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arXiv:gr-qc/9411066v128Nov1994Ontherelationbetweenmetricandspin–2formulationsoflinearizedEinsteintheoryJacekJezierskiDepartmentofMathematicalMethodsinPhysics,UniversityofWarsaw,ul.Ho˙za74,00-682Warsaw,PolandRef.No.4556AbstractAtwenty–dimensionalspaceofchargedsolutionsofspin–2equationsisproposed.Therelationwithextended(viadilatation)Poincar´egroupisanalyzed.Locally,eachsolutionofthetheorymaybedescribedintermsofapotential,whichcanbeinterpretedasametrictensorsatisfyinglinearizedEinsteinequations.Globally,thenon–singularmetrictensorexistsifandonlyif10amongtheabove20chargesdovanish.Thesituationisanalogoustothatinclassicalelectrodynamics,wherevanishingofmagneticmonopoleimpliestheglobalexistenceoftheelectro–magneticpotentials.ThenotionofasymptoticconformalYano–KillingtensorisdefinedandusedasabasicconcepttointroduceaninertialframeinGeneralRelativityviaasymptoticconditionsatspatialinfinity.Theintroducedclassofasymptoticallyflatsolutionsisfreeofsupertranslationambiguities.1IntroductionThelinearizedEinsteinequationsdescribingweakgravitationalfieldcanbeformulatedintermsofthemetrictensor(seeSection2)or,intermsoftheWeyltensor,asaspin–2field(seeSection3).Bothformulationsaregloballyequivalentifthetopologyofthespacetimeistrivial.However,thelinearizedEinsteintheorycanonlybeappliedintheasymptoticallyflatregion,whichhasanontrivialtopology(atubecontainingthestrongfieldregionhastoberemovedfromMinkowskispace).Inthiscasebothformulationsarenolongergloballyequivalent.Similarlyasinclassicalelectrodynamics,wherethevan-ishingofmagneticmonopolesintopologicallynon–trivialregionsimpliestheexistenceofmagneticvector–potential([1],[2]),thenecessaryandsufficientconditionfortheequiva-lenceofthetwoformulationsofthelinearizedgravityisthevanishingofcertainchargeswhichweintroduceinSection4.Thenewchargesresultinanaturalwayfromageometricformulationofthe“Gausslaw”forthegravitationalcharges,definedintermsoftheRiemanntensor.Wepresent1thisformulationinSection5.ItleadstothenotionoftheconformalYano–Killingtensor.AconformalYano–Killing(CYK)equation(30)possesestwenty–dimensionalspaceofsolutionsforflatMinkowskimetricinfour–dimensionalspacetime(n=4).Thereisnoobviouscorrespondencebetweenten–dimensionalasymptoticPoincar´egroupandthetwenty–dimensionalspaceofCYKtensors.Onlyhalfofthem(thefour–momentumvectorpμandtheangularmomentumtensorjμν)arePoincar´egenerators.Thissituationisanalogoustothatofelectrodynamics,where,intopologicallynon–trivialregions,wehavetwocharges(electric+magnetic)despitethefactthatthegaugegroupisone–dimensional.Letusnotice,thatforn=2(nisadimensionofspacetime)thespaceofsolutionsoftheequation(30)isinfiniteandforn=3thecorrespondingspaceisonlyfour–dimensional.Possibledimensionswesummarizeinatable:dimensionofspacetimen=2n=3n=4dimensionof(pseudo)euclideangroup3610dimensionofconformalgroup∞1015dimensionofspaceofCYKtensors∞420TheabovetableshowsthatthereisnoobviousrelationbetweenCYKtensorsandthegroup.Ontheotherhand,inthecasen=4,itispossibletoconnectCYKtensorswitheleven–dimensionalgroupofPoincar´etransformationsenlargedbydilatation(pseudo–si-milaritytransformations).Eleven–dimensionalalgebra(spaceofKillingvectors)ofthisgroupallowsustoconstruct(viathewedgeproduct)alltheCYKtensorsinMinkowskispacetime.Anaturalapplicationoftheaboveconstructiontothedescriptionofasymptoticallyflatspacetimesisproposedinsections6and7.Itallowsustodefineanasymptoticchargeatspatialinfinitywithoutsupertranslationambiguities.TheexistenceornonexistenceofthecorrespondingasymptoticCYKtensorscanbechosenasacriterionforclassificationofasymptoticallyflatspacetimes.Forexample,theTaub–NUTspacetimes[13]canbeexcludedassumingthatthecorrespondingconservedquantityvanishes.Similarly,theDemia´nskisolution[14]correspondstoanon–vanishingchargedandcanalsobeexcludedthisway(seeSection4).Westressthatthenew10chargesintroducedinthepresentpaperaredifferentfrom10exactgravitationally–conservedquantitiesatnullinfinity,consideredbyE.T.NewmanandR.Penrose[8].Thesituationatnullinfinity(whichwedonotanalyzeinourpaper)ismuchmoredelicatethantheoneatspatialinfinity.Toextendourdefinitiontothenullinfinity,wewillprobablyhavetoassumeadditionalasymptoticconditionswhichguaranteetheexistenceoftheasymptoticCYKtensor.22LinearizedgravityLinearizedEinsteintheory(seee.g.[5]or[4])canbeformulatedasfollows.Einsteinequation2Gμν(g)=16πTμνgive,afterlinearization:hμα;αν+hνα;αμ−hμν;αα−(ηαβhαβ);μν−ημν[hβα;αβ−hαα;ββ]=16πTμν(1)wherepseudoriemannianmetricgμν=ημν+hμν,ημνistheflatMinkowskimetric,“;”denotesfour–dimensionalcovariantderivativewithrespecttothemetricημν.Itisusefultodefinethefollowingobject:Hμανβ:=hανημβ+ηανhμβ−hμνηαβ−ημνhαβ+hγγ(ημνηαβ−ηανημβ)whichfulfillsthefollowingidentities:Hμανβ=Hνβμα=H[μα][νβ]Hμ[ανβ]=0Theequation(1)mayberewrittenas:Hμανβ;αβ=16πTμν(2)LetΣ={x0=const.}beaspacelikehyperplane.Wecandefinetheenergy–momentumvectorpμ:16πpμ:=16πZΣTμ0=ZΣHμα0β,αβ=ZΣHμα0k,αk=Z∂ΣHμα0k,αd2Sk(3)andtheangularmomentumtensorjμν:16πjμν:=16πZΣJμν0=16πZΣ(xμTν0−xνTμ0)==Z∂Σ(xμHνα0k,α−xνHμα0k,α+Hμk0ν−Hνk0μ)d2Sk(4)whereJμνκ:=xμTνκ−xνTμκ;Jμνκ,κ=0(5)Here(xμ)isaglobal(pseudo-)cartesia