On static shells and the Buchdahl inequality for t

arXiv:gr-qc/0605151v131May2006OnstaticshellsandtheBuchdahlinequalityforthesphericallysymmetricEinstein-VlasovsystemH˚akanAndr´eassonDepartmentofMathematics,Chalmers,S-41296G¨oteborg,Swedenemail:[e-mail]7,2008AbstractInapreviouswork[1]mattermodelssuchthattheenergydensityρ≥0,andtheradial-andtangentialpressuresp≥0andq,satisfyp+q≤Ωρ,Ω≥1,wereconsideredinthecontextofBuchdahl’sin-equality.ItwasprovedthatstaticshellsolutionsofthesphericallysymmetricEinsteinequationsobeyaBuchdahltypeinequalitywhen-everthesupportoftheshell,[R0,R1],R00,satisfiesR1/R01/4.Moreover,givenasequenceofsolutionssuchthatR1/R0→1,thenthelimitsupremumof2M/R1wasshowntobeboundedby((2Ω+1)2−1)/(2Ω+1)2.InthispaperweshowthatthehypothesisthatR1/R0→1,canberealizedforVlasovmatter,byconstructingasequenceofstaticshellsofthesphericallysymmetricEinstein-Vlasovsystemwiththisproperty.Wealsoprovethatforthissequencenotonlythelimitsupremumof2M/R1isbounded,butthatthelimitis((2Ω+1)2−1)/(2Ω+1)2=8/9,sinceΩ=1forVlasovmatter.Thus,staticshellsofVlasovmattercanhave2M/R1arbitrarycloseto8/9,whichisinterestinginviewof[3],wherenumericalevidenceispre-sentedthat8/9isanupperboundof2M/R1ofanystaticsolutionofthesphericallysymmetricEinstein-Vlasovsystem.1IntroductionUndertheassumptionofisotropicpressureandnon-increasingenergyden-sityoutwards,Buchdahl[6]hasprovedthatasphericallysymmetricfluidballsatisfies2MR1≤89,1whereMandR1isthetotalADMmassandtheouterboundaryofthefluidballrespectively.In[1]Buchdahl’sinequalitywasinvestigatedforsphericallysymmetricstaticshellswithsupportin[R0,R1],R00,forwhichneitherofBuchdahl’shypotheseshold.Werefertotheintroductionin[1]forareviewonpreviousresultsonBuchdahltypeinequalities.Themattermodelsconsideredin[1]wereassumedtohavenon-negativeenergydensityρandpressurep,andtosatisfythefollowinginequalityp+q≤Ωρ,Ω≥1,(1)whereqisthetangentialpressure.Itwasshownthatgivenǫ1/4,thereisaκ0suchthatanystaticsolutionofthesphericallysymmetricEinsteinequationssatisfies2MR1≤1−κ.Furthermore,givenasequenceofstaticsolutions,indexedbyj,withsupportin[Rj0,Rj1]whereRj1/Rj0→1asj→∞,itwasprovedthatlimsupj→∞2MjRj1≤(2Ω+1)2−1(2Ω+1)2,(2)whereMjisthecorrespondingADMmassofthesolutionwithindexj.Thelatterresultismotivatedbynumericalsimulations[3]ofthesphericallysym-metricEinstein-Vlasovsystem.ForVlasovmatterΩ=1andtheinequality(1)isstrict,andtheboundin(2)becomes8/9asinBuchdahl’soriginalwork.WewillseethatforVlasovmatter,asequencecanbeconstructedsuchthatRj1/Rj0→1,andsuchthatthevalue8/9of2M/R1isattainedinthelimit.Itshouldbeemphasizedthatthestaticsolutionwhichattainsthevalue8/9inBuchdahl’scaseisanisotropicsolutionwithconstanten-ergydensity,whereasthelimitstateofthesequencethatweconstructisaninfinitelythinshellwhichhaspj/qj→0asj→∞,whichmeansthatitishighlynon-isotropic.Beforedescribinginmoredetailthenumericalresultsin[3],whichpro-videsthemainmotivationforthiswork,letusfirstintroducethesphericallysymmetricEinstein-Vlasovsystem.Themetricofastaticsphericallysymmetricspacetimetakesthefollow-ingforminSchwarzschildcoordinatesds2=−e2μ(r)dt2+e2λ(r)dr2+r2(dθ2+sin2θdϕ2),2wherer≥0,θ∈[0,π],ϕ∈[0,2π].Asymptoticflatnessisexpressedbytheboundaryconditionslimr→∞λ(r)=limr→∞μ(r)=0,andaregularcentrerequiresλ(0)=0.Vlasovmatterisdescribedwithintheframeworkofkinetictheory.Thefundamentalobjectisthedistributionfunctionfwhichisdefinedonphase-space,andmodelsacollectionofparticles.Theparticlesareassumedtointeractonlyviathegravitationalfieldcreatedbytheparticlesthemselvesandnotviadirectcollisionsbetweenthem.ForanintroductiontokinetictheoryingeneralrelativityandtheEinstein-Vlasovsysteminparticularwereferto[2]and[15].ThestaticEinstein-VlasovsystemisgivenbytheEinsteinequationse−2λ(2rλr−1)+1=8πr2ρ,(3)e−2λ(2rμr+1)−1=8πr2p,(4)μrr+(μr−λr)(μr+1r)=4πqe2λ,(5)togetherwiththe(static)Vlasovequationwε∂rf−(μrε−Lr3ε)∂wf=0,(6)whereε=ε(r,w,L)=p1+w2+L/r2.Thematterquantitiesaredefinedbyρ(r)=πr2Z∞−∞Z∞0εf(r,w,L)dLdw,p(r)=πr2Z∞−∞Z∞0w2εf(r,w,F)dLdw,q(r)=πr4Z∞−∞Z∞0Lεf(r,w,L)dLdw.ThevariableswandLcanbethoughtofasthemomentumintheradialdirectionandthesquareoftheangularmomentumrespectively.LetE=eμε,3theansatzf(r,w,L)=Φ(E,L),(7)thensatisfies(6)andconstitutesanefficientwaytoconstructstaticsolutionswithfiniteADMmassandfiniteextension,cf.[14],[13].Itshouldbepointedoutthatsphericallysymmetricstaticsolutionswhichdonothavethisformgloballyexist,cf.[16],whichcontraststheNewtoniancasewhereallsphericallysymmetricstaticsolutionshavetheform(7),cf.[4].Asamatteroffact,thesolutionsweconstructinTheorem1belowaregoodcandidatesforsolutionswhicharenotgloballygivenby(7).HerethefollowingformofΦwillbeusedΦ(E,L)=(E0−E)k+(L−L0)l+,(8)wherel≥1/2,k≥0,L00,E00,andx+:=max{x,0}.IntheNewtoniancasewithl=L0=0,thisansatzleadstosteadystateswithapolytropicequationofstate.NotethatwhenL00therewillbenomatterintheregionrsL0(E0e−μ(0))2−1,(9)sincetherenecessarilyEE0andfvanishes.Theexistenceofsolutionssupportedin[R0,R1],R00,withfiniteADMmasshasbeengivenin[13],andweshallcallsuchconfigurationsstaticshellsofVlasovmatter.ItwillbeassumedthatΦisalwaysasabove,whichinparticularmeansthatL00,sothato