Spherically symmetric gravitating shell as a repar

arXiv:gr-qc/9708008v218Aug1997SphericallysymmetricgravitatingshellasareparametrizationinvariantsystemP.H´aj´ıˇcekInstituteforTheoreticalPhysicsUniversityofBernSidlerstrasse5,CH-3012Bern,SwitzerlandAugust1997AbstractThesubjectofthispaperaresphericallysymmetricthinshellsmadeofbarotropicidealfluidandmovingundertheinfluenceoftheirowngravita-tionalfieldaswellasthatofacentralblackhole;thecosmologicalconstantisassumedtobezero.Thegeneralsuper-Hamiltonianderivedinapreviouspaperisrewrittenforthissphericallysymmetricspecialcase.Thedepen-denceoftheresultingactiononthegravitationalvariablesistrivializedbyatransformationduetoKuchaˇr.Theresultingvariationalprincipledependsonlyonshellvariables,isreparametrizationinvariant,andincludesbothfirst-andsecond-classconstraints.Severalequivalentformsoftheconstrainedsys-temarewrittendown.Exclusionofthesecond-classconstraintsleadstoasuper-HamiltonianwhichappearstooverlapwiththatbyAnsoldietal.inaquarterofthephasespace.AsKuchaˇr’variablesaresingularatthehorizonsofbothSchwarzschildspacetimesinsideandoutsidetheshell,thedynamicsisfirstwell-definedonlyinsideof16disjointsectors.The16sectorsare,how-ever,showntobecontainedinasingle,connectedsymplecticmanifoldandtheconstraintsareextendedtothismanifoldbycontinuity.Poissonbracketbetweennotwoindependentspacetimecoordinatesoftheshellvanishatanyintersectionoftwohorizons.1IntroductionSphericallysymmetricthinshellsarepopularmodelsusedextensivelyinthestudyofanumberofphenomena:propertiesofclassicalgravitationalcollapse[1],proper-tiesofclassicalblackholes[2],quantumgravitationalcollapse[3],thedynamicsofdomainwallsinearlyUniverse[4]and[5],thebackreactioninHawkingeffect[6],entropyonblackholes[7]orquantumtheoryofblackholes[8],tomentionjustfewexamples.AttemptstoderiveaHamiltonianformalismforsuchshellsareforexampleRefs.[9],[6],[10],[11],[5]and[12].TheHamiltonian(orsuper-Hamiltonian)iseitherguesseddirectlyfromequationsofmotion(Refs.[9]and[11]),oritisderivedfromavariationalprincipleguessedforthesphericallysymmetricsystemconsistingofdustshellsandgravity(Refs.[6]and[10]),oritisderivedfromtheLagrangianformalismbasedonthesumoftheEinstein-Hilbertactionandanactionforidealfluideitherafterreducingtheactionbysphericalsymmetry[5]orwithoutanyassumptionaboutsymmetry[12].InRef.[12],bothsuper-HamiltonianandthesymplecticstructurearederivedfromtheEinstein-Hilbert-ideal-fluidvariationalprinciple.Inthissense,thesym-plecticstructureisunique;itcontainsaboundarytermatthehypersurfaceoftheshellanditturnsoutthatthemomentumconjugatetothesurfaceareaoftheshellisthe(hyperbolic)anglebetweentheshellandthefoliationhypersurface(‘Kijowskimomentum’,[13];cf.alsoRef.[14]).Thismomentumwillplayanimportantroleinourcalculations.Inthepresentpaper,weshallderiveasuper-Hamiltonianandasymplecticformforthesphericallysymmetricidealfluidshells,startingfromthegeneralformulaofRef.[12].Forthesakeofsimplicity,weshallalsoassumethatthecosmologicalconstantandallfieldsdifferentfromgravityarezero.Ourleadingprincipleisthereparametrizationinvariance.Thus,theresultmustbeasuper-HamiltonianratherthanaHamiltonian.Oneproblemisthenhowthevariablesdescribingthegravita-tionalfieldaroundtheshellcanbemadetodisappearfromtheactionsothatthefinalformalismcontainstheshellvariablesonly.Asmostofthesegravitationalvari-ablesjustdescribeagauge,onepossiblemethodistochooseagaugeandtoreducethesystem,asforinstanceinRefs.[10]and[15];then,however,thereparametriza-tioninvarianceislost.WefindasuitabletoolinatransformationduetoKuchaˇr[16].Thistransformationtrivializesthegravitationalpartoftheequationsofmotiontosuchanextentthattheydonotcontainanymoreinformationaboutthemotionoftheshell.TheboundarytermsthatresultfromKuchaˇr’transformationcontributetotheshellpartofthesymplecticform.TheynotonlymodifyKijowski’smomen-tumbutprovideadditionaltermssothatthispartitselfbecomesnon-degenerate;thus,thesymplecticstructureoftheshellemerges.Severalequivalentformsofthe1variationalprinciplecanbewrittendown.Forexample,oneoftheresultingphasespacesislocallydescribedbyfourpairsofconjugatequantities,namely(E+,T+),(E−,T−),(P+,R+)and(P−,R−),whereT±andR±aretheSchwarzschildcoordinates,E±theSchwarzschildmasses,andP±themodifiedKijowski’smomenta;thesign‘+’referstotheoutsideand‘−’totheinsideSchwarzchildspacetimes.Therearethenthreeconstraints:1)thesuper-HamiltonianconstraintCs=0is(roughly)thetime-timecomponentofIsrael’smatchingconditionattheshellanditisaprimaryconstraint,2)thecontinuityconditionR+−R−=0isanotherprimaryand3)thePoissonbracketχ:={Cs,R+−R−}failstovanish,soχ=0isasecondaryconstraint.ThetwoconstraintfunctionsχandR+−R−formasecond-classpair.Thesecondclassconstraintscanbesolvedfor[R]and¯P1,andthesolutioncanbesubstitutedbackintotheaction;inthismanner,apartiallyreducedsystemwiththreepairsofconjugatevariables(E+,T+),(E−,T−)and([P],¯R)andjustoneconstraintCrs=0isobtained.Infoursectorsofthephasespace,Crshasasimilarformasthesuper-HamiltonianofRef.[5],whichhasbeenderivedinacompletelydifferentway.Theoriginofthesecond-classconstraintsisintheadditionalconditionsbywhichthegeneralEinstein-Hilbert-ideal-fl