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物理学论文-The Equivalence Principle, the Covariance Principle and the Question of Self-Consistency in General Relativity .doc物理学论文-The Equivalence Principle, the Covariance Principle and the Question of Self-Consistency in General Relativity .doc -- 2 元

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物理学论文TheEquivalencePrinciple,theCovariancePrincipleandtheQuestionofSelfConsistencyinGeneralRelativityTheEquivalencePrinciple,theCovariancePrincipleandtheQuestionofSelfConsistencyinGeneralRelativityC.Y.LoAppliedandPureResearchInstitute17NewcastleDrive,Nashua,NH03060,USASeptember2001AbstractTheequivalenceprinciple,whichstatesthelocalequivalencebetweenaccelerationandgravity,requiresthatafreefallingobservermustresultinacomovinglocalMinkowskispace.Ontheotherhand,covarianceprincipleassumesanyGaussiansystemtobevalidasaspacetimecoordinatesystem.GiventhemathematicalexistenceofthecomovinglocalMinkowskispacealongatimelikegeodesicinaLorentzmanifold,acrucialquestionforasatisfactionoftheequivalenceprincipleiswhetherthegeodesicrepresentsaphysicalfreefall.Forinstance,ageodesicofanonconstantmetricisunphysicaliftheaccelerationonarestingobserverdoesnotexist.ThisanalysisismodeledafterEinsteinillustrationoftheequivalenceprinciplewiththecalculationoflightbending.Tojustifyhiscalculationrigorously,itisnecessarytoderivetheMaxwellNewtonApproximationwithphysicalprinciplesthatleadtogeneralrelativity.Itisshown,asexpected,thattheGalileantransformationisincompatiblewiththeequivalenceprinciple.Thus,generalmathematicalcovariancemustberestrictedbyphysicalrequirements.Moreover,itisshownthroughanexamplethataLorentzmanifoldmaynotnecessarilybediffeomorphictoaphysicalspacetime.Alsoobservationsupportsthataspacetimecoordinatesystemhasmeaninginphysics.Ontheotherhand,Pauliversionleadstotheincorrectspeculationthatingeneralrelativityspacetimecoordinateshavenophysicalmeaning1.Introduction.Currently,amajorproblemingeneralrelativityisthatanyRiemanniangeometrywiththepropermetricsignaturewouldbeacceptedasavalidsolutionofEinsteinequationof1915,andmanyunphysicalsolutionswereaccepted1.Thisis,inpart,duetothefactthatthenatureofthesourcetermhasbeenobscuresincethebeginning2,3.Moreover,themathematicalexistenceofasolutionisoftennotaccompaniedwithunderstandingintermsofphysics1,4,5.Consequently,theadequacyofasourceterm,foragivenphysicalsituation,isoftennotclear69.Pauli10consideredthathetheoryofrelativitytobeanexampleshowinghowafundamentalscientificdiscovery,sometimesevenagainsttheresistanceofitscreator,givesbirthtofurtherfruitfuldevelopments,followingitsownautonomouscourse.Thus,inspiteofobservationalconfirmationsofEinsteinpredictions,oneshouldexaminewhethertheoreticalselfconsistencyissatisfied.Tothisend,onemayfirstexaminetheconsistencyamongphysicalrincipleswhichleadtogeneralrelativity.Thefoundationofgeneralrelativityconsistsofathecovarianceprinciple,btheequivalenceprinciple,andcthefieldequationwhosesourcetermissubjectedtomodification3,7,8.Einsteinequivalenceprincipleisthemostcrucialforgeneralrelativity1013.Inthispaper,theconsistencybetweentheequivalenceprincipleandthecovarianceprinciplewillbeexaminedtheoretically,inparticularthroughexamples.Moreover,theconsistencybetweentheequivalenceprincipleandEinsteinfieldequationof1915isalsodiscussed.Theprincipleofcovariance2statesthathegenerallawsofnaturearetobeexpressedbyequationswhichholdgoodforallsystemsofcoordinates,thatis,arecovariantwithrespecttoanysubstitutionswhatevergenerallycovariant.Thecovarianceprinciplecanbeconsideredasconsistingoftwofeatures1themathematicalformulationintermsofRiemanniangeometryand2thegeneralvalidityofanyGaussiancoordinatesystemasaspacetimecoordinatesysteminphysics.Feature1waseloquentlyestablishedbyEinstein,butfeature2remainsanunverifiedconjecture.IndisagreementwithEinstein2,Eddington11pointedoutthatpaceisnotalotofpointsclosetogetheritisalotofdistancesinterlocked.EinsteinacceptedEddingtoncriticismandnolongeradvocatedtheinvalidargumentsinhisbook,heMeaningofRelativityof1921.EinsteinalsopraisedEddingtonbookof1923tobethefinestpresentationofthesubjecteverwrittenMoreover,incontrasttothebeliefofsometheorists14,15,ithasneverbeenestablishedthattheequivalenceofallframesofreferencerequirestheequivalenceofallcoordinatesystems9.Ontheotherhand,ithasbeenpointedoutthat,becauseoftheequivalenceprinciple,themathematicalcovariancemustberestricted8,9,16.Moreover,Kretschmann17pointedoutthatthepostulateofgeneralcovariancedoesnotmakeanyassertionsaboutthephysicalcontentofthephysicallaws,butonlyabouttheirmathematicalformulation,andEinsteinentirelyconcurredwithhisview.Pauli10pointedoutfurther,hegenerallycovariantformulationofthephysicallawsacquiresaphysicalcontentonlythroughtheprincipleofequivalence....Nevertheless,Einstein2arguedthat...thereisnoimmediatereasonforpreferringcertainsystemsofcoordinatestoothers,thatistosay,wearriveattherequirementofgeneralcovariance.Thus,Einsteincovarianceprincipleisonlyaninterimconjecture.Apparently,hecouldmeanonlytoamathematicalcoordinatesystemforcalculationsincehisequivalenceprinciple,amongothers,isanimmediatereasonforpreferringcertainsystemsofcoordinatesinphysics壯56.Notethatamathematicalgeneralcovariancerequires,asHawkingdeclared18,theindistinguishabilitybetweenthetimecoordinateandaspacecoordinate.Ontheotherhand,theequivalenceprincipleisrelatedtotheMinkowskispace,whichrequiresadistinctionbetweenthetimecoordinateandaspacecoordinate.Hence,themathematicalgeneralcovarianceisinherentlyinconsistentwiththeequivalenceprinciple.Althoughtheequivalenceprincipledoesnotdeterminethespacetimecoordinates,itdoesrejectphysicallyunrealizablecoordinatesystems9.WhereasinspecialrelativitytheMinkowskimetriclimitsthecoordinatetransformations,amonginertialframesofreference,totheLorentzPoincarétransformationsingeneralrelativitytheequivalenceprinciplelimitsthephysicalcoordinatetransformationstobeamongvalidspacetimecoordinatesystems,whichareinprinciplephysicallyrealizable.Thus,theroleoftheMinkowskimetricisextendedbytheequivalenceprincipleeventowheregravityispresent.Mathematically,however,theequivalenceprinciplecanbeincompatiblewithasolutionofEinsteinequation,evenifitisaLorentzmanifoldwhosespacetimemetrichasthesamesignatureasthatoftheMinkowskispace.IthasbeenproventhatcoordinaterelativisticcausalitycanbeviolatedforsomeLorentzmanifolds9,16.Unfortunately,duetoinadequatephysicalunderstanding,somerelativists1923believethatapropermetricsignaturewouldimplyasatisfactionoftheequivalenceprinciple.Themisconceptionthat,inaLorentzmanifold,areefallwouldautomaticallyresultinalocalMinkowskispace20,23,hasdeeprootedphysicalmisunderstandingsfrombelievinginthegeneralmathematicalcovarianceinphysics.Althoughtheequivalenceprincipleforaphysicalspacetime1isclearlystated,theconditionsforitssatisfactioninaLorentzmanifoldhavebeenmisleadinglyoversimplified.Thus,itisnecessarytoclarifyfirst,intermsofphysics,themeaningoftheequivalenceprincipleanditssatisfaction§2§3.Thecrucialconditionforasatisfactionoftheequivalenceprincipleisthatthegeodesicrepresentsaphysicalfreefall.ThemathematicalexistenceoflocalMinkowskispacesmeansonlymathematicalcompatibilityofthetheoryofgeneralrelativitytoRiemanniangeometry.Then,itbecomespossibletodemonstratemeaningfullythroughdetailedexamplesthatdiffeomorphiccoordinatesystemsmaynotbeequivalentinphysics§5§6.Moreover,toavoidprejudiceduetotheoreticalpreferences,thesedemonstrationsarebasedontheoreticalinconsistency.Tothisend,Einsteinillustrationoftheequivalenceprincipleinhiscalculationofthelightbendingisusedasamodelforthisanalysis.However,inhiscalculation,therearerelatedtheoreticalproblemsthatmustbeaddressed.First,thenotionofgaugeusedinhiscalculationisactuallynotgenerallyvalid9aswillbeshowninthispaper.Also,itisknownthatvalidityofthe1915Einsteinequationisquestionable7,8,2426.Foracompletetheoreticalanalysis,theseissuesshould,ofcourse,beaddressedthoroughly.Nevertheless,forthevalidityofEinsteincalculationonthelightbending2,itissufficienttojustifythelinearfieldequationasavalidapproximation.Forthispurpose,theMaxwellNewtonApproximationi.e.,thelinearfieldequationisderiveddirectlyfromthephysicalprinciplesthatleadtogeneralrelativity§4.Moreover,thereareintrinsicallyunphysicalLorentzmanifoldsnoneofwhichisdiffeomorphic21toaphysicalspacetime§7.Thus,toacceptaLorentzmanifoldasvalidinphysics,itisnecessarytoverifytheequivalenceprinciplewithaspacetimecoordinatesystemforphysicalinterpretations.Then,forthepurposeofcalculationonly,anydiffeomorphismcanbeusedtoobtainnewcoordinates.Itisonlyinthissensethatacoordinatesystemforaphysicalspacetimecanbearbitrary.Inthispaper,therequirementofageneralcovarianceamongallconceivablemathematicalcoordinatesystems2willbefurtherconfirmedtobeanoverextendeddemand9.NotethatEddington11didnotacceptthegaugerelatedtogeneralmathematicalcovariance.Analysisshowsthatasatisfactionoftheequivalenceprinciplerestrictedcovariance壯35.Afterthisnecessaryrectification,somecurrentlyacceptedwellknownLorentzmanifoldswouldbeexposedasunphysical§7.But,generalrelativityasaphysicaltheoryisunaffected9.Itishopedthatthisclarificationwouldhelpurtherfruitfuldevelopments,followingitsownautonomouscourse10.2.EinsteinEquivalencePrinciple,FreeFall,andPhysicalSpaceTimeCoordinatesInitiallybasedontheobservationthatthepassivegravitationalmassandinertialmassareequivalent,Einsteinproposedtheequivalenceofuniformaccelerationandgravity.In1916,thisproposalisextendedtothelocalequivalenceofaccelerationandgravity2becausegravityisingeneralnotuniform.Thus,ifgravityisrepresentedbythespacetimemetric,thegeodesicisthemotionofaparticleundertheinfluenceofgravity.Then,foranobserverinafreefall,thelocalmetricislocallyconstant.Tobeconsistentwithspecialrelativity,suchalocalmetricisrequiredtobelocallyaMinkowskispace2.Thus,acentralproblemingeneralrelativityiswhetherthegeodesicrepresentsaphysicalfreefall.However,validityofthisglobalpropertyisrealizedlocallythroughasatisfactionoftheequivalenceprinciple.Moreover,Eddington11observedthatspecialrelativityshouldapplyonlytophenomenaunrelatedtothesecondorderderivativesofthemetric.Thus,Einstein27addedacrucialphrase,tleasttoafirstapproximationontheindistinguishabilitybetweengravityandacceleration.TheequivalenceprinciplerequiresthatafreefallphysicallyresultinacomovinglocalMinkowskispace23.However,inaLorentzmanifold,althoughalocalMinkowskispaceexistsinareefallalongageodesic,theformationofsuchcomovinglocalMinkowskispacesmaynotbevalidinphysicssincethegeodesicmaynotrepresentaphysicalfreefall9,16.Inotherwords,giventhemathematicalexistenceoflocalMinkowskispacecomovingalongatimelikegeodesic,thecrucialphysicalquestionforthesatisfactionoftheequivalenceprincipleiswhetherthegeodesicrepresentsaphysicalfreefall.Einstein28pointedout,sfarastheprepositionsofmathematicsreferstoreality,theyarenotcertainandasfarastheyarecertain,theydonotrefertoreality.Thus,anapplicationofamathematicaltheoremshouldbecarefullyexaminedalthoughnecannotreallyarguewithamathematicaltheorem18.If,attheearlierstage,Einsteinargumentsarenotsoperfect,heseldomallowedsuchdefectsbeusedinhiscalculations.Thisisevidentinhisbook,heMeaningofRelativitywhichheeditedin1954.Accordingtohisbookandrelated
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