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INTRODUCTIONTOGENERALRELATIVITYGerard'tHooftInstituteforTheoreticalPhysicsUtrechtUniversityandSpinozaInstitutePostbox80.1953508TDUtrecht,theNetherlandse-mail:[e-mail]:~thooft/VersionNovember20101PrologueGeneralrelativityisabeautifulschemefordescribingthegravitational¯eldandtheequationsitobeys.Nowadaysthistheoryisoftenusedasaprototypeforother,moreintricateconstructionstodescribeforcesbetweenelementaryparticlesorotherbranchesoffundamentalphysics.Thisiswhyinanintroductiontogeneralrelativityitisofimportancetoseparateasclearlyaspossiblethevariousingredientsthattogethergiveshapetothisparadigm.Afterexplainingthephysicalmotivationswe¯rstintroducecurvedcoordinates,thenaddtothisthenotionofana±neconnection¯eldandonlyasalaterstepaddtothatthemetric¯eld.Onethenseesclearlyhowspaceandtimegetmoreandmorestructure,until¯nallyallwehavetodoisdeduceEinstein's¯eldequations.ThesenotesmaterializedwhenIwasaskedtopresentsomelecturesonGeneralRela-tivity.Smallchangesweremadeovertheyears.Idecidedtomakethemfreelyavailableontheweb,viamyhomepage.Somereadersexpressedtheirirritationoverthefactthatafter12pagesIswitchnotation:theiinthetimecomponentsofvectorsdisappears,andthemetricbecomesthe¡+++metric.Whythis\inconsistencyinthenotation?Thereweretworeasonsforthis.Thetransitionismadewhereweproceedfromspecialrelativitytogeneralrelativity.Inspecialrelativity,theihasaconsiderablepracticaladvantage:Lorentztransformationsareorthogonal,andallinnerproductsonlycomewith+signs.Noconfusionoversignsremain.Theuseofa¡+++metric,orworseeven,a+¡¡¡metric,inevitablyleadstosignerrors.Ingeneralrelativity,however,theiissuper°uous.Here,weneedtoworkwiththequantityg00anyway.Choosingittobenegativerarelyleadstosignerrorsorotherproblems.Butthereisanotherpedagogicalpoint.Iseenoreasontoshieldstudentsagainstthephenomenonofchangesofconventionandnotation.Suchtransitionsarenecessarywheneveroneswitchesfromone¯eldofresearchtoanother.Theybettergetusedtoit.Asforapplicationsofthetheory,theusualonessuchasthegravitationalredshift,theSchwarzschildmetric,theperihelionshiftandlightde°ectionareprettystandard.Theycanbefoundinthecitedliteratureifonewantsanyfurtherdetails.Finally,Idopayextraattentiontoanapplicationthatmaywellbecomeimportantinthenearfuture:gravitationalradiation.Thederivationsgivenareoftentedious,buttheycanbeproducedratherelegantlyusingstandardLagrangianmethodsfrom¯eldtheory,whichiswhatwillbedemonstrated.Whenteachingthismaterial,Ifoundthatthislastchapterisstillabittootechnicalforanelementarycourse,butIleaveitthereanyway,justbecauseitisomittedfromintroductorytextbooksabittoooften.IthankA.vanderVenforacarefulreadingofthemanuscript.1LiteratureC.W.Misner,K.S.ThorneandJ.A.Wheeler,\Gravitation,W.H.FreemanandComp.,SanFrancisco1973,ISBN0-7167-0344-0.R.Adler,M.Bazin,M.Schi®er,\IntroductiontoGeneralRelativity,Mc.Graw-Hill1965.R.M.Wald,\GeneralRelativity,Univ.ofChicagoPress1984.P.A.M.Dirac,\GeneralTheoryofRelativity,WileyInterscience1975.S.Weinberg,\GravitationandCosmology:PrinciplesandApplicationsoftheGeneralTheoryofRelativity,J.Wiley&Sons,1972S.W.Hawking,G.F.R.Ellis,\Thelargescalestructureofspace-time,CambridgeUniv.Press1973.S.Chandrasekhar,\TheMathematicalTheoryofBlackHoles,ClarendonPress,OxfordUniv.Press,1983Dr.A.D.Fokker,\Relativiteitstheorie,P.Noordho®,Groningen,1929.J.A.Wheeler,\AJourneyintoGravityandSpacetime,Scienti¯cAmericanLibrary,NewYork,1990,distr.byW.H.Freeman&Co,NewYork.H.Stephani,\GeneralRelativity:Anintroductiontothetheoryofthegravitational¯eld,CambridgeUniversityPress,1990.2Prologue1Literature2Contents1SummaryofthetheoryofSpecialRelativity.Notations.42TheEÄotvÄosexperimentsandtheEquivalencePrinciple.83Theconstantlyacceleratedelevator.RindlerSpace.94Curvedcoordinates.145Thea±neconnection.Riemanncurvature.196Themetrictensor.267TheperturbativeexpansionandEinstein'slawofgravity.318Theactionprinciple.359Specialcoordinates.4010Electromagnetism.4311TheSchwarzschildsolution.4512MercuryandlightraysintheSchwarzschildmetric.5213GeneralizationsoftheSchwarzschildsolution.5614TheRobertson-Walkermetric.5915Gravitationalradiation.6331.SummaryofthetheoryofSpecialRelativity.Notations.SpecialRelativityisthetheoryclaimingthatspaceandtimeexhibitaparticularsymmetrypattern.Thisstatementcontainstwoingredientswhichwefurtherexplain:(i)Thereisatransformationlaw,andthesetransformationsformagroup.(ii)Considerasysteminwhichasetofphysicalvariablesisdescribedasbeingacorrectsolutiontothelawsofphysics.Thenifallthesephysicalvariablesaretransformedappropriatelyaccordingtothegiventransformationlaw,oneobtainsanewsolutiontothelawsofphysics.Asaprototypeexample,onemayconsiderthesetofrotationsinathreedimensionalcoordinateframeasourtransformationgroup.Manytheoriesofnature,suchasNewton'slaw~F=m¢~a,areinvariantunderthistransformationgroup.WesaythatNewton'slawshaverotationalsymmetry.A\point-eventisapointinspace,givenbyitsthreecoordinates~x=(x;y;z),atagiveninstanttintime.Forshort,wewillcallthisa\pointinspace-time,anditisafourcomponentvector,x=0BB@x0x1x2x31CCA=0BB@ctxyz1CCA:(1.