The principle of equivalence and projective struct

arXiv:gr-qc/0703104v121Mar2007Theprincipleofequivalenceandprojectivestructureinspace-timesGSHall1andDPLonie21DepartmentofMathematicalSciences,UniversityofAberdeen,MestonBuilding,Aberdeen,AB243UE,Scotland,U.K.2108eAndersonDrive,Aberdeen,AB156BW,Scotland,U.K.E-mail:[e-mail],[e-mail]enttowhichonecandeterminethespace-timemetricfromaknowledgeofacertainsubsetofthe(unparametrised)geodesicsofitsLevi-Civitaconnection,thatis,fromtheexperimentalevidenceoftheequivalenceprinciple.Itisshownthat,ifthespace-timeconcernedisknowntobevacuum,thentheLevi-Civitaconnectionisuniquelydeterminedanditsassociatedmetricisuniquelydetermineduptoachoiceofunitsofmeasurement,bythespecificationofthesegeodesics.Itisfurtherdemonstratedthatiftwospace-timessharethesameunparametrisedgeodesicsandonlyoneisassumedvacuumthentheirLevi-Civitaconnectionsareagainequal(andsotheothermetricisalsoavacuummetric)andthefirstresultaboveisrecovered.PACSnumbers:04.20.-q,04.20.Cv,02.40.KySubmittedto:Class.QuantumGrav.Theprincipleofequivalenceandprojectivestructureinspace-times21.IntroductionInNewtoniangravitationaltheoryonecanconsideraparticleashaving,inprinciple,threetypesofmass;itsactivegravitationalmassmAG,itspassivegravitationalmassmPGanditsinertialmassmI.ThemassmAGisameasureoftheparticle’sabilitytogravitationallyattractanotherparticle,whilstthemassmPGisameasureofitssusceptibilitytobeinggravitationallyattractedbyanotherparticle.ThemassmIisameasureoftheparticle’sresistancetobeingacceleratedinaninertialframe.ThusfortwomutuallyattractingparticleslabelledmandMinaninertialframeandatdistancerapart,Newton’sthirdandsecondlawsgive,respectively,forthistwobodyproblemMAGmPG=MPGmAGGMAGmPG=r2mIa(1)whereGistheNewtoniangravitationalconstantandaisthemagnitudeoftheaccelerationofminthisframe.FromthefirstoftheseequationsonegetsMAG/MPG=mAG/mPG.Itfollowsthatonemaychooseunitswithwhichtomeasuretheactiveandpassivegravitationalmassessuchthat,foranyparticle,itsactiveandpassivegravitationalmassesareequal(andwrittenas,say,mG).Thesecondequationin(1)thenshowsthat,withinthegravitationalfieldofM,aG−1mI/mGisthesameforallparticlesatafixedevent.NewtoniantheorythenassumestheconstancyofGandacceptstheexperimentalresult,containedwithintheprincipleofequivalence,thataisthesameforallparticlesatafixedevent.ItfollowsthatmG/mIisparticleindependentandso,choosingappropriateunitsforthemeasurementofinertialmass,onemaytakeforanyparticlemG=mI.Theconclusionisthatonlyonemassparameterisneededforeachparticleandthat,fromthelinearityinNewtoniantheory,agivengravitationalfieldprovidesawell-definedgravitationalaccelerationateacheventandwhichisinheritedbyeachfreelyfallingparticleatthatevent,independentlyofitsmake-up.Itfollowsthatthepathofaparticlepassingthroughthateventdependsonlyonitsvelocityatthatevent.ThisconclusionisoneformoftheprincipleofequivalenceinNewtoniantheoryandappearsasaconsequenceofNewton’slawstogetherwiththeexperimentalresultsmentionedabove.Ifoneassumesfromtheoutsettheresultthattheinertial,activegravitationalandpassivegravitationalmassesareequal,thentheconstancyoftheaccelerationfollowsimmediatelyfromtheconstancyofG.InEinstein’sgeneralrelativitytheory,the(weak)principleofequivalencenowarises,basedontheexperimentalevidenceabove,asanassumptionregardingthepathsofsuchfreelyfallingparticlesatsomespace-timeevent,thisassumptionamountingtotheirdependence,asintheNewtoniancase,onlyontheparticlevelocityatthatevent.Thispathisrelatedtothegeometryofspace-timebybeingassumedtobe(partof)atimelikegeodesicoftheLevi-Civitaconnectionassociatedwiththespace-timemetric.[RegardingtheextenttowhichthisresultcanbeprovedfromtheotheraxiomsofEinstein’stheory,see[1].]Thispaperexaminestheextenttowhichonecanidentifythespace-timemetricingeneralrelativityfromaknowledgeofacertainsetofspace-timepathsrepresentingsuchfreelyfallingparticlesandwhichareassumedtobetimelikegeodesics.ItgeneralisesTheprincipleofequivalenceandprojectivestructureinspace-times3workinanearlierpaper[2]inwhichthisproblemwasbrieflyconsideredandprovidesthedetailsomittedinthatpaper.Inthispaper,toclarifynotation,thetermgeodesicwillbeusedinthemostgeneralsensewiththecurveparameterarbitrary.Thetermunparametrisedgeodesicissometimesusedinthissense.Ontheotherhand,anaffinelyparametrisedgeodesicwillbereferredtoasjustthat.LetMbeasmoothspace-timemanifoldwithallgeometricalobjectsdefinedonMsmooth.Considerthefollowinggeneralsituation.Letgandg′beLorentzmetricsonMwithsignatures(−,+,+,+)andwithassociatedLevi-Civitaconnections∇and∇′.Supposethatforeachp∈MthereisanopensubsetGp6=∅ofthetangentspaceTpMtoMatpsuchthatforeachp∈Mandforeachu∈Gp,uistimelikewithrespecttogandg′andthatthereexistsacurveinMcontainingpwhosetangentatpisuandwhichisanunparametrisedgeodesicwithrespecttoboth∇and∇′.Howare∇and∇′relatedandhowaregandg′related?Inotherwords,giventhattheprincipleofequivalencedetermineslocal(unparametrised)geodesicpathsforacertainfamilyofparticlesateachp∈M,howmuchdoesitsayabouttheLevi-CivitaconnectionandmetriconM?Theassumptionsmadesofarinthisparagraphwillbereferredtoas