Relativistic Conservation Laws on Curved Backgroun

arXiv:gr-qc/9911025v324Jun2001RELATIVISTICCONSERVATIONLAWSONCURVEDBACKGROUNDSANDTHETHEORYOFCOSMOLOGICALPERTURBATIONS*AlexanderN.Petrov†SternbergAstronomicalInstitute,Universitetskiiprospect13,Moscow119899,RussiaandJosephKatz‡TheRacahInstituteofPhysics,91904Jerusalem,IsraelABSTRACTWefirstconsidertheLagrangianformulationofgeneralrelativityforperturbationswithrespecttoabackgroundspacetime.WeshowthatbycombiningNœther’smethodwithBelinfante’s“symmetrization”procedureweobtainconservedvectorsthatareinde-pendentofanydivergenceaddedtotheperturbedHilbertLagrangian.Wealsoshowthatthecorrespondingperturbedenergy-momentumtensorissymmetricalanddivergencelessbutonlyonbackgroundsthatare“Einsteinspaces”inthesenseofA.Z.Petrov.deSitteroranti-deSitterandEinstein“spacetimes”areEinsteinspacesbutingeneralFriedmann-Robertson-Walkerspacetimesarenot.Eachconservedvectorisadivergenceofananti-symmetrictensor,a“superpotential”.WefindsuperpotentialswhichareageneralizationofPapapetrou’ssuperpotentialandarerigorouslylinear,evenforlargeperturbations,intermsoftheinversemetricdensitycomponentsandtheirfirstorderderivatives.Thesuper-potentialsgivecorrectgloballyconservedquantitiesatspatialinfinity.TheyresembleAb-bottandDeser’ssuperpotential,butgivecorrectlytheBondi-Sachstotalfour-momentumatnullinfinity.*AsummaryofthisworkhasbeenpresentedattheConferenceon“FundamentalInteractions:FromSymmetriestoBlackHoles”,Brussels,March25-27,1999.IthasbeenpublishedintheProceedingsoftheConferenceunderthetitleConservationLawsforLargePerturbationsonCurvedBackgroundswithref.:FundamentalInteractions:FromSymmetriestoBlackHoles(eds.:J.M.J.M.Fr´ere,M.Henneaux,A.Servin&Ph.Spindel),p.p.147-157,Brussels:Universit´eLibredeBruxelles(1999),seealsogr-qc/9905088.†E-mail:[e-mail]‡E-mail:[e-mail]1NextwecalculateconservedvectorsandsuperpotentialsforperturbationsofaFried-mann-Robertson-Walkerbackgroundassociatedwithits15conformalKillingvectorsgiveninaconvenientform.TheintegralofeachconservedvectorinafinitevolumeVatagivenconformaltimeisequaltoasurfaceintegralontheboundaryofVofthesuperpotential.Forgivenboundaryconditionseachsuchintegralispartofafluxwhosetotalthroughaclosedhypersurfaceisequaltozero.ForgivenboundaryconditionsonV,theintegralcanbeconsideredasan“integralconstraint”ondatainthevolumeandthisdataalwaysincludestheenergy-momentumperturbations.Wegiveexplicitlythese15integralcon-straintsandaddsomesimpleapplicationsofinterestincosmology.OfparticularinterestareTraschenintegralconstraintsinwhichthevolumeintegralcontainsonlythematterenergy-momentumtensorperturbationsandnotthefieldperturbations.Weshowthattheseparticularintegralconstraintsareassociatewithtimedependentlinearcombina-tionsofconformalKillingvectors.SuchlinearcombinationsareneitherKillingvectorsnorconformalKillingvectors.Wealsofindthatifweaddthe“uniformHubbleconstanthy-persurface”gaugeconditionofBardeen,thereexists14suchintegralconstraints.Theexceptionisassociatedwithconformaltimetranslations(k=±1)orconformaltimeac-celerations(k=0).AsanexamplewefindtheconstantsofmotionofaspacetimethatisasymptoticallySchwarzschild-deSitter(k=0).21.Introduction(i)Conservationlawsandcosmology.Conservationlawsassociatedwith“symmetric”infinitesimaldisplacementsinFried-man-Robertson-Walkerspacetimeshavebeenusedinrelativisticcosmologyonseveraloccasions.Infinitesimaldisplacementsarecharacterizedbyvectorfieldsand,asweshallsee,thevectorsusedinsomeapplicationsarenotalwaysKillingvectorsnorevenconformalKillingvectors.AnexampleinwhichnoKillingnorconformalKillingvectorsareusedhasbeengivenbyTraschen[1]whointroduced“integralconstraints”intermsof“integralconstraintsvectors”.TraschenandEardley[2]analyzedmeasurableeffectsofthecosmicbackgroundradiationduetospatiallylocalizedperturbations.Byusing“integralconstraints”theypointedtoanimportantreductionoftheSachs-Wolfe[3]effectonthemeansquareangularfluctuationsatlargeanglesofthecosmicbackgroundtemperatureduetolocalinhomo-geneities.Traschen’sintegralconstraintsvectorshaveasomewhatintriguingorigin[4].TheequationsforintegralconstraintvectorshavebeenstudiedbyTod[5].HeshowedthattheseequationsareconditionsforaspacelikehypersurfacetobeembeddableinaspacetimewithconstantcurvatureofwhichthesolutionsareKillingvectors.InKatz,Bi˘c´akandLynden-Bell[6],apaperreferredtoasKBL97,integralconstraintsappearasconservationlawswithKillingvectorsinadeSitterbackground;moreonthisbelow.Localdifferentialconservationlaws,ratherthanglobalones,havebeenusedbyVeer-araghavanandStebbin[7].Theyfoundandusedaconserved“energy-momentum”pseudo-tensorinanefforttointegrateEinstein’sequationswithscalarperturbationsandtopo-logicaldefectsinthelimitoflongwavelengthsonaFriedmann-Robertson-Walkerspace-timewithflatspatialsections(t=const,k=0).Uzan,DeruelleandTurok[8]realizedthattheseconservationlawsmightbeassociatedwiththeconformalKillingvectoroftimetranslationsandtheyextendedVeeraraghavanandStebbin’smethodtoFriedmann-Robertson-Walkerperturbedspactimeswithnon-flatspatialsections(k=±1).Moreonthisinsection4.InLynden-Bell,KatzandBiˇc´ak’s[9]studyofMach’sprinciplefromth