Relativistic conservation laws and integral constr

arXiv:gr-qc/0504041v19Apr2005RelativisticconservationlawsandintegralconstraintsforlargecosmologicalperturbationsJosephKatz∗,Jiˇr´ıBiˇc´ak†,andDonaldLynden-Bell‡InstituteofAstronomy,MadingleyRoad,CambridgeCB30HA,UnitedKingdomPublishedin:Phys.Rev.D55,5957(1997)(Received19July1996)February5,2008AbstractForeverymappingofaperturbedspacetimeontoabackgroundandwithanyvectorfieldξweconstructaconservedcovariantvectordensityI(ξ),whichisthedivergenceofacovariantantisymmetrictensordensity,a“superpotential”.I(ξ)islinearintheenergy-momentumtensorperturbationsofmatter,whichmaybelarge;I(ξ)doesnotcontainthesecondorderderivativesoftheperturbedmetric.Thesuperpotentialisidenticallyzerowhenperturbationsareabsent.ByintegratingconservedvectorsoverapartΣofahypersurfaceSofthebackground,whichspansatwo-surface∂Σ,weobtainintegralrelationsbetween,ontheonehand,initialdataoftheperturbedmetriccomponentsandtheenergy-momentumperturbationsonΣand,ontheotherhand,theboundaryvalueson∂Σ.Weshowthatthereareasmanysuchintegralrelationsastherearedifferentmappings,ξ’s,Σ’sand∂Σ’s.Forgivenboundaryvalueson∂Σ,theintegralrelationsmaybeinterpretedasintegralconstraintsonlocalinitialdataincludingtheenergy-momentumperturbations.ConservationlawsexpressedintermsofKillingfieldsξofthebackgroundbecome“physical”conservationlaws.Incosmology,toeachmappingofthetimeaxisofaRobertson-WalkerspaceonadeSitterspacewiththesamespatialtopologytherecorrespondtenconservationlaws.Theconformalmappingleadstoastraightforwardgeneralizationofconservationlawsinflatspacetimes.Othermappingsarealsoconsidered.Traschen’s“integralconstraints”forlinearizedspatiallylocalizedperturbationsoftheenergy-momentumtensorareexamplesofconservationlawswithpeculiarξvectorswhoseequationsarerederivedhere.InRobertson-Walkerspacetimes,the“integralconstraintvectors”aretheKillingvectorsofadeSitterbackgroundforaspecialmapping.[S0556-2821(97)00310-X]PACSnumber(s):04.20.Cv,98.80.Hw∗Permanentaddress:TheRacahInstituteofPhysics,91904Jerusalem,Israel.email:[e-mail]†Permanentaddress:InstituteofTheoreticalPhysics,FacultyofMathematicsandPhysics,CharlesUni-versity,VHoleˇsoviˇck´ach2,18000Prague8,CzechRepublic.email:[e-mail].mff.cuni.cz‡email:[e-mail]dcosmologyBackgroundspacetimesarecommonlyusedinperturbationtheoriesingeneralrelativity[1]andplayanessentialroleincosmology[2].One“puzzle”[3]inthetheoryofcosmologicalperturbationsisTraschen’s“integralconstraints”forspatiallylocalizedperturbations[4,5].TheseGauss-typerestrictionsontheenergy-momentumofmatterperturbationshavesignif-icanteffects[6]:TheypointtoanimportantreductionoftheSachs-Wolfe[7]effectonthemeansquareangularfluctuationsatlargeanglesofthecosmicbackgroundtemperatureduetolocalinhomogeneitiesintheuniverseforspatiallyisolatedperturbations.Traschen’srelationsremindusofBergmann’sstrongconservationlaws[8]appliedtoperturbationsofisolatedsystems.Suchconservationlaws,whichwereexploredindetailbyBergmannandSchiller[9],are,infact,identities.Theidentities,whichinvolveanarbitraryvectorξ,haveplayedabasicroleinthederivationofweakorNœtherconservedcurrentsingeneralrelativity[10]andarestillinuse[11].Wefounditthusinterestingtostudyconservationlawsonbackgroundspacetimes[12]inthecontextofcosmologicalperturbations.ConservationlawsareobtainedfromLagrangiansthatarescalardensitieswithnothigherthanfirstorderderivativesofthefields.Therearenosuchscalardensitiesforthemetricandthereforeconservationlawsingeneralrelativityarecoordinatedependent.Thecoordinatedependencecanbe“brushedundertherug”bymappingthespacetimeonaflatbackground[13].Thismethodoffers,forexample,theadvantageofmakingtheBondimass[14]calculablefromEinstein’spseudotensorinBondicoordinates[15]ratherthaninMinkowskicoordinates[16].Butbackgroundsaremorethanausefultoolinrelativisticcosmology;theyareinevitableinlinearandnonlinearperturbationtheories.Here,wederivestrongconservationlawswithrespecttocurvedbackgroundsalongthelineindicatedbyBergmann.WedefineaLagrangiandensityˆLGforthegravitationalfield,quadraticinthefirstordercovariantderivativesoftheperturbedmetric(thecaretmeans“density”,i.e.,multiplicationby√−g).ˆLGisnormalizedsothatˆLG=0whentherearenoperturbations.Perturbationsdonothavetobesmall.TheconservationlawsderivedfromˆLGareidenticallyconservedvectordensitiesˆIμ(ξ),thedivergencesofcovariantsuperpotentialdensitiesˆJμν:ˆIμ=∂νˆJμν,ˆJμν=−ˆJνμ.(1.1)TheˆIμ’sareidenticallyconservedindependentlyofwhetherEinstein’sequationsaresatisfiedornot.However,weconsideronlymetricsthatsatisfyEinstein’sequations.ˆIμ’sarelinearintheperturbedenergy-momentumtensor,andbothˆIμandˆJμνcontaintheperturbedmetricanditsfirst-ordercovariantderivatives(nosecond-orderderivatives);botharezerowhentherearenoperturbations.ItfollowsfromEq.(1.1)thatifΣisanypieceofahypersurfaceSwhichspansatwo-surface∂Σ,ZΣˆIμdΣμ=Z∂ΣˆIμνdΣμν.(1.2)TheseexactnonlinearintegralidentitiesrepresentglobalconservationlawsiftheintegrationisoverthewholehypersurfaceS.IfΣisonlyapieceofthetotal,onemay,inthemannerofPenrose[17],speakofquasi-localconservationlaws.No