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

Matterandenergy

• Thedistribution/flowsofmatterandenergyaredescribedbytheenergy-momentumtensor𝑇6C

Einstein’sEquation

• Einstein’sEquationrelatesspace-timecurvaturetotheenergy-momentumtensor

• Here,𝐺6C = 𝑅6C −X*𝑅𝑔6C istheEinsteintensor,whichis

againcompletelyspecificbythespace-timemetric

• 𝑅6C = 𝑅6SCS istheRiccitensor(averageoftheRiemanntensor)

• 𝑅 istheRicciscalar,𝑅 = 𝑅66 = 𝑔6C𝑅6C

SummaryofGR

Space-timetellsmatterhowtomove(geodesicequation)

Mattertellsspace-timehowtocurve

(Einstein’sequation)