!)Becauseinredheart‘srestframedx′=0,wedecidetore-labeldt′asdτ,andso 𝑐!𝑑𝜏! = 𝑐!𝑑𝑡! − 𝑑𝑥!.Herrocket’sspeedis𝑣 = 𝑑𝑥/𝑑𝑡asmeasuredbyusonEarth[and𝑢 = 𝑑𝑥/𝑑𝜏asmeasuredbyredheartherself—theverysamedistanceofcourse,but(asweshallbringout)aquitedifferenttimeinterval].Sowerearrange:
𝑐! !!!"
!= 𝑐! − 𝑣!,so
𝑑𝑡𝑑𝜏 =
1
1− 𝑣!
𝑐!
≡ 𝛾 = 1−𝑣!
𝑐!
!!!= 1+
12𝑣!
𝑐! +⋯
Thatlaststepisabinomialexpansion(inventedbyNewton).Sincenormallyvisverymuchsmallerthanc,allhigherorderterms(+⋯)canbeignored.Thefirstpartofourequationdemonstratestimedilation.Iftherocketspeedv(measuredfromEarth)iszero,ourequationsaysthat𝑑𝑡 = 𝑑𝜏andthetimeaboardthe(stationary)shipisthesameastimeonEarth.Butasvapproachesc,thecoefficientofdτapproachesinfinity,andsodτ,thetriptimeasmeasuredbyredheart,approacheszero.Timeslowsdownasoneapproachesthespeedoflight.Andtimedoesnotpassatallforaphoton!Photonsareabsorbedandceasetoexistthesamemomentthattheyarecreated,accordingtotheirownwrist-watches!Nowlet’sreturntoourequation 𝑚!𝑐!𝑑𝜏! = 𝑚!𝑐!𝑑𝑡! −𝑚!𝑑𝑥!whereIhavenowmultipliedbothsidesofourequationbym2(afterall,weseekE=mc2,som,themassofeitheryoursister,orofSisplusherrocket,hastobeintroducedsomehow,doesn’tit?)Fromthat,
𝑚!𝑐! = 𝑚!𝑐! !"!"
!−𝑚! !"
!"
!or𝑚!𝑐! = 𝑚𝑐𝛾 ! −𝑚𝑢!
Noticethatitisthevelocityuthatappearsinthisequation.Nowre-writethisinthisform:
𝑚!𝑐! = 𝑚𝑐 1+12𝑣!
𝑐! +⋯!
−𝑚!𝑢!
𝑚!𝑐! = !!𝑚𝑐! + !
!𝑚𝑣! +⋯
!− 𝑚𝑢 !or,ifwe,now,abandonNewtonandwe
redefinemomentumtobep=muandenergytobe𝐸 = 𝑚𝑐! + !!𝑚𝑣! +⋯then
𝐸! = 𝑐𝑝 ! + 𝑚𝑐! !
Ifredheartisnotmovingatall,hermomentumpiszero,andwehave,atlast:𝐸 = 𝑚𝑐!
