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