2018The Equivalence Principle， the Covariance Principle and the
作者： C. Y. Lo TheEquivalencePrinciple,theCovariancePrincipleand
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-timecoordinateshavenophysicalmeaninghttp://
1.Introduction.
Currently,amajorproblemingeneralrelativityisthatanyRiemanniangeometrywiththepropermetricsignaturewouldbeacceptedasavalidsolutionofEinstein?equationof1915,andmanyunphysicalsolutionswereaccepted.Thisis,inpart,duetothefactthatthenatureofthesourcetermhasbeenobscuresincethebeginning.Moreover,themathematicalexistenceofasolutionisoftennotaccompaniedwithunderstandingintermsofphysics.Consequently,theadequacyofasourceterm,foragivenphysicalsituation,isoftennotclearn bsp;.Pauliconsideredthat?hetheoryofrelativitytobeanexampleshowinghowafundamentalscientificdiscovery,sometimesevenagainsttheresistanceofitscreator,givesbirthtofurtherfruitfuldevelopments,followingitsownautonomouscourse."Thus,inspiteofobservationalconfirmationsofEinstein?predictions,oneshouldexaminewhethertheoreticalself-consistencyissatisfied.Tothisend,onemayfirstexaminetheconsistencyamongphysical?rinciples"whichleadtogeneralrelativity.
Thefoundationofgeneralrelativityconsistsofa)thecovarianceprinciple,b)theequivalenceprinciple,andc)thefieldequationwhosesourcetermissubjectedtomodification.Einstein?equivalenceprincipleisthemostcrucialforgeneralrelativity.Inthispaper,theconsistencybetweentheequivalenceprincipleandthecovarianceprinciplewillbeexaminedtheoretically,inparticularthroughexamples.Moreover,theconsistencybetweentheequivalenceprincipleandEinstein?fieldequationof1915isalsodiscussed.
Theprincipleofcovariancestatesthat?hegenerallawsofnaturearetobeexpressedbyequationswhichholdgoodforallsystemsofcoordinates,thatis,arecovariantwithrespecttoanysubstitutionswhatever(generallycovariant)."Thecovarianceprinciplecanbeconsideredasconsistingoftwofeatures:1)themathematicalformulationintermsofRiemanniangeometryand2)thegeneralvalidityofanyGaussiancoordinatesystemasaspace-timecoordinatesysteminphysics.Feature1)waseloquentlyestablishedbyEinstein,butfeature2)remainsanunverified conjecture.IndisagreementwithEinstein,Eddingtonpointedoutthat?paceisnotalotofpointsclosetogether;itisalotofdistancesinterlocked."EinsteinacceptedEddington?criticismandnolongeradvocatedtheinvalidargumentsinhisbook,?heMeaningofRelativity"of1921.EinsteinalsopraisedEddington?bookof1923tobethefinestpresentationofthesubjecteverwritten
Moreover,incontrasttothebeliefofsometheorists,ithasneverbeenestablishedthattheequivalenceofallframesofreferencerequirestheequivalenceofallcoordinatesystems.Ontheotherhand,ithasbeenpointedoutthat,becauseoftheequivalenceprinciple,themathematicalcovariancemustberestricted. 代写论文 http://
Moreover,Kretschmannpointedoutthatthepostulateofgeneralcovariancedoesnotmakeanyassertionsaboutthephysicalcontentofthephysicallaws,butonlyabouttheirmathematicalformulation,andEinsteinentirelyconcurredwithhisview.Paulipointedoutfurther,?hegenerallycovariantformulationofthephysicallawsacquiresaphysicalcontentonlythroughtheprincipleofequivalence...."Nevertheless,Einsteinarguedthat"...thereisnoimmediatereasonforpreferringcertainsystemsofcoordinatestoothers,thatistosay,wearriveatthenb sp;requirementofgeneralco-variance."
Thus,Einstein?covarianceprincipleisonlyaninterimconjecture.Apparently,hecouldmeanonlytoamathematicalcoordinatesystemforcalculationsincehisequivalenceprinciple,amongothers,isanimmediatereasonforpreferringcertainsystemsofcoordinatesinphysics(壯56).Notethatamathematicalgeneralcovariancerequires,asHawkingdeclared,theindistinguishabilitybetweenthetime-coordinateandaspace-coordinate.Ontheotherhand,theequivalenceprincipleisrelatedtotheMinkowskispace,whichrequiresadistinctionbetweenthetime-coordinateandaspace-coord inate.Hence,themathematicalgeneralcovarianceisinherentlyinconsistentwiththeequivalenceprinciple. 论文网 http://
Althoughtheequivalenceprincipledoesnotdeterminethespace-timecoordinates,itdoesrejectphysicallyunrealizablecoordinatesystems.WhereasinspecialrelativitytheMinkowskimetriclimitsthecoordinatetransformations,amonginertialframesofreference,totheLorentz-Poincarétransformations;ingeneralrelativitytheequivalenceprinciplelimitsthephysicalcoordinatetransformationstobeamongvalidspace-timecoordinatesystems,whichareinprinciplephysicallyrealizable.Thus,theroleoftheMinkowskimetricisextendedbytheequivalenceprincipleeventowheregravityispresent.
Mathematically,however,theequivalenceprinciplecanbeincompatiblewithasolutionofEinstein?equation,evenifitisaLorentzmanifold(whosespace-timemetrichasthesamesignatureasthatoftheMinkowskispace).IthasbeenproventhatcoordinaterelativisticcausalitycanbeviolatedforsomeLorentzmanifolds.Unfortunately,duetoinadequatephysicalunderstanding,somerelativistsbelievethatapropermetricsignaturewouldimplyasatisfactionoftheequivalenceprinciple.Themisconceptionthat,inaLorentzmanifold,a?reefall"wouldautomatically ;resultinalocalMinkowskispace,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,thenotionofgaugeusedinhiscalculationisactuallynotgenerallyvalidaswillbeshowninthispaper.Also,itisknownthatvalidityofthe1915Einsteinequationisquestionable.Foracompletetheoreticalanalysis,theseissuesshould,ofcourse,beaddressedthoroughly.Nevertheless,forthevalidityo fEinstein?calculationonthelightbending,itissufficienttojustifythelinearfieldequationasavalidapproximation.Forthispurpose,theMaxwell-NewtonApproximation(i.e.,thelinearfieldequation)isderiveddirectlyfromthephysicalprinciplesthatleadtogeneralrelativity(§4). 论文代写 http://
Moreover,thereareintrinsicallyunphysicalLorentzmanifoldsnoneofwhichisdiffeomorphictoaphysicalspace-time(§7).Thus,toacceptaLorentzmanifoldasvalidinphysics,itisnecessarytoverifytheequivalenceprinciplewithaspace-timecoordinatesystemforphysicalinterpretations.Then,forthepurposeofcalculationonly,anydiffeomorphismcanbeusedtoobtainnewcoordinates.Itisonlyinthissensethatacoordinatesystemforaphysicalspace-timecanbearbitrary.
Inthispaper,therequirementofageneralcovarianceamongallconceivablemathematicalcoordinatesystemswillbefurtherconfirmedtobeanover-extendeddemand.(NotethatEddingtondidnotacceptthegaugerelatedtogeneralmathematicalcovariance.)Analysisshowsthatasatisfactionoftheequivalenceprinciplerestrictedcovariance(壯3-5).Afterthisnecessaryrectification,somecurrentlyacceptedwell-knownLorentzmanifoldswouldbeexposedasunphysical(§7).But,generalrelativityasaphysicaltheoryisunaffected.Itishopedthatthisclarificationwoul dhelp?urtherfruitfuldevelopments,followingitsownautonomouscourse".
2.Einstein?EquivalencePrinciple,FreeFall,andPhysicalSpace-TimeCoordinates
Initiallybasedontheobservationthatthe(passive)gravitationalmassandinertialmassareequivalent,Einsteinproposedtheequivalenceofuniformaccelerationandgravity.In1916,thisproposalisextendedtothelocalequivalenceofaccelerationandgravitybecausegravityisingeneralnotuniform.Thus,ifgravityisrepresentedbythespace-timemetric,thegeodesicisthemotionofaparticleundertheinfluenceofgravity.Then,foranobserverinafreefall,thelocalmetricislocallyconstant.Tobeconsistentwithspecialrelativity,suchaloc almetricisrequiredtobelocallyaMinkowskispace.
Thus,acentralproblemingeneralrelativityiswhetherthegeodesicrepresentsaphysicalfreefall.However,validityofthisglobalpropertyisrealizedlocallythroughasatisfactionoftheequivalenceprinciple.Moreover,Eddingtonobservedthatspecialrelativityshouldapplyonlytophenomenaunrelatedtothesecondorderderivativesofthemetric.Thus,Einsteinaddedacrucialphrase,?tleasttoafirstapproximation"ontheindistinguishabilitybetweengravityandacceleration.
Theequivalenceprinciplerequiresthatafreefallphysicallyresultinaco-movinglocalMinkowskispace2).However,inaLorentzmanifold,althoughalocalMinkowskispaceexistsina?reefall"alongageodesic,theformationofsuchco-movinglocalMinkowskispacesmaynotbevalidinphysicssincethegeodesicmaynotrepresentaphysicalfreefall.Inotherwords,giventhemathematicalexistenceoflocalMinkowskispaceco-movingalongatime-likegeodesic,thecrucialphysicalquestionforthesatisfactionoftheequivalenceprincipleiswhether ;thegeodesicrepresentsaphysicalfreefall. 代写论文 http://
Einsteinpointedout,?sfarastheprepositionsofmathematicsreferstoreality,theyarenotcertain;andasfarastheyarecertain,theydonotrefertoreality."Thus,anapplicationofamathematicaltheoremshouldbecarefullyexaminedalthough?necannotreallyarguewithamathematicaltheorem".If,attheearlierstage,Einstein?argumentsarenotsoperfect,heseldomallowedsuchdefectsbeusedinhiscalculations.Thisisevidentinhisbook,?heMeaningofRelativity'whichheeditedin1954.Accordingtohisbooknbs p;andrelatedpapers,Einstein?viewpointsonspace-timecoordinatesare:
1)Aphysical(space-time)coordinatesystemmustbephysicallyrealizable(seealso2)3)below).
Einsteinmadeclearin?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-timecoordinatesystembypointingoutthatEinsteinalsowroteinhisbookthat?nananalogousway(toGaussiancurvilinearcoordinates)weshallintroduceinthegeneraltheoryofrelativityarbitraryco-ordinates,x1,x2,x3,x4,whichshallnumberuniquelythespace-timepoints,sothatneighboringeventsareassociatedwithneighboringvaluesofthecoordinates;otherwise,thechoiceofco-ordinateisarbitrary."But,Einsteinqualifiedthiswithaphysicalstatementthatnbs p;?ntheimmediateneighborofanobserver,fallingfreelyinagravitationalfield,thereexistsnogravitationalfield."Thisstatementwillbeclarifiedlaterwithademonstrationoftheequivalenceprinciple(seeeqs.).
3)Theequivalenceprinciplerequiresnotonly,ateachpoint,theexistenceofalocalMinkowskispace2)
ds2=c2dT2-dX2-dY2-dZ2,(1)http://
butafreefallmustresultinaco-movinglocalMinkowskianspace(seealso).Notethattheequivalenceprinciplerequiresthatsuchalocalcoordinatetransformationbeduetoaspecificphysicalaction,accelerationinthefreefallalone.Einsteinwrote,"Forthispurposewemustchoosetheaccelerationoftheinfinitelysmall(?ocal")systemofco-ordinatessothatnogravitationalfieldoccurs;thisispossibleforaninfinitelysmallregion."
Also,foraLorentzmanifold,ifa?reefall"resultsinalocalconstantmetric,whichisdifferentfromMinkowskimetric,thentheequivalenceprincipleisnotsatisfiedintermsofphysics.Einsteinwrote,"...inordertobeabletocarrythroughthepostulateofgeneralrelativity,ifthespecialtheoryofrelativityappliestothespecialcaseoftheabsenceofagravitationalfield."
AccordingtoEinstein,thebodytowhicheventsarespatiallyreferrediscalledthecoordinatesystem.Tobemoreprecise,aspatialcoordinatesystemattachedtoabody(i.e.,norelativemotionnoracceleration)isits?rameofreference".Thesecoordinatestogetherwiththetime-coordinateformthespace-timecoordinatesystem.Aframeofreferencecanbechosenphysicallyand,duetotheequivalenceprinciple,thetime-coordinateisdeterminedaccordingly(壯56).Thus,onemaycalllooselytheframeofreferenceasacoordinatesystem.Inthispaper, forthepurposeofconsideringasatisfactionoftheequivalenceprinciple,aframeofreferenceandarelatedspace-timecoordinatesystem,aredistinguishedasabove.http://
Toclarifythetheory,Einsteinwrote,?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".
Einstein?equivalenceprincipleisverydifferentfromtheversionformulatedbyPauli,?oreveryinfinitelysmallworldregion(i.e.aworldregionwhichissosmallthatthespace-andtime-variationofgravitycanbeneglectedinit)therealwaysexistsacoordinatesystemK0(X1,X2,X3,X4)inwhichgravitationhasnoinfluenceeitherinthemotionofparticlesoranyphysicalprocess."NotethatinPauli?misinterpretation,gravitationalaccelerationasaphysicalcauseisnotmentioned,andthusPauli?version3),whichisnowcommonlybut ;mistakenlyregardedasEinstein?versionoftheprinciple,actuallyisnotaphysicalprinciple.BasedonPauli?version,itwasbelievedthatingeneralrelativityspace-timecoordinateshavenophysicalmeaning.Inturn,diffeomorphiccoordinatesystemsareconsideredasequivalentinphysicsnotjustincertainmathematicalcalculations.However,accordingtoEinstein?calculations,thisissimplynottrue(seesection3). 论文网 http://
Theinitialformoftheequivalenceprincipleisarelationbetweenaccelerationandgravity.However,intheaboveclarification,theroleofaccelerationisnotexplicitlyshown.Onemayaskifaccelerationdoesnotexistforastaticobject,wouldtheequivalenceprinciplebesatisfied?Onemustbecarefulbecauseageodesicmaynotrepresentaphysicalfreefall.
TherearethreephysicalaspectsinEinstein?equivalenceprincipleasfollows:
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
Einsteinillustratedhisequivalenceprincipleinhiscalculationofthelightbendingaroundthesun.(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,itisnecessarytojustifyeq.(4a)again.Also,thenotionofweakgravitymaynotbecompatiblewiththeprincipleofgeneralcovariance.Inthenextsection,eq.(4a)willbejustifieddirectlyandisindependentofthedetailsofhigherordertermsofanexactEinsteinequation.Forthisreasonandthedynamicalincompatibilitywitheq.(3),eq.(4a)iscalledtheMaxwell-NewtonApproximation.Inotherwords,eq.(4a)shouldbevalidfordynamicproblems.Also,ani mplicitassumptionofEinstein?calculationisthatthegravitationaleffectsduetothelightitself,isnegligible.Toaddressthisissuetheoretically,wouldbecomplicatedandisbeyondthescopeofthispaper.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.).Thisisthetimedilationofmetric(4b).Eq.(6)showsthatthegravitationalredshiftsarerelatedtogtt,andiscompatiblewithhis1911derivation.Moreover,sincethespacecoordinatesareorthogonaltodt,at(x0,y0,z0,t0),forthesameds2,eq.(6)implieshttp://
(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.
Einsteinwrote,?ecanthereforedrawtheconclusionfromthis,thatarayoflightpassingnearalargemassisdeflected."Thus,Einsteinhasdemonstratedthattheequivalenceprinciplerequiresthataspace-timecoordinatessystemmusthaveaphysicalmeaning;andaspace-timecoordinatesystemcannotbejustanyGaussiancoordinatesystem.Itseems,Einsteinchosethiscalculationmethodtoclarifyhisstatementsontheequivalenceprinciple.Inmanytextbooks,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,ithasbeenproventhateq.(4a)isdynamicallyincompatiblewitheq.(3).Thebinarypulsarexperimentsmakeitnecessarytomodifyeq.(1)toa1995updateversion, 论文网 http://
Gab(Rab-Rgab=-K.(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-NewtonApproximationandisequivalenttoeq.(4a).
Theabovemodificationisbasedonthefactsthat,asafirstorderapproximation,eq.(10c)issupportedbyexperimentsandthatitisthenaturalextensionfromNewtoniantheory.However,onemayarguethatthisisnotyetentirelysatisfactorysinceithasnotbeenshownrigorouslythateq.(10c)iscompatiblewithgeneralrelativity.Inparticular,onemightstillarguethatthewavecomponentingat(fora=x,y,z,t)asartificiallyinducedbytheharmonicgauge.
ItwillbeshownthattheMaxwell-NewtonApproximation(10c)canberigorouslyderivedfromtheequivalenceprincipleandrelatedphysicalprinciplesthatleadtogeneralrelativity.Sincelineareq.(10c)issupportedbyexperiments,toreaffirmthevalidityofgeneralrelativity,onemustshowclearlythateq.(10c)iscompatiblewiththetheoreticalframeworkofrelativity.Thus,suchaproofofeq.(10c)notonlyprovidesatheoreticalfoundationforeq.(10)butalsoreaffirmsgeneralrelativity.
Ingeneralrelativitytherearethreebasicassumptionsnamely: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.IncomparisonwithNewton?theory,Einsteinobtainsthegravitationalpotential, 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,ithasbeenproventhatthecurvaturetensor?((((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).
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)iscompatiblewiththeequivalenceprincipleasdemonstratedbyEinsteininhiscalculationofthebendingoflight.Thus,thederivationisself-consistent.
OnemightarguethatEinsteinequation(3)couldbe?erived"fromalinearequationmoregeneralthaneq.(12a),ifoneregardsthegravitationalfieldasaspin-2fieldcoupledtotheenergytensor.However,sucha?ure"theoreticalapproachisnotreallyconsistentwithNewton?theoryandrelatedobservationsbecausethenotionofgaugeisused.Moreover,insucha?roof",theexistenceofboundeddynamic2)solutionsforeq.(3)mustbeinvalidlyassumed.
NotethatEinsteinobtainedthesamevaluesfor(and(byconsideringeq.(13)afterassumingX(2)ab=0.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.However,Pauliwasthefirsttopointoutexplicitlythepossibilityofsuchanantigravitycoupling.Moreover,theexistenceofsuchacouplingis,inaway,implicitlysuggestedbythesingularitytheorems,whichshowthatifallthecouplingsareofthesamesign,theexistenceofunrealisticspacetimesingularitieswouldbinevitable.Theneedofananti-gravitycouplingwasfirstdiscoveredincalculatingthegravityofannb sp;electromagneticwave.
Moreover,itwasEinsteinandRosenwhofirstdiscoverthatthe1915equationmaynothaveapropagatingwavesolution.In1953Hogarthconjecturedthatthisequationdoesnothaveadynamicsolution.Adefinitiveindicationofthisisthenon-existenceoftheplane-wavesolution.Notethatthe?lanewaves"proposedbyBondi,Pirani,andRobinson,areactuallyunboundedalthoughtheybelievethataplane-waveisanidealizationofaweakwavefromadistantisolatedsource.Theseunboundedsolutionssatisfytheconditionofplanenessbutare notrelatedtoanyweakwave.Also,althoughMisner,ThorneWheelerconcludecorrectlythattheplane-wavesarebounded,theirequationforplane-wavesactuallyhasnoboundedwavesolution.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.
ItwillbeshownthatnotallmathematicalcoordinatesystemsareequivalentinphysicsasclaimedbyBergmannandLiu.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)canbetransformedtohttp://
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=0implythatthelightspeedishttp://
==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/2,whereC=constant.(22a)
Then
ds2=2dT2-dx2-dy2-dz2,(22b) 论文网 http://
isthemetrictransformedfromtheMinkowskimetric.Ifmetric(22b)isrealizable,accordingtods2=0,themeasuredlightspeedwouldbe/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}.http://
Itfollowseq.(23a)thatoneobtains,forsomeconstantk 代写论文 http://
dT/ds=k-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,theyalsodisagreewithEinsteinandregardthatcoordinatelightspeedsasmeaningless.
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)toanotherconstantLorentzmetrichttp://
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)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.
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,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. 论文网 http://
Forinstance,anacceptedsolutionofmetricforanelectromagneticplanewaveis 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).Thus,themanifold(29)cannotbediffeomorphictoaphysicalspacesinceadiffeomorphismcannoteliminatetheparameters,whichviolatetheprincipleofcausality.
8.ConclusionsandDiscussions
Einsteinproposedtheequivalenceprincipleforthereality,whichhemodelsasaRiemannianphysicalspace-time.However,Pauli?versionimpliesthattheequivalenceprinciplewouldbesatisfiedeventhoughthecoordinatesystemisnotphysicallyrealizable.NowitisclarifiedthatEinsteincorrectlyobjectedPauli?versionasamisinterpretation.Also,itisproventhattheequivalenceprincipleissatisfiedifandonlyifamanifoldisphysicallyrealizable. 论文代写 http://
Ingeneralrelativity,theMinkowskimetricinspecialrelativityisobviouslyaspecialcase.However,itwasnotclearthatthe?rinciples"whichleadtogeneralrelativityarecompatiblewitheachothereveninthisspecialcase.SometheoristsbelieveincorrectlythattheGalileantransformationwerevalidforgeneralrelativity,althoughEinsteinhasmadeclear,?pecialtheoryofrelativityappliestothespecialcaseoftheabsenceofagravitationalfield".Torectifythis,itisshowndirectlythat,duetotheequivalenceprinciple,theMinkowskimetricnbs p;istheonlyvalidconstantspace-timemetric(§6).
Toestablishspecialrelativity,theGalileantransformationisproventobeunrealizablebyexperiments.Thus,theoreticallyaGalileantransformationshouldbeincompatiblewiththeequivalenceprinciple,whichisapplicabletoonlyaphysicalspace.Thismeans,incontrastofthebeliefofsometheorists,thattheequivalenceofallframesofreferenceisnotthesameasthephysicalequivalenceofallmathematicalcoordinatesystems.Infact,itisinvalidinphysicstoextendthespace-timephysicalcoordinatesystemtoanarbitraryGaussiansystem.Forinstance, nbsp;thetimecoordinateisnotarbitrary.http://
TheGalileantransformationimpliedthatthereisnolimitonthevelocityoflight.This,inprinciple,disagreeswiththenotionofinvariantlightspeed.However,duetoentrenchedmisconceptionsoncovariance,thisproblemwasnotevenrecognizedforfurtherinvestigations.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.Thisisanotherexamplethatamisunderstandingoftheequivalenceprinciplecanle adstodisagreementswithexperiments.
Perhaps,duetoconfusingmathematicaltheoremswithEinstein?equivalenceprincipleasPaulidid,thisprincipleisoftennotexplainedadequatelyinsomebooks.Todealwithallthetheoreticalinconsistencesuperficially,sometheoristsclaimedthatthespace-timecoordinateshavenophysicalmeaningingeneralrelativity.Suchaspeculationdisagreeswiththefactthattherearenon-scalarsinphysics.Thedeflectionoflightisrelatedtothelightraybeingobservedasanalmoststraightlineawayfromthesun,andgravitationalredshiftsarerelatedtogtt- nbsp;thetime-timecomponentofspace-timemetric.http://
Nevertheless,basedonsuchanabsurdclaim,Hawkingdeclares,?ngeneralrelativity,thereisnorealdistinctionbetweenthespaceandtimecoordinates,justasthereisnorealdifferencebetweenanytwospacecoordinates."Ontheotherhand,Hawkingalsobelieved,?narrowoftime,somethingthatdistinguishedpastfromfuture,givingadirectionoftime".Apparently,hedidnotseethatthereisaninconsistencybetweenthesetwostatements.Moreover,likeothers,Hawkingacceptedthedeflectionoflight.Heprobablydidnotrealizethatthede flectionanglecouldbedefinedonlyinacertaintypeofphysicalcoordinatesystems,wherethetrajectoryofalightray,whenfarawayfromthesun,isapproximatelyastraightline.Notethatsuchlogicaldeficiencyisacommonproblemamongthoseso-called?tandard"relativists.
TheoristssuchasSynge,Fockandmorerecently,Hawking,Ohanian,Ruffini,andWheeler,whodonotunderstandEinstein?equivalenceprincipleforvariousreasonsincludinginadequateunderstandingofphysicsormathematicsatthefundamentallevelordeficiencyinlogic,advocatedessentiallythatthebasisofgeneralrelativityshouldbetheEinsteinfieldequationalone.However,experimentallytheunrestrictedvalidityofEinstein?equationhasnotyetbeenestablishedbeyondreasonabledoubt.
TheoreticallythereisnosatisfactoryproofofrigorousvalidityofEinstein?fieldequation(e.g.,theinadequatesourcetermmentionedin§1,isthecauseoftheunphysicalsolution(29)).Infact,in1953Hogarthconjecturedthatthe1915Einsteinequationisinvalidforadynamictwo-bodyproblem;andEinsteinhimselfhadpointedoutthathisequationmightnotbevalidformatterofveryhighdensity.Moreover,ithasbeenprovenbythebinarypulsarexperimentthatEinstein?equationmustbemodifiedandYilmazpointed outthatEinstein?equationof1915isonlyatestparticletheory.Moreover,intermsofphysics,astaticsolutionisonlyanapproximationforsomedynamicalproblems.Thismeansthat,tosupportEinstein?illustrationoftheequivalenceprinciplewithcalculationsonthelightbending,itisnecessarytoshowthathislinearequationisjustifiablefordynamicalproblems.Thus,itisnecessarytoderivetheMaxwell-NewtonApproximationindependentlyfromphysicalprinciples(§4)sincethe1915Einsteinequationisvalidforstaticproblemsonly.
Asimpledynamicalproblemwouldbethegravityduetotheinteractionoftwomassiveparticles.Then,Einstein?conditionofweakgravity,whichisalsoduetotheequivalenceprinciple,requiressuchasolutionofgravitymustbebounded.Thus,satisfyingboundednessofgravityduetoaweaksourceisindependentofthefieldequation.But,nosuchsolutionhaseverbeenproventobeinexistence.Beingawareoftheunboundednessofcylindricalandspherical?aves",afterprovingtheexistenceofCauchysolutions,Bruhatremarkedthatthen bsp;physicalvalidityofanyCauchysolutionisuptotheexperimentstodecide.Whileherclarificationisreasonableforamathematician,physicistsshouldhaveknownphysicsbetter. 代写论文 http://
Theequivalenceprincipleremainsindispensablebecauseofitssolidexperimentalfoundationsuchasgravitationalredshiftsandtheblendingoflight.Theoretically,asillustrated,itsfailureisalwaysaccompaniedwithaviolationofanotherphysicalrequirement.Thus,asWeinbergpointsout,?tismuchmoreusefultoregardgeneralrelativityaboveallasatheoryofgravitation,whoseconnectionwithgeometryarisesfromthepeculiarempiricalpropertiesofgravitation,propertiessummarizedbyEinstein?PrincipleoftheEquivalenceofGravitationandInertia."
Thelong-standingerrorsingeneralrelativityhaveprofoundhistoricalreasons.Mostphysicistsareusedtolinearequations,andunavoidablytheywouldapplytechniques,whicharevalidforlinearequations.But,innonlinearfieldequation,asecondordertermfromtheviewpointofphysics,maybecrucialfortheexistenceofaboundedphysicalsolution.Inotherwords,onemaynottakeitforgranted(i.e.,withoutaproof)thataphysicalrequirementiscompatiblewithafieldequation.EvenwellknownphysicistssuchasEinsteinandmorerecentlyFeymann madesuchmistakesingeneralrelativity.http://
Ingeneralrelativity,assumingtheexistenceofaboundeddynamicsphysicalsolution,hasneverbeenproven,butithasbeenusedwithblindfaith.Thisisessentiallywheremanyofthemathematicalerrorscomefrom.Amosttellingevidenceisthatthe?lane-waves"proposedbyBondietal.arenotboundedalthoughtheybelievethattheyare.Furthermore,ithasbeenproventhattherearenoboundedplane-wavesforthe1915Einsteinequation.Then,thenon-existenceofadynamicsolutionformassivematterisprovenbecauseexperimentsnbs p;supporttheMaxwell-NewtonApproximation.ThisapproachofproofhasbeencompletedsincetheMaxwell-NewtonApproximationcanbederivedfromphysicalprinciples(§4).
Currently,relativistsoftenignorephysicalrequirementsbecausetheymisunderstoodtheequivalenceprincipleandacceptedthecovarianceprinciple,whichwasrejectedbyEddington.Historically,afterearlyobservationalconfirmationsofEinstein?predictions,Einsteindeclaredlogicalcompletenessofhistheoryalthoughsuchconfirmationsverifyonlythetheoreticalframeworkofgeneralrelativity.Subsequently,ablindfaithonthetheoreticalself-consistencyofgeneralrelativitywasdeveloped.Manyphysicistsworkingongeneralrelativity,inspiteofwarningsfromGullstrand,BohrKlein,andotherphysicists,tendtohaveoverconfidenceonEinstein?equation(exceptafewsuchasN.Rosen). 论文代写 http://
Adynamicphysicalsolution,aspointedoutbyLow,isnotjustatime-dependentsolution,whichcanbeobtainedfromtheMinkowskimetricbymakingacoordinatetransformation.Inphysics,suchadynamicsolutionmustberelatedtothedynamicsofsourcematterandgravitationalradiation.Nevertheless,ChristodoulouandKlainermanclaimedtheexistenceofdynamicalsolutionsbytheirconstructionalthoughsuch?olutions"areunrelatedtodynamicalsourcesorradiation.Itisalsosurprisingthattheirmainmathematicalmistakesareactuallyatthefundamentallevel.AspointedoutbyKrameretal.,manyrelativistshaveaproblemindistinguishingaphysicallyvalidsolutionfrommathematicalsolutions.Bonnoretalfurtherconfirmthisproblembypointingoutthatitisnotpossibletohaveaconsistentphysicalinterpretation.
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,Einsteinconsidersthefour-dimensionalspace-timerealityasaphysicalspace-timemodeledasaRiemannianspace-time(M,g).TheRiemannianspaceMischaracterizedbyaspace-timemetricgikthatcanbedeterminedbyphysicalconsiderationssuchasthedistributionofmatter.In?elativityandtheproblemofspace",Einsteinwrote,
?orthefunctionsgikdescribenotonlythefield,butatthesametimealsothetopologicalandmetricalstructuralpropertiesofthemanifold....Thereisnosuchthingasanemptyspace,i.e.,aspacewithoutfield.Space-timedoesnotclaimexistenceonitsown,butonlyasastructuralqualityofthefield."
Moreover,sincesuchaRiemannianspace-timemodelsreality,allthephysicalrequirementsmustbesufficientlysatisfied.
2)AlocalMinkowskianspaceisashorthandtoexpressthatspecialrelativityislocallyvalid,exceptforphenomenainvolvingthespace-timecurvature.
3)Forexample,theWheeler-HawkingSchoolfollowsPauli?misinterpretation,andthus,theirtheoriesaredifferentfromgeneralrelativity.They,differentfromEinstein,believethatspace-timecoordinateshavenophysicalmeaning.HawkingmakesnosecretofhisdisagreementswithEinstein.Morerecently,basedonmisinterpretationsofFock,Ohanian,Ruffini,andWheeleropenlycriticizedEinstein?theoryasconfusingandhisprinciplesinvalid.http://
4)Sometheoristsbelievethatthesolutionofgravityforaweaksourceneednotbebounded.However,ithasbeenshownthattheequivalentprincipleimpliescompatibilitywithEinstein?notionofweakgravity.
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