2018The Equivalence Principle， the Covariance Principle and the

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　　作者： C. Y. Lo TheEquivalencePrinciple,theCovariancePrinciple
and
theQuestionofSelf-ConsistencyinGeneralRelativity
C.Y.Lo
AppliedandPureResearchInstitute
17NewcastleDrive,Nashua,NH03060,USA
September2001  代写论文 http://
Abstract
Theequivalenceprinciple,whichstatesthelocalequivalencebetweenaccelerationandgravity,requiresthatafreefallingobservermustresultinaco-movinglocalMinkowskispace.Ontheotherhand,covarianceprincipleassumesanyGaussiansystemtobevalidasaspace-timecoordinatesystem.Giventhemathematicalexistenceoftheco-movinglocalMinkowskispacealongatime-likegeodesicinaLorentzmanifold,acrucialquestionforasatisfactionoftheequivalenceprincipleiswhetherthegeodesicrepresentsaphysicalfreefall.Forinstance,ageodesicofanon-constant metricisunphysicaliftheaccelerationonarestingobserverdoesnotexist.ThisanalysisismodeledafterEinstein?illustrationoftheequivalenceprinciplewiththecalculationoflightbending.Tojustifyhiscalculationrigorously,itisnecessarytoderivetheMaxwell-NewtonApproximationwithphysicalprinciplesthatleadtogeneralrelativity.Itisshown,asexpected,thattheGalileantransformationisincompatiblewiththeequivalenceprinciple.Thus,generalmathematicalcovariancemustberestrictedbyphysicalrequirements.Moreover,itisshownthroughanexamplethatan bsp;Lorentzmanifoldmaynotnecessarilybediffeomorphictoaphysicalspace-time.Alsoobservationsupportsthataspacetimecoordinatesystemhasmeaninginphysics.Ontheotherhand,Pauli?versionleadstotheincorrectspeculationthatingeneralrelativityspace-timecoordinateshavenophysicalmeaning  http://

1.Introduction.
Currently,amajorproblemingeneralrelativityisthatanyRiemanniangeometrywiththepropermetricsignaturewouldbeacceptedasavalidsolutionofEinstein?equationof1915,andmanyunphysicalsolutionswereaccepted[1].Thisis,inpart,duetothefactthatthenatureofthesourcetermhasbeenobscuresincethebeginning[2,3].Moreover,themathematicalexistenceofasolutionisoftennotaccompaniedwithunderstandingintermsofphysics[1,4,5].Consequently,theadequacyofasourceterm,foragivenphysicalsituation,isoftennotclearn bsp;[6-9].Pauli[10]consideredthat?hetheoryofrelativitytobeanexampleshowinghowafundamentalscientificdiscovery,sometimesevenagainsttheresistanceofitscreator,givesbirthtofurtherfruitfuldevelopments,followingitsownautonomouscourse."Thus,inspiteofobservationalconfirmationsofEinstein?predictions,oneshouldexaminewhethertheoreticalself-consistencyissatisfied.Tothisend,onemayfirstexaminetheconsistencyamongphysical?rinciples"whichleadtogeneralrelativity.
Thefoundationofgeneralrelativityconsistsofa)thecovarianceprinciple,b)theequivalenceprinciple,andc)thefieldequationwhosesourcetermissubjectedtomodification[3,7,8].Einstein?equivalenceprincipleisthemostcrucialforgeneralrelativity[10-13].Inthispaper,theconsistencybetweentheequivalenceprincipleandthecovarianceprinciplewillbeexaminedtheoretically,inparticularthroughexamples.Moreover,theconsistencybetweentheequivalenceprincipleandEinstein?fieldequationof1915isalsodiscussed.  
Theprincipleofcovariance[2]statesthat?hegenerallawsofnaturearetobeexpressedbyequationswhichholdgoodforallsystemsofcoordinates,thatis,arecovariantwithrespecttoanysubstitutionswhatever(generallycovariant)."Thecovarianceprinciplecanbeconsideredasconsistingoftwofeatures:1)themathematicalformulationintermsofRiemanniangeometryand2)thegeneralvalidityofanyGaussiancoordinatesystemasaspace-timecoordinatesysteminphysics.Feature1)waseloquentlyestablishedbyEinstein,butfeature2)remainsanunverified conjecture.IndisagreementwithEinstein[2],Eddington[11]pointedoutthat?paceisnotalotofpointsclosetogether;itisalotofdistancesinterlocked."EinsteinacceptedEddington?criticismandnolongeradvocatedtheinvalidargumentsinhisbook,?heMeaningofRelativity"of1921.EinsteinalsopraisedEddington?bookof1923tobethefinestpresentationofthesubjecteverwritten
Moreover,incontrasttothebeliefofsometheorists[14,15],ithasneverbeenestablishedthattheequivalenceofallframesofreferencerequirestheequivalenceofallcoordinatesystems[9].Ontheotherhand,ithasbeenpointedoutthat,becauseoftheequivalenceprinciple,themathematicalcovariancemustberestricted[8,9,16]. 代写论文 http://

Moreover,Kretschmann[17]pointedoutthatthepostulateofgeneralcovariancedoesnotmakeanyassertionsaboutthephysicalcontentofthephysicallaws,butonlyabouttheirmathematicalformulation,andEinsteinentirelyconcurredwithhisview.Pauli[10]pointedoutfurther,?hegenerallycovariantformulationofthephysicallawsacquiresaphysicalcontentonlythroughtheprincipleofequivalence...."Nevertheless,Einstein[2]arguedthat"...thereisnoimmediatereasonforpreferringcertainsystemsofcoordinatestoothers,thatistosay,wearriveatthenb sp;requirementofgeneralco-variance."
Thus,Einstein?covarianceprincipleisonlyaninterimconjecture.Apparently,hecouldmeanonlytoamathematicalcoordinatesystemforcalculationsincehisequivalenceprinciple,amongothers,isanimmediatereasonforpreferringcertainsystemsofcoordinatesinphysics(壯56).Notethatamathematicalgeneralcovariancerequires,asHawkingdeclared[18],theindistinguishabilitybetweenthetime-coordinateandaspace-coordinate.Ontheotherhand,theequivalenceprincipleisrelatedtotheMinkowskispace,whichrequiresadistinctionbetweenthetime-coordinateandaspace-coord inate.Hence,themathematicalgeneralcovarianceisinherentlyinconsistentwiththeequivalenceprinciple. 论文网 http://
Althoughtheequivalenceprincipledoesnotdeterminethespace-timecoordinates,itdoesrejectphysicallyunrealizablecoordinatesystems[9].WhereasinspecialrelativitytheMinkowskimetriclimitsthecoordinatetransformations,amonginertialframesofreference,totheLorentz-Poincarétransformations;ingeneralrelativitytheequivalenceprinciplelimitsthephysicalcoordinatetransformationstobeamongvalidspace-timecoordinatesystems,whichareinprinciplephysicallyrealizable.Thus,theroleoftheMinkowskimetricisextendedbytheequivalenceprincipleeventowheregravityispresent.
Mathematically,however,theequivalenceprinciplecanbeincompatiblewithasolutionofEinstein?equation,evenifitisaLorentzmanifold(whosespace-timemetrichasthesamesignatureasthatoftheMinkowskispace).IthasbeenproventhatcoordinaterelativisticcausalitycanbeviolatedforsomeLorentzmanifolds[9,16].Unfortunately,duetoinadequatephysicalunderstanding,somerelativists[19-23]believethatapropermetricsignaturewouldimplyasatisfactionoftheequivalenceprinciple.Themisconceptionthat,inaLorentzmanifold,a?reefall"wouldautomatically ;resultinalocalMinkowskispace[20,23],hasdeep-rootedphysicalmisunderstandingsfrombelievinginthegeneralmathematicalcovarianceinphysics. 代写论文 http://
Althoughtheequivalenceprincipleforaphysicalspace-time1)isclearlystated,theconditionsforitssatisfactioninaLorentzmanifoldhavebeenmisleadinglyoversimplified.Thus,itisnecessarytoclarifyfirst,intermsofphysics,themeaningoftheequivalenceprincipleanditssatisfaction(§2§3).Thecrucialconditionforasatisfactionoftheequivalenceprincipleisthatthegeodesicrepresentsaphysicalfreefall.ThemathematicalexistenceoflocalMinkowskispacesmeansonlymathematicalcompatibilityofthetheoryofgeneralrelativitytoRiemanniangeometry.n bsp;Then,itbecomespossibletodemonstratemeaningfullythroughdetailedexamplesthatdiffeomorphiccoordinatesystemsmaynotbeequivalentinphysics(§5§6).Moreover,toavoidprejudiceduetotheoreticalpreferences,thesedemonstrationsarebasedontheoreticalinconsistency.
Tothisend,Einstein?illustrationoftheequivalenceprincipleinhiscalculationofthelightbendingisusedasamodelforthisanalysis.However,inhiscalculation,therearerelatedtheoreticalproblemsthatmustbeaddressed.First,thenotionofgaugeusedinhiscalculationisactuallynotgenerallyvalid[9]aswillbeshowninthispaper.Also,itisknownthatvalidityofthe1915Einsteinequationisquestionable[7,8,24-26].Foracompletetheoreticalanalysis,theseissuesshould,ofcourse,beaddressedthoroughly.Nevertheless,forthevalidityo fEinstein?calculationonthelightbending[2],itissufficienttojustifythelinearfieldequationasavalidapproximation.Forthispurpose,theMaxwell-NewtonApproximation(i.e.,thelinearfieldequation)isderiveddirectlyfromthephysicalprinciplesthatleadtogeneralrelativity(§4). 论文代写 http://
Moreover,thereareintrinsicallyunphysicalLorentzmanifoldsnoneofwhichisdiffeomorphic[21]toaphysicalspace-time(§7).Thus,toacceptaLorentzmanifoldasvalidinphysics,itisnecessarytoverifytheequivalenceprinciplewithaspace-timecoordinatesystemforphysicalinterpretations.Then,forthepurposeofcalculationonly,anydiffeomorphismcanbeusedtoobtainnewcoordinates.Itisonlyinthissensethatacoordinatesystemforaphysicalspace-timecanbearbitrary.
Inthispaper,therequirementofageneralcovarianceamongallconceivablemathematicalcoordinatesystems[2]willbefurtherconfirmedtobeanover-extendeddemand[9].(NotethatEddington[11]didnotacceptthegaugerelatedtogeneralmathematicalcovariance.)Analysisshowsthatasatisfactionoftheequivalenceprinciplerestrictedcovariance(壯3-5).Afterthisnecessaryrectification,somecurrentlyacceptedwell-knownLorentzmanifoldswouldbeexposedasunphysical(§7).But,generalrelativityasaphysicaltheoryisunaffected[9].Itishopedthatthisclarificationwoul dhelp?urtherfruitfuldevelopments,followingitsownautonomouscourse[10]".
2.Einstein?EquivalencePrinciple,FreeFall,andPhysicalSpace-TimeCoordinates  
Initiallybasedontheobservationthatthe(passive)gravitationalmassandinertialmassareequivalent,Einsteinproposedtheequivalenceofuniformaccelerationandgravity.In1916,thisproposalisextendedtothelocalequivalenceofaccelerationandgravity[2]becausegravityisingeneralnotuniform.Thus,ifgravityisrepresentedbythespace-timemetric,thegeodesicisthemotionofaparticleundertheinfluenceofgravity.Then,foranobserverinafreefall,thelocalmetricislocallyconstant.Tobeconsistentwithspecialrelativity,suchaloc almetricisrequiredtobelocallyaMinkowskispace[2].
Thus,acentralproblemingeneralrelativityiswhetherthegeodesicrepresentsaphysicalfreefall.However,validityofthisglobalpropertyisrealizedlocallythroughasatisfactionoftheequivalenceprinciple.Moreover,Eddington[11]observedthatspecialrelativityshouldapplyonlytophenomenaunrelatedtothesecondorderderivativesofthemetric.Thus,Einstein[27]addedacrucialphrase,?tleasttoafirstapproximation"ontheindistinguishabilitybetweengravityandacceleration.
Theequivalenceprinciplerequiresthatafreefallphysicallyresultinaco-movinglocalMinkowskispace2)[3].However,inaLorentzmanifold,althoughalocalMinkowskispaceexistsina?reefall"alongageodesic,theformationofsuchco-movinglocalMinkowskispacesmaynotbevalidinphysicssincethegeodesicmaynotrepresentaphysicalfreefall[9,16].Inotherwords,giventhemathematicalexistenceoflocalMinkowskispaceco-movingalongatime-likegeodesic,thecrucialphysicalquestionforthesatisfactionoftheequivalenceprincipleiswhether ;thegeodesicrepresentsaphysicalfreefall. 代写论文 http://
Einstein[28]pointedout,?sfarastheprepositionsofmathematicsreferstoreality,theyarenotcertain;andasfarastheyarecertain,theydonotrefertoreality."Thus,anapplicationofamathematicaltheoremshouldbecarefullyexaminedalthough?necannotreallyarguewithamathematicaltheorem[18]".If,attheearlierstage,Einstein?argumentsarenotsoperfect,heseldomallowedsuchdefectsbeusedinhiscalculations.Thisisevidentinhisbook,?heMeaningofRelativity'whichheeditedin1954.Accordingtohisbooknbs p;andrelatedpapers,Einstein?viewpointsonspace-timecoordinatesare:
1)Aphysical(space-time)coordinatesystemmustbephysicallyrealizable(seealso2)3)below).
Einstein[29]madeclearin?hatistheTheoryofRelativity?(1919)'that?nphysics,thebodytowhicheventsarespatiallyreferrediscalledthecoordinatesystem."Furthermore,Einsteinwrote?fitisnecessaryforthepurposeofdescribingnature,tomakeuseofacoordinatesystemarbitrarilyintroducedbyus,thenthechoiceofitsstateofmotionoughttobesubjecttonorestriction;thelawsoughttobeentirelyindependentofthischoice(generalprincipleofrelativity)".Thus,Einstein?coordinatesystemhasastateofmotionand ;isusuallyreferredtoaphysicalbody.Sincethetimecoordinateisaccordinglyfixed,choosingaspace-timesystemisnotonlyamathematicalbutalsoaphysicalstep. 论文网 http://
2)AphysicalcoordinatesystemisaGaussiansystemsuchthattheequivalenceprincipleissatisfied.
OnemightattempttojustifytheviewpointofacceptinganyGaussiansystemasaspace-timecoordinatesystembypointingoutthatEinstein[3]alsowroteinhisbookthat?nananalogousway(toGaussiancurvilinearcoordinates)weshallintroduceinthegeneraltheoryofrelativityarbitraryco-ordinates,x1,x2,x3,x4,whichshallnumberuniquelythespace-timepoints,sothatneighboringeventsareassociatedwithneighboringvaluesofthecoordinates;otherwise,thechoiceofco-ordinateisarbitrary."But,Einstein[3]qualifiedthiswithaphysicalstatementthatnbs p;?ntheimmediateneighborofanobserver,fallingfreelyinagravitationalfield,thereexistsnogravitationalfield."Thisstatementwillbeclarifiedlaterwithademonstrationoftheequivalenceprinciple(seeeqs.[6][7]).
3)Theequivalenceprinciplerequiresnotonly,ateachpoint,theexistenceofalocalMinkowskispace2)  
ds2=c2dT2-dX2-dY2-dZ2,(1)  http://

butafreefallmustresultinaco-movinglocalMinkowskianspace(seealso[10-13]).Notethattheequivalenceprinciplerequiresthatsuchalocalcoordinatetransformationbeduetoaspecificphysicalaction,accelerationinthefreefallalone.Einstein[2]wrote,"Forthispurposewemustchoosetheaccelerationoftheinfinitelysmall(?ocal")systemofco-ordinatessothatnogravitationalfieldoccurs;thisispossibleforaninfinitelysmallregion."
Also,foraLorentzmanifold,ifa?reefall"resultsinalocalconstantmetric,whichisdifferentfromMinkowskimetric,thentheequivalenceprincipleisnotsatisfiedintermsofphysics.Einstein[2]wrote,"...inordertobeabletocarrythroughthepostulateofgeneralrelativity,ifthespecialtheoryofrelativityappliestothespecialcaseoftheabsenceofagravitationalfield."
AccordingtoEinstein,thebodytowhicheventsarespatiallyreferrediscalledthecoordinatesystem.Tobemoreprecise,aspatialcoordinatesystemattachedtoabody(i.e.,norelativemotionnoracceleration)isits?rameofreference"[2,3].Thesecoordinatestogetherwiththetime-coordinateformthespace-timecoordinatesystem.Aframeofreferencecanbechosenphysicallyand,duetotheequivalenceprinciple,thetime-coordinateisdeterminedaccordingly(壯56).Thus,onemaycalllooselytheframeofreferenceasacoordinatesystem.Inthispaper, forthepurposeofconsideringasatisfactionoftheequivalenceprinciple,aframeofreferenceandarelatedspace-timecoordinatesystem,aredistinguishedasabove.  http://
Toclarifythetheory,Einstein[3]wrote,?ccordingtotheprincipleofequivalence,themetricalrelationoftheEuclideangeometryarevalidrelativetoaCartesiansystemofreferenceofinfinitelysmalldimensions,andinasuitablestateofmotion(freefalling,andwithoutrotation)."Thus,atanypoint(x,y,z,t)ofspace-time,a?reefalling"observerPmustbeinaco-movinglocalMinkowskispaceLas(1),whosespatialcoordinatesareattachedtoP,whosemotionisgovernedbythegeodesic, 论文代写 http://
=0,where,(2) 论文代写 http://
ds2=g((dx(dx(andg((isthespace-timemetric.Theattachmentmeansthat,betweenPandL,thereisnorelativemotionoracceleration.Thus,whenaspaceshipisundertheinfluenceofgravityonly,thelocalspace-timeisautomaticallyMinkowski.Notethatthefreefallimpliesbutisbeyondjusttheexistenceof?rthogonaltetradofarbitrarilyacceleratedobserver"[4].
Einstein?equivalenceprincipleisverydifferentfromtheversionformulatedbyPauli[10,p.145],?oreveryinfinitelysmallworldregion(i.e.aworldregionwhichissosmallthatthespace-andtime-variationofgravitycanbeneglectedinit)therealwaysexistsacoordinatesystemK0(X1,X2,X3,X4)inwhichgravitationhasnoinfluenceeitherinthemotionofparticlesoranyphysicalprocess."NotethatinPauli?misinterpretation,gravitationalaccelerationasaphysicalcauseisnotmentioned,andthusPauli?version3),whichisnowcommonlybut ;mistakenlyregardedasEinstein?versionoftheprinciple[30],actuallyisnotaphysicalprinciple.BasedonPauli?version,itwasbelievedthatingeneralrelativityspace-timecoordinateshavenophysicalmeaning.Inturn,diffeomorphiccoordinatesystemsareconsideredasequivalentinphysics[21]notjustincertainmathematicalcalculations.However,accordingtoEinstein?calculations[2,3],thisissimplynottrue(seesection3). 论文网 http://
Theinitialformoftheequivalenceprincipleisarelationbetweenaccelerationandgravity.However,intheaboveclarification,theroleofaccelerationisnotexplicitlyshown.Onemayaskifaccelerationdoesnotexistforastaticobject,wouldtheequivalenceprinciplebesatisfied?Onemustbecarefulbecauseageodesicmaynotrepresentaphysicalfreefall.
TherearethreephysicalaspectsinEinstein?equivalenceprincipleasfollows[3]:
1)Inaphysicalspace,themotionofafreefallingobserverisageodesic.
2)Theco-movinglocalspace-timeofanobserverisMinkowski,when1)istrue.
3)Aphysicaltransformationtransformsthemetrictotheco-movinglocalMinkowskispace.
Point3)mustindicatethatthisphysicallocalcoordinatetransformationisduetothefreefallalone.Inotherwords,thephysicalvalidityofthegeodesic1)isaprerequisiteforthesatisfactionoftheequivalenceprinciple,andvalidityof3)isanindicationofsuchasatisfaction.Thus,asatisfactionoftheequivalenceprincipleisbeyondthemathematicaltangentspace(壯5-7).
Perhaps,thisinadequateunderstandingis,inpart,duetothefactthatitisoftendifficulttoseethephysicalvalidityofpoint2)directly,i.e.,howthemetrictransformedautomaticallytoalocalMinkowskispace.Tothisend,examiningpoint1)and/orpoint3)wouldbeuseful.Point1)isaprerequisiteoftheequivalenceprinciplepoint2).ForPoint1)tobevalid,i.e.,thegeodesicrepresentingaphysicalfreefall,itisrequiredthatthemetricofsuchamanifoldshouldsatisfyallphysicalprinciples.Needlesstosay,suchametric ;mustbephysicallyrealizable.Ifpoint1)isvalidinphysics,point3)shouldproducevalidphysicalresults.Thus,onecancheckpoint3)todeterminethevalidityofpoint1)orviceversa.

Themathematicalexistenceofaco-movingLocalMinkowskispacealonga?reefall"geodesicimpliesonlythatRiemanniangeometryiscompatiblewiththeequivalenceprinciple.Thephysicsiswhethertheexistenceofaphysicallocaltransformationwhichtransformsthemetrictotheco-movinglocalMinkowskispace.Thisispossibleonlyifthegeodesicrepresentsaphysicalfreefall,i.e.,theequivalenceprincipleensurestheexistenceofsuchaphysicaltransformation.Thus,onemustcarefullydistinguishmathematicalpropertiesofaLorentzmetricfromphysicalrequirements.Apparently, adiscussiononthepossiblefailureofsatisfyingtheequivalenceprinciplewasover-lookedbyEinsteinandothers(see壯67).
3.Einstein?IllustrationoftheEquivalenceprinciple
Einstein[3]illustratedhisequivalenceprincipleinhiscalculationofthelightbendingaroundthesun.(Note,theothermethoddoesnothavesuchabenefit.)His1915equationforthespace-timemetricg((is  
G(((R((-Rg((=-KT(m)(((3)  http://

whereKisthecouplingconstant,T(m)((istheenergy-stresstensorformassivematter,R((istheRiccicurvaturetensor,andR=R((g((,whereg((,istheinversemetricofg((.Now,heconsideredacoordinatesystemS(x,y,z,t)withthesunattachedtothespatialorigin.Basedoneq.(3),andthenotionofweakgravity,Einstein?ustified"thelinearequation, 论文网 http://
=2K(T((-g((T),where(((=g((-(((,(4a) 论文网 http://
(((istheflatmetric,andT=T((g((.Then,fromeq.(4a),andTtt=(,otherwiseT((iszero,byusingtheasymptoticallyflatofthemetric,Einsteinobtained,toasufficientlycloseapproximation,themetricforcoordinatesystemS  
ds2=c2(1-)dt2-(1+)(dx2+dy2+dz2),(4b)  http://

where(isthemassdensityandr2=x2+y2+z2.
However,sinceeq.(3)itselfisquestionablefordynamicproblems[7,8,24-26],itisnecessarytojustifyeq.(4a)again.Also,thenotionofweakgravitymaynotbecompatiblewiththeprincipleofgeneralcovariance[3].Inthenextsection,eq.(4a)willbejustifieddirectlyandisindependentofthedetailsofhigherordertermsofanexactEinsteinequation.Forthisreasonandthedynamicalincompatibilitywitheq.(3),eq.(4a)iscalledtheMaxwell-NewtonApproximation[7].Inotherwords,eq.(4a)shouldbevalidfordynamicproblems.Also,ani mplicitassumptionofEinstein?calculationisthatthegravitationaleffectsduetothelightitself,isnegligible.Toaddressthisissuetheoretically,wouldbecomplicatedandisbeyondthescopeofthispaper[9].Here,thisnegligibilityisjustifiedfromtheviewpointofpracticalobservationsonly.
Now,accordingtothegeodesiceq.(2),onehasd2x/ds2=0forx((=x,y,z)since(gtt/(x((0.Thus,thegravitationalforceisnon-zero,andtheequivalenceprinciplewouldbeapplicable.(Forthenon-applicablecases,pleasesee壯5-7.)ConsideranobserverPat(x0,y0,z0,t0)ina?reefalling"state,  http://
dx/ds=dy/ds=dz/ds=0.(5) 论文代写 http://
Accordingtotheequivalenceprincipleandeq.(1),state(5)impliesthetimedtanddTarerelatedby 论文代写 http://
c2(1-)dt2=ds2=c2dT2(6)  http://

sincethelocalcoordinatesystemisattachedtotheobserverP(i.e.,dX=dY=dZ=0ineq.[1]).Thisisthetimedilationofmetric(4b).Eq.(6)showsthatthegravitationalredshiftsarerelatedtogtt,andiscompatiblewithhis1911derivation[2].Moreover,sincethespacecoordinatesareorthogonaltodt,at(x0,y0,z0,t0),forthesameds2,eq.(6)implies[3]  http://

(1+)(dx2+dy2+dz2)=dX2+dY2+dZ2.(7) 论文网 http://
Ontheotherhand,thelawofthepropagationoflightischaracterizedbythelight-conecondition, 论文代写 http://
ds2=0.(8) 论文代写 http://
Then,tothefirstorderapproximation,thevelocityoflightisexpressedinourselectedcoordinatesSby  
=c(1-).(9) 代写论文 http://
Itiscrucialtonotethatthelightspeed(9),foranobserverP1attachedtothesystemSat(x0,y0,z0),issmallerthanc;andthisconditionisrequiredbythecoordinaterelativisticcausalityforaphysicallyrealizablespace-timecoordinatesystem(see§6).ObserverP1sharesthesameframeofreferencewiththesun,andthevelocityoflightisclearlyframe-dependent,butrestricted.
Thisdifferencefromcisduetogravity(orthecurvedspace)togetherwiththeequivalenceprinciple.TheobserverPisinafreefallingframeofreferenceandthuswouldnotexperiencethegravitationalforceasP1.Notethateq.(9)isconsistentwitheqs.(6)and(7)whichareduetotheequivalenceprinciple.Areasonforderivingeq.(6)andeq.(7)isthatifthemetricofamanifolddoesnotsatisfytheequivalenceprinciple,ds2=0wouldleadtoanincorrectlightvelocity(see§5-7).Thus,notonlyeq.(6),whichlead stogravitationalredshifts,butalsoeq.(9)isatestoftheequivalenceprinciple.
Einstein[3]wrote,?ecanthereforedrawtheconclusionfromthis,thatarayoflightpassingnearalargemassisdeflected."Thus,Einsteinhasdemonstratedthattheequivalenceprinciplerequiresthataspace-timecoordinatessystemmusthaveaphysicalmeaning;andaspace-timecoordinatesystemcannotbejustanyGaussiancoordinatesystem.Itseems,Einstein[2]chosethiscalculationmethodtoclarifyhisstatementsontheequivalenceprinciple.Inmanytextbooks[12,13,21-23],derivationofthecoordinatelightspeediscircumvented,andthedeflectionangleisobt aineddirectly.But,suchamanipulationhasnotreallyachievedaderivationindependentofthecoordinatesystemsinceaparticulartypeisneededtodefinetheangle. 论文网 http://
However,althoughEinsteinemphasizedtheimportanceofsatisfyingtheequivalenceprinciple,hedidnotdiscusswhatcouldgowrong.Forinstance,iftherequirementofasymptoticallyflatwerenotused,onecouldobtainasolution,whichdoesnotsatisfytheequivalenceprinciple.Anotherinterestingquestioniswhethertheequivalenceprincipleissatisfiedif((tt=0((=x,y,z).Whathasbeenmissingisadiscussiononthevalidityofthegeodesicrepresentingaphysicalfreefall.Understandably,suchadiscussionwasnotprovidedsincethevalidityof(4b) canbedecidedonlythroughobservations.Thisillustratesalsothattoseewhethertheequivalenceprincipleissatisfied,onemustconsiderbeyondtheEinsteinequation(see§5).
4.DerivationoftheMaxwell-NewtonApproximationforMassiveMatter
Formassivematter,ithasbeenproven[7]thateq.(4a)isdynamicallyincompatiblewitheq.(3).Thebinarypulsarexperiments[31]makeitnecessarytomodifyeq.(1)toa1995updateversion, 论文网 http://

Gab(Rab-Rgab=-K[T(m)ab-t(g)ab].(10a)
and
(cT(m)cb=(ct(g)cb=0,(10b)  http://
wheret(g)abisthegravitationalenergy-stresstensor.Thefirstorderapproximationofeq.(10a)is 论文代写 http://
(c(cab=-KT(m)ab(10c)  http://

Eq.(10c)iscalledtheMaxwell-NewtonApproximation[7]andisequivalenttoeq.(4a).
Theabovemodificationisbasedonthefactsthat,asafirstorderapproximation,eq.(10c)issupportedbyexperiments[7,19]andthatitisthenaturalextensionfromNewtoniantheory.However,onemayarguethatthisisnotyetentirelysatisfactorysinceithasnotbeenshownrigorouslythateq.(10c)iscompatiblewithgeneralrelativity.Inparticular,onemightstillargue[32]thatthewavecomponentingat(fora=x,y,z,t)asartificiallyinducedbytheharmonicgauge.
ItwillbeshownthattheMaxwell-NewtonApproximation(10c)canberigorouslyderivedfromtheequivalenceprincipleandrelatedphysicalprinciplesthatleadtogeneralrelativity.Sincelineareq.(10c)issupportedbyexperiments,toreaffirmthevalidityofgeneralrelativity,onemustshowclearlythateq.(10c)iscompatiblewiththetheoreticalframeworkofrelativity.Thus,suchaproofofeq.(10c)notonlyprovidesatheoreticalfoundationforeq.(10)butalsoreaffirmsgeneralrelativity.
Ingeneralrelativity[2]therearethreebasicassumptionsnamely:1)theprincipleofequivalence;2)theprincipleofcovariance(aswillbeshownnecessarilyberestrictedtospace-timecoordinatesystemswhicharecompatiblewiththeequivalenceprinciple.)and3)thefieldequationwhosesourcecanbemodified.Notethateq.(10c)isinvariantwithrespecttotheLorentztransformations.Moreover,eq.(10c)iscompatiblewiththenotionofweakgravity.Thus,eq.(10c)asanapproximationforaspecifiedcoordinatesystem,iscompatiblewiththerequirementofcovarianc eandcompatibilitywithweakgravity.Itremainstoshowthateq.(10c)isderivablefromtheequivalenceprinciple.  http://

Theequivalenceprincipleandtheprincipleofgeneralrelativityimplythatthegeodesicequation(2)istheequationofmotionforaneutralparticle[2,3].IncomparisonwithNewton?theory,Einstein[2]obtainsthegravitationalpotential, 代写论文 http://
("c2g00/2.(11)  
Since(satisfiesthePoissonequation((=4(((,accordingtothecorrespondenceprinciple,onehasthefieldequation,(g00/2=4((c-2T00,whereT00"(,themassdensityand(isthecouplingconstant.
Then,accordingtospecialrelativityandtheLorentzinvariance,onehas 论文代写 http://
(c(cgab=(c(c(ab=-4((c-2((T(m)ab+((m)(ab(,(12a)
where
(+(=1,(m)=(cdT(m)cd,(12b)  http://
T(m)abisthetensorformassivematter,(abistheMinkowskimetric,and(and(areconstants.Eq.(12)isafieldequationforthefirstorderapproximation(asassumed)forweakgravityofmovingparticles.Animplicitgaugeconditionisthattheflatmetric(abistheasymptoticlimitatinfinity.Tohavetheexactequation,sincethelefthandsideofeq.(12a)doesnotsatisfythecovarianceprinciple,onemustsearchforatensorwhosedifferencefrom(c(c(ab/2isofsecondorderin(c-2.
InRiemanniangeometry,ithasbeenproven[12]thatthecurvaturetensor?((((istheonlytensorthatcanbeconstructedfromthemetrictensoranditsfirstandsecondderivatives,andislinearinthesecondderivatives."EinsteinidentifiedtheRiccicurvaturetensorRab((R(a(b)astherequiredtensor.IfRabincludesnootherfirstordersum,theexactequationwouldbe 论文代写 http://
Rab=X(2)ab-4((c-2((T(m)ab+(T(m)gab,(13) 论文网 http://

whereT(m)(=gcdT(m)cd)isthetrace,X(2)abisasecondorderunknowntensorchosenbyEinsteintobezero.However,anon-zeroX(2)abmaybeneededtoensureeq.(12)asanapproximationofeq.(13)[7].
Now,letusexamineRabfurtherwhethertheabovephysicalrequirementcanbevalid.Letusdecompose  
Rab=R(1)ab+R(2)ab,(14a)
where
R(1)ab=(c(c(ab-((c((b(ac((a(bc(((a(b(,(14b) 论文代写 http://
andR(2)abconsistsofhigherorderterms.Ifeq.(12)providesthefirstorderapproximation,thesumofotherlineartermsmustbeofsecondorder.Tothisend,letusconsidereq.(12a),andobtainK=8((c-2and 论文网 http://
(c(c((a(ab)=(K(((aT(m)ab+((b(m)(.(15a)

From(cT(m)cb=0,itisclearthatK(cT(m)cbisofsecondorderbutK(b(m)isnot.However,onemayobtainasecondordertermbyasuitablelinearcombinationof(c(cband(b(.From(15a),onehas 代写论文 http://
(c(c((a(ab(C(b()=(K(((aT(m)ab+((+4C(+C()(b(m)(.(15b) 论文代写 http://
Thus,simplychoosingtheharmoniccoordinates(i.e.,((a(ab((b(/2"0),canleadtoinconsistency.Itfollowseq.(14b)andeq.(12b)that,fortheothertermstobeofsecondorder,onemusthave 论文代写 http://
((4C(+C(=0,2C+1=0,and(+(=1.(15c)  http://

Thesolutionofeq.(15c)isC=-1/2,(=2,and(=-1.Thus,forthefirstorderapproximation, 论文网 http://
(c(c(ab=(K(T(m)ab+(m)(ab(,(16) 论文网 http://
whichisequivalenttoeq.(10c),hasbeendeterminedtobethefieldequationofmassivematter.
ThisderivationisindependentoftheexactformofanEinsteinequation.Animplicitgaugeconditionisthattheflatmetric(abistheasymptoticlimit.Eq.(16)iscompatiblewiththeequivalenceprincipleasdemonstratedbyEinstein[2]inhiscalculationofthebendingoflight.Thus,thederivationisself-consistent.
OnemightarguethatEinsteinequation(3)couldbe?erived"fromalinearequationmoregeneralthaneq.(12a),ifoneregardsthegravitationalfieldasaspin-2fieldcoupledtotheenergytensor[19,33].However,sucha?ure"theoreticalapproachisnotreallyconsistentwithNewton?theoryandrelatedobservationsbecausethenotionofgaugeisused.Moreover,insucha?roof",theexistenceofboundeddynamic2)solutionsforeq.(3)mustbeinvalidlyassumed.
NotethatEinsteinobtainedthesamevaluesfor(and(byconsideringeq.(13)afterassumingX(2)ab=0[34].Thepresentapproachmakesitpossibletoobtainfromeq.(13)anequationwithanadditionalsecondorderterm,i.e., 论文网 http://
Gab(Rab(gabR=-K(T(m)ab(Y(1)ab(,(17)
where
KY(1)ab=X(2)ab-gab(X(2)cdgcd( 代写论文 http://
isofsecondorder.Theconservationlaw(cT(m)cb=0and(cGcb(0impliesalso(aY(1)ab=0.IfY(1)abisidentifiedasthegravitationalenergytensort(g)ab,eq.(10)isreaffirmed.
Theanti-gravitycouplingoft(g)abthatexplainsthedynamicalfailureofeq.(1),isduetoEinstein?radiationformula[7].However,Pauli[10]wasthefirsttopointoutexplicitlythepossibilityofsuchanantigravitycoupling.Moreover,theexistenceofsuchacouplingis,inaway,implicitlysuggestedbythesingularitytheorems,whichshowthatifallthecouplingsareofthesamesign[21],theexistenceofunrealisticspacetimesingularitieswouldbinevitable.Theneedofananti-gravitycouplingwasfirstdiscoveredincalculatingthegravityofannb sp;electromagneticwave[6].
Moreover,itwasEinsteinandRosen[35]whofirstdiscoverthatthe1915equationmaynothaveapropagatingwavesolution.In1953Hogarth[36]conjecturedthatthisequationdoesnothaveadynamicsolution.Adefinitiveindicationofthisisthenon-existenceoftheplane-wavesolution[8].Notethatthe?lanewaves"proposedbyBondi,Pirani,andRobinson[37],areactuallyunboundedalthoughtheybelievethataplane-waveisanidealizationofaweakwavefromadistantisolatedsource.Theseunboundedsolutionssatisfytheconditionofplanenessbutare notrelatedtoanyweakwave.Also,althoughMisner,ThorneWheeler[13]concludecorrectlythattheplane-wavesarebounded,theirequationforplane-wavesactuallyhasnoboundedwavesolution[8].Thisillustratesthatoverconfidencemayleadtocareless,andresultininconsistency.  http://
Inshort,thetheoreticalframeworkofgeneralrelativitypermitanadditionaltermY(1)ab(0whoseexistenceisrequiredbythedynamiccases.The1915equationisonlyanoversimplifiedspecialchoiceofEinstein.Note,however,suchachoiceisconsistentwiththeequivalenceprincipleisknownonlyforthestaticcase.
5.ValidityofaSpace-TimeMetricandtheEquivalencePrinciple
Einsteinproposedthattheequivalenceprincipleissatisfiedinaphysicalspace-time1).Infact,theequivalenceprincipleissatisfied,ifandonlyifthespace-timemanifoldisphysicallyrealizable,sinceasatisfactionoftheequivalenceprinciplerequiresthatthegeodesicrepresentaphysicalfreefall.Thus,althoughdefiningacoordinatesystemforthepurposeofcalculationisonlyamathematicalstep,choosingaspace-timecoordinatesystemrequiresphysicalconsiderations.
ItwillbeshownthatnotallmathematicalcoordinatesystemsareequivalentinphysicsasclaimedbyBergmann[14]andLiu[15].Forclarity,thiswillbeillustratedwithafewsimpleLorentzmetricswithoutgravity.
Example1.Toseetheneedofconsideringbeyondthemetricsignature,considertheartificialmetric, 代写论文 http://
ds2=(2dt2-dx2-dy2-dz2,(18a)  
thetimeunitoftissecond,thespaceunitiscm,and(((2c,c=3x1010cm/sec).Iftheequivalenceprinciplewerevalid,ds2=0wouldimplythelightspeedtobe(.Immediately,thereisacontradiction,andthustheequivalenceprinciplecannotbevalid.
Nevertheless,onemightarguethatmetric(18a)canbetransformedto  http://
ds2=c2dt'2-dx'2-dy'2-dz'2,(18b)  
bythefollowingdiffeomorphism, 论文网 http://
x'=x,y'=y,z'=z,andt'=t(/c.(19a) 代写论文 http://
Eq.(19a)implies,however,thattheunitoft'isc/((sec).Thelightspeedinthex'-directionis 论文网 http://
=c()=(((cm/sec).(19b) 论文网 http://
Thus,thelightspeedremains(.If(=2c,Metric(19a)impliesthatthelightspeedwouldbe6x1010cm/sec;andmetric(19b)impliesthatthelightspeedis3x1010cm/half-sec.
Intheliterature,theunitsofthecoordinatesareusuallynotspecified.Then,thedistinctmetric(18a)wouldbeconfusedwitharescalingoftheMinkowskimetric.Thiscreatesaillusionthatallconstantmetricswereequivalenceinphysics.Thisexampleillustratesalsothatitisinvalidto?efine"lightspeedintermsoflocalMinkowskispacesinamanifold.
Moreover,iftheequivalenceprinciplewerevalid,accordingtoEinstein?approach,onewouldobtain 论文代写 http://

c2dT2=(2dt2,and(dX2+dY2+dZ2)=(dx2+dy2+dz2),(20a) 论文网 http://
forarestingobserveratapoint(x0,y0,z0,t0).Eq.(20a)andds2=0implythatthelightspeedis  http://
==1(20b) 论文代写 http://
Eq.(20b)implies,however,thatthelightspeediscinthelocalMinkowskicoordinate,butis(((2c)inthe(x,y,z,t)space.Ontheotherhand,sincethereisnogravitationalforceforthiscase,wecanhavealso 论文网 http://
x=X,y=Y,andz=Z(21)  
Eq.(20b)andeq.(21)absurdlymeanthatforthesameframeofreference,wehavedifferentlightspeeds.Thisisindisagreementwiththeprincipleofuniquenessforaphysicalmeasurement.
Example2,considertheMinkowskiflatmetricandthetransformation,whichisadiffeomorphism,  
t=C[exp(T/C)-exp(-T/C)]/2,whereC=constant.(22a)
Then
ds2=[exp(T/C)+exp(-T/C)]2dT2-dx2-dy2-dz2,(22b) 论文网 http://
isthemetrictransformedfromtheMinkowskimetric.Ifmetric(22b)isrealizable,accordingtods2=0,themeasuredlightspeedwouldbe[exp(T/C)+exp(-T/C)]/2.
From(22b),theChristoffelsymbols((,((arezerosexcept(t,tt=(tgtt/2.Then,accordingtothegeodesicequation,theequationofmotionforaparticleat(x,y,z,T)is  
+(t,tt=0,and===0(23a)
where
(t,tt={ln[exp(T/C)+exp(-T/C)]}.  http://
Itfollowseq.(23a)thatoneobtains,forsomeconstantk 代写论文 http://
dT/ds=k[exp(T/C)+exp(-T/C)]-1anddx(/ds=Constant(23b) 论文网 http://
Now,considerthecasedx/dT=dy/dT=dz/dT=0.Forthiscase,onehasdx/ds=dy/ds=dz/ds=0anddx2/dT2=dy2/dT2=dz2/dT2=0.Thus,insucha?reefall",thereisnochangeinthespatialpositionoracceleration.Physically,thismeansthatsuchanobserverwouldhavethesameframeofreference,whether?reefall"ornot.Thus,hewouldabsurdlyhavetwodifferentlightspeedsfromthesameframeofreference.Accordingly,theequivalenceprincipleisnotsatisfiedandmetric(22)isnotrealizable.Notethat,fr ommetric(22),thereisnoaccelerationforastaticparticle.
Nevertheless,sometheoristswoulddisregardalltheseinconsistencybecausetheybelievethatspace-timecoordinateshavenophysicalmeaning.Therefore,theyalsodisagreewithEinstein[2,3]andregardthatcoordinatelightspeedsasmeaningless.
6.IncompatibilityoftheGalileanTransformationtotheEquivalencePrinciple
ItwillbeshownthataGalileantransformation,whichisunrealizable,isincompatiblewiththeequivalenceprinciple.Sometheorists,however,consideredincorrectlythattheGalileantransformationwouldleadtoaspace-timecoordinatesystem.Therootoftheirproblemisthattheymistakentheexistenceofthetetradasequivalenttoasatisfactionoftheequivalenceprinciple.Theydonotunderstandthatasatisfactionofthisprinciplerequiresthegeodesicof?reefall"mustbevalidinphysics.  
ConsidertheGalileantransformationfrom(x,y,z,t)totheK'coordinates, 论文代写 http://
t=t',x=x',y=y',andz=z'-vt',(24a)  
wherevisaconstant.Eq.(24a)transformsmetric(10a)toanotherconstantLorentzmetric  http://
ds2=((c-v)dt'+dz'(((c+v)dt'-dz'(-dx'2-dy'2,(24b) 论文网 http://
Metric(24b)isaspecialcaseofaspacewithanindefinitemetric.Then,forlightraysinthez'-direction,ds2=0wouldimplyatanypointthelightspeedswere 论文代写 http://
dz'/dt'=c+v,ordz'/dt'=-c+v.(25) 代写论文 http://
Clearly,?ightspeed"(25)violatescoordinaterelativisticcausality(i.e.nocauseeventcanpropagatefasterthanthevelocityoflightinavacuum).Thus,metric(24b)isnotphysicallyrealizable,andthosein(25)arenotcoordinatelightvelocities.
Moreover,accordingtothegeodesicequation(2),metric(24b)impliesd2x'(/ds2=0,andthus 论文网 http://
dx'(/ds=constant.wherex'((=x',y',z',ort')(26a)  http://
atanypoint.Now,accordingtometric(24),considerthecaseofa?reefall"at(x'0,y'0,z'0,t'0) 论文代写 http://
dx'/ds=dy'/ds=dz'/ds=0,anddt'/ds=(c2-v2)-1/2(26b)  
Sincethereisnoaccelerationormotion,sucha?reefalling"observerP'carrieswithhimtheframeofreferenceK'.Sincea?reefall"doesnotautomaticallyobtainalocalMinkowskispace,point2)oftheequivalenceprincipleisviolated.Also,forobserverP',accordingeq.(1)themeasuredlightspeedisc,butaccording(24b)thelightspeedinthex-directionis(c2-v2)1/2.Thisinconsistencyalsoimpliesthatpoint3)oftheequivalenceprincipleisnotsatisfiedinK'.
Nevertheless,mathematicsensurestheexistenceofalocalMinkowskispace,whichcanbeobtainedbychoosingfirstthepathofaparticletobethetimecoordinateandthentheotherthreespacecoordinatesbyusingorthogonality.Accordingtocondition(26b),thetimecoordinatewouldremainthesamedt'.But,thecoordinatedz'isnotorthogonaltodt'.Now,letusworkoutthelocalorthogonaltetradofP',whosedirectionvP'is(0,0,0,dt').Then,theorthonormalvectorsofthetetradare 论文网 http://
a1=(1,0,0,0),a2=(0,1,0,0),a3=(0,0,(,(),andbp'=(0,0,0,()(27a)
where
(=(-1,(=-(v/c2,and(=(c2-v2)-1/2.  http://
Thecorrespondingtransformationsisasfollows: 论文代写 http://
dt'=((dT-v/c2dZ),dz'=(-1dZ,dx'=dX,anddy'=dY.(27b)  http://
Thus,(dx',dy',dz')and(dX,dY,dZ)sharethesameframeofreferencesincethereisnoacceleration.But,thereisaspacemeasurementchangeinthez-direction.Metric(24b)doesnotsatisfypoint2)oftheequivalenceprinciplesincethereisnophysicalcausefortransformation(27b).Inrelativity,suchaphysicaltransformationhappensonlywhenthereisrelativemotionoracceleration.But,P'isrestatK'.Thus,(27b)illustratesalsothatgeodesic(26)doesnotrepresentaphysicalfreefall.
Amisunderstandingoftheequivalenceprinciple,asYu(p.42of[23])believed,isthatatanyspace-timepoint,itisalwayspossibletoestablishalocalMinkowskispace,whichisrelatedtoa?reefall".However,thisisnecessarybutinsufficient.Forinstance,atanyspace-timepointofmanifold(18a),(22b)or(24b),thereisalocalMinkowskispace,whichisco-movingwitha?reefalling"observerinthemanifold.But,thegeodesicdoesnotrepresentaphysicalfreefall.NotethatYu?interpretationisessentiallyrephrasingPauli?misinterpretatio n[3,p.145].
TheGalileantransformationisanunphysicaltransformation,anditsimplytakesanotherunphysicaltransformationtocancelouttheunphysicalpropertiessointroduced.Infact,(24a),and(27b)imply 代写论文 http://
dt=((dT-v/c2dZ),anddZ=(dz'=((dz+vdt).(27c)  http://
Transformation(27c)isjustaLorentz-Poincarétransformation.(27b)completesthetransformation(27c)startingform(24a).
Ithasbeenshownindifferentapproachesthatmetric(24b)isincompatiblewithphysicsandinparticulartheequivalenceprinciple.Since(24a)isaGalileantransformation,theGalileantransformationisalsonotphysicallyvalidingeneralrelativity.ThefailureofsatisfyingtheequivalenceprincipleshouldbeexpectedsincetheGalileantransformationisexperimentallynotrealizable.ThisanalysisshowsalsothattheMinkowskimetricisonlyvalidconstantmetricinphysics.Infact,ageneralresultisthatif((tt=0for(=x,y,orz,thentheequivalen ceprincipleissatisfiedonlyifthemetricisMinkowski.
Anotherconsequenceisthereaffirmationofcoordinaterelativisticcausalityinvacuum.Thatthespeedoflightcouldbelargerthancthroughacoordinatetransformationisinconsistentwiththenotionthatthelightspeedcisthemaximumpossiblespeed.Theequivalenceprinciplerulesoutsuchapossibility.Itthusfollowsthatphysicallythespeedoflightcannotbelargerthancatthepresenceofgravity.Infact,observationconfirmsthatgravityonlyleadstoareductionofthelightspeed. 论文网 http://
IthasbeenillustratedthattheGalileantransformationisincompatiblewiththeequivalenceprincipleintheabsenceofgravity.Infact,theincompatibilityisalsotrueevenwhengravityispresent.Toillustratesthis,letusconsiderphysicalmetric(4b)andthephysicalsituationthataparticleat(0,0,z0,t0)movingwithvelocityvatthez-direction.TheGalileantransformation(24a)transformsmetric(4b)to  
ds2=c2(1-)dt'2-(1+)(dx'2+dy'2+[dz'-vdt?2(28a)  
Ifmetric(28a)hadaphysicalrealizablecoordinatesystemS',theparticlewouldbeat(0,0,z'0,t'0)inthestate(0,0,0,dt')andthelocalspatialcoordinatesdx',dy',anddz'wouldbeattachedtotheparticleattheinstancet'0.Theproblemcanbereducedtopreviouscasebyconsideringthelimits(?0.
Moreover,accordingtoEinstein[3],theequivalenceprincipleisvalidonlyifds2=0producesthecorrectlightspeeds.Thus,ifS'wererealizable,thelightspeedsinthez-directionwouldbe  
=c(1-()+v,or=-c(1-()+v,where(=(28b)  http://
accordingtometric(28a).Thus,coordinaterelativisticcausalityisviolatedforsufficientlylarger.Inotherwords,point1)oftheequivalenceprinciplecannotbesatisfiedandmetric(28a)isnotrealizable.
Thisillustratesthattheequivalenceprincipleisarequirementforavalidphysicalspace-timecoordinatesystem.
7.RestrictionofCovariance,andIntrinsicallyUnphysicalLorentzManifolds
Einsteinproposedthattheequivalenceprincipleissatisfiedinaphysicalspace-time1).Moreover,theequivalenceprincipleissatisfiedonlyinaphysicalspace-timesincetheexistenceofalocalMinkowskispacehasbeenprovenbymathematicsandasatisfactionoftheequivalenceprinciplerequiressufficientsatisfactionsofallphysicalconditions.Forexample,whencoordinaterelativisticcausalityisnotsatisfied,theequivalenceprincipleisprovendirectlytobenotvalidforthismanifold.Thecurrentconfusionwasduetothattheequivalenceprinciplehasnotbeenunderstoodnb sp;correctlyfromtheviewpointofphysics.
However,onemaystillwonderwhetheraLorentzmanifoldisalwaysdiffeomorphictoaphysicalspace.Ifthisweretrue,thenthemetricsignaturewouldbeessentiallyequivalenttotheequivalenceprinciple.But,thereareLorentzmanifoldsanyofwhichcannotbediffeomorphictoaphysicalspace.Inviewofthis,suchmisunderstandingofrelativitymustberectified.SincethebeliefthataLorentzmanifoldwerediffeomorphictoaphysicalspace,hasneverbeenproven;theburdenofproofisonsuchbelievers.Nevertheless,itisdesirabletogiveanexample ofanintrinsicallyunphysicalLorentzmanifold.ThiscanevenbeasolutionofEinstein?equationifitfailsaphysicalrequirement,whichisindependentofacoordinatesystem[8,9,16]. 论文网 http://
Forinstance,anacceptedsolutionofmetricforanelectromagneticplanewave[38]is 代写论文 http://
ds2=dudv+Hdu2-dxidxj,whereH=hij(u)xixj,hii(u)0,hij=hji,(29) 论文代写 http://
u=t-z,v=t+z.ThisisaLorentzmanifoldsinceitseigenvaluesareH((H2+1)1/2,-1,and-1.However,sincethecondition1((1+H)/(1-H)maynotbevalid,metric(29)doesnotsatisfycoordinaterelativisticcausalityandthereforetheequivalenceprinciple.Moreover,sinceHcanbearbitrarilylarge,metric(29)isincompatiblewithEinstein?notionofweakgravity4)andthecorrespondenceprinciple.Also,inthelightbendingexperiment,thegravitationaleffectofthelightisimplicitlyassumedtobenegligible.Thus,n bsp;metric(29)cannotbevalidinphysics.
Nevertheless,toshowthatmetric(29)cannotbediffeomorphictoaphysicalspace,needsmorework.Thegravitationalforce(relatedto(ztt=(1/2)((hijxixj)/(thasarbitraryparameters(thecoordinateorigin).Thisarbitrarinessinthemetricviolatestheprincipleofcausality(i.e.,thecausesofphenomenaareidentifiable)[8,11].Thus,themanifold(29)cannotbediffeomorphictoaphysicalspacesinceadiffeomorphismcannoteliminatetheparameters,whichviolatetheprincipleofcausality.
8.ConclusionsandDiscussions
Einstein[2,3]proposedtheequivalenceprincipleforthereality,whichhemodelsasaRiemannianphysicalspace-time.However,Pauli?[10,p.145]versionimpliesthattheequivalenceprinciplewouldbesatisfiedeventhoughthecoordinatesystemisnotphysicallyrealizable.NowitisclarifiedthatEinsteincorrectlyobjectedPauli?versionasamisinterpretation[30].Also,itisproventhattheequivalenceprincipleissatisfiedifandonlyifamanifoldisphysicallyrealizable. 论文代写 http://
Ingeneralrelativity,theMinkowskimetricinspecialrelativityisobviouslyaspecialcase.However,itwasnotclearthatthe?rinciples"whichleadtogeneralrelativityarecompatiblewitheachothereveninthisspecialcase.SometheoristsbelieveincorrectlythattheGalileantransformationwerevalidforgeneralrelativity,althoughEinstein[2]hasmadeclear,?pecialtheoryofrelativityappliestothespecialcaseoftheabsenceofagravitationalfield".Torectifythis,itisshowndirectlythat,duetotheequivalenceprinciple,theMinkowskimetricnbs p;istheonlyvalidconstantspace-timemetric(§6).
Toestablishspecialrelativity,theGalileantransformationisproventobeunrealizablebyexperiments.Thus,theoreticallyaGalileantransformationshouldbeincompatiblewiththeequivalenceprinciple,whichisapplicabletoonlyaphysicalspace.Thismeans,incontrastofthebeliefofsometheorists[14,15],thattheequivalenceofallframesofreferenceisnotthesameasthephysicalequivalenceofallmathematicalcoordinatesystems.Infact,itisinvalidinphysicstoextendthespace-timephysicalcoordinatesystemtoanarbitraryGaussiansystem[9].Forinstance, nbsp;thetimecoordinateisnotarbitrary[2,3].  http://
TheGalileantransformationimpliedthatthereisnolimitonthevelocityoflight.This,inprinciple,disagreeswiththenotionofinvariantlightspeed.However,duetoentrenchedmisconceptionsoncovariance,thisproblemwasnotevenrecognizedforfurtherinvestigations[20,23].Moreover,somesupportedsuchamisconceptionswithothererrorsandmisunderstandings.Inotherwords,suchcurrent?heories"arecharacterizedandmaintainedwithasystemoferrors.Thus,itisnecessarytocalculateexamplesthatdirectlydemonstrateaviolationoftheequivalenceprinciple.
Sometheoristsincorrectlyclaimedthattheequivalenceprincipleisequivalenttothemathematicalexistenceofthetetrad.TheyoversimplifiedEinstein?principlemerelyasthemathematicalexistenceofaco-movinglocalMinkowskispacealongatime-likegeodesic.However,thephysicsisnotonlyjustsuchanexistence,butalsotheformationofsuchlocalspacebythefreefallalone.Forinstance,thelocalspace-timeofaspaceshipundertheinfluenceofonlygravityisalocalMinkowskispace.Thus,therealquestionfortheequivalenceprincipleiswhetherthen bsp;geodesicrepresentsaphysicalfreefall.  http://

Thefactthatthereisadistinctionbetweentheequivalenceprincipleandthepropermetricsignaturewouldimplyalsothatthecovarianceprinciplemustberestricted.AnimportantfunctionoftheequivalenceprincipletestistoeliminateunphysicalLorentzmanifolds(see§7).Forexample,thefactthatmetric(29)isintrinsicallyunphysicalresolvesitsseeminglyparadoxwiththelightbendingcalculationinwhichthegravityduetothelightisimplicitlyassumedtobenegligible[2,3].Thisisanotherexamplethatamisunderstandingoftheequivalenceprinciplecanle adstodisagreementswithexperiments.
Perhaps,duetoconfusingmathematicaltheoremswithEinstein?equivalenceprincipleasPaulidid,thisprincipleisoftennotexplainedadequatelyinsomebooks[21-23].Todealwithallthetheoreticalinconsistencesuperficially,sometheoristsclaimedthatthespace-timecoordinateshavenophysicalmeaningingeneralrelativity.Suchaspeculationdisagreeswiththefactthattherearenon-scalarsinphysics.Thedeflectionoflightisrelatedtothelightraybeingobservedasanalmoststraightlineawayfromthesun,andgravitationalredshiftsarerelatedtogtt- nbsp;thetime-timecomponentofspace-timemetric.  http://

Nevertheless,basedonsuchanabsurdclaim,Hawking[18]declares,?ngeneralrelativity,thereisnorealdistinctionbetweenthespaceandtimecoordinates,justasthereisnorealdifferencebetweenanytwospacecoordinates."Ontheotherhand,Hawking[18]alsobelieved,?narrowoftime,somethingthatdistinguishedpastfromfuture,givingadirectionoftime".Apparently,hedidnotseethatthereisaninconsistencybetweenthesetwostatements.Moreover,likeothers,Hawkingacceptedthedeflectionoflight.Heprobablydidnotrealizethatthede flectionanglecouldbedefinedonlyinacertaintypeofphysicalcoordinatesystems,wherethetrajectoryofalightray,whenfarawayfromthesun,isapproximatelyastraightline.Notethatsuchlogicaldeficiencyisacommonproblemamongthoseso-called?tandard"relativists.
TheoristssuchasSynge[19],Fock[39]andmorerecently,Hawking[18,40],Ohanian,Ruffini,andWheeler[22],whodonotunderstandEinstein?equivalenceprincipleforvariousreasonsincludinginadequateunderstandingofphysicsormathematicsatthefundamentallevelordeficiencyinlogic,advocatedessentiallythatthebasisofgeneralrelativityshouldbetheEinsteinfieldequationalone.However,experimentallytheunrestrictedvalidityofEinstein?equationhasnotyetbeenestablishedbeyondreasonabledoubt[7,25,26,41].  
TheoreticallythereisnosatisfactoryproofofrigorousvalidityofEinstein?fieldequation[42-44](e.g.,theinadequatesourcetermmentionedin§1,isthecauseoftheunphysicalsolution(29)[6,8]).Infact,in1953Hogarth[36]conjecturedthatthe1915Einsteinequationisinvalidforadynamictwo-bodyproblem;andEinsteinhimselfhadpointedoutthathisequationmightnotbevalidformatterofveryhighdensity[3].Moreover,ithasbeenprovenbythebinarypulsarexperimentthatEinstein?equationmustbemodified[7]andYilmaz[45]pointed outthatEinstein?equationof1915isonlyatestparticletheory.Moreover,intermsofphysics,astaticsolutionisonlyanapproximationforsomedynamicalproblems.Thismeansthat,tosupportEinstein?illustrationoftheequivalenceprinciplewithcalculationsonthelightbending,itisnecessarytoshowthathislinearequationisjustifiablefordynamicalproblems.Thus,itisnecessarytoderivetheMaxwell-NewtonApproximationindependentlyfromphysicalprinciples(§4)sincethe1915Einsteinequationisvalidforstaticproblemsonly.
Asimpledynamicalproblemwouldbethegravityduetotheinteractionoftwomassiveparticles.Then,Einstein?conditionofweakgravity,whichisalsoduetotheequivalenceprinciple[46],requiressuchasolutionofgravitymustbebounded.Thus,satisfyingboundednessofgravityduetoaweaksourceisindependentofthefieldequation.But,nosuchsolutionhaseverbeenproventobeinexistence.Beingawareoftheunboundednessofcylindricalandspherical?aves"[47,48],afterprovingtheexistenceofCauchysolutions,Bruhat[4]remarkedthatthen bsp;physicalvalidityofanyCauchysolutionisuptotheexperimentstodecide.Whileherclarificationisreasonableforamathematician,physicistsshouldhaveknownphysicsbetter. 代写论文 http://

Theequivalenceprincipleremainsindispensablebecauseofitssolidexperimentalfoundationsuchasgravitationalredshiftsandtheblendingoflight[7,20].Theoretically,asillustrated,itsfailureisalwaysaccompaniedwithaviolationofanotherphysicalrequirement.Thus,asWeinberg[5]pointsout,?tismuchmoreusefultoregardgeneralrelativityaboveallasatheoryofgravitation,whoseconnectionwithgeometryarisesfromthepeculiarempiricalpropertiesofgravitation,propertiessummarizedbyEinstein?PrincipleoftheEquivalenceofGravitationandInertia."
Thelong-standingerrorsingeneralrelativityhaveprofoundhistoricalreasons.Mostphysicistsareusedtolinearequations,andunavoidablytheywouldapplytechniques,whicharevalidforlinearequations.But,innonlinearfieldequation,asecondordertermfromtheviewpointofphysics,maybecrucialfortheexistenceofaboundedphysicalsolution.Inotherwords,onemaynottakeitforgranted(i.e.,withoutaproof)thataphysicalrequirementiscompatiblewithafieldequation.EvenwellknownphysicistssuchasEinstein[49]andmorerecentlyFeymann [33]madesuchmistakesingeneralrelativity.  http://
Ingeneralrelativity,assumingtheexistenceofaboundeddynamicsphysicalsolution,hasneverbeenproven,butithasbeenusedwithblindfaith[50-53].Thisisessentiallywheremanyofthemathematicalerrorscomefrom.Amosttellingevidenceisthatthe?lane-waves"proposedbyBondietal.[37]arenotboundedalthoughtheybelievethattheyare.Furthermore,ithasbeenproventhattherearenoboundedplane-waves[8]forthe1915Einsteinequation.Then,thenon-existenceofadynamicsolutionformassivematterisproven[7]becauseexperimentsnbs p;supporttheMaxwell-NewtonApproximation.ThisapproachofproofhasbeencompletedsincetheMaxwell-NewtonApproximationcanbederivedfromphysicalprinciples(§4).
Currently,relativistsoftenignorephysicalrequirements[1,5]becausetheymisunderstoodtheequivalenceprincipleandacceptedthecovarianceprinciple,whichwasrejectedbyEddington.Historically,afterearlyobservationalconfirmationsofEinstein?predictions,Einsteindeclaredlogicalcompletenessofhistheory[34]althoughsuchconfirmationsverifyonlythetheoreticalframeworkofgeneralrelativity.Subsequently,ablindfaithonthetheoreticalself-consistencyofgeneralrelativitywasdeveloped.Manyphysicistsworkingongeneralrelativity,inspiteofwarningsfromGullstrand[25,26],BohrKlein[40 ],andotherphysicists[10,37],tendtohaveoverconfidenceonEinstein?equation(exceptafewsuchasN.Rosen[35]). 论文代写 http://
Adynamicphysicalsolution,aspointedoutbyLow[54],isnotjustatime-dependentsolution,whichcanbeobtainedfromtheMinkowskimetricbymakingacoordinatetransformation.Inphysics,suchadynamicsolutionmustberelatedtothedynamicsofsourcematterandgravitationalradiation.Nevertheless,ChristodoulouandKlainerman[55]claimedtheexistenceofdynamicalsolutionsbytheirconstructionalthoughsuch?olutions"areunrelatedtodynamicalsourcesorradiation.Itisalsosurprisingthattheirmainmathematicalmistakesareactuallyatthefundamentallevel[5 6-58].AspointedoutbyKrameretal.[1],manyrelativistshaveaproblemindistinguishingaphysicallyvalidsolutionfrommathematicalsolutions.Bonnoretal[5]furtherconfirmthisproblembypointingoutthatitisnotpossibletohaveaconsistentphysicalinterpretation.
9.Acknowledgments
ThispaperisdedicatedtomygrandfatherLuZhuQiu.TheauthorgratefullyacknowledgesstimulatingdiscussionswithProfessorsC.Au,C.L.Cao,S.-J.Chang,A.J.Coleman,Li-ZhiFang,L.Ford,R.Geroch,J.E.Hogarth,LiuLiao,F.E.Low,P.Morrison,A.Napier,H.C.Ohanian,R.M.Wald,ErickJ.Weinberg,J.A.Wheeler,ChuenWong,H.Yilmaz,YuYun-qiang,andY.Z.Zhang.ThisworkissupportedinpartbyInnotecDesign,Inc.,U.S.A. 论文代写 http://

ENDNOTES
1)Ingeneralrelativity,Einstein[2,3]considersthefour-dimensionalspace-timerealityasaphysicalspace-timemodeledasaRiemannianspace-time(M,g).TheRiemannianspaceMischaracterizedbyaspace-timemetricgikthatcanbedeterminedbyphysicalconsiderationssuchasthedistributionofmatter.In?elativityandtheproblemofspace",Einstein[27]wrote,
?orthefunctionsgikdescribenotonlythefield,butatthesametimealsothetopologicalandmetricalstructuralpropertiesofthemanifold....Thereisnosuchthingasanemptyspace,i.e.,aspacewithoutfield.Space-timedoesnotclaimexistenceonitsown,butonlyasastructuralqualityofthefield."
Moreover,sincesuchaRiemannianspace-timemodelsreality,allthephysicalrequirementsmustbesufficientlysatisfied.
2)AlocalMinkowskianspaceisashorthandtoexpressthatspecialrelativityislocallyvalid,exceptforphenomenainvolvingthespace-timecurvature.
3)Forexample,theWheeler-HawkingSchool[13,18,40]followsPauli?misinterpretation,andthus,theirtheoriesaredifferentfromgeneralrelativity.They,differentfromEinstein[2,3],believethatspace-timecoordinateshavenophysicalmeaning.Hawking[18]makesnosecretofhisdisagreementswithEinstein[2,3].Morerecently,basedonmisinterpretationsofFock[39],Ohanian,Ruffini,andWheeler[22]openlycriticizedEinstein?theoryasconfusingandhisprinciplesinvalid.  http://
4)Sometheoristsbelievethatthesolutionofgravityforaweaksourceneednotbebounded[38].However,ithasbeenshownthattheequivalentprincipleimpliescompatibilitywithEinstein?notionofweakgravity[46].
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