Self-gravitating stationary spherically symmetric

arXiv:gr-qc/0609074v425Sep2006Self-gravitatingstationarysphericallysymmetricsystemsinrelativisticgalacticdynamicsMikaelFj¨allborg1∗,J.MarkHeinzle2†,andClaesUggla3‡1DepartmentofMathematics,UniversityofKarlstadS-65188Karlstad,SwedenandDepartmentofMathematics,ChalmersUniversityofTecnologyS-41296G¨oteborg,Sweden2InstituteforTheoreticalPhysics,UniversityofViennaBoltzmanngasse5,A-1090Vienna,Austria3DepartmentofPhysics,UniversityofKarlstadS-65188Karlstad,SwedenAbstractWestudyequilibriumstatesinrelativisticgalacticdynamicswhicharede-scribedbystationarysolutionsoftheEinstein-Vlasovsystemforcollisionlessmat-ter.Werecasttheequationsintoaregularthree-dimensionalsystemofautonomousfirstorderordinarydifferentialequationsonaboundedstatespace.Basedonadynamicalsystemsanalysiswederivenewtheoremsthatguaranteethatthesteadystatesolutionshavefiniteradiiandmasses.PACSnumber(s):02.90.+p,04.40.-b,98.20.-d.∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]‡Electronicaddress:[e-mail]11INTRODUCTION21IntroductionTheEinstein-Vlasovsystemdescribesacollisionlessgasofparticlesthatonlyinteractviathesmoothedrelativisticgravitationalfieldtheygeneratecollectivelythroughtheiraveragedstress-energy.Inthisarticlewestudytheequilibriumstatesofthespheri-callysymmetricEinstein-Vlasovequations.Thistopicisofconsiderablephysicalandmathematicalinterest:Fackerell,Ipser,andThornehaveusedthesemodelsasmodelsforrelativisticstarclusters[1,2,3,4,5,6](seealsothesereferencesforreferencestoearlierliterature);Martin-GarciaandGundlachstudiedthesemodelsinthecontextofcriticalcollapse[7];ReinandRendallapproachedtheareafromamoremathematicalpointofview[8]—thispaperwillbetakenasthestartingpointforthepresentwork.Itisstraightforwardtoworkwithparticleswithdifferentmasses,seee.g.[5,9],butforsimplicitywewillrestrictourselvestoparticleswithasinglemass,m0≥0,whichyieldsthesamemathematicalproblem.Acollisionlessgasischaracterizedbyanon-negativephasespacedistributionf(xμ,pμ)definedonthefuturemassshellPMofthetangentspaceTMofthespacetime,i.e.,xμarespacetimecoordinatesandpμarelocalcoordinatesofthefour-momentumwithrespecttothecoordinatebasisxμsuchthatpμpμ=−m20,p00,wherepμ=gμνpν,andwheregμνisthespacetimemetricwithsignature(−,+,+,+);throughoutthisarticlewesetc=1=G,wherecisthespeedoflightandGisthegravitationalconstant.Forstaticsphericallysymmetricmodels,themetriccanbewrittenasds2=−e2μ(r)dt2+e2λ(r)dr2+r2(dθ2+sin2θdφ2),(1)andtheVlasovequationasv√1+v2·∇xf−p1+v2μ′xr·∇vf=0,(2)wherevisdefinedbelow.Thenon-negativeenergy-densityandradial-andtangentialpressuresaregivenbyρ=Zf(r,v)p1+v2dv,(3a)prad=Zf(r,v)x·vr2dv√1+v2,(3b)pT=12Zf(r,v)|x×v|2r2dv√1+v2.(3c)TheVlasovequationadmitsastationarysolutiongivenbyf(E,L2),whereEandLareconservedquantities,interpretedasparticleenergyandangularmomentum,perunitmass;theyaredefinedbyE:=eμ(r)p1+v2,L2=r2v2T,(4)wherewehavefollowed[8]anddefinedv2asv2:=hμνuμuν,whereuμ=pμ/m0andhμν:=nμnν+gμν,wherenμ=gμνnνandnμ=e−μ(∂t)μ;v2Tisdefinedasv2T=v2−v2r,1INTRODUCTION3wherevr=eλpr/m0andwherepristheradialcoordinatecomponentofpμ.Notethattheaboveobjectsareassociatedwiththespatialprojectionofthefour-velocity:vμ=hμνuνandthattheythusdifferfromthephysicalthree-velocity,Vμ,whichisdefinedbyuμ=Γ(nμ+Vμ)andnμVμ=0,whereΓ=1/√1−V2;hence,e.g.,v2=V2/(1−V2)andE=Γeμ(r).Usingthatf=f(E,L2)yields:ρ=2πre−3μ∞ZeμL2maxZ0f(E,L2)E2dL2dEpr2(e−2μE2−1)−L2,(5a)prad=2πe−μr3Z∞eμZL2max0f(E,L2)pr2(e−2μE2−1)−L2dL2dE,(5b)pT=πe−μr3Z∞eμZL2max0f(E,L2)L2dL2dEpr2(e−2μE2−1)−L2,(5c)whereL2max=r2(e−2μE2−1).LetR∈(0,∞]denotetheradiusofsupportofthesystem,i.e.,ρ(r)=0whenr≥R.WeareinterestedingravitationallyboundsystemsinequilibriumthateitherhavefiniteradiiRorpossessafunctionρthatdecreasessufficientlyrapidlytowardsinfinitysothatinbothcaseslimr→Rμ(r)=μR∞.Becauseoftheequilibriumassumption,itfollowsthatvroftheindividualparticleshastosatisfylimr→Rvr=0.Wethereforerequire,sinceE=eμ(r)p1+v2r+L2/r2,thatf(E,L2)=0whenE≥E0=eμR,i.e.,weassumethatthereexistsacut-offenergythatpreventsparticlesfromescapingthegravitationalfieldtheycreatecollectively.ThebasicremainingequationsoftheEinstein-Vlasovsystemaregivenbye−2λ2dλdξ−1+1=8πr2ρ,(6a)e−2λ2dμdξ−1−1=8πr2prad,(6b)whereξ:=logr.Wedefinethemassfunction,m(r),accordingtoe−2λ:=1−2m(r)r,(7)which,togetherwith(6a),yieldsdmdξ=4πr3ρ.(8)Consequently,Eq.(6b)takestheformdμdξ=e2λmr+4πr2prad.(9)SincefisassumedtobepositivewhenEE0,itfollowsthatρ0whenrR,andthatμisamonotonicallyincreasingfunctioninr(orξ=logr);italsofollowsthat0m(r)≤M:=m(R)when0r≤R,whereMdenotestheADMmass.1INTRODUCTION4Inthispaperwewillconsiderdistributionfunctionsofthetypef=φ(E)L2l,(10)whereφ(E)isanon-negativefunctionsuchthatφ(E)=0whenE≥E0;wealsoassumethat−1l∞.Letusmakethebasicdefinitionsη:=ln(E0)−μ=μR−μ,E:=E/E0.(11)Rewriting(9)intermsofηresultsindηdξ=−1−2mr−1mr+4πr2prad.(12)Notethatηiscloselyrelatedtotheredshift,z,measuredatthesurfaceR:z=eη−1.Fordistributionsofthetype(10)thedensityandtheradialpressuretakethefollowingform:prad=ar2le(2l+4)ηgl+3/2(η),(13a)ρ=(l+32)ar2le(2l+4)η[2gl+3/2(η)+e−2ηgl+1/2(η)],(13b)wherethecons