ONINCOMPATIBILITYOFGRAVITATIONALRADIATIONWITHTHE1915EINSTEINEQUATION
发布时间：2003-10-23作者：佚名
AppliedandPureResearchInstitute
17NewcastleDrive,Nashua,NH03060
PhysicsEssays,vol.13,no.4,2000
Abstract
Itisshownthatthe1915Einsteinequationisincompatiblewiththephysicalnotionthatawavecarriesawayenergy-momentum.ThisproofiscompatiblewiththatMaxwell-NewtonApproximation(thelinearfieldequationforweakgravity),andissupportedbythebinarypulsarexperiments.Fordynamicproblems,thelinearfieldequationisindependentof,andfurthermoreincompatiblewiththeEinsteinequation.Thelinearequation,asafirst-orderapproximation,requirestheexistenceoftheweakgravitationalwavesuchthatitmustbeboundedinamplitudeandberelatedtothe;dynamicsofthesourceofradiation.Duetoneglectingthesecrucialphysicalassociations,inadditiontoinadequateunderstandingoftheequivalenceprinciple,unphysicalsolutionsweremistakenasgravitationalwaves.Itisconcludedtheoreticallythat,asEinsteinandRosensuggested,aphysicalgravitationalwavesolutionforthe1915equationdoesnotexist.Thisconclusionisgivenfurthersupportsbyanalyzingtheissueofplane-wavesversusexact"wave"solutions.Moreover,theapproachesofDamourandTaylorfortheradiationofbinarypulsarswouldbevalidonlyiftheyareasanapproximationoftheequationof1995update.Inaddition,theupdateequationshowsthatthesingularitytheoremsproveonlythepeakingdownofWheeler-Hawkingtheories,butnotgeneralrelativity.ItispointedoutthatsomeLorentzmanifoldsareamongthosethatactuallydisagreewithknownexperimentalfacts.
KeyWords:compatibility,dynamicsolution,gravitationalradiation,principleofcausality,plane-wave,Wheeler-Hawkingtheories
1.Introduction
Inphysics,theexistenceofawaveisduetothefact,asrequiredbyspecialrelativity,thataphysicalcausemustpropagatewithafinitespeed[1].Thisimpliesalsothatawavecarriesenergy-momentum.Thus,thefieldequationforgravitymustbeabletoaccommodatethegravitationalwave,whichcarriesawaygravitationalenergy-momentum.Inthispaper,itwillbeshownthattheEinsteinequationof1915failsthis.
Ingeneralrelativity,theEinsteinequationof1915[2]forgravityofspace-timemetricg((is
G(((R((-g((R=-KT(m)((,(1)
whereG((istheEinsteintensor,R((istheRiccicurvaturetensor,T(m)((istheenergy-stresstensorformassivematter,andK(=8((c-2,and(istheNewtoniancouplingconstant)isthecouplingconstant1).Thus,
G(((R((-g((R=0,orR((=0,(1")
atvacuum.However,(1")alsoimpliesnogravitationalwavetocarryawayenergy-momentum.
AnincompatibilitywithradiationwasfirstdiscoveredbyEinstein&Rosen[3,4]in1936.However,duetoconceptualandmathematicalerrorsthen,theirdiscoverywasnotaccepted.Theseerrorsformthebasisoftheso-calledgeometricviewpointoftheWheeler-Hawkingschool[5,6](seealsoSection4).Anobviousproblemoftheirviewpointisthatonecannotdistinguishaphysicalsolutionamongmathematicalsolutions[7].
Conceptually,onewouldargueincorrectlythat(1")carriesenergy-momentumbecause
G(((G(1)((+G(2)(((2a)
whereG(1)((consistsofthelinearterms(ofthedeviation(((=g((-(((fromtheflatmetric((()inG((,andG(2)((consistsoftheothers.SinceG(2)((hasbeenidentifiedasequivalenttothegravitationalenergy-stressofEinstein"snotion[8],itseemedobviousthatG(2)((carriestheenergy-momentum.However,unless(1)canaccommodateaphysicalgravitationalwave,suchanargumenthasnomeaning.Moreover,nowavesolutionhaseverbeenobtainedforequation(1).Infact,thisisimpossible(seeSection2).
Thereareso-called"wavesolutions"for(1"),buttheyareactuallyinvalidinphysics(see§§3&5)sincephysicalrequirements(suchastheprincipleofcausality2),theequivalenceprinciple,andsoon)arenotsatisfied.Infact,someofthemhavebeenproventobeindisagreementwithexperiments[9,10].Theirinvalidacceptanceisduetotheincorrectbelief3)thattheequivalenceprincipleweresatisfiedbyanyLorentzmanifold[11].
Moreover,Einstein"snotioncannotbeexact,sinceitisnotlocalizable[12].Inafieldtheory,acentralproblemistheexchangeofenergybetweenaparticleandthefieldwheretheparticleislocated[13].Therefore,thegravitationalenergy-stressmustbeatensor(seealsoSection4).
2.TheGravitationalWaveandNonexistenceofDynamicSolutionsforEinstein"sEquation
First,amajorproblemisamathematicalerrorontherelationshipbetween(1)andits"linearization".ItwasincorrectlybelievedthatthelinearMaxwell-NewtonApproximation[13]
(c(c((=-KT(m)((,where((=(((-(((((cd(cd)(3a)
and
(((xi,t)=-(T(((yi,(t-R)]d3y,whereR2=(xi-yi)2.(3b)
alwaysprovidesthefirst-orderapproximationforequation(1).Thisbeliefwasverifiedforthestaticcaseonly.
Foradynamic4)case,however,thisisnolongervalid.WhiletheCauchydatacanbearbitraryfor(3a),butnotfor(1).TheCauchydataof(1)mustsatisfyfourconstraintequations,G(t=-KT(m)(t((=x,y,z,t)sinceG(tcontainsonlyfirst-ordertimederivatives[8].Thisshowsthat(3a)wouldbedynamicallyincompatible5)withequation(1)[10].Furtheranalysisshowsthat,intermsofboththeory[11]andexperiments[13],thismathematicalincompatibilityisinfavorof(3),insteadof(1).
In1957,Fock[14]pointedoutthat,inharmoniccoordinates,therearedivergentlogarithmicdeviationsfromexpectedlinearizedbehavioroftheradiation.ThiswasinterpretedtomeanmerelythatthecontributionofthecomplicatednonlineartermsintheEinsteinequationcannotbedealtwithsatisfactorilyfollowingthismethodandthatotherapproachisneeded.Subsequently,vacuumsolutionsthatdonotinvolvelogarithmicdeviation,werefoundedbyBondi,Pirani&Robinson[15]in1959.Thus,theincorrectinterpretationappearstobejustifiedandthefaithonthedynamicsolutionsmaintained.Itwasnotrecognizeduntil1995[13]thatsuchasymptomofdivergenceactuallyshowstheabsenceofboundedphysicaldynamicsolutions.
Inphysics,theamplitudeofawaveisgenerallyrelatedtoitsenergydensityanditssource.Equation(3)showsthatagravitationalwaveisboundedandisrelatedtothedynamicofthesource.Theseareusefultoprovethat(3),asthefirst-orderapproximationforadynamicproblem,isincompatiblewithequation(1).Itsexisting"wave"solutionsareunboundedandthereforecannotbeassociatedwithadynamicsource[11].Inotherwords,thereisnoevidencefortheexistenceofaphysicaldynamicsolution.
WiththeHulse-Taylorbinarypulsarexperiment[16],itbecameeasiertoidentifythattheproblemisin(1).Subsequently,ithasbeenshownthat(3),asafirst-orderapproximation,canbederivedfromphysicalrequirementswhichleadtogeneralrelativity[11].Thus,(3)isonsolidtheoreticalgroundandgeneralrelativityremainsaviabletheory.Note,however,thattheproofofthenonexistenceofboundeddynamicsolutionsfor(1)isessentiallyindependentoftheexperimentalsupportsfor(3).
Toprovethis,itissufficienttoconsiderweakgravitysinceaphysicalsolutionmustbecompatiblewithEinstein"s[2]notionofweakgravity(i.e.,iftherewereadynamicsolutionforafieldequation,itshouldhaveadynamicsolutionforarelatedweakgravity[11]).Tocalculatetheradiation,considerfurther,
G(((G(1)((+G(2)((,whereG(1)((=(c(c((+H(1)((,(2b)
H(1)(((-(c((((c+(((c(+((((c(dcd,and?(((?<<1.(2c)
G(2)((isatleastofsecondorderintermsofthemetricelements.Foranisolatedsystemlocatedneartheoriginofthespacecoordinatesystem,G(2)(tatlarger(=(x2+y2+z2(1/2)isofO(K2/r2)(5,8,17(.
Onemayobtainsomegeneralcharacteristicsofadynamicsolutionforanisolatedsystemasfollows:
1)Thecharacteristicsofsomephysicalquantitiesofanisolatedsystem:
Foranisolatedsystemconsistingofparticleswithtypicalmass,typicalseparation,andtypicalvelocities,Weinberg(8(estimated,thepowerradiatedatafrequency(oforder/willbeoforder
P"((/)624orP"8/,
since(/isoforder2.ThetypicaldecelerationradofparticlesinthesystemowingthisenergylossisgivenbythepowerPdividedbythemomentum,orrad"7/.ThismaybecomparedwiththeaccelerationscomputedinNewtonianmechanics,whichareoforder2/,andwiththepost-Newtoniancorrectionof4/.Sinceradiationreactionissmallerthanthepost-Newtonianeffectsbyafactor3,if((c,thevelocityoflight,theneglectofradiationreactionisperfectlyjustified.Thisallowsustoconsiderthemotionofaparticleinanisolatedsystemasalmostperiodic.
Consider,forinstance,twoparticlesofequalmassmwithanalmostcircularorbitinthex-yplanewhoseoriginisthecenterofthecircle(i.e.,theorbitofaparticleisacircleifradiationareneglected).Thus,theprincipleofcausality[9,10]impliesthatthemetricg((isweakandveryclosetotheflatmetricatdistancefarfromthesourceandthatg(((x,y,z,t")isanalmostperiodicfunctionoft"(=t-r/c).
2)Theexpansionofaboundeddynamicsolutiong((foranisolatedweakgravitationalsource:
According(3),afirst-orderapproximationofmetricg(((x,y,z,t")isboundedandalmostperiodicsinceT((is.Physically,theequivalenceprinciplerequiresg((tobebounded[11],andtheprincipleofcausalityrequiresg((tobealmostperiodicintimesincethemotionofasourceparticleis.Suchametricg((isasymptoticallyflatforalargedistancer,andtheexpansionofaboundeddynamicsolutionis:
g(((nx,ny,nz,r,t")=(((+(((k)(nx,ny,nz,t")/rk,wheren(=x(/r.(4)
3)Thenon-existenceofdynamicsolutions:
Itfollowsexpansion(4)thatthenon-zerotimeaverageofG(1)(twouldbeofO(1/r3)dueto
((n(=((((+n(n()/r,(5)
sincethetermofO(1/r2),beingasumofderivativeswithrespecttot",canhaveazerotime-average.IfG(2)(tisofO(K2/r2)andhasanonzerotime-average,consistencycanbeachievedonlyifanothertermoftime-averageO(K2/r2)atvacuumbeaddedtothesourceof(1).Notethatthereisnoplane-wavesolutionfor(1")[9,18].
Itwillbeshownbycontradictionthatthereisnodynamicsolutionfor(1)withamassivesource.Letusdefine
(((=((1)((+((2)((;(i)((=((i)((-((((((i)cd(cd),wherei=1,2;
and
(((((1)((=-KT(m)((.(6)
Then(1)((isofafirst-order;and((2)((isfinite.Ontheotherhand,from(1),onehas
(((((2)((+H(1)((+G(2)((=0.(7)
Notethat,foradynamiccase,equation(7)maynotbesatisfied.If(6)isafirst-orderapproximation,G(2)((hasanonzerotime-averageofO(K2/r2)(8(;andthus(2)((cannothaveasolution.
However,if(2)((isalsoofthefirst-orderofK,onecannotestimateG(2)((byassumingthat(1)((providesafirst-orderapproximation.Forexample,(6)doesnotprovidethefirstapproximationforthestaticSchwarzschildsolution,althoughitcanbetransformedtoaformsuchthat(6)providesafirst-orderapproximation[11(.Accordingto(7),(2)((willbeasecondordertermifthesumH(1)((isofsecondorder.From(2c),thiswouldrequire((((beingofsecondorder.Forweakgravity,itisknownthatacoordinatetransformationwouldturn(((( toasecondorderterm(canbezero)(8,14,17(.(Eq.[7]impliesthat(c(c(2)((-(c((((c+(((c(wouldbeofsecondorder)Thus,itisalwayspossibletoturn(6)tobecomeanequationforafirst-orderapproximationforweakgravity.
Fromtheviewpointofphysics,sinceithasbeenproventhat(3)necessarilygivesafirst-orderapproximation[11],afailureofsuchacoordinatetransformationmeansonlythatsuchasolutionisnotvalidinphysics.Moreover,forthedynamicofmassivematter,experiment[16]supportsthefactthatMaxwell-NewtonApproximation(3)isrelatedtoadynamicsolutionofweakgravity[13].Otherwise,notonlyisEinstein"sradiationformulanotvalid,butthetheoreticalframeworkofgeneralrelativity,includingthenotionoftheplane-waveasanidealization,shouldbe re-examined(seeSection3).Inotherwords,theoreticalconsiderationsinphysicsaswellasexperimentseliminateotherunverifiedspeculationsthoughttobepossiblesince1957.
Asshown,thedifficultycomesfromtheassumptionofboundedness(Section3),whichallowstheexistenceofaboundedfirst-orderapproximation,whichinturnimpliesthatatime-averageoftheradiativepartofG(2)((isnon-zero(7(.ThepresentmethodhasanadvantageoverFock"sapproachtoobtaininglogarithmicdivergence[13,14(forbeingsimpleandclear.
Inshort,accordingtoEinstein"sradiationformula,atimeaverageofG(2)(tisnon-zeroandofO(K2/r2)[13(.Although(3)impliesG(1)(tisoforderK2,itstermsofO(1/r2)canhaveazerotimeaveragebecauseG(1)(tislinearonthemetricelements.Thus,(1")cannotbesatisfied.Nevertheless,astaticmetriccansatisfy(1),sincebothG(1)((andG(2)((areofO(K2/r4)invacuum.Thus,thatagravitationalwavecarriesenergy-momentumdoesnotfollowfromthefactthatG(2)((canbeidentifiedwithagravitationalenergy-stress(8,17(.JustasG((,G(2)((shouldbeconsideredonlyasageometricpart.NotethatG(t=-KT(m)(tareconstraintsontheinitialdata.
Inconclusion,indisagreementwiththephysicalrequirement,assumingtheexistenceofdynamicsolutionsofweakgravityfor(1)[14,15,19-24(isinvalid.Thismeansthatthecalculations[25,26(onthebinarypulsarexperimentsshould,inprinciple,bere-addressed[12(.ThisexplainsalsothatanattemptbyChristodoulouandKlainerman[26(toconstructbounded"dynamic"solutionsforG((=0failstorelatetoadynamicsourceandtobecompatiblewith(3)[28]althoughtheirsolutionsdonotimplythatagravitationalwavecarriesenergy-momentum.
Foraproblemsuchasscattering,althoughthemotionoftheparticlesisnotperiodic,theproblemremains.Thiswillbeexplained(seeSection4)intermsofthe1995updateoftheEinsteinequation,duetothenecessaryexistenceofgravitationalenergy-momentumtensortermwithanantigravitycouplinginthesource.Toestablishthe1995updateequation,thesupportsofbinarypulsarexperimentsfor(3)areneeded[13].
3.GravitationalRadiations,BoundednessofPlane-Waves,andtheMaxwell-NewtonApproximation
Anadditionalpieceofevidenceisthatthereisnoplane-wavesolutionfor(1).Aplane-waveisaspatial-localidealizationofaweakwavefromadistantsource.Theplane-wavepropagatinginthez-directionisaphysicalmodelalthoughitstotalenergyisinfinite[8,10].Accordingto(3),onecansubstitute(t-R)with(t-z)andtheotherdependenceonrcanbeneglectedbecauserisverylarge.Thisresultsin(((xi,t)becomingaboundedperiodicfunctionof(t-z).SincetheMaxwell-NewtonApproximationprovidesthefirst-order,theexact plane-waveasanidealizationisaboundedperiodicfunction.Sincethedependenceof1/risneglected,oneconsidersessentiallytermsofO(1/r2)inG(2)((.Infact,thenon-existenceofboundedplane-waveforG((=0,wasprovendirectlyin1991[9,18].
Inshort,Einstein&Rosen[4,29]isessentiallyright,i.e.,therearenowavesolutionsforR((=0.Thefactthattheexisting"wave"solutionsareunboundedalsoconfirmsthenonexistenceofdynamicsolutions.Thefailuretoextendfromthelinearizedbehavioroftheradiationisduetothefactthatthereisnoboundedphysicalwavesolutionfor(1)andthusthisfailureisindependentofthemethodused.
NotethattheEinsteinradiationformuladependson(3)asafirst-orderapproximation.Thus,metricg((mustbebounded.OtherwiseG((=0canbesatisfied.Forexample,themetricofBondietal.[15]is
ds2=exp(2()(d(2-d(2)-u2(ch2((d(2+d(2)+sh2(cos2((d(2-d(2)-2sh2(sin2(d(d((,(8)
where(,(,(arefunctionsofu(=(-().Itsatisfiesthedifferentialequation(i.e.,theireq.(2.8(),
2("=u(("2+("2sh2(2).(9)
However,metric(8)isnotbounded,becausethiswouldrequiretheimpossibilityofu2<constant.Notethatanunboundedfunctionofu,f(u)growsanomalylargeastime(goesby.
Itshouldbenotedalsothatmetric(8)isonlyaplane,butnotaperiodicfunctionbecauseasmoothperiodicfunctionmustbebounded.Thisunboundednessisasymptomofunphysicalsolutionsbecausetheycannotberelatedtoadynamicsource(seealso[9,11]).Notethatsolution(8)canbeusedtoconstructasmoothone-parameterfamilyofsolutions[11]althoughsolution(8)isincompatiblewithEinstein"snotionofweakgravity[2].
In1953,questionswereraisedbySchiedigger[30]astowhethergravitationalradiationhasanywell-definedexistence.ThefailureofrecognizingG((=0asinvalidforgravitationalwavesisduetomistaking(3)asafirst-orderapproximationof(1).Thus,inspiteofEinstein"sdiscovery[3]andHogarth"sconjecture6)[31]ontheneedofmodification,theincompatibilitybetween(1)and(3)wasnotprovenuntil1993[13]afterthenon-existenceoftheplane-wavesforG((=0,hasbeenproven[9,18].
4.GravitationalRadiationandthe1995updateoftheEinsteinEquation
Ingeneral,(3)isactuallyanapproximationofthe1995updateoftheEinsteinequation[13],
G(((R((-g((R=-K(T(m)((-t(g)(((,(10)
wheret(g)((istheenergy-stresstensorsforgravity.Then,
((T(m)((=0,and((t(g)((=0.(11)
Equation(11)impliesthattheequivalenceprinciplewouldbesatisfied.From(10),theequationinvacuumis
G(((R((-g((R=Kt(g)((.(10")
Notethatt(g)((isequivalenttoG(2)(((andEinstein"sgravitationalpseudotensor)intermsofhisradiationformula.Thefactthatt(g)((andG(2)((arerelatedundersomecircumstancesdoesnotcauseG(2)((tobeanenergy-stressnort(g)((ageometricpart,justasG((andT((mustbeconsideredasdistinctin(1).
Whengravitationalwaveispresent,thegravitationalenergy-stresstensort(g)((isnon-zero.Thus,agravitationalradiationdoescarryenergy-momentumasphysicsrequires.Thisexplainsalsothattheabsenceofananti-gravitycouplingwhichisdeterminedbyEinstein"sradiationformula,isthephysicalreasonthatthe1915Einsteinequation(1)isincompatiblewithradiation.
Notethattheradiationofthebinarypulsarcanbecalculatedwithoutdetailedknowledgeoft(g)((.From(10"),theapproximatevalueoft(g)((atvacuumcanbecalculatedthroughG((/Kasbeforesincethefirst-orderapproximationofg((canbecalculatedthrough(3).InviewofthefactsthatKt(g)((isofthefifthorderinapost-Newtonianapproximation,thatthedecelerationduetoradiationisofthethreeandahalforderinapost-Newtonianapproximation[8]andthattheperihelionofMercurywassuccessfullycalculatedwiththesecond-orderapproximationfrom(1),theorbitsofthebinarypulsarcanbecalculatedwiththesecond-orderpost-Newtonianapproximationof(10)byusing(1)(seealsoSection6).Thus,thecalculationapproachesofDamourandTaylor[25,26]wouldbeessentiallyvalidexceptthattheydidnotrealizethecrucialfactthat(3)isactuallyanapproximationoftheupdateequation(10)[13].
Inlightoftheabove,theHulse-Taylorexperimentssupporttheanti-gravitycouplingbeingcrucialtotheexistenceofthegravitationalwave[10,13],and(3)beinganapproximationofweakwavesgeneratedbymassivematter.Thus,ithasbeenexperimentallyverifiedthat(1)isincompatiblewithradiation.
Itshouldbenotedalsothattheexistenceofananti-gravitycoupling7)meanstheenergyconditionsinthesingularitytheorems[6,17]arenotvalidatleastforadynamicsituation.Thus,theexistenceofsingularityisnotcertain,andtheclaimofinevitablypeakingofgeneralrelativityisactuallybaselesssincethesesingularitytheoremshavebeenproventobeunrealisticinphysics.AspointedoutbyEinstein[2],hisequationmaynotbevalidforveryhighdensityoffieldandmatter.Inshort,thesingularitytheoremsshowonlythepeakingdown;oftheoriesoftheWheeler-Hawkingschool,whichareactuallydifferent3)fromgeneralrelativity.
Thetheoriesofthisschool,inadditiontomakingcrucialmistakesinmathematicsasshowninthispaper(seealso[11,28]),differfromgeneralrelativityinatleastthefollowingimportantaspects:
1)Theyrejectananti-gravitycoupling7),whichisconsideredashighlyprobablebyEinsteinhimself.
2)TheyimplicitlyreplacedEinstein"sequivalenceprincipleinphysics3)withmerelythemathematicalrequirementoftheexistenceoflocalMinkowskispaces[5,6].
3)They,donotconsiderphysicalprinciples[9-11,28](seealsoSection5),suchastheprincipleofcausality,thecoordinaterelativisticcausality,thecorrespondenceprincipleandetc.ofwhichthesatisfactionisvitalforaphysicalspace,whichmodelsreality,suchthatEinstein"sequivalenceprinciplecanbeapplicable.
Thus,inspiteofcurrentlydeclaringtheirtheoriesasthedevelopmentofgeneralrelativity,thesetheoriesactuallycontradictcrucialfeaturesthatareindispensableinEinstein"stheoryofgeneralrelativity.Moreimportantly,inthedevelopmentoftheirso-called"orthodoxtheory,"theyimplicitlyviolatephysicalprinciplesthattookgenerationstoestablish.Asaresult,Einstein"stheoryhasbeenunfairlyconsideredasirrelevantintheeyesofmanyphysicists.
Ofcourse,theexactformoft(g)((isimportantfortheinvestigationofhighdensityoffield.However,itseems,thephysicsofveryhighdensityoffieldandmatterisnotyetmatureenoughatpresenttoallowadefinitiveconclusion.Forinstance,itisunclearwhatinfluencethediscoveryofquarksandgluonsinparticlephysicswouldhaveontheevolutionofstars.Itisknownthatatomicphysicssupportsthenotionofwhite-dwarfstars,andthatnuclearphysicsleadstothenotionofneutronstars.
5.PhysicallyInvalidUnbounded"GravitationalWaves"andthePrincipleofCausality
"Tomymindtheremustbeatthebottomofitall,notanequation,butanutterlysimpleidea.Andtomethatidea,whenwefinallydiscoverit,willbesocompelling,soinevitable,thatwewillsaytooneanother,"Oh,howbeautiful.Howcouldithavebeenotherwise?""--J.A.Wheeler[32].
Itseems,theprincipleofcausality2)(i.e.,phenomenacanbeexplainedintermsofidentifiablecauses)[9,10]wouldbequalifiedasWheeler"sutterlysimpleidea.Beingaphysicist,hisnotionofbeautyshouldbebasedoncompellingandinevitability,butwouldnotbebasedonsomeperceivedmathematicalideas.Itwillbeshownthattheprincipleofcausalityisusefulinexaminingvalidityofaccepted"wave"solutions.
Accordingtotheprincipleofcausality,awavesolutionmustberelatedtoadynamicsource,andthereforeisnotjustatime-dependentmetric.Atime-dependentsolution,whichcanbeobtainedsimplybyacoordinatetransformation,maynotberelatedtoadynamicsource8)[33].Eveninelectrodynamics,satisfyingthevacuumequationcanbeinsufficient.Forinstance,theelectromagneticpotentialsolutionA0[exp(t-z)2](A0isaconstant),isnotvalidinphysicsbecauseonecannotrelatesuchasolutiontoadynamicsource.Thus,asshowninSection4,asolutionfreeofsingularitiesmaynotbephysicallyvalid.
Amajorproblemingeneralrelativityisthattheequivalenceprinciplehasnotbeenunderstoodadequately[11,34].SinceaLorentzmanifoldwasmistakenasalwaysvalid,physicalprincipleswereoftennotconsidered.Forinstance,theprincipleofcausalitywasneglectedsuchthatagravitationalwavewasnotconsideredasrelatedtoadynamicsource,whichmaynotbejustthesourceterminthefieldequation[8,35].
Sincetheprincipleofcausalitywasnotunderstoodadequately,solutionswitharbitrarynonphysicalparameterswereacceptedasvalid[34].Similarly,Misner,Thorne&Wheeler[5],assumedthatthemetricduetoanelectromagneticplane-waveisinvariantwithrespecttoarotationwhoseaxisisinthedirectionofpropagation.Consequently,inadditiontothefactthatthepolarizationisincorrect,Misneretal.werenotawareofthat,indisagreementwithwhattheystated,suchametriccannotbebounded.Suchunboundedsolutionsdisagreewithexperiments[10,11].
Amongtheexistingso-calledwavesolutions,notonlyEinstein"sequivalenceprinciplebuttheprincipleofcausalityisnotsatisfiedbecausetheycannotberelatedtoadynamicsource.(However,asourceterminanequation,thoughrelatedto,maynotnecessarilyrepresentthephysicalcause[9,34].)Here,examplesofaccepted"gravitationalwaves"areshownasactuallyinvalidinphysics.
1.LetusexaminethecylindricalwavesofEinstein&Rosen[29].Incylindricalcoordinates,(,(,andz,
ds2=exp(2(-2()(dT2-d(2)-(2exp(-2()d(2-exp(2()dz2(12)
whereTisthetimecoordinate,and(and(arefunctionsof(andT.Theysatisfytheequations
(((+(1/()((-(TT=0,((=([((2+(T2],and(T=2((((T.(13)
Rosen[36]considertheenergy-stresstensorT((thathascylindricalsymmetry.Hefoundthat
T44+t44=0,andT4l+t4l=0(14)
wheret((isEinstein"sgravitationalpseudotensor,t4lismomentumintheradialdirection.
However,Weber&Wheeler[37]arguedthattheseresultsaremeaninglesssincet((isnotatensor.Theyfurtherpointedoutthatthewaveisunboundedandthereforetheenergyisundefined.Moreover,theyclaimedmetric(12)satisfyingtheequivalenceprincipleandspeculatedthattheenergyfluxisnon-zero.
Theirclaimshowsaninadequateunderstandingoftheequivalenceprinciple.Tosatisfythisprinciplerequiresthatatime-likegeodesicmustrepresentaphysicalfreefall.Thismeansthatall(notjustsome)physicalrequirementsarenecessarilysatisfied.Thus,theequivalenceprinciplemaynotbesatisfiedinaLorentzManifold[11,35],whichimpliesonlythenecessaryconditionofthemathematicalexistenceofaco-movinglocalMinkowskispacealongatime-likegeodesic.Itwillbeshownthatmanifold(12)cannotsatisfycoordinaterelativisticcausality.Moreover,aspointedoutearlier,anunboundedwaveisunphysical.
WeberandWheeler"sargumentsforunboundednessarecomplicated,andtheyagreedwithFierz"sanalysis,basedon(13),that(isastrictlypositivewhere(=0[37].However,itispossibletoseethatthereisnophysicalwavesolutioninasimplerway.Gravitationalredshiftsimplythatgtt(1[2];and
-g(((gtt,-g((/(2(gtt,and-gzz(gtt,(15a)
areimpliesbycoordinaterelativisticcausality.Thus,accordingtotheseconstraints,frommetric(12)onehas
exp(2()(1andexp(2()(exp(4().(15b)
Equation(15)impliesthatgtt(1andthat((0.However,thisalsomeansthatthecondition(>0cannotbemet.Thus,thisshowsagainthatthereisnophysicalwavesolutionforG((=0.
WeberandWheelerareprobablytheearliesttoshowtheunboundednessofawavesolutionforG((=0.Nevertheless,duetotheirinadequateunderstandingoftheequivalenceprinciple,theydidnotreachavalidconclusion.ItisironicthattheythereforecriticizedRosenwhocometoavalidconclusion,thoughwithdubiousreasoning.
2.RobinsonandTrautman[38]dealtwithametricofspherical"gravitationalwaves"forG((=0.However,theirmetrichasthesameproblemofunboundednessandhavingnodynamicsourceconnection.Thisconfirmsfurtherthatthecauseofthisproblemisintrinsicallyphysicalinnature.Theirmetrichasthefollowingform:
ds2=2d(d(+(K-2H(-2m/()d(2-(2p-2{[d(+((q/(()d(]2+[d(+((q/(()d(]2},(16a)
wheremisafunctionof(only,pandqarefunctionsof(,(,and(,
H=p-1(p/((+p(2p-1q/((((-pq(2p-1/((((,(16b)
andKistheGaussiancurvatureofthesurface(=1,(=constant,
K=p2((2/((2+(2/((2)lnp.(16c)
Forthismetric,theempty-spaceconditionG((=0reducesto
(2q/((2+(2q/((2=0,and(2K/((2+(2K/((2=4p-2((/((-3H)m.(17)
Toseethismetrichasnodynamicconnection,letusexaminetheirspecialcaseasfollows:
ds2=2d(d(-2Hd(2-d(2-d(2,and(H/((=(2H/((2+(2H/((2=0.(18)
Thisisaplane-fronted"wave"[39]derivedfrommetric(16)byspecializing
p=1+((2+(2)K(()/4.(19a)
substituting
(=(-2+(-1,(=(,(=(2,(=(2,q=(4,(19b)
where(isconstant,andtakingthelimitas(tendstozero[38].Although(18)isaLorentzmetric,thereisasingularityoneverywavefrontwherethehomogeneityconditions
(3H/((3=(3H/((3=0.(20)
areviolated[38].Obviously,thisisalsoincompatiblewithEinstein"snotionofweakgravity[2].Aproblemincurrenttheoryisitsratherinsensitivitytowardtheoreticalself-consistency[9,13,35,40-42].
3.Toillustratethenon-existenceofaboundedradiatingphysicalsolutionfurther,letusexaminearecentsolutionofR((=0,thecylindricalsymmetrysolutionofAu,Fang&To[43].Theirmetricis
ds2=N2(c2dt2-dz2)-L2d(2-M2(2d(2(21)
where
N2=(-4exp(-4((d()exp(2n1),L2=(-8(1+(()2exp(-6((d(),
and
M2=exp(2((d()wheren1=n1(ct-z),and(=((()
arerespectivelyarbitraryfunctionsof(ct-z)andof(.Thefunctionn1(ct-z)makesN2apropagatingwave.Ifsolution(21)wereaphysicalsolution,Mshouldbeaboundedfunctionof(,i.e.,
exp(2((d()<C12(22)
forsomeconstantC1.However,thisalsomeansthatNisnotboundedforsmall(.Moreover,iflightvelocityisnotlargerthanitsvacuumvelocityc,oneshouldhaveN2/L2andN2/M2(1.Itthusfollowsthat
(1+(()2((4exp(2((d()exp(2n1),andexp(6((d()(exp(2n1)(-4.(23)
Hence,
(1/(+()2((2/3exp(8n1/3)and(2>(O((2/3).(24)
Thus,condition(24)isalsoinconsistentwithcondition(22).Insummary,solution(21)isalsonotaphysicalsolutionandisunboundedincontrasttoasrequiredbytheprincipleofcausality.
4.Toillustrateaninvalidsourceandanintrinsicnon-physicalspace,considerthefollowingmetric,
ds2=dudv+Hdu2-dxidxi,whereH=hij(u)xixj(25)
whereu=ct-z,v=ct+z,x=x1andy=x2,hii(u)(0,andhij=hji[44].Thismetricsatisfiestheharmonicgauge.Thecauseofmetric(25)canbeanelectromagneticplanewave.Metric(25)satisfies
((((((((tt=-2{hxx(u)+hyy(u)}where(((=g((-(((.(26)
However,thisdoesnotmeanthatcausalityissatisfiedalthoughmetric(25)isrelatedtoadynamicsource.Itwillbeshownthat(25)isnotaphysicalsolutionbecausephysicalprinciplesareviolated.
Alighttrajectorysatisfiesds2=0[2].Foralightinthez-direction(i.e.dx=dy=0),oneobtains
dz/dt=cor-c(1+H)/(1-H);butH(0(27)
wouldfailsincehii(u)(0;andsocoordinaterelativisticcausalitywouldalsofail.Thus,aformalsatisfactionoftheconservationlawdueto((G(((0,isinadequatetoensurethevalidityof(1).
Moreover,thegravitationalforceisrelatedto(ztt=(1/2)(H/(t.Therearearbitrarynon-physicalparameters(thechoiceoforigin)thatareunrelatedtothecause(aplanewave).Apparently,believingthatanyLorentzmanifoldisvalidinphysics,Penrose[44]over-lookedthephysicalrequirements,inparticulartheprincipleofcausality.Experimentally,beingunbounded,metric(25)isalsoincompatiblewiththecalculationoflightbendingandclassicalelectrodynamics.
Theseexamplesconfirmthatthereisnoboundedwavesolutionfor(1)althougha"time-dependent"solutionneednotbelogarithmicdivergent[14].Afundamentalreasonfortheboundednessofadynamicsolutionforgravity,istheequivalenceprinciple[11].ThiswouldmeanthatthehyperboloidsolutioninFriedmann"stheorymightnotbevalidingeneralrelativity(seeAppendix).
6.ConclusionsandDiscussions
Ingeneralrelativity,theexistenceofgravitationalwaveisacrucialtestofthefieldequation.Thus,animportantquestionis:whatdoesthegravitationalfieldofaradiatingasymptoticallyMinkowskiansystemlooklike?Withoutexperimentalinputs,toanswerthisquestionwouldbeverydifficult.
Einstein[2]proposedthelinearizedtheoryforaweakradiatinggravitationalfield.But,Bondi[24]commented,"itisneverentirelyclearwhethersolutionsderivedbytheusualmethodoflinearapproximationnecessarilycorrespondineverycasetoexactsolutions,orwhethertheremightbespuriouslinearsolutionswhicharenotinanysenseapproximationstoexactones."Thus,incalculatinggravitationalwavesfromtheEinsteinequation,problemsareconsideredasduetothemethodratherthaninherentintheequations.
Physically,itisnaturaltocontinueassumingEinstein"snotionofweakgravityisvalid.(Boundedness,thoughaphysicalrequirement,maynotbemathematicallycompatibletoanonlinearfieldequation.But,nooneexceptperhapsGullstrand[40,41],expectedthenonexistenceofdynamicsolutions.)ThecomplexityoftheEinsteinequationmakesitverydifficulttohaveacloseform.Thus,itisnecessarythatamethodofexpansionshouldbeusedtoexaminetheproblemofweakgravity,ifoneexpectssuchanexpansiontobevalid.
Afactorwhichcontributestothisfaithisthat((G(((0implies((T(m)((=0,theenergy-momentumconservationlaw.However,thisisonlynecessarybutnotsufficientforadynamicsolution.Althoughthe1915equationgivesanexcellentdescriptionofplanetarymotion,includingtheadvanceoftheperihelionofMercury,thisisessentiallyatest-particletheory,inwhichthereactionofthetestparticleisneglected.Thus,thesoobtainedsolutionsarenotdynamicsolutions.AspointedoutbyGullstrand[41,45]suchasolutionmaynotbeobtainableasalimitofadynamicsolution.Nevertheless,Einstein,Infeld,andHoffmann[22]incorrectlyassumedtheexistenceofboundeddynamicsolutionanddeducedthegeodesicequationfromthe1915equation.Recently,Feymann[23]madethesameincorrectassumptionthataphysicalrequirementwouldbeunconditionallyapplicabletoamathematicalequation.
ThenonlinearnatureofEinsteinequationcertainlygivessurprises.In1959,Fock[14]pointedoutthat,inharmoniccoordinates,therearedivergentlogarithmicdeviationsfromexpectedlinearizedbehavioroftheradiation.Afterthediscoverythatsomevacuumsolutionsarenotlogarithmicdivergent[15],theinadequacyofEinstein"sequationwasnotrecognized.Instead,themethodofcalculationwasmistakenastheproblem.
Toavoidtheappearanceoflogarithms,Bondietal.[24]andSachs[46]introducedanewapproachtogravitationalradiationtheory.Theyusedaspecialtypeofcoordinatesystem,andinsteadofassuminganasymptoticexpansioninthegravitationalcouplingconstant(,theyassumetheexistenceofanasymptoticexpansionininversepowerofthedistancer(fromtheoriginwheretheisolatedsourceislocatedinr(a,whichisapositiveconstant).TheapproachofBondi-Sachswasclarifiedbythegeometrical"conformal"reformulationofPenrose[47].
However,thisapproachisunsatisfactory,"becauseitrestsonasetofassumptionsthathavenotbeenshowntobesatisfiedbyasufficientlygeneralsolutionoftheinhomogeneousEinsteinfieldequation[48]."Inotherwords,thisapproachprovidesonlyadefinitionofaclassofspace-timesthatonewouldliketoassociatetoradiativeisolatedsystems,neithertheglobalconsistencynorthephysicalappropriatenessofthisdefinitionhasbeenproven.Moreover,perturbationcalculationshavegivensomehintsofinconsistencybetweentheBondi-Sachs-Penrosedefinitionandsome approximatesolutionofthefieldequation.Notlessimportant,itseemsaprioridifficulttorelatetothesourcelocatedwithinr(a[48].
Therearetwoothermainclassesofapproach:1)thepost-Newtonianapproaches(1/cexpansions)andthepost-Minkowskianapproaches(Kexpansions).Thepost-Newtonianapproachesarefraughtwithseriousinternalconsistencyproblems[48]becausetheyoftenlead,inhigherapproximations,todivergentintegrals.Thepost-Minkowskianapproachisanextensionofthelinearization,onemayexpectthattherearesomeproblemsrelatedtodivergentlogarithmicdeviations[14].Moreover,ithasunexpectedlybeenfoundthatperturbativecalculationsonradiationactuallydependontheapproachchosen[49].Mathematically,thisnon-uniquenessshows,indisagreementwith(3),thatadynamicsolutionof(1)isunbounded.
Basedonthebinarypulsarexperiments,itisproventhattheEinsteinequationdoesnothaveanydynamicsolutionevenforweakgravity[13].Mathematically,however,theproofthatisaimeddirectlytothenonexistenceofadynamicsolutionisindependentoftheexperimentalsupportsfor(3).Thislongprocessis,inpart,duetotheoreticalconsistencywereinadequatelyconsidered[9,10,13,35].Moreover,itwasnotrecognizedthatboundednessofawaveiscrucialforitsassociationwithadynamicsource.Theseinadequaciesallowedacceptanceofunphysical"time-dependent"solutionsasphysicalwaves(Sections3&5).
Althoughnon-linearityofthe1915Einsteinequationwasnew,inviewofimpressiveobservationalconfirmations,itseemednaturaltoassumethatgravitationalwaveswouldbeproduced.Moreover,gravitationalradiationisoftenconsideredasduetotheaccelerationinageodesicalone[50-52].Itisremarkablethatin1936EinsteinandRosen[4]arethefirsttodiscoverthisproblemofexcludingthegravitationalwave.However,withoutclearexperimentalevidence,itwasdifficulttomakeanappropriatemodification.
Fromstudyingthegravityofelectromagneticwaves,itwasalsoclearthatEinsteinequationmustbemodified[11,18].However,theHulseandTaylorbinarypulsarexperiments,whichconfirmHogarth"s1953conjecture6)[31,35],areindispensableforverifyingthenecessityoftheanti-gravitycouplingingeneralrelativity[10,13].Inadditiontoexperimentalsupports,theMaxwell-NewtonApproximationcanbederivedfromphysicalprinciples,andtheequivalenceprinciplealsoimpliesboundednessofanormalizedmetricingeneralrelativity[11].Aperturbativeapproachcannotbefullyestablishedfor (1)simplybecausetherearenoboundeddynamicsolutions10),whichmust,owingtoradiation,beassociatedwithananti-gravitycoupling.
Nevertheless,ChristodoulouandKlainerman[27]claimedtohaveconstructedboundedgravitational(unverified)waves.Obviously,theirclaimisincompatiblewiththefindingsofothers.Furthermore,theirpresumedsolutionsareincompatiblewithEinstein"sradiationformulaandareunrelatedtodynamicsources[10,11].Thus,theysimplyhavemistaken5)anunphysicalassumption(whichdoesnotsatisfyphysicalrequirements)asawave[28].
Withinthetheoreticalframeworkofgeneralrelativity,however,thegravitationalfieldofaradiatingasymptoticallyMinkowskiansystemisgivenbytheMaxwell-NewtonApproximation[13].Withtheneedofrectifyingthe1915Einsteinequationestablished,theexactformoft(g)((intheequationof1995update[13]isanimportantproblemsinceadynamicsolutionthatgivesanapproximationfortheperihelionofMercuryremainsunsolved[41].Moreover,theupdateequationshowsthatthesingularitytheoremsproveonlythepeakingdownoftheoriesoftheWheeler-Hawkingschool3),but notgeneralrelativity(seeSection4).Experimentally,theMaxwell-NewtonApproximationwouldbefurthertestedbytheGravityProbe-Bgyroscopes[53]ontheprecessions.ThisanalysissuggeststhatfurtherconfirmationofthisApproximationandthustheequivalenceprincipleisexpected.
Appendix:DynamicSpace-Time,Space-TimeCoordinateSystem,andtheBigBangTheory
Theequivalenceprinciple,inacertainsense,isanon-localproperty,sinceitsphysicsiswhetherthegeodesicrepresentsaphysicalfreefall[11].Thus,onemustconsiderbeyondthemathematicaltangentspace,thatis,mathematicallocalMinkowskispaces.Todeterminewhetheramanifoldsolutioncanbediffeomorphictoaphysicalspaceisadifficultproblemandphysicalrequirementsareneeded[10].
Inphysics,theframeofreferenceisoftenchosentobebestfortheproblem.Ifavalidphysicalsolutioncannotbefound,thedifficultisusuallynotduetothecoordinates.Inaddition,asapracticalapproximatemeans,aGalileantransformationcanbeusedinsomeclassofproblems.Thus,thatacertaincoordinatesystemisusefulforsomecalculationsdoesnotmeanthatthecoordinatesystemis,inprinciple,realizable.
Forapracticalproblem,inspiteoftalksaboutcoordinatescannotbechosenapriori,generalrelativityisactuallynotanexception11).Forinstance,intheSchwarzschildstaticsolution,theframeofreferenceischosenaprioriandtheradialris(x2+y2+z2)1/2.Thisframeofreferenceisusedtoaccesstheamountoflightbending.Intheproblemoflightbending,thetotalfield(space-timemetric)shouldbetime-dependent,butrasavariablewouldbethesameiftheframeofreferencedoesnotchange.
Nevertheless,incosmology,therearetime-dependentsolutionsthatdonotinvolveacoordinatesystemchosenapriori,norgravitationalradiation.However,oneshouldnotealsothatallthecosmologicalmodelsarebasedonidealizationsthathavenotbeenestablishedbeyondreasonabledoubt[32,54].Forthisreasonalone,suchexamplesareunsuitableforourdiscussiononafundamentalproblemofrealisticsituations.However,somediscussionsonthissubjectareneeded,sinceitisclaimedthatthebigbangtheoryisbasedongeneralrelativity[32,55].
Itisgenerallyassumed[55]"thattheenergy-momentumtensorintheuniversetodayisthatofauniformgaswithzeropressure.Thegalaxiesmayberegardedasthe"particles"outofwhichthisgasismade,andsincethevelocitiesofthegalaxiesdonotdeviatemuchfromuniformexpansion,wecanneglectthe"pressure"ofthegasofgalaxies...."TheFriedmannmodelsassumedhomogeneous,isotropicmodelsoftheuniversewithmassdensitybutwithzeropressure.Adifficultincosmologyisthatmanyusualphysicalrequirements,onwhich ajudgmentofphysicalvaliditydepends,areprobablynotapplicable.
Nevertheless,somediscussionsmaybehelpfulinclarifyingcoordinaterelativisticcausality.TodiscusstheFriedmannmodel,onemustfirstacceptessentiallybyfaiththatthemassdistributionofthewholeuniverseishomogeneousandisotropic.OnemustdecidealsomodelingagalaxyasaparticleisconsistentwiththenormalunderstandingofEinstein"sequivalenceprinciple.Then,inCartesiancoordinates,
ds2=d(2-2((){dx2+dy2+dz2},(A1)
theRobertson-Walkergeometry,isbelievedtobeappropriate.Then,theEinsteinequation(1)withsourceenergytensorT((=u(u(+P(u(u(-g(()leadstothefollowinggeneralevolutionequations[17]:
3=-4(((+3P)(A2)
and
32/2=8((-3k/2,(A3)
where(isthemassdensity,andPisthepressure.Fordifferentvaluesofk,therearedifferenttypesofsolutions:k=+1forthe3-sphere,k=0fortheflatspace,andk=-1forthehyperboloid.Fork=-1,2(()isunbounded[17]andisthereforeincompatiblewiththeequivalenceprinciple[11].
TherateofchangeofR(thedistancebetweentwoisotropicobserversattime()is
v=HR,(A4)
whereH(()=/isidentifiedwithHubble"sconstant.Thismeans,however,theconstantistime-dependent.Note,however,theobservedredshiftsmaynotbeduetotheDopplereffectalone[11,54,56].
However,withintheaboveconstraint,amodel-independentfeatureof(()is
(()((((=0;(A5a)
and
((()n(()=constant,wheren(3(A5b)
Ontheotherhand,ds2=0couldimplythatthelightspeedinthex-directionwouldbe
(A6)
Thus,(A5a)and(A6)leadtoaresultthatthelightspeedcouldbelargerthanc.Thus,itseems,eitherthatcoordinaterelativisticcausalitycouldbeviolatedormetric(A1)wouldbeinvalid.
Nevertheless,onemustbecarefulbecausethingsarenotthatsimple.Fords2=0leadstoalightspeedinvacuum.However,intheFriedmannmodel,whena(()isverysmall,accordingto(A5b),notonlythereisnovacuumbutthemassdensity((()wouldbetoolargeforthelighttogothrough.Thus,theargumentthatleadsto(A6)peaksdown.Moreover,tojustifytheRobertson-Walkergeometry,theeffectsofgravitationalradiationshouldhavebeenshowntobenegligibleatleastfortheassumedearlyuniverse.Theexistenceofgravitationalradiation,aspointedoutbyLorentzandWheeler[1],isduetothetheoryofrelativity.Thus,itisalsonotclearthatFriedmann"ssolutionmustbededucedfromgeneralrelativity.
Inreality,agalaxyisnotaparticle,themassdistributionisnothomogeneous,andalightspeedhasnothingtodowithFriedmann"smodeling.Thus,itisclearlyunsuitableforadiscussiononfundamentalquestions.Now,itshouldbeclearalsothattheBigBangtheory,thoughcanberelatedto(1),dependsontoomanydubiousassumptions(seealso[32,54])fortheclaimofbeingaconsequenceofgeneralrelativity.(Also,inviewoftheidealizations,thepossibilityofderivingeqs.(A2)and(A3)fromanotherequationcannotberuleout.)Nevertheless,thisdiscussionillustratesalsotheimportanceoftheequivalenceprinciple.
Acknowledgments
ThispaperisdedicatedtoProfessorJ.E.HogarthofQueen"sUniversity,Kingston,Ontario,Canada,whoconjecturedin1953thenonexistenceofdynamicsolutionsforthe1915Einsteinequation.TheauthorwishestoexpresshisappreciationtoProfessorXinYuforthehospitalityoftheHongKongPolytechnicUniversitywheresubstantialofthisworkwasdonein1995.TheauthorgratefullyacknowledgesstimulatingdiscussionswithDr.H.C.Chan,ProfessorC.Au,ProfessorJ.E.Hogarth,ProfessorS.A.Lamb,ProfessorP.Morrison,andProfessorH.Nicolai.Theauthor;wishestothanktherefereesforvaluablecommentsandpointingoutusefulliterature;andMs.P.MafortheFrenchabstract.TheauthorisindebtedtoMr.DavidP.ChanandMr.RichardC.Y.HuifortheirsupportsandhospitalitywhileinHongKong.ThispublicationissupportedbyInnotecDesign,Inc.,U.S.A.
ENDNOTES
1)Someauthorsprefer,differentfromEinstein,todefineK=8((c-4[55].Then,thefourvelocityu(wouldbedefinedascdx(/ds,whereds2=g((dx(dx(suchthatequation(1)remainsthesame.
2)Thetime-testedassumptionthatphenomenacanbeexplainedintermsofidentifiablecausesiscalledtheprincipleofcausality.Thisisthebasisofrelevanceforallscientificinvestigations.Theprincipleofcausalityimpliesthatanyparameterinaphysicalsolutionmustberelatedtosomephysicalcauses.
3)ThisexplicitreinterpretationofEinstein"sequivalenceprinciple(basedonPauli"smisinterpretationthatEinsteinobjected[57])asjustthesignatureofLorentzmetricwasadvocatedbySynge[58]earlierandFriedman9)currently.Recently,ithasbeenproventhatsuchareductionisinconsistentwithEinstein"sowninterpretationandphysicalprinciples[11,35,57]aswellasindisagreementwithexperimentsincludingtheMichelson-Morleyexperiment[59].However,theadvocatesdisregardalltheseinconsistenciesbecause,owingtotheirinadequateunderstandingofphysicsatthefundamentallevel,theybelievethatacoordinatesystem(includingitsmetric)hasnophysicalmeaning[60].(Moreover,followingthestepofFock[61],Ohanian,andRuffiniopenlydeclaredintheirbook[55],whichisendorsedbyWheeler,thatbothofEinstein"sequivalenceprincipleandtheprincipleofrelativityareinvalid.)Nevertheless,thisseeminglyexceedinglyingeniousdefensecollapsesbecausetheobservedgravitationalredshiftsunequivocallyimplythattheirinterpretationisinvalidinphysics.
4)Adynamicmetricsolutioningravityisrelatedtothedynamicsofitssourcematter.Adynamicsource,accordingtorelativity,wouldgenerategravitationalradiation[1].FortheperihelionofMercuryandthedeflectionoflight,themetricisastaticsolutionalthoughsolutionsofthetestparticlesarecalculated.Itwasbelievedthattheinfluenceofatestparticletothemetriccouldalsobecalculatedwith(1).However,assuspectedbyGullstrand[40,41]andconjecturedbyHogarth6)[31],thetruthistheopposite.
5)K.Kuchar[62]claimedtohaveprovedthattheinitialconditionofEinstein"sequation(1)canbeapproximatedbytheinitialconditionofthelinearequation(3)byusingapowerseriesexpansion.Note,however,thattheonlyvalidcaseofsuchapowerseriesexpansionisanon-dynamicsolution(seeSections2-4).Thus,hehasprovenonlythatthepropertiesaretrueinanunintendedvoidset.Suchabasicmistakeisessentiallyrepeated20yearslaterbyChristodoulouandKlainerman[27]forclaimingtheexistenceofboundedradiativesolutions(seeSection6).Nevertheless,theEditorialBoardofQuantumandClassicalGravity[63],unlikethebookreview[64]andtheEditorofGRG[65],consideredtheseinvalidclaimsas"proofs".Moreover,asolutionrelatingtoadynamicsourcebyanequationalone,assuggestedbyKlainermanandNicolò[66],isinsufficientbecausesuchasolutionmaystillviolateotherphysicalrequirements(seeSection5).
6)Hogarthconjecturedthat,foranexactsolutionofthetwo-particleproblem,theenergy-momentumtensordidnotvanishinthesurroundingspaceandwouldrepresenttheenergyofgravitationalradiation.
7)Thepossibilityofhavingananti-gravitycouplingwasformallymentionedbyPauli[12].Inadifferentway,suchapossibilitywasactuallyfirstmentionedbyEinstein[67]in1921.Hewrotein"GeometryandExperience,""But,iftheuniverseisfinite,thereisaseconddeviationfromNewtoniantheory,which,inthelanguageofNewtoniantheory,maybeexpressedthus:thegravitationalfieldissuchasifitwereproduced,notonlybytheponderablemasses,butinadditionbyamass-densityofnegativesign,distributeduniformlythroughoutspace."Healsofirmlybelievedinsuchapossibility.However,itwasnotrecognizedthatananti-gravitycouplingiscrucialforadynamicsolution[9,13].Ontheotherhand,HawkingandPenrose[6,17]hadimplicitlyassumed,intheirsingularitytheorems,theimpossibilityofananti-gravitycoupling.Arathercommonerroneousgroundtorejecttheexistenceanantigravitycouplingisduetoamisinterpretationoftheequivalenceofmassandenergyintheenergy-massconservationlawE=mc2[68].Forinstance,Fock[61]claimed,"WesawthattoanyenergyE oneshouldascribeamassm=E/c2andtoeverymassoneshouldascribeanenergyE=mc2."However,thisisinconsistentwithgeneralrelativitywithatensorfield.AccordingtoEinstein[69],onlythelatterisvalid.Einsteinstated,"NowwecanreversetherelationandsaythatanincreaseofEintheamountofenergymustbeaccompaniedbyanincreaseofE/c2inthemass.Icaneasilysupplyenergytothemass-forinstance,ifIheatitbytendegrees."Healsowrote"Foramassincreasetobemeasurable,thechangeofenergypermassunitmustbeenormouslylarge."ThekeywordinEinstein"sstatementsis"increase".Thus,E/c2isrelatedtoanincrementofmasstomassivematter.However,thisdoesnotmeanthatingeneralanykindofenergyEhasarelatedmassE/c2,asFockclaimed.Healsoremarked,"Also,thelawpermitsustocalculateinadvance,frompreciselydeterminedatomicweights,justhowmuchenergywillbereleasedwithanyatomicdisintegrationwehaveinmind.Thelawsaysnothing,ofcourse,astowhether-orhow-thedisintegrationreactioncanbepoughtabout."
8)AtraditionalviewpointofthePhysicsDepartmentofMITisthatgeneralrelativitymustbeunderstoodintermsofphysics[8].
9)JohnL.Friedman,DivisionalAssociateEditorofPhys.Rev.Letts.,officiallyclaims"TheexistenceoflocalMinkowskispacehasreplacedtheequivalenceprinciplethatinitiallymotivatedit."NotethathealsoclarifiesthatthetheoryoftheWheeler-Hawking[5,6]schoolisnotreallygeneralrelativitybyusingtheword"replaced"(Feb.17,2000).
10)MaxPlanckonceremarked,"Anewscientifictruthdoesnottriumphbyconvincingitsopponentsandmakingthemseethelight,butratherbecauseitsopponentseventuallydie,andanewgenerationgrowsupthatisfamiliarwithit."Fortunately,itseems,mathematicsisanexceptiontohisrule.
11)Einstein[70]onceremarked,"Ifyouwanttofindoutanythingfromthetheoreticalphysicistsaboutthemethodstheyuse,Iadviseyousticktooneprinciple,don"tlistentotheirwords,fixyourattentionontheirdeeds."
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Résumé
Ilestdémontréquel"équation1915d"Einsteinestincompatibleaveclanotionphysiqueoùuneondeemportelavitesseàl"énergie.Cettepreuveestcompatibleavecl"approximationMaxwell-Newton,quiformulequel"équationduchamplinéairepourlagravitéfaible,estsoutenueparlesexpériencesbinairespulsars.Pourlesproblèmesdynamiques,l"équationduchamplinéaireestindépendanteonplusd"êtreincompatibleavecl"équationd"Einstein.L"équationlinéaire,commepremièreapproximation,nécessitel"existencedel"ondegravitationnellefaible.Ilfautqu"ellesoitliéedansl"amplitudeetsoitenrapportaveclesdynamiquesdelasourcedelaradiation.Enraisondelanégligenceauxassociationsphysiquesimportantes,etunecompréhensioninsuffisanteduprinciped"équivalence,lessolutionsnonphysiquesauraientcauséesdesinterprétationserronéesdesondesgravitationnelles.Théoriquement,ettelquesuggéréparEinsteinetRosen,ilestconcluquelasolutionconcernantuneondephysiquegravitationnellepourl"équation1915n"existepas.Cetteconclusionestsoutenueparlesanalysesdesondesplatescontrelessolutionsondesexactes.Deplus,lesconceptionsdelaradiationdespulsarsbinairesdeDamouretdeTaylorseraientvalidesseulementsiellessontuneapproximationdel"équationmiseàjourde1995.L"équationmiseàjourmontrequelesthéorèmessingulierspeuventseulementl"analysedesthéoriesdeWheeler-Hawking,maispaslarelativitégénérale.Deplus,ilestàremarquerquequelquesattestationsdeLorentzsontparmicellesquinesontpasd"accordavecdesfaitsexpérimentauxbienconnus.
Subj:Re:CQG/104867/PAPandCommentsonChristodoulou&Kla
Date:10/6/012:13:33PMEasternDaylightTime
From:Chungylo
To:[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected],[email protected]
file:C&K.ZIP(96253bytes)
DepaWills
PublishingAdministrator
ClassicalandQuantumGravity
E-mail:[email protected]
DearMs.Wills:
Ihaveinformedyoutwoyearsagothatyourboardreportwillberespondedinmypublishedpapers.Now,thetwopapersconcernedyourboardreporthavebeenpublished.Theyare:C.Y.Lo,PhysicsEssays,13(1),109-120(March2000);andC.Y.Lo,PhysicsEssays,13(4),527-539(Dec.2000).
Inthefirstpaper,Ishowthatthe"solutions"constructedbyChristodoulouandKlainermanareproventobephysicallyincorrect.Mathematically,theirclaimsareinvalidsimplybecausethe"proof"isincomplete.YourJournalisreferredtoinreference[44].ThesecondpaperpointedoutthattheclaimsofKuchararealsoincorrect,andthe"proof"isinvalidbecauseitisnotapplicabletoadynamicalcase.Also,themorerecentpaperofKlainermanandNicolopublishedinyourjournalisalsocommented.Myoverallcriticismtoyourjournal ontheseissuesare(inendnote[5])thatyourjournalconsideredtheseinvalidclaimsas"proofs".
Iassumethatyourboardwouldbeinterestedinreadingmypapers.Foryourconvenience,theelectronicfilesofthesetwopapersareattached.Anycommentstheboardofyourjournalmayhavewillbegreatlyappreciated.Thankyou.
Sincerelyyours,
C.Y.Lo