Classical General Relativity

arXiv:gr-qc/0506065v218Aug2005Toappearin:HandlbookofthePhilosophyofPhysics,eds.J.ButterfieldandJ.Earman,ElsevierClassicalRelativityTheory∗Version2.4DavidB.MalamentDepartmentofLogicandPhilosophyofScience3151SocialSciencePlazaUniversityofCalifornia,IrvineIrvine,[e-mail]first(section2),Igiveabriefaccountofthestructureofclassicalrelativitytheory.Inthesecond(section3),Idiscussthreespecialtopics:(i)thestatusoftherelativesimultaneityrelationinthecontextofMinkowskispacetime;(ii)the“geometrized”versionofNewtoniangravitationtheory(alsoknownasNewton-Cartantheory);and(iii)thepossibilityofrecoveringtheglobalgeometricstructureofspacetimefromits“causalstructure”.Keywords:relativitytheory;spacetimestructure;simultaneity;Newto-niangravitationtheory∗IamgratefultoJeremyButterfield,ErikCuriel,andJohnEarmanforcommentsonearlierdrafts.1Contents1Introduction32TheStructureofRelativityTheory42.1RelativisticSpacetimes........................42.2ProperTime.............................92.3Space/TimeDecompositionataPointandParticleDynamics..132.4MatterFields.............................162.5Einstein’sEquation..........................212.6CongruencesofTimelikeCurvesand“PublicSpace”.......282.7KillingFieldsandConservedQuantities..............323SpecialTopics363.1RelativeSimultaneityinMinkowskiSpacetime..........363.2GeometrizedNewtonianGravitationTheory............443.3RecoveringGlobalGeometricStructurefrom“CausalStructure”5221IntroductionTheessaythatfollowsisdividedintotwoparts.Inthefirst,Igiveabriefaccountofthestructureofclassicalrelativitytheory.1Inthesecond,Idiscussthreespecialtopics.Myaccountinthefirstpart(section2)islimitedinseveralrespects.Idonotdiscussthehistoricaldevelopmentofclassicalrelativitytheory,northeevidencewehaveforit.Idonottreat“specialrelativity”asatheoryinitsownrightthatissupersededby“generalrelativity”.AndIdonotdescribeknownexactsolutionstoEinstein’sequation.(Thislistcouldbecontinuedatgreatlength.2)Instead,Ilimitmyselftoafewfundamentalideas,andpresentthemasclearlyandpreciselyasIcan.Theaccountpresupposesagoodunderstandingofbasicdifferentialgeometry,andatleastpassingacquaintancewithrelativitytheoryitself.3Insection3,IfirstconsiderthestatusoftherelativesimultaneityrelationinthecontextofMinkowskispacetime.Atissueiswhetherthestandardrelation,theonepickedoutbyEinstein’s“definition”ofsimultaneity,isconventionalincharacter,orisratherinsomesignificantsenseforcedonus.ThenIdescribethe“geometrized”versionofNewtoniangravitationtheory(alsoknownasNewton-Cartantheory).Itisincludedherebecauseithelpstoclarifywhatisandisnotdistinctiveaboutclassicalrelativitytheory.Finally,Iconsidertowhatextenttheglobalgeometricstructureofspacetimecanberecoveredfromits“causalstructure”.41Ispeakof“classical”relativitytheorybecauseconsiderationsinvolvingquantummechan-icswillplaynorole.Inparticular,therewillbenodiscussionofquantumfieldtheoryincurvedspacetime,orofattemptstoformulateaquantumtheoryofgravitation.(Forthelatter,seeRovelli(thisvolume,chapter12).)2TwoimportanttopicsthatIdonotconsiderfigurecentrallyinothercontributionstothisvolume,namelytheinitialvalueformulationofrelativitytheory(Earman,chapter15),andtheHamiltonianformulationofrelativitytheory(Belot,chapter2).3Areviewoftheneededdifferentialgeometry(and“abstract-indexnotation”thatIuse)canbefound,forexample,inWald[1984]andMalament[unpublished].(Sometopicsarealsoreviewedinsections3.1and3.2ofButterfield(thisvolume,chapter1).)Inpreparingpart1,Ihavedrawnheavilyonanumberofsources.AtthetopofthelistareGeroch[unpublished],HawkingandEllis[1972],O’Neill[1983],SachsandWu[1977a,1977b],andWald[1984].4Furtherdiscussionofthefoundationsofclassicalrelativitytheory,fromaslightlydifferentpointofview,canbefoundinRovelli(thisvolume,chapter12).32TheStructureofRelativityTheory2.1RelativisticSpacetimesRelativitytheorydeterminesaclassofgeometricmodelsforthespacetimestruc-tureofouruniverse(andsubregionsthereofsuchas,forexample,oursolarsys-tem).Eachrepresentsapossibleworld(orworld-region)compatiblewiththeconstraintsofthetheory.Itisconvenienttodescribethesemodelsinstages.Westartbycharacterizingabroadclassof“relativisticspacetimes”,anddis-cussingtheirinterpretation.LaterweintroducefurtherrestrictionsinvolvingglobalspacetimestructureandEinstein’sequation.Wetakearelativisticspacetimetobeapair(M,gab),whereMisasmooth,connected,four-dimensionalmanifold,andgabisasmooth,semi-RiemannianmetriconMofLorentzsignature(1,3).5WeinterpretMasthemanifoldofpoint“events”intheworld.6Thein-terpretationofgabisgivenbyanetworkofinterconnectedphysicalprinciples.Welistthreeinthissectionthatarerelativelysimpleincharacterbecausetheymakereferenceonlytopointparticlesandlightrays.(Theseobjectsalonesuf-ficetodeterminethemetric,atleastuptoaconstant.)Inthenextsection,welistafourththatconcernsthebehaviorof(ideal)clocks.Stillotherprinciplesinvolvinggenericmatterfieldswillcomeuplater.Webeginbyreviewingafewdefinitions.Inwhatfollows,let(M,gab)beafixedrelativisticspacetime,andlet∇abethederivativeoperatoronMdeter