HONOURS:GENERALRELATIVITYWORKBOOK
RelativisticPhysics
Class1:SpecialRelativity
A)LORENTZTRANSFORMATIONSEinsteinpostulatedthatthespeedoflightisthesameinallinertialreferenceframes,regardlessofthemotionofthesource.Considertwoinertialreferenceframes𝑆,recordingeventswithspace-timecoordinates(𝑐𝑡, 𝑥, 𝑦, 𝑧),and𝑆′,withco-ordinates(𝑐𝑡′, 𝑥′, 𝑦′, 𝑧′).Let’ssendalightsignaloutfromtheorigin,when𝑆and𝑆′coincide.AccordingtoEinstein’spostulate,eventsalongthelightsignalmustberelatedin𝑆and𝑆′by:
𝑥+ + 𝑦+ + 𝑧+ = 𝑐𝑡 +
𝑥′+ + 𝑦′+ + 𝑧′+ = 𝑐𝑡′ +
Showthatthisrequirementissatisfiedinbothframesifeventstransformfrom𝑆to𝑆′accordingtotheLorentztransformations:
𝑐𝑡. = 𝛾(𝑐𝑡 − 12𝑥)
𝑥. = 𝛾(𝑥 − 12𝑐𝑡)
𝑦. = 𝑦
𝑧. = 𝑧
where𝛾 = 1/ 1 − 𝑣+/𝑐+.
B)SPACE-TIMEDIAGRAMSLet’sdrawsomespace-timediagramsinframe𝑆foreventswithspacecoordinate𝑥andtimecoordinate𝑡.Onagraphof𝑐𝑡against𝑥:a)Drawthepathofalightray,andthepathaparticletravellingwithspeed𝑣 < 𝑐.b)Anevent𝐸occursat𝑥 = 0,𝑡 = 0.Drawthelocusofeventsin𝑆whichoccur(i)1secondofpropertimeafter𝐸,(ii)1secondofpropertimebefore𝐸,(iii)1light-secondofproperdistanceawayfrom𝐸.Whichoftheseeventscanbecausedby𝐸?c)Howdotheselociofeventslookinthespace-timediagramof𝑐𝑡′against𝑥′inframe𝑆′?d)Inthespace-timediagramforframe𝑆,drawthelociofeventswhichoccuratconstant𝑥′,andatconstant𝑡′,inthecoordinatesystemof𝑆′.
C)RELATIVISTICMECHANICSConservationofNewtonianmomentum𝑝 = 𝑚𝑣isinconsistentwithspecialrelativity.Here’sasimpleexampletoshowwhy,andtoillustratethefix.Inframe𝑆,considertwoidenticalparticles,𝐴and𝐵,ofrestmass𝑚>withequalandoppositevelocities±𝑣,collidingandstickingtogethertoformaparticleofmass2𝑚>.Nowconsiderthecollisionasviewedfromframe𝑆′,travellingwithparticle𝐵.a)InaGalileantransformationofvelocities,whatistheinitialvelocityofparticle𝐴in𝑆′?Showthatmomentum𝑝 = 𝑚>𝑣isconservedinframe𝑆′.b)InaLorentztransformationofvelocities,whatistheinitialvelocityofparticle𝐴in𝑆′?[Youwillneedtousethe“additionofvelocities”formula:𝑢. = (𝑢 + 𝑣)/(1 + B1
2C)].Show
thatmomentum𝑝 = 𝑚>𝑣isnotconservedin𝑆′,butwedoconservemomentumifwemodifythedefinitionofmasstodependonvelocitysuchthat
𝑚 𝑣 =𝑚>
1 − 𝑣+
𝑐+
D)“PARADOX”OFSPECIALRELATIVITYAnalysisofeventsinspecialrelativitycanbeillustratedbycertainapparent“paradoxes”.Afamousexampleisthe“twinparadox”,inthisactivityweconsideranotherexample.Abarnhasproperlength𝐿.Apole,alsoofproperlength𝐿,iscarriedtowardsthebarnbyafast-movingrunner.Intherestframeofthebarn,𝑆,thepoleisobservedascontractedtolength𝐿/𝛾,soshouldfitinsidethebarn.However,intherunner’sframe,𝑆.,thebarnappearscontractedtolength𝐿/𝛾,sothepolecannotfitinside.Explainwhythissituationisnotaparadoxbydrawingspace-timediagramsin𝑆and𝑆′,markingin4events:
𝐸E:thefrontendofthepolepassesthefrontdoorofthebarn𝐸+:thefrontendofthepolepassesthereardoorofthebarn𝐸F:therearendofthepolepassesthefrontdoorofthebarn𝐸G:therearendofthepolepassesthereardoorofthebarn
Usingyourspace-timediagrams,inwhatorderdotheseeventsoccurin𝑆and𝑆′?
Class2:IndexNotation
A)PRODUCING4-VECTORSA4-vectorisagroupoffourphysicalquantitieswhosevaluesindifferentinertialframesarerelatedbytheLorentztransformations.Theprototypical4-vectoristhespace-timecoordinatesofanevent𝑥H = (𝑐𝑡, 𝑥, 𝑦, 𝑧).Thesumordifferenceoftwo4-vectorsisalsoa4-vector.Hence,takingthedifferencebetweentwoneighbouringevents,wefindthe4-vector𝑑𝑥H = (𝑐𝑑𝑡, 𝑑𝑥, 𝑑𝑦, 𝑑𝑧).New4-vectorsmayalsobeproducedbymultiplyingordividingbyaninvariant.a)Divide𝑑𝑥Hbytheinvariantpropertimeinterval𝑑𝜏toobtainthecomponentsofthe4-velocityofaparticle𝑣H = 𝑑𝑥H/𝑑𝜏intermsofitsvelocity𝑣 = (𝑣K, 𝑣L, 𝑣M).b)ByapplyingtheLorentztransformationstothe4-velocity,findarelationbetweenthe𝑥-componentsofvelocityofaparticleinframes𝑆and𝑆′.c)Multiply𝑣Hbytheinvariantrestmass𝑚>toobtainthecomponentsofthe4-momentumofaparticle𝑝H = 𝑚>𝑣Hintermsofitsenergy𝐸andmomentum𝑝 = (𝑝K, 𝑝L, 𝑝M).d)Nowconsiderapplyingtheresultsofpartsb)andc)toaphoton,whichhaszerorestmass.If𝑣K = 𝑐,whatis𝑣K.?Whatis𝑝Hforaphoton?
B)INDEXNOTATIONPRACTICEWeintroducethe“down4-vector”withloweredindex,wherewechangethesignofthefirstcomponent,suchthat𝑥H = (−𝑐𝑡, 𝑥, 𝑦, 𝑧).Wecanthenwritethespace-timeintervalas
𝑑𝑠+ = 𝑑𝑥H𝑑𝑥HF
HO>
Inindexnotationwedon’twritethesummation,sothisequationreads𝑑𝑠+ = 𝑑𝑥H𝑑𝑥H.Wheneverwehaveapairofraised/loweredindices,asumoverthatindexisimplied.Theprocessofconvertingfroman“up”toa“down”4-vectorcanbewrittenas𝑥H = 𝜂HQ𝑥Q,
where𝜂HQ =−1 00 1
0 00 0
0 00 0
1 00 1
.Likewise,𝑥H = 𝜂HQ𝑥Q,where𝜂HQ =−1 00 1
0 00 0
0 00 0
1 00 1
.
a)TheLorentztransformationsmaybewritten𝑥′H = 𝐿HQ𝑥Q.Whatisthematrix𝐿HQ?b)WriteanexpressioninindexnotationfortheinverseLorentztransformationsofanup4-vector.Whatmatrixcarriesoutthetransformation?c)Whatisthematrix𝐿HR = 𝜂RQ𝐿HQ?d)Wehaveseenthat𝑑𝑥H𝑑𝑥Hisaninvariant.Whatarethevaluesoftheinvariantquantities𝑣H𝑣H,𝑝Q𝑝Q,𝑣S𝑝S,𝜂TR𝜂TRand𝐿SU𝐿SU?
C)4-CURRENTANDCONSERVATIONLAWSThespace-timevolumeelement𝑑𝑉𝑑𝑡isaLorentzinvariant(since𝑑𝑥. = 𝛾𝑑𝑥and𝑑𝑡. =𝑑𝑡/𝛾,then𝑑𝑥.𝑑𝑡. = 𝑑𝑥𝑑𝑡).Ifasmallregionofspace-timecontainselectriccharge𝑑𝑄,wemayhenceconstructa4-vector,
𝐽H =𝑑𝑄𝑑𝑥H
𝑑𝑉𝑑𝑡
a)Let𝑑𝑥Hrepresentthespace-timedisplacementofallthechargesintheregion,insomesmallinterval.Usethecomponentsof𝑥H,andthedefinitionofcurrent,toshowthatthecomponentsofthis4-vectorare𝐽H = (𝜌𝑐, 𝐽K, 𝐽L, 𝐽M)intermsofchargedensity𝜌andspatialcurrentdensity𝐽 = (𝐽K, 𝐽L, 𝐽M).b)Chargeconservationinelectromagnetismisexpressedby∇. 𝐽 = −𝜕𝜌/𝜕𝑡.Showthatthisrelationmaybewritteninindexnotationas𝜕H𝐽H = 0.
D)ENERGY-MOMENTUMTENSORNowsupposeasmallregionofspace-timecontainsmomentum𝑑𝑝H.Wedefinetheenergy-momentumtensoras,
𝑇HQ =𝑑𝑝H𝑑𝑥Q
𝑑𝑉𝑑𝑡
AsinActivityC,suppose𝑑𝑥Hrepresentsthespace-timedisplacementofallthematter-energyintheregion,insomesmallinterval.a)Usethecomponentsof𝑝Hand𝑥Htoshowthat𝑇>>representstheenergydensityinthisregion.b)Showthat𝑇>^ isthefluxofenergyinthe𝑥^-direction(𝑖 > 0).c)Showthat𝑇^a isthefluxof𝑖-momentuminthe𝑥a-direction(𝑖, 𝑗 > 0).
Class3:Electromagnetism
A)MAXWELL’SEQUATIONSRE-VISITEDElectromagnetismmaybedescribedintermsoftheMaxwellfieldtensor
𝐹HQ = 𝜕H𝐴Q − 𝜕Q𝐴H
Inthisequation,𝜕H =E2dde, ddK, ddL, ddM
–toobtain𝜕Hyouwouldraisetheindex–and𝐴H =𝑉/𝑐, 𝐴K, 𝐴L, 𝐴M istheelectromagneticpotential4-vector,where𝑉istheelectrostaticpotentialand𝐴isthemagneticvectorpotential.Substitutingintherelationsforthe
electricfield𝐸 = −∇𝑉 − dfdeandmagneticfield𝐵 = ∇×𝐴,wefind:
𝐹HQ =
0 𝐸K/𝑐−𝐸K/𝑐 0
𝐸L/𝑐 𝐸M/𝑐𝐵M −𝐵L
−𝐸L/𝑐 −𝐵M−𝐸M/𝑐 𝐵L
0 𝐵K−𝐵K 0
a)Intensornotation,twoofMaxwell’sEquationscanbewrittencompactlyas
𝜕H𝐹HQ = −𝜇>𝐽Q
where𝜇>isthepermeabilityoffreespace,and𝐽H = (𝜌𝑐, 𝐽K, 𝐽L, 𝐽M)isthecurrent4-vector.Byconsideringcases𝜈 = 0and𝜈 = 1,showthatyourecovertwoofMaxwell’sequations.b)Showthat𝜕R𝐹HQ + 𝜕H𝐹QR + 𝜕Q𝐹RH = 0.c)Byconsideringcases 𝜆, 𝜇, 𝜈 = (0,1,2)and(1,2,3)intherelationinpartb),showthatyourecovertheothertwoofMaxwell’sEquations.
B)MAGNETICFIELDOFACURRENTByapplyingtheLorentztransformationtotheMaxwellfieldtensor,wecandeducehowelectromagneticfieldstransformbetweentwoframes𝑆and𝑆′:
𝐹′HQ = 𝐿HT𝐿QR𝐹TR
where𝐿HT =
𝛾 − 12𝛾
− 12𝛾 𝛾
0 00 0
0 00 0
1 00 1
.
a)UsetheLorentztransformationtoshowthat:
𝐵K. = 𝐵K𝐵L. = 𝛾 𝐵L + 𝑣𝐸M/𝑐+
𝐵M. = 𝛾 𝐵M − 𝑣𝐸L/𝑐+
b)Considerastaticlineofchargeinframe𝑆,suchthat𝐵 = 0.Gauss’sLawshowsthat𝐸L =𝜆/2𝜋𝜀>𝑦(at𝑧 = 0)in𝑆,where𝜆isthechargeperunitlength.Inframe𝑆′,thelineofchargebecomesacurrent.UsetheLorentztransformationtorecovertheexpectedmagneticfieldstrengthatdistance𝑑fromacurrent𝐼,whichat𝑧 = 0is𝐵M′ = 𝜇>𝐼/2𝜋𝑦.
C)ELECTROMAGNETICENERGYDENSITYANDFLOWTheenergy-momentumtensor𝑇HQforelectromagnetismis:
𝑇HQ =1𝜇>
𝐹HR𝐹QR −14 𝜂
HQ𝐹TR𝐹TR
Recallthat𝐹HR = 𝜂RQ𝐹HQand𝐹TR = 𝜂TH𝜂RQ𝐹HQ,where𝜂HQ =−1 00 1
0 00 0
0 00 0
1 00 1
.
Let’sfirstconsidertheenergydensitycomponent,𝑇>>.a)Showthat𝐹TR𝐹TR = 2 𝐵+ − 𝐸+/𝑐+ and𝐹>R𝐹>R = 𝐸+/𝑐+.b)Henceshowthattheenergydensityinelectromagneticfieldsis𝑇>> = E
+𝜀>𝐸+ + 𝐵+/2𝜇>.
Nowconsidertheflowofenergyineachdirection,𝑇>^.c)Showthat𝑇>K = (𝐸L𝐵M − 𝐸M𝐵L)/𝜇>𝑐.
Thisisthe𝑥-componentofthePoyntingvector EHp
q×r2.
Class4:AcceleratedMotion
A)WORLD-LINEOFACCELERATINGOBJECTConsideranobjectmovingwithconstantproperacceleration𝛼.Thismeansthat
𝛼 = t1.tu=constant
where𝑑𝜏isthepropertimeelapsedinasmallinterval,and𝑑𝑣′isthemomentaryincreaseinspeedfromrestin𝑆′.Assumetheobjectisatrestin𝑆at𝜏 = 0.a)Usetherelativisticadditionofvelocitiesformulatoshowthattheincreaseinvelocityin𝑆is𝑑𝑣 ≈ 𝑑𝑣′ 1 − 1C
2C.
b)Hencebysubstitutingin𝑑𝑣. = 𝛼𝑑𝜏,showthat𝑣 = 𝑐 tanh Su
2atpropertime𝜏.
c)Use𝛾 = 1/ 1 − 𝑣+/𝑐+ = cosh Su
2,andthetimeintervalin𝑆,𝑑𝑡 = 𝛾𝑑𝜏,toshowthat
thetimecoordinate𝑡in𝑆isrelatedtothepropertime𝜏by𝑡 = 2Ssinh Su
2.
d)Startingfromtherelationforthespace-timeinterval,for𝑑𝜏intermsof𝑑𝑡and𝑑𝑥,showthatthe𝑥-coordinateoftheobjectin𝑆isgivenby𝑥 = 2C
Scosh Su
2.
e)Drawtheworldlineoftheobjectin𝑆onaspace-timediagramof𝑐𝑡against𝑥.
GravityandCurvature
Class5:EquivalencePrinciple
A) GRAVITATIONALBENDINGOFLIGHTAconsequenceoftheEquivalencePrincipleisthatlightwillbebentinagravitationalfield.HowmuchbendingshouldweseeinalaboratoryatrestontheEarth’ssurface?AccordingtotheEquivalencePrinciple,insuchalaboratoryonewouldobservethesameeffectsasinalaboratoryacceleratingindeepspacewithauniformaccelerationof𝑔 = 9.8𝑚𝑠�+.Imaginethatalaseratoneendofthelaboratoryemitsabeamoflightthatoriginallytravelsparalleltothelaboratoryfloor.Thelightshinesontheoppositewallofthelaboratory,atahorizontaldistanceof𝑑 = 3.0𝑚.a)Whatisthemagnitudeoftheverticaldeflectionofthelightbeam?b)Whatisthemagnitudeofthisdeflectionifthelaboratorysitsonthesurfaceofaneutronstar,whichhasamass𝑀 = 3.0×10F>𝑘𝑔andradius𝑅 = 12𝑘𝑚?(Forthepurposesofthisquestion,neglectstrong-fieldeffectsandcalculate𝑔usingNewtonianmethods!)
B) THEGLOBALPOSITIONINGSYSTEMTheGlobalPositioningSystem(GPS)isanetworkofsatellitesthatallowsanyone,withtheaidofasmalldevice(receiver),todetermineexactlywheretheyareontheEarth’ssurface.Eachsatellitecontainsaverypreciseclockandmicrowavetransmitter.a)SupposetheclocksontheGPSsatellitescontainaverysmallerror,suchthattheydriftby“only”1partin10billion.Whatdistanceerrorwouldaccumulateeveryday?Thepropertimeinterval𝑑𝜏between2events,intermsoftheco-ordinatetimeinterval𝑑𝑡,is𝑑𝜏 = 𝑑𝑡 1 + 2𝜙/𝑐+,where𝜙isthegravitationalpotential.b)AssumingtheweakfieldexpressionforthegravitationalpotentialneartheEarth,𝜙 =−𝐺𝑀/𝑟,andconsideringforthemomentthattheclocksareatrestinthegravitationalfield,whatfractionaltimingerroriscausedbythedifferencein𝜙betweentheEarth’ssurfaceandthesatellites?(Estimateorlookupthedatayouneed).DothesatelliteclocksrunfastorslowcomparedtoEarthclocks?c)TheGPSsatellitesareinmotion,orbitingtheEarth.ForthepurposesofthispartofthequestionwewillassumethatthesatellitesandEarthobserversareinthesameinertialreferenceframe.Estimatethevelocityofthesatellitesintheirorbit,andhenceusetimedilationinSpecialRelativitytodeterminethefractionaltimingerrorcausedbythemotionofthesatellitesinorbit.DothesatelliteclocksrunfastorslowcomparedtoEarthclocks?
Class6:CurvedSpaceandMetrics
A)GEOMETRYONACURVEDSURFACEThenormalgeometricrelationsinflatspacedonotapplyinacurvedspace.Considera2Dsphericalsurfacewithco-ordinates(𝜃, 𝜙).Tomakeiteasytovisualize,we’llconsiderthesurfaceofa3Dsphereofradius𝑅.a)Showthatthedistancemetriconthesurfaceofthesphereis
𝑑𝑠+ = 𝑅+𝑑𝜃+ + 𝑅 sin 𝜃 +𝑑𝜙+b)StartingfromtheNorthPole,moveasmallconstantdistance𝜀inalldirections,forminga“circle”inthecurvedspace.Showthatthecircumferenceofthiscircleisnottheflat-spacerelation2𝜋𝜀,butrather,
𝐶 ≈ 2𝜋𝜀 1 −𝜀+
6𝑅+
c)Theareaelementofa2Dco-ordinatespacewithmetric𝑑𝑠+ = 𝑔HQ𝑑𝑥H𝑑𝑥Qis𝑑𝐴 =|𝑔|𝑑𝑥>𝑑𝑥E.Usingthemetricofparta),showthattheareaelementofasphericalsurface
is𝑑𝐴 = 𝑅+ sin 𝜃 𝑑𝜃𝑑𝜙.d)Showthattheareaofthecircleinpartb)isnottheflat-spacerelation𝜋𝜀+,butrather,
𝐴 ≈ 𝜋𝜀+ 1 −𝜀+
12𝑅+
B)METRICSIN2DThemetricdeterminesthegeometryofspace.Butthegeometrydoesnotuniquelydeterminethemetric,becausewemayalwaystransformco-ordinates.a)Whataresomegeometricalmethodswecouldusetodeterminewhetheragivenco-ordinatespaceisflatorcurved?b)MotivatedbytheresultofActivityA,partd),wecandefinethecurvatureofa2Dsurfaceatapointbytherelation
𝑘 = lim�→>
12𝜀+ 1 −
𝐴𝜋𝜀+
where𝐴istheareaenclosedbymovingasmallconstantdistance𝜀.Whatisthecurvatureofthe2DsphericalsurfacefromActivityA?c)Considertwodistancemetricsforco-ordinates(𝑟, 𝜃).Thefirstisapolarco-ordinatesystem,with𝑑𝑠+ = 𝑑𝑟+ + 𝑟+𝑑𝜃+.Thesecondisamodifiedco-ordinatesystemwithmetric
𝑑𝑠+ =𝑑𝑟+ + 𝑟+𝑑𝜃+
1 + 𝑟+
Usetheformulainpartb)tofindthecurvatureat𝑟 = 0ofthesetwospaces.Dotheyrepresentflatorcurvedspace?
Class7:Geodesics
A)GEODESICSONASPHEREAsinClass6,we’llconsidera2Dsphericalsurfacewithco-ordinates(𝜃, 𝜙),byembeddingasphereofradius𝑅ina3Dspace.a)Writedownthemetricelements𝑔��,𝑔��,𝑔��and𝑔��onthesurfaceofthesphere.b)Whatarethevaluesof𝑔��and𝑔��?c)UsetherelationfortheChristoffelsymbols,ΓTR
H = E+𝑔HQ 𝜕R𝑔QT + 𝜕T𝑔RQ − 𝜕Q𝑔TR ,to
showthatthenon-zerosymbolsareΓ��� = −sin 𝜃 cos 𝜃andΓ��� = Γ��
� = cot 𝜃.
d)HenceshowthatthegeodesicequationstCK�
tuC+ ΓTR
H tK�
tutK�
tu= 0onthesurfaceare:
𝑑+𝜃𝑑𝜏+ − sin 𝜃 cos 𝜃
𝑑𝜙𝑑𝜏
+
= 0𝑑+𝜙𝑑𝜏+ + 2 cot 𝜃
𝑑𝜃𝑑𝜏
𝑑𝜙𝑑𝜏 = 0
e)ConsiderthegeodesicbetweentwopointsAandBonthesphere.Withoutlossofgenerality,wecanrotatethecoordinatesystemsuchthatthetwopointsareontheequator,𝜃 = 𝜋/2.Inthiscase,findthegeodesicandexplainwhyitisa“greatcircle”.
B) MOTIONINAWEAKFIELDThespace-timemetricofaweak,staticgravitationalfieldis
𝑔HQ 𝑥^ = 𝜂HQ + ℎHQ 𝑥^
where𝜂HQ =−1 00 1
0 00 0
0 00 0
1 00 1
isthemetricforflatspace-time,and|ℎHQ| ≪ 1isasmall
perturbationwhichdependsonlyonspatialco-ordinates𝑥^ = (𝑥, 𝑦, 𝑧),nottime.
a)ParticlesmovealonggeodesicswhichsatisfytCK�
tuC+ ΓTR
H tK�
tutK�
tu= 0.Ifaparticleisslowly
moving,thentK�
tu≪ tK�
tu.Explainwhythisimpliesthat,intermsofco-ordinatetime𝑡,
𝑑+𝑥H
𝑑𝑡+ ≈ −𝑐+ΓeeH
b)UsetherelationfortheChristoffelsymbols,ΓTR
H = E+𝑔HQ 𝜕R𝑔QT + 𝜕T𝑔RQ − 𝜕Q𝑔TR ,to
showthat,for𝑖 = (𝑥, 𝑦, 𝑧),
Γee^ ≈ −12𝜕ℎee𝜕𝑥^
c)Newton’sLawsrelatetheaccelerationofaparticletothegravitationalpotential𝜙(𝑥)viatCKteC
= −∇𝜙.Usetheresultsforpartsa)andb)todemonstratethattheweak-fieldmetricis
𝑔ee ≈ −1 −2𝜙𝑐+
d)Henceforaclockatrestinaweakgravitationalfield,showthataco-ordinatetimeinterval𝑑𝑡isrelatedtothepropertimeinterval𝑑𝜏by
𝑑𝑡 =𝑑𝜏
1 + 2𝜙/𝑐+
Class8:Space-timeGeometry
A)RIEMANNTENSORONASPHEREInthisActivity,wewillcomputeasanexampletheRiemanncurvaturetensorona2Dsphericalsurface,withmetric:
𝑑𝑠+ = 𝑅+𝑑𝜃+ + 𝑅 sin 𝜃 +𝑑𝜙+
TheRiemanntensormaybedeterminedfromtheChristoffelsymbolsusingtherelation,
𝑅TRHQ = 𝜕HΓRQT − 𝜕QΓRHT + ΓHST ΓRQS − ΓQST ΓRHS
Inthepreviousclass,wesawthattheonlynon-zeroChristoffelsymbolsforthismetricareΓ��� = −sin 𝜃 cos 𝜃andΓ��
� = Γ��� = cot 𝜃.
a)Fora2Dspace,theRiemanntensorhasonly1independentcomponent.Showthatthiscomponentmaybewritten
𝑅���� = sin 𝜃 +b)UsetherelationfortheRiemanntensorintermsoftheChristoffelsymbolstoshowthat
𝑅���� = 1c)Sincethereisonly1independentcomponent,wemustbeabletodeduce𝑅����from𝑅����!WecanshowfromthedefinitionoftheRiemanntensorthattwosymmetriesare:
𝑅RTHQ = −𝑅TRHQ𝑅HQTR = 𝑅TRHQ
Usethesesymmetriestodeducetheresultofpartb)fromparta).
BlackHolesandtheUniverse
Class9:BlackHoles
A)THESCHWARZSCHILDRADIUSa)TheSchwarzschildradiusofanobjectofmass𝑀is𝑅� = 2𝐺𝑀/𝑐+.Ablackholeisanobjectwhichhasaradius𝑟 < 𝑅�.Determinetheminimumdensityofanobjectwhichsatisfiesthisrequirementif(1)𝑀 = 1𝑀⨀,(2)𝑀 = 10E>𝑀⨀.b)InClass4werelatedthechangeintheclockrate𝐶withproperdistance𝐿totheproperacceleration𝛼,whichisequivalenttothegravitationalfield.
𝑑𝐶𝐶 =
𝛼𝑑𝐿𝑐+
IntheSchwarzschildmetrictheclockrate𝐶 ∝ 1 − 𝑅�/𝑟,andproperdistanceinterval𝑑𝐿isrelatedtoco-ordinatedistanceinterval𝑑𝑟as𝑑𝐿 = 𝑑𝑟/ 1 − 𝑅�/𝑟.Showthat:
𝛼 = −𝐺𝑀
𝑟+ 1 − 𝑅�/𝑟
Whatarethevaluesof𝛼at𝑟 ≫ 𝑅�and𝑟 = 𝑅�?
B)RADIALPLUNGEINTOABLACKHOLETheSchwarzschildspace-timemetricaroundablackholeis
𝑑𝑠+ = − 1 −𝑅�𝑟 𝑐+𝑑𝑡+ +
𝑑𝑟+
1 − 𝑅�𝑟+ 𝑟+ 𝑑𝜃+ + sin 𝜃 +𝑑𝜙+
intermsoftheSchwarzschildradius𝑅�.Freely-fallingobservershaveworld-lines𝑥H(𝜏)followinggeodesicst
CK�
tuC+ ΓTR
H tK�
tutK�
tu= 0,whereΓTR
H = E+𝑔HQ 𝜕R𝑔QT + 𝜕T𝑔RQ − 𝜕Q𝑔TR .
a)Writing𝐴 = 1 − 𝑅�/𝑟,showthatΓ�ee =
E+f
tft�.Hencedemonstratethatthe𝜇 = 𝑡
geodesicequationmaybewrittenintheform
𝑑𝑑𝜏 𝐴
𝑑𝑡𝑑𝜏 = 0
andhencethat𝑑𝑡/𝑑𝜏 = 𝐾/𝐴,where𝐾isaconstant.b)Writing𝑑𝑠+ = −𝑐+𝑑𝜏+,usetheoriginalequationforthemetrictodemonstratethat,foranobjectradiallyplungingintoablackhole(suchthat𝑑𝜃 = 𝑑𝜙 = 0),
−𝐴𝑑𝑡𝑑𝜏
+
+1𝐴𝑐+
𝑑𝑟𝑑𝜏
+
+ 1 = 0c)Consideranobjectwhichisatrest(𝑑𝑟/𝑑𝜏 = 0)at𝑟 = ∞.Whatisthevalueof𝐾?Showthatthepropertimerequiredtotravelfrom𝑟 = 𝑅>to𝑟 = 0is
∆𝜏 =2𝑅>F/+
3𝑐𝑅£E/+
d)Whatistheco-ordinatetimeinterval∆𝑡requiredtoreach𝑟 = 𝑅�?
C)ORBITSAROUNDABLACKHOLEa)Lightraysmovethroughspace-timesuchthat𝑑𝑠 = 0.UsetheSchwarzschildmetrictoshowthatforaradially-movinglightraynearablackhole,
1𝑐𝑑𝑟𝑑𝑡 = 1 −
𝑅�𝑟
Whydoesthisequationimplythatalightrayemittedfrom𝑟 < 𝑅�cannotescapetheblackhole?Whathappenstoalightrayemittedat𝑟 = 𝑅�?Nowconsideralightrayinacircularorbitaroundablackhole,suchthat𝑟 =constant.Wecanchoosetheorbitinthe𝜙direction,suchthat𝜃 = 90°.b)Usethecondition𝑑𝑠 = 0forthisorbittoshowthattheangularvelocityofthelightrayis
1𝑐𝑑𝜙𝑑𝑡 =
1 − 𝑅�/𝑟𝑟
c)Usethe𝜇 = 𝑟componentofthegeodesicequationtCK�
t¥C+ ΓTR
H tK�
t¥tK�
t¥= 0toshowthat
𝑐+Γee�𝑑𝑡𝑑𝑝
+
+ 2𝑐Γe��𝑑𝑡𝑑𝑝
𝑑𝜙𝑑𝑝 + Γ���
𝑑𝜙𝑑𝑝
+
= 0
d)UsetheresultsΓee� =
E+𝐴 tft�,Γe�� = 0andΓ��� = −𝐴𝑟 sin 𝜃 +,intermsof𝐴 = 1 − 𝑅£/𝑟,
toshowthatweobtainasecondrelationfortheangularvelocity,
1𝑐𝑑𝜙𝑑𝑡 =
𝑅�2𝑟F
e)Byequatingtheresultsofpartsb)andd),findtheradiusoforbitoflightraysinacircularorbitaroundablackhole.
Class10:Einsteinequation
A)THENEWTONIANLIMITInClass7,ActivityB,weshowedthatthefirstentryofthespace-timemetricforaweakgravitationalfieldwas𝑔ee ≈ −1 − 2𝜙/𝑐+,intermsofNewtoniangravitationalpotential𝜙(𝑥^).WealsocalculatedtheChristoffelsymbolΓee^ ≈
E2C
d�tK�
.TheRiccitensor𝑅HQisrelatedtotheChristoffelsymbolsby
𝑅HQ = 𝜕RΓHQR − 𝜕QΓHRR + ΓTRT ΓHQR − ΓQRT ΓHTR
a)Showthat𝑅ee ≈ ∇+𝜙/𝑐+.(Hint:sincethisisaweakfield,wecanneglectthelast2termsbecausetheyareproductsofsmallquantities).b)TheEinsteinequationrelatesspace-timecurvaturetomatter-energyby𝑅HQ −
E+𝑅𝑔HQ =
¦§¨2©𝑇HQ,wheretheRicciscalar𝑅maybecalculatedas𝑅 = 𝑔HQ𝑅HQ.Byapplying𝑔HQtoboth
sidesoftheequation,showthattheEinsteinequationmaybere-writtenintheform
𝑅HQ =8𝜋𝐺𝑐G 𝑇HQ −
12𝑇𝑔HQ
where𝑇 = 𝑔HQ𝑇HQ.c)Explainwhythematter-energytensor𝑇HQforslowly-movingmatterwithmassdensity𝜌is𝑇ee ≈ 𝜌𝑐+,𝑇�ª£e ≈ 0.Henceshowthat,foraweakfield,𝑇 ≈ −𝜌𝑐+.d)Usetheaboveresultstodemonstratethat,intheweak-fieldlimit,theEinsteinequationisconsistentwiththeNewtonianrelationforthegravitationalpotential,∇+𝜙 = 4𝜋𝐺𝜌.
Class11:Cosmology
A)LIGHTRAYSINEXPANDINGSPACEThemetricofahomogeneousUniverseofcurvature𝑘,expandingwithscalefactor𝑎(𝑡),intermsofspace-timeco-ordinates(𝑡, 𝑟, 𝜃, 𝜙)is:
𝑑𝑠+ = −𝑐+𝑑𝑡+ + 𝑎(𝑡)+𝑑𝑟+
1 − 𝑘𝑟+ + 𝑟+ 𝑑𝜃+ + sin 𝜃 +𝑑𝜙+
a)UsetherelationfortheChristoffelsymbolsΓTR
H = E+𝑔HQ 𝜕R𝑔QT + 𝜕T𝑔RQ − 𝜕Q𝑔TR toshow
thatforthisspace-timemetric,
Γ��e =𝑎𝑎𝑔��𝑐
(theothersymbolsΓ�ª£e = 0).b)Lightrayswith4-vector𝑘H = 𝑑𝑥H/𝑑𝑝satisfythegeodesicequationt¬
�
t¥+ ΓTR
H 𝑘T𝑘R = 0andtherelationforzerospace-timeinterval,𝑔HQ𝑘H𝑘Q = 0.Usetheserelationstogetherwiththeresultfromparta)toshowthat,foraradiallypropagatinglightray,
𝑑𝑘e
𝑑𝑝 +𝑎𝑎 𝑘e + = 0
c)Thefrequency𝜔ofthelightrayisgivenby𝑘e = 𝜔/𝑐.Showthattheresultofpartb)impliesthatthefrequencyofalightrayinanexpandingUniversechangessuchthat
𝜔 ∝1𝑎
B)THEFRIEDMANNEQUATIONThenon-zeroelementsoftheRiccitensor𝑅HQofthespace-timemetricinActivityAare:
𝑅ee = −3𝑐+𝑎𝑎
𝑅^^ =𝑎𝑎 + 2
𝑎𝑎
+
+2𝑘𝑐+
𝑎+𝑔^^𝑐+
a)ShowthattheRicciscalar𝑅 = 𝑔HQ𝑅HQis
𝑅 =6𝑐+
𝑎𝑎 +
𝑎𝑎
+
+𝑘𝑐+
𝑎+
b)HenceshowthatthefirstcomponentoftheEinsteintensor𝐺HQ = 𝑅HQ −
E+𝑅𝑔HQis,
𝐺ee =3𝑐+
𝑎𝑎
+
+𝑘𝑐+
𝑎+
c)Usethe𝜇 = 𝑡,𝜈 = 𝑡componentoftheEinsteinequations,𝐺HQ =
¦§¨2©𝑇HQ,andthe
energy-momentumtensorforslowly-movingmatter,𝑇ee = 𝜌(𝑡)𝑐+and𝑇�ª£e = 0,toshowthatthescalefactoroftheexpandingUniversesatisfiestheFriedmannequation
𝑎𝑎
+
=8𝜋𝐺𝜌(𝑡)
3 −𝑘𝑐+
𝑎+
C)LIGHTTRAVELINANEXPANDINGUNIVERSELetuscombinetheresultsofActivitiesAandBtodetermine,ifarayoflightreachesourtelescopeswithredshift𝑧,howlonghasitbeentravellingthroughtheexpandingUniverse?We’llsupposethattheUniversetoday(𝑡 = 𝑡>)haszerocurvature(𝑘 = 0)andaspecialmatterdensitycalledthecriticaldensity,
𝜌 𝑡> =3𝐻>+
8𝜋𝐺
where𝐻>isthevalueof𝑎/𝑎intoday’sUniverse,knownastheHubbleconstant.Thematterdensityatotherscalefactorsisthen,
𝜌 𝑡 =𝜌(𝑡>)𝑎F
a)UsetheFriedmannequationtoshowthattheevolutionofthescalefactorisgovernedby
𝑑𝑎𝑑𝑡 =
𝐻>𝑎
b)Henceshowthatarayoflightwithredshift𝑧hasbeentravellingthroughtheUniverseforco-ordinatetime
𝑡 =23𝐻>
1 −1
1 + 𝑧 F/+
c)Whatistheradialco-ordinateoftheobjectthatemittedthislightray?Usethefactthatlightraystravelwith𝑑𝑠 = 0toshowthat,inthisUniversewithzerocurvature,
𝑑𝑟𝑑𝑡 =
𝑐𝑎
d)Henceshowthat
𝑟 =2𝑐𝐻>
1 −11 + 𝑧
