In this class we will summarize the conceptual principles ...
Transcript of In this class we will summarize the conceptual principles ...
Class12:SummaryofGR
InthisclasswewillsummarizetheconceptualprinciplesofGeneralRelativity,linkingthemto
themathematicsofspace-timegeometry
• Gravitationalmassisequaltoinertialmass
• Considertwochargesinteractingelectrostatically:
• Considertwomassesinteractinggravitationally:
• Remarkably,𝑚" equals𝑚# to13significantfigures!
Whyisgravityspecial?
𝑄 𝑞 𝐹 =𝑘𝑄𝑞𝑟* = 𝑚"𝑎𝑟
(inertialmass𝑚")
𝑀 𝑚# 𝐹 =𝐺𝑀𝑚#𝑟* = 𝑚"𝑎𝑟
(inertialmass𝑚")
Whyisgravityspecial?
• Allobjectsexperiencethesameacceleration inagravitationalfield
• Iflaunchedwiththesameposition/velocity,allobjectsfollowthesametrajectoryinagravitationalfield
• Whatpropertyofemptyspacedeterminesthetrajectoryofallobjects?
Geodesics
• Inageneralspace,auniquetrajectorybetweentwopointscanbedefinedbytheshortestpathbetweenthepoints,knownasageodesic
• Thegeometriccharacteristicsofaspacedefineuniquegeodesicpathsinthatspace
Geodesics
• ThegeodesichypothesisofGeneralRelativitystatesthatafreeparticlefollowsageodesicinspace-time
• Thegravitationalfieldshapesspace-time,specifyingthepathsthatparticlesmustfollow
• Notethatthisappliesto4Dspace-time,not3Dspace(throwingaballandalightrayacrossaroomlooksdifferent)
Geodesics
• Thegeodesichypothesisexplainswhygravitationalandinertialmassarethesamething
• IfIdropanobjectattheEarth’ssurface,itacceleratesdownwardat9.8𝑚/𝑠*,followingageodesic
• Toholdtheobjectatrest,Imustapplyanupwardforcetoaccelerateitat9.8𝑚/𝑠* relativetothisgeodesic– intheNewtonianview,thisforceisdeterminedbyitsinertialmass
• IntheNewtonianview,therequiredforceisequaltotheweightoftheobjectasdeterminedbyitsgravitationalmass
• IntheviewofGeneralRelativity,measuringgravitationalweightisthesameasmeasuringresistancetoacceleration
TheEquivalencePrinciple
• Newtonsaid:“anobjectremainsatrestorinuniformmotionunlessactedonbyaforce”
• Thislawappliesininertialframes– inotherco-ordinateframeswefeel“fictitiousforces”
• Ifgravitynolongercountsasaforce(becauseit’sproducedbyspace-time),thenaframeatrestonthesurfaceoftheEarthisnolongeraninertialframe
TheEquivalencePrinciple
• EinsteinformulatedthisideaastheEquivalencePrinciples…
• Afreely-fallingframeislocallyequivalenttoaninertialframe
• Agravitationalfieldislocallyequivalenttoanacceleration
• “Equivalent”meansthatperforminganyexperimentwouldyieldthesameresults
TheEquivalencePrinciple
• TheEquivalencePrincipleisdirectlytestable…
• ConsideralaboratoryonthesurfaceoftheEarth
• Shinearayoflightfromtheceilingtothefloor
• Measurethefrequencyofthelightrayatthefloor
• Ifthelaboratoryisaninertialframe,thefrequencywouldbeunchanged
TheEquivalencePrinciple
• TheEquivalencePrincipleisdirectlytestable…
• BytheEquivalencePrinciple,theexperimentisthesameasifthelaboratorywereacceleratingupward
• Bythetimethelightrayreachesthefloor,thefloorhasarelativeupwardspeed
• BytheDopplereffect,thefrequencyofthelightwouldchange(→ non-inertialframe)
TheEquivalencePrinciple
• Wecancarryoutthisexperiment,andthefrequencyoflightchangesasitmovesthroughagravitationalfield
• Sincefrequencyislikeaclock,thisimpliesthattimerunsdifferentlyateachpointinagravitationalfield
• Thepresenceofgravityinaframeimpliesthatitisnon-inertial
TheEquivalencePrinciple
• TheEquivalencePrincipleisconsistentwiththenotionthatspace-timeis“curved”
• BackinthelaboratoryonthesurfaceoftheEarth,shinethelightrayhorizontally
• Theresultisthesameasifthelaboratorywereacceleratingupward
• Inthatview,thelightraywouldappeartobend
TheEinsteinEquation
• Thefinalpieceistopredicthowagravitatingbodyaffectsthecurvatureofspace-time
• Einstein’sfamousequationlinksthecurvatureofspace-timetothematter-energywithinit…
Thespace-timemetric
• Withtheseconceptsinplace,wecannowsummarizethemathematicaldetail
• Thegeometricalpropertiesofthecurvedspace-time𝑥6 aredescribedbythespace-timemetric:
• Forexample,thepropertimeintervals𝑑𝜏 tickedbyafreely-fallingclockisrelatedtotheinterval𝑑𝑡 whenit’satrestinaframecontaininggravityby𝑑𝜏 = −𝑔<<� 𝑑𝑡
• Inaweakgravitationalfield,𝑔<< �⃗� = −1 − 2𝜙/𝑐*
𝑑𝑠* = 𝑔6C𝑑𝑥6𝑑𝑥C
Thespace-timemetric
• Thespace-timemetricfarfromanygravitatingmassistheMinkowski metric𝑔6C = 𝜂6C:
• Thespace-timemetricintheemptyspacearoundablackhole istheSchwarzschildmetric:
• Thespace-timemetricofahomogeneous/isotropicUniverse istheFRWmetric:
𝑑𝑠* = − 1 −𝑅F𝑟 𝑐𝑑𝑡 * +
𝑑𝑟*
1 − 𝑅F𝑟+ 𝑟* 𝑑𝜃* + sin 𝜃 𝑑𝜙 *
𝑑𝑠* = −𝑐*𝑑𝑡* + 𝑎(𝑡)*𝑑𝑟*
1 − 𝐾𝑟* + 𝑟* 𝑑𝜃* + sin 𝜃 *𝑑𝜙*
𝑑𝑠* = −𝑐*𝑑𝑡* + 𝑑𝑥* + 𝑑𝑦* + 𝑑𝑧*
Geodesics
• Freely-fallingobjectsfollowgeodesics 𝑥6(𝜏) ofthespace-time,whicharecompletelyspecifiedbythespace-timemetric:
• Geodesicsarepathswhichextremize thepropertimeforanobjecttomovebetweentwopoints
𝑑*𝑥6
𝑑𝜏* + ΓRS6 𝑑𝑥R
𝑑𝜏𝑑𝑥S
𝑑𝜏 = 0
ΓRS6 =
12𝑔
6C 𝜕S𝑔CR + 𝜕R𝑔SC − 𝜕C𝑔RS
Space-timecurvature
• Thecurvature ofspace-timeisdescribedbytheRiemanntensor,whichisagaincompletelyspecifiedbythemetric:
𝑅S6CR = 𝜕6ΓSC
R − 𝜕CΓS6R + Γ6VR ΓSC
V − ΓCVR ΓS6V
• TheRiemanntensordescribestherotationofavectorduringparalleltransport,orthedivergenceofgeodesics
Einstein’sEquation
• Einstein’sEquationrelatesspace-timecurvaturetotheenergy-momentumtensor
• Here,𝐺6C = 𝑅6C −X*𝑅𝑔6C istheEinsteintensor,whichis
againcompletelyspecificbythespace-timemetric
• 𝑅6C = 𝑅6SCS istheRiccitensor(averageoftheRiemanntensor)
• 𝑅 istheRicciscalar,𝑅 = 𝑅66 = 𝑔6C𝑅6C