物理學論文-TheEquivalencePrinciple,theCovariancePrincipleandtheQuestionofSelf-ConsistencyinGeneralRelativityTheEquivalencePrinciple,theCovariancePrincipleandtheQuestionofSelf-ConsistencyinGeneralRelativityC.Y.LoAppliedandPureResearchInstitute17NewcastleDrive,Nashua,NH03060,USASeptember2001AbstractTheequivalenceprinciple,whichstatesthelocalequivalencebetweenaccelerationandgravity,requiresthatafreefallingobservermustresultinaco-movinglocalMinkowskispace.Ontheotherhand,covarianceprincipleassumesanyGaussiansystemtobevalidasaspace-timecoordinatesystem.Giventhemathematicalexistenceoftheco-movinglocalMinkowskispacealongatime-likegeodesicinaLorentzmanifold,acrucialquestionforasatisfactionoftheequivalenceprincipleiswhetherthegeodesicrepresentsaphysicalfreefall.Forinstance,ageodesicofanon-constantmetricisunphysicaliftheaccelerationonarestingobserverdoesnotexist.ThisanalysisismodeledafterEinsteinillustrationoftheequivalenceprinciplewiththecalculationoflightbending.Tojustifyhiscalculationrigorously,itisnecessarytoderivetheMaxwell-NewtonApproximationwithphysicalprinciplesthatleadtogeneralrelativity.Itisshown,asexpected,thattheGalileantransformationisincompatiblewiththeequivalenceprinciple.Thus,generalmathematicalcovariancemustberestrictedbyphysicalrequirements.Moreover,itisshownthroughanexamplethataLorentzmanifoldmaynotnecessarilybediffeomorphictoaphysicalspace-time.Alsoobservationsupportsthataspacetimecoordinatesystemhasmeaninginphysics.Ontheotherhand,Pauliversionleadstotheincorrectspeculationthatingeneralrelativityspace-timecoordinateshavenophysicalmeaning1.Introduction.Currently,amajorproblemingeneralrelativityisthatanyRiemanniangeometrywiththepropermetricsignaturewouldbeacceptedasavalidsolutionofEinsteinequationof1915,andmanyunphysicalsolutionswereaccepted1.Thisis,inpart,duetothefactthatthenatureofthesourcetermhasbeenobscuresincethebeginning2,3.Moreover,themathematicalexistenceofasolutionisoftennotaccompaniedwithunderstandingintermsofphysics1,4,5.Consequently,theadequacyofasourceterm,foragivenphysicalsituation,isoftennotclear6-9.Pauli10consideredthathetheoryofrelativitytobeanexampleshowinghowafundamentalscientificdiscovery,sometimesevenagainsttheresistanceofitscreator,givesbirthtofurtherfruitfuldevelopments,followingitsownautonomouscourse.Thus,inspiteofobservationalconfirmationsofEinsteinpredictions,oneshouldexaminewhethertheoreticalself-consistencyissatisfied.Tothisend,onemayfirstexaminetheconsistencyamongphysicalrincipleswhichleadtogeneralrelativity.Thefoundationofgeneralrelativityconsistsofa)thecovarianceprinciple,b)theequivalenceprinciple,andc)thefieldequationwhosesourcetermissubjectedtomodification3,7,8.Einsteinequivalenceprincipleisthemostcrucialforgeneralrelativity10-13.Inthispaper,theconsistencybetweentheequivalenceprincipleandthecovarianceprinciplewillbeexaminedtheoretically,inparticularthroughexamples.Moreover,theconsistencybetweentheequivalenceprincipleandEinsteinfieldequationof1915isalsodiscussed.Theprincipleofcovariance2statesthathegenerallawsofnaturearetobeexpressedbyequationswhichholdgoodforallsystemsofcoordinates,thatis,arecovariantwithrespecttoanysubstitutionswhatever(generallycovariant).Thecovarianceprinciplecanbeconsideredasconsistingoftwofeatures:1)themathematicalformulationintermsofRiemanniangeometryand2)thegeneralvalidityofanyGaussiancoordinatesystemasaspace-timecoordinatesysteminphysics.Feature1)waseloquentlyestablishedbyEinstein,butfeature2)remainsanunverifiedconjecture.IndisagreementwithEinstein2,Eddington11pointedoutthatpaceisnotalotofpointsclosetogether；itisalotofdistancesinterlocked.EinsteinacceptedEddingtoncriticismandnolongeradvocatedtheinvalidargumentsinhisbook,heMeaningofRelativityof1921.EinsteinalsopraisedEddingtonbookof1923tobethefinestpresentationofthesubjecteverwrittenMoreover,incontrasttothebeliefofsometheorists14,15,ithasneverbeenestablishedthattheequivalenceofallframesofreferencerequirestheequivalenceofallcoordinatesystems9.Ontheotherhand,ithasbeenpointedoutthat,becauseoftheequivalenceprinciple,themathematicalcovariancemustberestricted8,9,16.Moreover,Kretschmann17pointedoutthatthepostulateofgeneralcovariancedoesnotmakeanyassertionsaboutthephysicalcontentofthephysicallaws,butonlyabouttheirmathematicalformulation,andEinsteinentirelyconcurredwithhisview.Pauli10pointedoutfurther,hegenerallycovariantformulationofthephysicallawsacquiresaphysicalcontentonlythroughtheprincipleofequivalence.Nevertheless,Einstein2arguedthat.thereisnoimmediatereasonforpreferringcertainsystemsofcoordinatestoothers,thatistosay,wearriveattherequirementofgeneralco-variance.Thus,Einsteincovarianceprincipleisonlyaninterimconjecture.Apparently,hecouldmeanonlytoamathematicalcoordinatesystemforcalculationsincehisequivalenceprinciple,amongothers,isanimmediatereasonforpreferringcertainsystemsofcoordinatesinphysics(壯5&6).Notethatamathematicalgeneralcovariancerequires,asHawkingdeclared18,theindistinguishabilitybetweenthetime-coordinateandaspace-coordinate.Ontheotherhand,theequivalenceprincipleisrelatedtotheMinkowskispace,whichrequiresadistinctionbetweenthetime-coordinateandaspace-coordinate.Hence,themathematicalgeneralcovarianceisinherentlyinconsistentwiththeequivalenceprinciple.Althoughtheequivalenceprincipledoesnotdeterminethespace-timecoordinates,itdoesrejectphysicallyunrealizablecoordinatesystems9.WhereasinspecialrelativitytheMinkowskimetriclimitsthecoordinatetransformations,amonginertialframesofreference,totheLorentz-Poincartransformations；ingeneralrelativitytheequivalenceprinciplelimitsthephysicalcoordinatetransformationstobeamongvalidspace-timecoordinatesystems,whichareinprinciplephysicallyrealizable.Thus,theroleoftheMinkowskimetricisextendedbytheequivalenceprincipleeventowheregravityispresent.Mathematically,however,theequivalenceprinciplecanbeincompatiblewithasolutionofEinsteinequation,evenifitisaLorentzmanifold(whosespace-timemetrichasthesamesignatureasthatoftheMinkowskispace).IthasbeenproventhatcoordinaterelativisticcausalitycanbeviolatedforsomeLorentzmanifolds9,16.Unfortunately,duetoinadequatephysicalunderstanding,somerelativists19-23believethatapropermetricsignaturewouldimplyasatisfactionoftheequivalenceprinciple.Themisconceptionthat,inaLorentzmanifold,areefallwouldautomaticallyresultinalocalMinkowskispace20,23,hasdeep-rootedphysicalmisunderstandingsfrombelievinginthegeneralmathematicalcovarianceinphysics.Althoughtheequivalenceprincipleforaphysicalspace-time1)isclearlystated,theconditionsforitssatisfactioninaLorentzmanifoldhavebeenmisleadinglyoversimplified.Thus,itisnecessarytoclarifyfirst,intermsofphysics,themeaningoftheequivalenceprincipleanditssatisfaction(2&3).Thecrucialconditionforasatisfactionoftheequivalenceprincipleisthatthegeodesicrepresentsaphysicalfreefall.ThemathematicalexistenceoflocalMinkowskispacesmeansonlymathematicalcompatibilityofthetheoryofgeneralrelativitytoRiemanniangeometry.Then,itbecomespossibletodemonstratemeaningfullythroughdetailedexamplesthatdiffeomorphiccoordinatesystemsmaynotbeequivalentinphysics(5&6).Moreover,toavoidprejudiceduetotheoreticalpreferences,thesedemonstrationsarebasedontheoreticalinconsistency.Tothisend,Einsteinillustrationoftheequivalenceprincipleinhiscalculationofthelightbendingisusedasamodelforthisanalysis.However,inhiscalculation,therearerelatedtheoreticalproblemsthatmustbeaddressed.First,thenotionofgaugeusedinhiscalculationisactuallynotgenerallyvalid9aswillbeshowninthispaper.Also,itisknownthatvalidityofthe1915Einsteinequationisquestionable7,8,24-26.Foracompletetheoreticalanalysis,theseissuesshould,ofcourse,beaddressedthoroughly.Nevertheless,forthevalidityofEinsteincalculationonthelightbending2,itissufficienttojustifythelinearfieldequationasavalidapproximation.Forthispurpose,theMaxwell-NewtonApproximation(i.e.,thelinearfieldequation)isderiveddirectlyfromthephysicalprinciplesthatleadtogeneralrelativity(4).Moreover,thereareintrinsicallyunphysicalLorentzmanifoldsnoneofwhichisdiffeomorphic21toaphysicalspace-time(7).Thus,toacceptaLorentzmanifoldasvalidinphysics,itisnecessarytoverifytheequivalenceprinciplewithaspace-timecoordinatesystemforphysicalinterpretations.Then,forthepurposeofcalculationonly,anydiffeomorphismcanbeusedtoobtainnewcoordinates.Itisonlyinthissensethatacoordinatesystemforaphysicalspace-timecanbearbitrary.Inthispaper,therequirementofageneralcovarianceamongallconceivablemathematicalcoordinatesystems2willbefurtherconfirmedtobeanover-extendeddemand9.(NotethatEddington11didnotacceptthegaugerelatedtogeneralmathematicalcovariance.)Analysisshowsthatasatisfactionoftheequivalenceprinciplerestrictedcovariance(壯3-5).Afterthisnecessaryrectification,somecurrentlyacceptedwell-knownLorentzmanifoldswouldbeexposedasunphysical(7).But,generalrelativityasaphysicaltheoryisunaffected9.Itishopedthatthisclarificationwouldhelpurtherfruitfuldevelopments,followingitsownautonomouscourse10.2.EinsteinEquivalencePrinciple,FreeFall,andPhysicalSpace-TimeCoordinatesInitiallybasedontheobservationthatthe(passive)gravitationalmassandinertialmassareequivalent,Einsteinproposedtheequivalenceofuniformaccelerationandgravity.In1916,thisproposalisextendedtothelocalequivalenceofaccelerationandgravity2becausegravityisingeneralnotuniform.Thus,ifgravityisrepresentedbythespace-timemetric,thegeodesicisthemotionofaparticleundertheinfluenceofgravity.Then,foranobserverinafreefall,thelocalmetricislocallyconstant.Tobeconsistentwithspecialrelativity,suchalocalmetricisrequiredtobelocallyaMinkowskispace2.Thus,acentralproblemingeneralrelativityiswhetherthegeodesicrepresentsaphysicalfreefall.However,validityofthisglobalpropertyisrealizedlocallythroughasatisfactionoftheequivalenceprinciple.Moreover,Eddington11observedthatspecialrelativityshouldapplyonlytophenomenaunrelatedtothesecondorderderivativesofthemetric.Thus,Eins