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# Black Hole Evaporation: Validity of Quasi-Static Approximation

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 9 CHAPTER 1INTRODUCTION .................................. 10 1.1ClassicalBlackHoles .............................. 10 1.1.1GravitationalCollapse ......................... 11 1.1.2Theorems ................................ 14 1.1.3BlackHoleDynamics .......................... 15 1.2HawkingRadiation ............................... 18 1.2.1ParticleCreation ............................ 18 1.2.2BackScattering ............................. 23 1.2.3QuantumStressTensor ......................... 25 1.3BackReaction .................................. 28 1.3.1Quasi-static(Qs)Approximation .................... 30 1.3.2MotivationForTheProblem:IsTheQsApproximationValid? ... 31 1.3.3ViolationOfWEC ........................... 32 1.3.4MethodOfApproach .......................... 34 2MODEL ........................................ 37 2.1ClassicalMetric ................................ 37 2.1.1SchwarzschildExterior ......................... 40 2.1.1.1JumpconditionsontheNullshell ............. 40 2.1.1.2Gaugechoice ......................... 43 2.1.2GeneralExterior ............................. 45 2.2QuantumField ................................. 46 2.2.1hTiInTwoDimensions ........................ 48 2.2.24D-Model ................................ 50 2.2.3StressTensorInSchwarzschildExterior. ................ 52 3QUASI-STATICAPPROXIMATION ........................ 54 3.1ConstructingTheMetric ............................ 56 3.2ComputingTheEnergyFlux .......................... 59 3.3Analysis ..................................... 62 3.4TheCompleteQuasi-StaticGeometry ..................... 66 5

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............................ 69 4.1Algorithm .................................... 71 4.1.1InitialData ............................... 71 4.1.2ConstraintEquations .......................... 71 4.1.3Algorithm ................................ 73 4.1.4CriticalRadius ............................. 75 4.1.5AdaptiveMeshing ............................ 76 4.2TestingTheCode ................................ 77 4.2.1AccumulatedError ........................... 77 4.2.2ConstraintViolation .......................... 78 4.2.3CorrespondenceCheckWithSchwarzschildGeometry ........ 79 4.2.4ValidityOfGaugeChoiceAtTheSurfaceS2 80 4.3RetrievingOutput ............................... 80 4.3.1TraceAConstantrSurface ...................... 81 4.3.2ApparentHorizon ........................... 81 4.3.3NegativeEnergyDensity ........................ 81 4.3.4EnergyFlux ............................... 82 5RESULTSANDDISCUSSION ........................... 87 5.1PositionOfApparentHorizon ......................... 88 5.2ViolationOfWEC ............................... 90 5.3ComparisonOfMassLoss ........................... 91 5.4ApparentHorizonmass ............................. 97 6SUMMARYANDCONCLUSION .......................... 102 6.1FutureDirections ................................ 105 APPENDIX AINTERPOLATION .................................. 106 BDERIVATIVETERMS ................................ 114 B.1NonlinearEquations .............................. 114 B.2LinearizedEquations .............................. 116 B.3SchwarzschildDerivatives ........................... 119 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 126 6

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Table page 5-1PositionoftheBulge ................................. 90 5-2Positionofmarkers .................................. 92 5-3P()andS() .................................... 99 7

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Figure page 1-1Gravitationalcollapse ................................ 12 1-2Penrosediagramforacollapsegeometry ...................... 13 1-3Geometricopticsapproximation ........................... 23 1-4Evaporationgeometry ................................ 29 1-5Quasistaticgeometry1 ................................ 32 1-6CollapseandEvaporationwithoutsingularity ................... 34 3-1Quasi-staticgeometry2 ............................... 58 3-2Quasi-staticgeometry3 ............................... 67 4-1Algorithmforevolution ............................... 74 4-2AdaptiveMesh .................................... 77 5-1ConstantrcurvesofSchwarzschild ......................... 88 5-2ConstantrcurvesofevaporatingBH ........................ 89 5-3AHandTvv ..................................... 91 5-4Massloss=5 ..................................... 93 5-5Massloss=4 ..................................... 94 5-6Massloss=3 ..................................... 95 5-7Massloss=2 ..................................... 96 5-8Massloss=1 ..................................... 97 5-9Massloss=0.5 .................................... 98 5-10P()plot ....................................... 100 5-11S()plot ....................................... 100 5-12Bondimass ...................................... 101 8

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c2T: TherearesomeveryinterestingsolutionstoEinstein'seldequationsknownasblackholes.Thereisavarietyofsuchsolutions.Butallthesesolutionscarryacommonproperty,namely,asectionofthespacetimemanifold(sayBH)iscausallydisconnectedfromtherestofthemanifold(sayExt),thatis,noinformationfromthesectionBHcanleakouttotheothersectionExt.Inotherwords,notimelikeornullcurvecanoriginateinsideBHandenterExt,butthereexisttimelikeandnullcurvesthatoriginateintheExtandenterBH.TheboundarybetweenExtandBHiscommonlyknownastheEventhorizon.Anothercommonfeatureoftheseblackholesolutionsistheexistenceofsingularities.Apointinthespacetimeissupposedtobesingularifthereexistsageodesicwhichcannotbecontinuedthroughthatpoint.Inmostsituations,thishappensatapointwherethecurvaturetensordiverges. 10

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1.1.1 ,wewillbrieygooverthekeypointsinvolvedinagravitationalcollapse.Inthisrespect,wewillalsointroducethenotionofPenrosediagrams.RepresentingthecausalstructureofthespacetimegeometryusingPenrosediagramsisveryusefulininterpretingmanyofthephysicalpropertiesofthegeometry,andtheywillbefrequentlyusedinthiswork. Tobetterunderstandthesingularities,blackholesandthecollapseprocessthatgivesrisetoblackholes,thereisanumberoftheoremsweneedtobefamiliarwith.WewillspendSection 1.1.2 andSection 1.1.3 doingthat.InSection 1.1.2 ,wewillbrieydiscusstherelevanttheoremswhichcategorizedthevariousblackholesolutions,andinSection 1.1.3 ,wewillgoovertheclassicallawsofblackholemechanicswhichresemblethelawsofthermodynamics. ThereisasimpletheorembyBirkhowhichisveryusefulinthecontextofcollapse.Itstatesthatthemetriconthemanifoldisstaticanduniquelydeterminedifweassumethatthespacetimeissphericallysymmetricandthatthereisnomatter(T=0).Inparticular,whenasphericalstarcollapsessymmetrically,itsexteriorgeometryhastobestatic(independentoftime)andthemetricoutsidethesurfaceofthestarisgivenbytheSchwarschildsolution.ds2=(12M r)dt2+1 (12M r)dr2+r2d2

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Gravitationalcollapseofsphericalstarwhichresultsinformationofablackhole. AthoroughdescriptionoftheSchwarzschildgeometryandthevariouscoordinatesystemsgenerallyusedtodescribeitcanbefoundinanystandardgeneralrelativitytextbook(foreg.[ 23 ],[ 34 ]).Henceletusnotdiscussthesedetails.Nevertheless,wewouldliketoexplicitlymentiononeoftheimportantpropertiesofSchwarzschildgeometry.Ther=constantsurfacesaretimelikesurfacesoutsidetheeventhorizon(r=2M)andtheyarespacelikesurfacesinsidetheeventhorizon.Thisimpliesthatanytimelikecurveinsidethehorizonshouldreachthesingularityatr=0.Henceanyr=constantsurfaceinsidethehorizonisaTrappedsurface,basicallybecauseevenlightcannotmoveoutinthedirectionofincreasingr.Inageneralgeometry,aTrappedsurfaceisdenedasaclosedspacelike2-surfaceonwhichallnullgeodesiccongruenceshaveapositiveconvergence. Comingbacktothecollapseofasphericalstar,Birkho'stheoremtellsusthatthegeometryoutsidethesurfaceofthestaristheSchwarzschildgeometry.Thismeans,oncethesurfaceofthestarshrinkstoasizelessthanitsgravitationalradius(r=2M),aneventhorizonformsandoutgoinglightraysfromthesurfaceofthestarwillmoveinthedirectionofdecreasingr.InFig. 1-1 ,wetracetheoutgoinglightraysfromthepointr=0 12

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Penrosediagramrepresentingagravitationalcollapsewhichresultsinformationofablackhole. atvariousstagesofcollapse.Thex-axisisthercoordinateandthey-axiscorrespondstosometimecoordinatesuchthateachofthet=constantsurfacesarespacelikesurfaces.Wecanclearlyseethattheraysthatcomeoutofr=0afteraparticulartime,nevergetouttoinnity.Theyreachamaximumvalueofr,thentheyturnbackandmoveinthedirectionofdecreasingr.ThenullraywhichstaysataconstantvalueofrafterleavingthesurfaceofthestaristheEventhorizon.Theformationofasingularityisshowninthegurebythickeningther=0lineafteraparticulartime. Letusnowlookatthebehaviorofr=constantsurfacesinthecollapsegeometry.Letusrstdrawtheconformaldiagram(Penrosediagram)forthesphericallysymmetriccollapsegeometry.LookatFig. 1-2 .Ateverypoint,thenullconesare45olines.ThesurfaceofthestarR(t)isshowninred.Thevariousr=constantcurvesareshowninthediagram.Theboundaryofthediagramisthefutureandpastnullinnities,I+andIrespectively.Thepointwherer=0linebecomeshorizontalisthebeginningofthesingularity.Theeventhorizon,denotedbyH,coincideswiththecurver=2Mintheexteriorofthestarsurface.Theradiusofthestar'ssurfacedecreasesfrominnityandcrossestheeventhorizon.Observethatther=constantcurvesoutsidethesurfaceof 13

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28 ].Itstatesthatasingularitywillnecessarilyformifthefollowingthreeconditionsaresatised,(i)WeakEnergyconditionissatised,(ii)Thereexistsanon-compactCauchysurfaceinthespace-timemanifold,(iii)thereexistsatrappedsurfaceinthemanifold. Thisisaverypowerfultheoreminthecontextofcollapse,foritstatesthatthemomentthersttrappedsurfaceformswithinacollapsingstar,itsfateisdeterminedtobesingular. 29 ].Theyarebrieysummarizedbelow. 1. Thesurfacegravityofastationaryblackhole 2;;.ForSchwarzschildblackhole,=1=4Mg.Theanalogousstatementinthermodynamicsisthatasysteminequilibriumhasuniformtemperature.Thus,wemakeaconnectionbetweensurfacegravityandtemperature. 15

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LetusnowtakeasmalldigressiontoemphasizethatthereexistatleasttwowellknownprocessesbywhichenergycanbeextractedoutofaKerrblackholeuntilitloosesallitsangularmomentumandbecomesaSchwarzschildblackhole.Fordetails,referto[ 23 ]. 17

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@tuj(t;~x)=i!juj(t;~x);!j>0: 2. Wethenexpandtheeldoperator(x)withrespecttothemodefunctionsintermsofcreationandannihilationoperatorsas(x)=i(aiui(x)+ayiui(x)): Now,thecanonicalquantizationprocedureleadstothefollowingcommutationrelationsbetweenthecreationandannihilationoperators.[ai;ayj]=ijand[ai;aj]=[ayi;ayj]=0.Notethat,thequantizationprocedureverymuchdependsuponourchoiceofthemodefunctions. 4. LetusnowconstructtheFockspace,whichisdenedasthespaceofallpossiblestatesofthequantumeld. Thevacuumstatejvaciorthezeroparticlestateistheonewhichisannihilatedbyalltheannihilationoperators,thatis,ajvaci=0: 2 19

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Letusnowchooseanothersetofmodefunctions,sayfvj;vjgandcallthecreationandannihilationoperatorswithrespecttothesemodefunctionsasbjandbyj,sothattheeldexpansionis,(x)=j(bjvj(x)+byjvj(x)): 11 ],[ 4 ].Itturnsoutthat,hvacjjbyjbjjvaci=ijjijj2; Givenabackgroundgeometryandtwosetsofmodefunctions,wenowknowtheproceduretocalculatetheparticlecreation.Attherstglance,theconceptofparticlecreationseemstobeapurelymathematicalartifact,becauseitultimatelyjustdependsonthechoiceofthemodefunctions,andthereisnonaturalchoiceofmodefunctionsexceptinatspacetime.Butthatisnottrue,therearesomesituationswheretheconceptofparticlecreationdoeshaveaphysicalmeaning.Tounderstandthis,weshouldrstnoteanimportantpropertyofthemodefunctions:ifthemodefunctionsarexedonanyoneCauchysurface,theeldequationsdeterminethemeverywhereinspacetime.Letusnowaskhowweshouldchoosethemodefunctionsonasurfacesothatcorrespondingstates 20

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Letusnowconsideraspacetimewhichisalmostatatearlytimes(t=)andlatetimes(t=1).ConsidertwospacelikeCauchysurfacesatthesetwotimesand1respectively.Thenaturalchoiceofmodefunctionsonthesetwosurfacesmaynotmatch,becausethespacetimeisnotateverywhere.Thatis,whenonesetofmodefunctionsispropagatedthroughtimefromoneCauchysurfacetotheother,theyneednotmatchtheothersetofmodefunctions.ThismightleadtoanonzeroBugolubovcoecientij.Hence,thevacuumstatewithrespecttoanobserveratearlytimesonthesurfacewouldbeseenasanonzeroparticlestatebyanobserveratlatetimesonthesurface1. Avarietyofexamplesofparticleproductionhasbeenworkedout.Particlecreationeectsondierentcosmologicalbackgroundshavebeencalculated[ 4 ].Unruhcalculatedtheparticleproductioninatspacetimewithrespecttoanacceleratedobserver[ 32 ].Itturnsoutthattheacceleratedobserverwouldseeathermalbathofparticles.Hawkingappliedtheconceptofparticlecreationtothegravitationalcollapsegeometry,andhefoundthatanobserveratlatetimeswouldobserveparticlesinaccordancetoathermalspectrum[ 16 ],[ 17 ].LetusnowbrieydiscussHawking'sresult. InarealcollapsegeometryasinFig. 1-2 ,wehavetwoasymptoticallyatsurfacesfIgandfI+g,andhencetwosetsofinertialatspaceobservers.fIgisaCauchysurfacebyitself,andfI+;HgtogetherformaCauchysurface.TheobserveratIwillchoosehisnaturalsetofmodefunctionsf!lmsuchthatf!lmei!v

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OuraimistondthenumberofparticlesobservedatI+asafunctionoftheretardedtimeu.Sincethemodefunctionsp!donothaveacompactsupportatI+,whenwecalculatetheparticlecreation,theparticleswillbedelocalizedeverywhereonI+.TocalculatethenumberofparticlesobservedatI+asafunctionoftime,themodefunctionsp!shouldnotbeused.Wehavetoconstructacompletesetofwavepacketsoutofthesemodefunctions.LetususePu0!0todenoteawavepacketofcharacteristicfrequency!0andpeakedaroundu0withawidthofu0.Wewillnotgetintoconstructionofthesewavepackets,adetaileddescriptioncanbefoundin[ 11 ]. LetusconsidertheeldtobeinavacuumstatewithrespecttotheobserveratI.TocomputethenumberofparticlesemittedatI+inatimeintervalu0aroundthetimeu0,withfrequency!0,weneedtocalculatetheBugolubovcoecientu0!0!=(Pu0!0;fw),whichinvolvesevaluatingtheinnerproductintegralonaCauchysurface.WewillevaluatetheinnerproductonI.Forthis,werstneedtoknowthebehaviorofthemodesPu0!atI. ThewaytoobtainthefunctionalformofPu0!atIistobackpropagateitfromI+usingtheclassicaleldequations.Whenwebackpropagatethemodesfromlatetimesu0,thesemodeswouldtravelveryclosetothetheeventhorizonr2M,andarethehighlyblue-shifted.WenowinvokeGeometricopticsapproximation[ 23 ],[ 34 ],whichsaysthat,iftheeectivewavelengthofthewaveisveryshortcomparedtothecurvaturescaleofthegeometry,thesurfacesofconstantphaseofawavecanbeapproximatedbynullrays. 22

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DiagramrepresentingGeometricopticsapproximationbywhichmodesatI+aretracedbacktoI. Hence,wecanbackpropagatethePu0!modesusingnullrays.ThisispictoriallyshownintheFig. 1-3 Ifweneglectthepotential,thenwecanfreelypropagatethewavethroughspacetimetogetthefunctionalformofPu0!atI(thatis,wedonotevenneedthehelpofGeometricopticsapproximation).ByevaluatingtheinnerproductintegralatI,wecancomputetheBugolubovcoecientu0!0!,andhencethenumberofparticlesemitted.Itturnsoutthatatsucientlylatetimes(largeu0),theuxofparticles(numberofparticlesemittedperunittime)atI+isaconstant(independentofthetimeu0).n!=1 23

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2Z10~! e8!M1d!=(2l+1)~ Letusnowincludetheeectsofthepotentialforeachlmode.Since,thepotentialtermislargerforahighervalueofl,thehigherlmodeswillbebackscatteredmore.ItturnsoutthatincludingtheeectofbackscatteringmodiestheresultsofparticleuxandluminositybyjustamultiplicativefactorT!l. Foreachmode!;l,wecannumericallycomputethetransmissioncoecientbysolvingtheKlien-Gordonwaveequationswithappropriateboundaryconditions.Foragivenvalueof!,T!l!0aslbecomeslarge.Ifwecomputethetotalluminosityofthe 24

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25 ].Explicitnumericalcomputationshavebeenperformed[ 10 ]anditturnsoutthattheluminosityofthel=0modeis InGeneralrelativity,thecovariantstressenergytensorofthemattereldatanypointinspacetimecontainsallthephysicallyimportantinformationabouttheenergyuxesandmomentumuxes.ThequantumeldwhichapparentlycontributestoanenergyuxatI+(asseeninSection1.2.2),shouldhaveacorrespondingstressenergytensor.FromtheLagrangianoftheKlein-Gordoneld,weseethatclassically,ithas 25

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8 ],...Adetaileddescriptionofallthesemethodsaregiven[ 4 ],[ 13 ].Withoutgettingintothedetailsoftheseprocedures,weshallgoovertheresultsrelevanttoblackholeradiation. TocalculatehTi,weneedtoknowthestateofthequantumeldandthebackgroundgeometrycompletely.Wedonotknowthecollapsegeometrycompletely,allweknowisthattheexteriorgeometryisSchwarzschild.ItturnsoutthatispossiblecomputehTiintheexteriorSchwarzschildgeometry.TheexpressionforhTicontainsmessyintegrals.UsingaGaussianpathintegralapproximation,anapproximateanalyticexpression[ 24 ]foritcanbederived.Butacompletecomputationcanbeperformedonlynumerically[ 5 ],[ 6 ],[ 21 ]. LetusnowbrieydiscussthepropertiesofhTiintheexteriorsectionofSchwarzschildgeometryforvariousstatesofthequantumeld.TheregionunderconsiderationisboundedbyH;I(thepasthorizonandthepastasymptoticnullinnity)andH+;I+(thefuturehorizonandthefutureasymptoticnullinnity).Thestateoftheeldwillbedescribedwithrespecttoinertialobserversonthesesurfaces. 27

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FurtherdiscussionontheQuantumstresstensorgiveninChapter2. 28

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Penrosediagramrepresentingagravitationalcollapse,formationofablackholeandevaporationoftheblackhole. 1.2.3 ,wedonothaveagenericexpressionforthequantumstresstensorintermsofthemetric,wecannotsolvetheaboveequationtogetasolutiontotheevaporatingblackholegeometry.Beforeconsideringwaystotacklethisproblem,letuslookattheconformaldiagramoftheevaporatingblackholegeometry. Weshallassumethattheblackholeevaporatescompletelysothatitssingularitynallydisappearsandthatafteralltheradiationhasescaped,thegeometryisat.TheconformaldiagramrepresentingsuchasituationisshowninFig. 1-4 .r=constantcurvesofthisgeometryareshowninblack.Thelargercurvesareeverywheretimelikeasexpected.Forsmallvaluesofr,includingr=0,ther=constantcurvesareinitiallytimelike,thenbecomespacelike,andonceagainbecometimelike.Thepointwherer=0curveturnsspacelikeindicatestheformationofsingularity.Thepointwherether=0curveturnsbacktoatimelikecurveindicatesthecompletionofevaporation.Eventhorizon(EH)istheoutgoingnullray(inred)thatstartsfromtheinitialtimelikepieceofther=0 29

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Acrucialpointtonoteisthattheapparenthorizonisnotconnedwithintheeventhorizon.Thatistosaythatthetrappedregionextendsbeyondtheeventhorizon.WeshallcallthisregiontheBulge.Notethat,inacollapsegeometrywithoutevaporationasinFig. 1-2 ,thereexistsnosuchbulge. dt~ 30

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1-5 .Weseethat,atanyinstant(aspecicsnapshot),theapparenthorizonandtheeventhorizoncoincideoutsidethesurfaceofthestarandtheradiationcomingouttoanobserverfarawayisgeneratedfromaSchwarzschildexteriorlikeregion. Letustakealookatthefullybackreactinggeometry.Weseethatthereisaregionwheretheapparenthorizonbulgesoutoftheeventhorizon.Variousarguments[ 33 ], 31

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Inthiscoordinatesystem,theWeakenergyconditioncanbeexpressedasTuv(TuuTvv)1=2;Tuu0;Tvv0 FromtheEinsteinequations,wehaveTvv=r;vv Wecanthenconcludethatthereexistsaregionaroundthatsectionoftheapparenthorizon(whered2r=dv20),whereintheweakenergyconditionshouldbeviolated.Inparticular,wewillhaveTvv<0. WhatdoesitphysicallymeantohaveaWECviolation?Itmeansthat(i)thereexistsinertialobserversforwhomthelocalenergydensityisnegativeand(ii)anyinertialobserverintheseregionswillmeasureanonzeroenergyux,inthiscase,anegativeenergyuxgoestowardsthecenter. Theviolationofweakenergyconditioninthisregionpromptsustoconsideraninterestingalternative,wherethereisactuallynosingularityoreventhorizonbecausethesingularitytheoremsarenotapplicable.TheconformaldiagramforsuchasituationisshowninFig. 1-6 .Thetrappedregionformsanddisappearsinaniteamountoftimeandtheweakenergyconditionisviolatedinaneighborhoodoftheapparenthorizon.Eventhoughthereisnoeventhorizon,anexternalobserverwillmeasuresamekindsofradiationandeventheclassicalgravitationaleectswouldbethesameasaconventional 33

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Diagramrepresentingcollapsewithoutsingularity.Thecirclerepresentstheapparenthorizon. blackhole.So,observationallytherewillnotbeanydierenceuntilnalstagesofevaporation.AninterestingmodelofformationandevaporationofsuchasingularityfreeblackholeisgivenbyHayward[ 19 ]. 36 ]calculatedthebackreactioneectofthequantumeldinHartle-HawkingstateontheSchwarzschildbackground.HeconsideredthegeometrytobeaperturbationabouttheSchwarzschildgeometryandusedlinearizedEinsteinequationstosolveforthemetricperturbations.Heusedananalyticapproximation[ 24 ]tothequantumstresstensortobethesourceforthelinearizedEinsteinequations.Hefoundthatthemetricperturbationsdivergeatlargevaluedistances.Thatis,thereisregimebeyondwhichthelinearperturbationtheoryfails.Henallyconcludedthatablackholecanbeinthermalequilibriumwiththequantumeldsonlyifweapplyreasonableboundaryconditionsatsomenitelargedistance,and 34

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Letusnowfocusonthesituationwheretheblackholeformsandevaporates.Tipler[ 31 ]addressedthequestionofwhethersphericallysymmetricblackholesreallyevaporatequasistatically.Byanalyzingthenullgeodesicsveryclosetotheeventhorizon,heconcludedthattheeventhorizonissounstablethatevenasolarmassblackholewouldevaporateawayin1sec.IsraelandHajicek[ 14 ]usedamodeltoshowthatthereexistsevaporatingblackholesolutionswithastablehorizon,andhencedisprovedTipler'sarguments.LaterBardeen[ 1 ]generalizedtheiranalysisandpointedoutexactlywhereTiplerwentwrong.BardeenshowedthatifthereisanoutgoinguxfromtheblackholeatlargedistancessayLHandifthestressenergytensorontheapparenthorizonisregular,thentherateatwhichtheblackholeloosesitsmassisgivenbyLH,thatisdM=dt=LH,whereMistheapparenthorizonmassatanyinstant.ForfurtherdiscussiononBardeen'sanalysis,seeChapter5.Weknowthatthestresstensoronthehorizonisregularwhennobackreactionisconsidered(asseeninUnruhvacuum).ForBardeen'sanalysistobeapplicabletoarealisticsituation,weneedtoknowifthestresstensoronthehorizonisregularevenwhenthebackreactiononthegeometryisconsidered.ThiswasshowntobetruebyMassar[ 22 ].ThusBardeen'sdescriptionofblackholeevaporationisjustied. Bardeen'sanalysisisapplicableforanykindofLH.Weknowthat,iftheevaporationisquasi-static,thenLH1=M2.ThepointofthisthesisistoexamineifLH1=M2.Thewayinwhichweaddressthisproblemisthefollowing: Weconsideranullshellcollapsingtoformablackhole.Thequantumeldsareinthe\in"vacuumstate.ThepresenceofquantumeldschangesthegeometryfromSchwarzschild.Sincewedonothaveanexplicitexpressionfortheexpectationvalueofthequantumstresstensor,weuseasimplemodelforthequantumstresstensorwhichisinspiredfromthetwodimensionalmodelofUnruh,FullingandDavies[ 9 ].Weexplainthis 35

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rdv2+2edvdr+r2d:(2{1) Theangularpartofthemetricd2+sin2d2isrepresentedasd.Thequantitiesandmarefunctionsofrandv.Intheabovemetric,v=constisaningoingradialnullrayandtheoutgoingnullrayisgivenbydr dv=e r: rdt2+12m r1dr2+r2d:(2{2) OnemaindrawbackoftheusualSchwarzschildcoordinatesystem(t;r)isthecoordinatesingularitythatoccursatr=2m(eventhorizon),meaningthesecoordinatescannotbecontinuouslyextendedacrossthehorizon.Hence,wehavetousetwodierentpatchesofcoordinates,oneinsidethehorizonandtheotheroutside.Butnosuchproblemexistswiththecoordinatesof( 2{1 ).Hence,thiscoordinatesystem( 2{1 )willbeverygoodtodealwithsituationswhereastarcollapsestoformablackhole.Bardeen'suseofthiscoordinatesystem[ 1 ]inanalyzingthechangeinpropertiesofthenullgeodesicsclosetothehorizonduetothebackreactionofastresstensoronthegeometryisaclassicexample. TheEinsteinequationsforthisgeometry( 2{1 )canbewrittenoutinanextremelysimpleform, @v=4r2Trv;@m @r=4r2Tvr;@ @r=4r2Trr:(2{3) 38

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Forthemodelwearegoingtoconstruct,wewillnotbeusingrasacoordinate.Wewilluseadoublenullcoordinatesystemoralightconesystem(u;v),uistheadvancedtimecoordinateandvistheretardedtimecoordinate.Inthiscoordinatesystem,wecanwritethemostgeneralsphericallysymmetricmetricintermsoftwoindependentfunctionsr(u;v)andf(u;v). Thenullconesinthiscoordinatesystemareu=const(outgoing)andv=const(ingoing).Observethat,anycoordinatere-denitionU(u)orV(v)isjustagaugechoice,meaningthenewcoordinates(U;V)wouldalsocarrytheinterpretationofnullcoordinates.So,anysuchtransformationcanbedoneforconvenience. TheEinsteintensorGof( 2{4 )canbecalculatedtobeGuu=2 @u22@f @u@r @u;Guv=1 2r2e2f+4@r @u@r @v+4r@2r @u@v;Gvv=2 @v22@f @v@r @v;G=1 sin2G=4 @u@v+r@2f @u@v: Iftheregionoutsidetheshell(v>v0)isvacuum,thenBirkho'stheoremtellsusthattheexteriorregionisnothingbutaSchwarzschildgeometry.But,thepresenceofaquantum 39

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2{2 )intermsofthestandardnullcoordinatesusedintheliterature, 2{2 )asu=tr;v=t+r: 2 ]. whereSisinterpretedasthesurfacestressenergytensor.Thequestionnowis,cantheinteriorEq. 2{5 andtheexteriorEq. 2{6 geometriesbestitchedsmoothlysothattheEinsteinequationsG=8Tarevalidacrosstheshell?Clearly,theLHSandRHSoftheEinsteinequationsdivergeontheshellbecauseofthedeltafunction.But,thatisalright.WhatweneedtoknowisifitispossibletostartfromonesideoftheshellwithappropriateinitialconditionsandintegratetheEinsteinequationswiththegivenstressenergytensorEq. 2{7 andreachtheothersideoftheshelltoobtainthecorrectgeometry.Thisquestioniswellposedbecausethedeltafunctioninthestresstensoriswellbehavedunderintegration.Toanswerthisquestionwehavetorstconstructasingle 40

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2 ](aclearerdescriptionisavailablein[ 29 ]),itturnsoutthat,fortheEinsteinequationstoconsistentlyholdupacrosstheshell,thefollowingtwoconditionsshouldbesatised. 1. Theinducedmetricontheshellfromeithersidesofit(interiorandtheexterior)shouldbethesame. 2. WithrespecttothecoordinatesystemofEq. 2{5 andEq. 2{6 ,thesurfacestresstensorisgivenbyS=M wecanrewritetheequationr(~u;v0)=r(u;v0),toobtainaonetoonerelationbetweenthecoordinatesu(exterior)and~u(interior), 41

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du=14M v0eu=12M r0(u);(2{10) wherer0(u)istheradiusoftheshellwhenitisatthepoint(u;v0). Wecannowexpresstheexteriormetric( 2{6 )intermsofthecoordinate~uinsteadofu.ds2=(12M=r)du d~ud~udv+r2(u(~u);v)d; ds2=12M=r(u(~u);v) 12M=r0(u(~u))d~udv+r2(u(~u);v)d:(2{11) BycomparingtheexteriormetricintheaboveformEq. 2{11 withtheinteriormetricEq. 2{5 ,weseethatthemetriciscontinuousacrosstheshell.Hence,weobservethatthecoordinatesystem(~u;v)isacontinuouscoordinatesystem,sointegratingafunctiondenedonthemanifoldismeaningfulwithrespecttothiscoordinatesystem.Notethatthecoordinatevisthestandardatspaceretardedtimecoordinateintheinterior(vv0).Anyothercoordinatesystem(U(~u);V(v))willalsobeacontinuouscoordinatesystemiftherstderivativesofU(~u)andV(v)arecontinuousacrosstheshell. Fromthesecondcondition,weseethattheonlynonvanishingcomponentofthestressenergytensorfortheshellwithrespectto(~u;v)coordinatesystemisTvv=M dV2=M dV2=M dVdv dV2;TVV=M dV: 42

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IfwewishtoevolvetheEinsteinequationsinthiscoordinatesystemintheregionexteriortotheshell,wecannotstartwithinitialconditionsinsidetheshell,wewillhavetostartwithinitialconditionsonoroutsidetheshell.Insuchacase,thesimplestplacetolaytheinitialconditionsareonthesurfacesS1andS2. LetusnowrewritetheexteriorSchwarzschildmetric( 2{6 )intermsofournewcoordinates(U;V),equivalently(~u;~v). Thefunctionaldependenceofronthecoordinates~uand~vcanbeinferredthefollowingway.Thedependenceofronthecoordinatesuandvarealreadyknown. 4M=r WecanrewriteEq. 2{9 andEq. 2{12 ,toobtain 4M=e~v~u0 4M~v~u0 UsingtheabovethreeEqs. 2{16 2{17 and 2{15 ,wehavee(vu) 4Meu0 4M=e~v~u 4Mv0~u 4M=e~v~u v0~u0 Solvingtheaboveequationgivesusr(~u;~v).Comparingthemetric( 2{14 )with( 2{6 ),weseethate2f(~u;~v)=(12M=r)du d~udv d~v

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2{10 )and( 2{13 ),wecanseethat veu0114M v0eu1:(2{19) Thisequationappearstohaveanindeterminateformat~u=v04M.ButwecanuseEq. 2{18 torewriteitsothatitiswelldenedeverywhere. re~v~u2r v0~u0 OnecaneasilyseefromEq. 2{19 that,onthesurfacesS1andS2, v0eu01(2{21) isaconstant.Oneimportantreasonforchoosingthisgaugeisthatthemetric( 2{14 )takesaverysimpleformonthesurfacesS1andS2,risgivenby(~v~u)=2andfisaconstantgivenbytheaboveequation.Hence,specifyinginitialconditionsonthesesurfacesinordertoevolvetheEinsteinequationsbecomesverysimple. IfwearetoevolvetheEinsteinequationsfromtheinitialdataonaspacelikehypersurfaceSinaspeciedcoordinatesystem(gauge),theinitialdatashouldcompriseofthemetricandthenormalderivativesofthemetriconS.Theresultoftheevolutionwouldthengiveusthegeometry(eectivelythemetric,sincethecoordinatesystem(gauge)isalreadyxed)everywhereinthedomainofdependenceofS.ButifwearetoevolvetheEinsteinequationsfromtheinitialdataonalightconedenedbyaningoingnullsurfaceS1andanoutgoingnullsurfaceS2,weonlyneedthemetricontheinitialsurfaces. Intheappendix,wecalculatealltherelevantderivativesofthefunctionsr(~u;~v)andf(~u;~v).ThesewillbeusedinthealgorithmfornumericallyevolvingtheEinsteinequationsinChapter4. 45

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Areasonableassumptionisthatthequantumeldsdonothaveanysignicanteectonthegeometryatveryearlytimesofcollapse,thenwecanassumethatonasurfaceS2closetoI,theexteriorgeometryisapproximatelySchwarzschild. 2. AnotherreasonableassumptionisthatthemetricontheshellS1isSchwarzschild. Bychoosing~u0tobealargenegativenumber,S2canbetakenclosetoI.WewillusethesamegaugeasinSection2.1.1.2.Theexteriormetricisagaingivenby Thefunctionsf(~u;~v)andr(~u;~v)arenotgivenbyEq. 2{20 andEq. 2{18 anymore.TheytakethosevaluesonlyonthesurfacesS1andS2. Ifthequantumstresstensorisknowninthiscoordinatesystem,thenwehavealltheinitialdatarequiredtoevolvethesemi-classicalEinsteinequationsG=8hTi: Thestresstensorofthiseldisthefollowingoperator 2g@@:(2{23) Classically,theeldtakesavaluezeroeverywhere,hencetheclassicalstresstensoroftheeldiszero.Butthequantumuctuationsatanypointinspacetimewouldresultinanonzerostressenergytensorproportionalto~.Ifweknowthestateofthequantumeld,thenthemeasuredstresstensorcanbefoundfromtheexpectationvalueofthestress 46

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Theproceduresforrenormalizationareverytediousandinmostofthecasestheydonotgiveananalyticexpressionfortherenomalizedstresstensor.Thatis,forageneralmetricg,thereisnogenericexpressionforhT(g)i,exceptwhenthereisahighdegreeofsymmetryinthemattereldandthespacetimeg.Thistypicallyhappensinconformallyatspacetimesandformattereldsforwhichtheclassicalactionisinvariantunderconformaltransformations. Thisisabadnewsforusbecause,ingeneralasphericallysymmetricgeometryasinEq. 2{4 isnotconformallyat.NoteventhespecicgeometryofSchwarzschildexterior( 2{14 )isconformallyat.Butthingsgetbetterwhenwegodowntotwodimensions. 9 ]usingcovariantgeodesicPointsplitregularizationmethod[ 7 ].Theresultsoftheircalculationcanbesummarizedinthefollowingway. Considerthetwodimensionalconformallyatgeometrygivenintermsofthenull(conformal)coordinates.ds2=e2f(u;v)dudv: 48

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@u2@f @u2#; @v2@f @v2#;(2{24)hjTuvji=~ @u@v:

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2d2f=dx2 2{24 withureplacedby~uandvreplacedby~vtogettheexpectationvalueofthequantumstresstensorinthejinivacuumstate. 2{24 cannotbeobtainedin4dimensions,wecreateamodelforthequantumstresstensorin4dimensionsbymultiplyingafactor1=4r2tothestresstensorobtainedfromthetwodimensionalmodelEq. 2{24 ,soastomakephysicalsenseinfourdimensions.Accordingtothismodel, @u2@f @u2i;hTuvi= @u@v;hTvvi= @v2@f @v2i;hTi=hTi=0:(2{25) Theconstantdeterminesthestrengthofthequantumeld.Asinglequantumeldwouldcorrespondto=~=12.Onecaneasilyverifythatinagenericsphericallysymmetricgeometry( 2{4 ),theexpectationvalueoftheabovestresstensorisconserved,rhTi=0.Hence,itisphysicallyconsistenttosolvethesemi-classicalEinstein 50

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Nevertheless,weshallacceptthismodelbecauseitappearsthateventhoughneglectingthebackscatteringeectschangestheluminositybyafactorof5orso,itsqualitativebehavior,namelyL~=M2isstillretained. 2{25 ,letuscomputethequantumStresstensorinSchwarzschildbackgroundbyplugginginthefunctionalformoff(~u;~v)fromEq. 2{20 .Sincetheexpectationvalueofthequantumstresstensorisatensorquantity,wehavehTuui=hT~u~uid~u du2;hTvvi=hT~v~vid~v dv2;hTuvi=hT~u~vid~u dud~v dv: 2{10 and~vandvisgivenbyEq. 2{13 r3+3 2M2 r3+3 2M2 rM r3;(2{28) 52

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53

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whereM(u)istheBondimass.NotethattheonlynonvanishingcomponentoftheEinsteintensorfortheabovemetricisGuu=dM(u) Letusnowconstructthequasi-staticmetric.Thequasi-staticmetricshouldbeparameterizedbythepropertime,suchthatanyinstant(characterizedbytheproper 57

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SequenceofsnapshotsofSchwarzschildgeometrieswithvaryingmass. timeoftheobserver),themetricshouldbeapureSchwarzschildmetricofmassM().Henceweshallerectaninstantaneousnullcoordinatecoordinatesystem(u;v)andaninstantaneoustimelikecoordinatet,satisfyingthefollowingconditions. 3-1 representstheconstructionofthequasi-staticmetric.Atvariousmoments,theobserverseesaSchwarzschildexteriorofdierentmass. Atanyinstant,thequasi-staticmetricwouldthentaketheform ords2=12M() 58

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22r0+v0;(3{3)vp=t+r0=12M() 2+v0: Weshallnowinvokethequasi-staticapproximationtocalculatetherateofchangeofmassoftheblackholeasseenbytheobserver.Theenergyuxcrossingtheobservercanbecalculatedfromthestressenergytensorofthequantumeld.Theinstantaneousenergyuxmeasuredbytheobserveracrossthesurfacer=r0isgivenbyTrU.Weknowthattheenergycrossingtheobservergetsredshiftedasitgoesouttoinnitybyafactorp d_M()=4r2E~rp _M()=4r2p 59

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r3+3 2M2 r3+3 2M2 rM r3;(3{7) wherethecoordinatesuandvarethestandardSchwarzschildcoordinatesandthecoordinate~uisthestandardatspaceadvancedtimecoordinate.Therelationbetweenuand~uisgivenby du=14M v0~u:(3{8) Sincewehaveconstructedthequasi-staticmetricandtheinstantaneousnullcoordinatesystem(Section3.1)insuchawaythatitmatcheswithaSchwarzschildspacetimeofxedmassM(),weshalladaptthesameformofstressenergytensorforthequasi-staticmetricwithMreplacedbyM()intheEqs. 3{5 3{6 and 3{7 Sincethenullcoordinatesaredenedateveryinstantinsuchawaythatu=trandv=t+r,wehave@u @t=@v @t=1and@v @r=@u @r=1 12M=r: @t@u @r+Tvv@v @t@v @r+Tuv@u @t@v @r+@v @t@u @r;=(TvvTuu) 12M=r: Ttr==4r2 60

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_M()= (~up()v0)3+12M2() (~up()v0)4#:(3{10) Thefunctionalformof~up()canbecalculatedimplicitlyfromourknowledgeofup(),Eq. 3{3 andu(~u),Eq. 3{8 22[r0+2M()ln(r0=2M()1)];(3{11) 4M()1:(3{12) Dierentiatingtheaboveexpressionswithrespecttothepropertimegives _up=12M r01 2+_M r012M r03 24_Mln(r0=2M1)+4r0_M=(r02M);(3{13) _up=_~up4_Mlnv0~up NotethatwehavesuppressedthepropertimedependenceofthefunctionsM(),up()and~up()intheaboveequationsforbrevity.Eq. 3{14 canberewrittenas_~up1 1+4M=~up=_up+4_Mln(~up=4M1)+~up_M=M _~up=(1+4M=~up)(_up+4_Mln(~up=4M1))4_M;(3{15) where_upisgivenbyEq. 3{13 61

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3{10 togetherwithEq. 3{15 istantamounttosolvingthequasi-staticproblem.ThisinvolvessolvingtworstordernonlinearcoupleddierentialequationsinvolvingM()and~up().Weneedtwoinitialconditionstosolvetheseequations. Itisnotpossibletosolvetheseequationscompletelyanalytically.HencewesolvetheseequationsnumericallyusingthemathematicalsoftwareMaple. ThesesolutionsaregiveninChapter5andarealsocomparedwiththesolutionsfromcompletenumericalevolutionofEinstein'sequations,whicharediscussedinChapter4. 3{10 and 3{15 analytically,variouslimitsandanalyticpropertiesofthesolutioncanbediscussed. Sincetheobserveratanyinstant,isalwaysoutsidetheeventhorizonwithrespecttotheinstantaneouslydenedmetric,wehave~up()v0<4M().Wecanhencedeneapositivequantity"()suchthat~up()v0=4M()"(): Notethatthequantityx()isalwayspositiveandinitialvalueofxat=0isgivenbyx(=0)=r0=2M01:

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3{10 cannowberewrittenintermsofxas_M()= 16M2(1+x)3+3 64M2(1+x)4:(3{17) Thenwecanreplace~upinEq. 3{15 byxtoobtain,4_M(1+x)4M_x=_~up=(x FromEq. 3{17 ,observethat_M()<0,whichjustmeansthatthemassoftheblackholemonotonicallydecreases. FromEq. 3{13 ,weseethatforreasonablylarger0,_up()>0.Thereasonforsuchaconclusionisthefollowing:Ifatall_up()shouldbecomenegative,itcanhappenonlywhenr0=_M.Buttheevaporationwouldhavereachedacompletionlongbeforethistimebecause,evapM=_Misthetimescaleforevaporation.Forlarger0,wecanclearlyconcludethatfortherelevantevaporationscenario,wealwayshave_up()>0.Moreprecisely,byretainingonlythedominanttermsinEq. 3{13 ,wehave _up()'14_Mln(r0=2M1):(3{19) FromEq. 3{18 ,itispossibletoseethattherighthandsideisalwayspositive,whichmeans_xisalwaysnegative.Toseehowthisworksout,weshallcategorizethetimescaleintothreecategories,namely 1. Earlytimes,x()1.Themasslossduetoevaporationisnegligibleduringthisperiod.FromEq. 3{17 ,itisclearthat_M 3{18 ,namely4_Mx,scalesasx2.Hencetheterm_updominatestheRHSofEq. 3{18 ,makingitpositive.Thusweconcludethat,duringearlytimes_xisnegative. 63

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Intermediatetimes,x()1.Themasslossduetoevaporationissmallbutisnotnecessarilynegligible.Thestrengthofevaporation,,determineswhetherthemasslossinthisstageisnegligibleornot.TocheckiftheRHSofEq. 3{18 ispositiveduringthistime,wecanplugintheapproximateformof_upfromEq. 3{19 .TheRHSofEq. 3{18 takestheformx 3. Latetimes,x()1.Mostoftheevaporationtakesplaceinthisstage.Inthisstage,Eq. 3{17 canbeTaylorexpandedtogive_M()= 16M213x+6x2+:::+3 64M214x+10x2+::: 64M2+1 ItisstraightforwardtoseethattherighthandsideofEq. 3{18 ispositiveinthisstageofevaporation.Thisisbecause,forsmallx,ln(x)isabignegativenumber,_upisstillapositivenumberand_Misanegativenumber.Thusduringthelatetimesstageofevaporation_xisnegative.OnecanalsoneglectxinEq. 3{20 andsolveforM().WithasuitabledenitionofQ(),itturnsoutthatthelatetimessolutiontoEq. 3{20 canbeexpressedas 35r30241s r0+3M2() 2r2015M3() Wehavesofarseenthatduringallstagesofevaporation,_M()and_x()arebothnegative,whichimpliesthattheyarebothmonotonicallydecreasingfunctions.Letussaythattheevaporationcomestoanendatatime=end,thatisM(end)=0.FromthedenitionofxinEq. 3{16 ,weknowthatxispositivethatx6=0aslongasM6=0.ButwedonotknowthelimitxreachesasMgoestozero,thatiswedonotknowthevalueofx(end).Ifx(end)isnonzero,thentheevaporationwouldbeoverbeforeitreachesthelatetimes(stage3).Letusnowassumethattheevaporationreachesthelatetimesstageand 64

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3{18 tondx(end).WeshallrstrewriteEq. 3{18 intothefollowingform.4d dMx=x 3{19 for_upintheaboveequation.Thisgivesus4d dMx=x r0)+4_Mln(M r0)+4_Mln(x): dMx=x_Mln(Mx r0): 3{20 for_M,wehave dMx= r0):(3{22) ExpandingEq. 3{21 intermsofM=r0andconsideringtermsonlyuptotheleadingorder,wegetanexpressionforM(). [1+O(M=r0)]M3()=3 WecannowrewriteEq. 3{22 insuchawaythatthequantityMxgetsgroupedtogether. dMx=Mxln(Mx r0)=3(end):(3{24) Solvingthisequationyields,lnln(Mx r0)=1 3ln(end)+C;ln(Mx r0)=A(end)1=3: r0)goesto.Hence,theconstantAshouldbenegative.Now,wecangetasolutionforxasafunction 65

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3{23 4M:(3{25) ThisclearlyshowsthatasMgoestozero,xalsogoestozero.Thusweconcludethatx(end)=0. FromEq. 3{3 andEq. 3{8 ,wehavethe 3{16 ,weseethat~up!v0.Then,theaboveequationtakestheform, r0#: 3{1 totheatspacegeometryatthesurfaceu=upend.SeeFig. 3-2 .Applyingthejunctionconditions[ 2 ]onthissurface,weseethatthegeometriescanbesmoothlymappedonlyifdM(u)=duiszero.LetusnowshowthatdM(u)=duisindeedzero. Sinceweknowthat_xand_Marealwaysnegative,fromthedenitionofxEq. 3{16 ,wecanconcludethat_~upshouldalwaysbepositive.~up=4M(x+1))_~up=4d d(Mx)4_M:

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Penrosediagramrepresentingaquasistaticallyevaporatingblackhole. Moreover,neartheendofevaporation,bydierentiatingEq. 3{25 ,weobtain d(Mx)=MxA(3)1=3 d(Mx)=x M3!0as!end: 3{16 andtherelationshipbetweenuand~u,wehaved~up v0~up=11 x+1!0;as!end: 3{13 ,wecanseethatat_upend=1+end_M r0+4_Mln(2M=r0)_Mend

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LetusrstrecapitulatethehighlightsofthemodelconstructedinChapter2.Weconsideranullshellcollapsingtoformablackhole.Forconvenience,wewilldenotethe~u;~vcoordinateconstructedinChapter2asu;v.Themetricoutsidetheshelliswrittenasds2=e2f(u;v)dudv+r2(u;v)d: @u2@f @u2i;hTuvi= @u@v;hTvvi= @v2@f @v2i:(4{1) Theconstantdeterminesthestrengthofthequantumeld.Forasinglequantumeld=~=12.Wecanverifythattheexpectationvalueofthisstresstensorisconserved,rhTi=0. TheEinsteintensorGoftheabovegeometryis @v22@f @v@r @vi;Guu=2 @u22@f @u@r @uiGuv=1 2r2he2f+4@r @u@r @v+4r@2r @u@viG=1 sin2G=4 @u@v+r@2f @u@vi:(4{2) Theeldequations,G=8T,thengiveusfourdierentialequations. 2fururuu= rfuuf2u;(4{3) 1 4e2f+rurv+ruvr=fuv;(4{4) 2fvrvrvv= rfvvf2v;(4{5) 69

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FromEq. 4{4 andEq. 4{6 ,wecanobtain 4e2f+rurv InthisChapter,wewilldescribetheprocedureusetosolvetheEinsteinequationswithappropriateinitialconditions,soastoobtainthegeometryeverywhere.Thatis,wewillnumericallysolvethedierentialequationsEqs. 4{3 4{4 4{5 and 4{6 togetfunctionsfandreverywhere.Thenwewillcalculatesomephysicallyimportantquantitieslikeapparenthorizon,energyux,etc...onthisgeometry,soastocomparewiththecorrespondingresultsobtainedfromthequasi-staticapproximationwediscussedinChapter3. 70

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4.1.1InitialData U01(4{8) ThischoicebasicallymeansthatthegeometryonthesurfacesS1andS2isSchwarzschildgeometry.InChapter2,Section2.1.2,wejustifythischoice.Allthederivativesoffandrvanishonthesetwosurfacesexceptthefollowing,ru=1=2onS1andrv=1=2onS2.Itisnowpossibletoevolvetheseequationsnumericallyinsidethefuturelightcone. Tosolvethedierentialequationsnumerically,wehavetoconverttheequationstoanitedierenceequation.Todothis,letusstartbylayingdownatwodimensionalgridwithastepsizeuintheu-directionandvinthev-direction. 4{3 isaconstraintequationontheinitialdataonthesurfaceS1(v=v0)andtheEq. 4{5 isaconstraintequationontheinitialdataonthesurfaceS2(u=u0).Theinitialdatar(u;v)andf(u;v)shouldsatisfytheseconstraintsontheinitialdatasurface(thechosenlightcone)tohaveameaningfulevolution.Fromtheinitialconditions,onecanclearlycheckthattheseconstraintequationsaresatised. Letusnowevolvethefunctionsf(u;v)andr(u;v)fromthesurfacev=v0toasurfacev=v0+vbyusingthedynamicalequationsEq. 4{4 andEq. 4{6 ,oreectivelyEq. 4{7 andcheckiftheconstraintequationEq. 4{3 holdsonthisevolvedsurface 71

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@vv)2@ @u(f+@f @vv)@ @u(r+@r @vv)== r+@r @vv"@2 @vv)@ @u(f+@f @vv)2#: rfuuf2u+ rfuvu2fufuvrv(fuuf2u)=rv: 4{3 ,thetermsindependentofvcanceleachother.Letuscheckifthetermslinearinvsatisfytheequation.[ruv;u2furuv2rufuv]= rfuv;u2fufuvrv(fuuf2u)=r 4{7 forruvandfuv,andthenusingEq. 4{3 toeliminatefuu,wecanverifytheaboveequationafterabitoftediousalgebra. Now,letusevolvethefunctionsf(u;v)andr(u;v)fromthesurfaceu=u0toasurfaceu=u0+ubyusingthedynamicalequationsEq. 4{4 andEq. 4{6 (oreectivelyEq. 4{7 )andcheckiftheconstraintequationEq. 4{5 holdsonthisevolvedsurfaceu=u0+u.Thatis,wewanttocheckif@2 @uu)2@ @v(f+@f @uu)@ @v(r+@r @uu)== r+@r @uu"@2 @uu)@ @v(f+@f @uu)2#: rfvvf2v rfuvv2fvfuvru(fvvf2v)=ru:

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4{5 ,thetermsindependentofucanceleachother.Letuscheckifthetermslinearinusatisfytheequation.[ruv;v2fvruv2rvfuv]= rfuv;v2fvfuvru(fvvf2v)=r: 4{7 forruvandfuv,andthenusingEq. 4{5 toeliminatefvv,wecanverifytheaboveequationafterabitoftediousalgebra. Thus,wehaveshownthat,iftheconstraintequationsEq. 4{3 andEq. 4{5 aresatisedontheinitialsurface,thenwecanuseEq. 4{7 toevolvethefunctionsf(u;v)andr(u;v),andautomaticallytheconstraintequationswillbesatisedeverywhere. 4{7 isusedtoevolvethefunctionsandtheirderivativesfromonepointonthegridtothenextpoint. Ourstrategyistostartonav=constantsurface,andevaluateallthederivativesoffunctionsateverygridpointbymovingintheu-direction.Oncewehavethetheentiresolutiononthissurface,wejumptoaparallelsurfacegivenbyv+v=constant.WeshallcalltherstsurfaceS1andthenextparallelsurface WestartwiththeassumptionthatonthesurfaceS1,alltheuderivativesareknown,andatthepointp,wherethissurfacemeetswiththeinitialdatasurfaceS2,bothuandvderivativesareknown.Atanypointp,ifweknowallthepurederivativesofthefunctions,thenallthemixedderivativescanbecalculatedfromtheevolutionequations.Explicitcalculationofthesemixedderivativesareperformedintheappendix.Thesemixedderivativeswillbeusedtocomputethepurederivativesataneighboringpointofp.WeshallexplainthiswiththehelpofFig. 4-1 1. WerstcomputethevderivativesofthefunctionsfandronthesurfaceS1.Sincethesevaluesareknownatthepointp,wewillTaylorexpandthesefunctionstoobtainthevaluesattheimmediateneighboringpointqonS1,rqv=rpv+rpuv(u)+1 2rpu2v(u)2+1 6rpu3v(u)3

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Diagramrepresentingtheu-vgridonwhichthefunctionsrandfareevolved. 2rpu2v2(u)2+1 6rpu3v2(u)3rqv3=rpv3+rpuv3(u)+1 2rpu2v3(u)2+1 6rpu3v3(u)3 2fpu2v(u)2+1 6fpu3v(u)3fqv2=fpv2+fpuv2(u)+1 2fpu2v2(u)2+1 6fpu3v2(u)3fqv3=fpv3+fpuv3(u)+1 2fpu2v3(u)2+1 6fpu3v3(u)3 Sincewealreadyhavetheu-derivativesofthefunctionsatthepointq;andnowwealsoknowthev-derivativesofthefunctionsatthepointq,wecancomputeallthemixedderivativesatthepointq.Thisisexplicitlyworkedoutintheappendix.Onceweknowallthederivativesatq,weautomaticallyknowtheEinsteintensorGandthestresstensorTfromEqs. 4{1 and 4{2 3. Withtheknowledgeofallthederivativesatq,wecanproceedtocomputeallthederivativesatthenextpointonS1byrepeatingthesteps1and2.Inthesameway,wecancontinuetondallthederivativesateverypointonthesurfaceS1. 4. Ournextstepistocomputethefunctionalvaluesandtheu-derivativesonthesurface 74

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5. Sinceweknowthefunctionsandtheu-derivativeseverywhereonthesurface TheabovevestepsprescribethealgorithmforevolvingtheinitialconditionsonthenullsurfacesS1andS2intotheentirefuturelightcone. 4{7 .Wehaveruv=1 4e2f+rurv 26 ].Thedenominatorgoestozeroasrgoestorc.Ifthenumeratoralsogoestozerosucientlyfastsothattheruvisniteinthelimitr!rc,thenwecanjumpoverthispointintheevolution.Butunfortunately,thisisnotthecase.ItiseasytocheckthisontheinitialsurfaceS1,wheref=f0andru=1=2.Inordertohavethenumeratorvanish 75

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Whendowechoosetointerpolate?Weinterpolatewhenwesuspectthattheerrorsarebeyondsometolerancelevel.Sincethealgorithmisthirdorder,thatis,itincludesthirdordertermsintheTaylorexpansion,ajustiablecriterionisthat,weshouldinterpolatewheneverthecontributionfromthethirdderivativesinanyfunctionissignicantlycomparabletothesecondderivativecontribution.Inthealgorithm,we 76

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Thenumberofstepsneededtoreachthepoint(u;v)isinverselyproportionaltothestepsize.Thatis,Nu1=(u)andNv1=(v).Hence,wehave wherecuandcvarenumbersthatdonotdependonthestepsizesuandv.Theorderofconvergenceofanalgorithmwithrespecttoastepsize,canbecalculatedinthefollowingway wherenistheorderofconvergence.Withrespecttothisdenition,weexpectathirdorderconvergenceinbothu-directionandv-direction.Weverifythisonanarbitrarilychoseningoingnulllineandonanarbitrarilychosenoutgoingnullline.Wedoobservethatn=3inbothuandvdirections.Moreover,wecancalculatethecoecientscuandcv,anditturnsoutthatcvismorethan100timessmallerthancu.Hence,foraneectiveerrorcontrol,havingasmallstepsizeinudirectionuismuchmoreimportantthanhavingasmallstepsizeinvdirection,v. 4{7 areused.Buttheerroraccumulatedinthefunctionswillcontributetoviolationoftheconstraints.Wehavetokeepthe 78

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TocomparetheLHSandRHSoftheconstraintequations,wedeneaquantitycalledconstraintviolatingpercentage FromtheSection4.2.1,weknowthateachofthequantitiesGuu;Tuu;Gvv;Tvvshowsathirdorderconvergence.Hence,theseconstraintviolatingpercentagesshouldalsoshowathirdorderconvergence.AnappropriatestepsizecannowbechosenwiththecriteriathatbothCuandCvateverygridpointislesserthansay0.1%. Weknowthatifweturnotheevaporationandthenevolvetheequations,thatiswith=0,weshouldgettheSchwarzschildgeometry.Atanygivenpointinourcoordinatesystem,wecanexplicitlycalculatethefunctionsf;randalltheirderivativesforthepureSchwarzschildexteriorgeometry.Thisisexplicitlycalculatedintheappendix.Now,wecancomparethenumericallyevolvedsolutionsAtotheexactsolutionA0.Specically,wechooseanoutgoingnullray,notveryclosetotheeventhorizon,andwecompareAandA0forvariousstepsizestoseeifEq. 4{9 issatised. Thiscorrespondencecheckworksne,hencewehavecondenceinournumericalrecipe. 79

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FromChapter2,Section1.1.1,weseethatthechoiceofU0doesnotaecttheucoordinatechoice,itonlyaectsthevcoordinatechoice.Hence,anyoutgoingnullsurface FordierentchoicesofU0,thesurface 80

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d;dv d;U=e2f(u;v) d;du d: ddv d=e2f(u;v): d=ru d: d=ef(u;v)r d=ef(u;v)r rurv(rv;ru):(4{17) TheenergyuxwithrespecttoourobserverisgivenbyE=TU.Specically,weareinterestedincalculatingtheenergyuxacrossaconstantrsurface.LetusrstcalculatethequantityEr,andlaterunderstandhowexactlythisquantityisconnectedtotheenergyuxacrossaconstantrsurface.Er=Euru+Evrv;

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rurvTuur2vTvvr2u(4{18)HowisthequantityErrelatedtotheenergyuxthroughther=r0surface? v=ef(t+x);u=ef(tx):(4{19) Themetrictakestheform,ds2=e2fdudv+r2d=dt2+dx2+r2d: Ourobserverisnotatrestwithrespecttothiscoordinatesystem(t;x).Sincetheobservermovesonaconstantrsurface,wehavetoboostthecoordinatesystem(t;x)to(~t;~x)usingtheappropriatefourvelocityU.ThemetricisstilllocallyMinkowskiands2=dt2+dx2+r2d=d~t2+d~x2+r2d;

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@~t=@x @~x=Ut;@x @~t=@t @~x=Ux:(4{20) Theenergyuxacrossaconstant~xsurfacemeasuredbyanobserveratrestwithrespecttotheboostedcoordinatesystemisgivenbyT~t~x=T~t~x.FromthedenitionE(TU),itisclearthatE~x=T~t~x,istheenergyuxmeasuredbyalocallyinertialobserveratrestwithrespecttothecoordinatesystem(~t;~x).T~t~x=Ttt@t @~t@t @~x+Txx@x @~t@x @~x+Ttx@t @~t@x @~x+@x @~t@t @~x=Ttx(Ut)2+(Ux)2+(Txx+Ttt)UtUx: @u+Uv@t @v=(Uv+Uu)ef @u+Uv@x @v=(UvUu)ef HowdowecompareEq. 4{21 withEq. 4{18 ?Forthis,werstndtherelationbetweenrandthecoordinate~x,thatis,expressthefunctionr(u;v)intermsofthelocallyinertialcoordinates(~t;~x). FromthecoordinatetransformationsEq. 4{20 andEq. 4{19 ,wendthat@u @~t=@u @t@t @~t+@u @x@x @~t=ef(UtUx)=Uu;@v @~t=@v @t@t @~t+@v @x@x @~t=ef(Ut+Ux)=Uv;

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@~x=@u @t@t @~x+@u @x@x @~x=ef(UxUt)=Uu;@v @~t=@v @t@t @~x+@v @x@x @~x=ef(Ux+Ut)=Uv: 4{17 ,weget@ @~tr(~t;~x)=ru@u @~t+rv@v @~t=ruUu+rvUv=0;@ @~xr(~t;~x)=ru@u @~x+rv@v @~x=ruUu+rvUv=ef rurv(2rurv)=2efp rurv: @~xr(~t;~x): 4{21 .Aftereverystepofevolution,thisquantityisalsocalculatedandstored.WeshallrefertothisfunctionasE(u;v).ThisfunctionE(u;v)isapositivefunctionwhichmeansthatthereisanetenergyuxintheoutwarddirection.Massloss 85

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4{16 .Wewillset=0attheinstanttheobservercrossestheshell(v=0). ThedierentialequationEq. 4{16 canbeconvertedtonitedierenceequationandsolvednumericallyastheevolutionproceeds.=ef(u;v)r M=4r20p andhencethemassatanyinstantisM(+)=M()+M:

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InthisChapter,wewilldiscusstheresultsobtainedfromthenumericalcomputationdescribedinChapter4andcomparethemasslossduetoevaporationoftheblackholeobtainedfromtheexactnumericalcomputationstothemasslossoftheblackholeobtainedfromthequasi-staticapproximationschemedescribedinChapter3. Thestressenergytensorofourmodel(Eq. 2{25 )hasaparameterinit.Forasinglequantumeld,isproportionalto~.Toconsiderthepresenceofmorethanoneeld,wehavetomultiplybyanappropriatenumber.Inasense,determinesthestrengthofthequantumeld.Intermsofgeometrizedunits(whichwehavebeenusingthroughout),massandtimehasthesamedimensionaslengthandhasthedimensionoflength2.ThedimensionlessquantitythatcharacterizestheevaporationofablackholeofinitialmassM,isthen=M2.Tohaveaperspectiveonthesizeofthisparameter,itisusefultonotethat Sincethisquantityisverysmallforrealisticblackhole,itwouldbeimpossibletocarryoutnumericalcomputations.Weperformthenumericalcomputationsintheregimewhere=M21.LetusnotthinkofthisasaPlanckscaleblackholeevaporationproblembecauseourapproachofusingsemi-classicalEinsteinequationsmightnotbevalidinthisregime,becauseimportantquantumgravityeects(iftheyexist)willhavetobeincluded.Instead,wewillthinkofthelargevalueofasanoutcomeofaverylargenumberofquantumeldsworkingtogethertoincreasethestrengthofevaporation. AlldatathatwillbeshowninthisChapterwillcorrespondtoachoiceofM=2. 87

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Aplotofr=constantcurvesofSchwarzschildgeometry.Thesixcurvescorrespondtor=8,r=6,r=4.5,r=4,r=3.5,r=3 5-1 whichshowsther=constant(sayr0)curvesobtainedfromtheSchwarzschildgeometry.Weseethatthecurver0=(2M=4)isastraightlineatconstantu=8.Allthecurveswithr0>4startatv=0withauvaluelessthan8andasvincreases,umonotonicallyincreasestoasymptoticallyreachtheu=8line.Thisshowsthatallthesecurvesaretimelike.Anotherwaytoseethisis,onallthesecurvesthequantityrvispositive.Thecurveswithr0<4startatv=0withauvaluegreaterthan8andasvincreases,umonotonicallydecreasestoasymptoticallyreachtheu=8line.Thisshowsthatallthesecurvesarespacelike.Anotherwaytoseethisis,onallthesecurvesthequantityrvisnegative. LetusnowconsiderFig. 5-2 whichshowsther=constant(r0)curvesobtainedfromtheevaporatinggeometryfor=5.ArescaledversionisshowninFig. 5-3 .Thereexistsanrap0suchthat,forr0>rap0,thesecurvesarealwaystimelike(rv>0).Thecurver=rap0

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Aplotofr=constantcurvesofevaporatinggeometrywith=5.Thesixcurvescorrespondtor=8,r=6,r=4.5,r=4,r=3.5,r=3 startsoutatv=0withrv=0;andthenattainsapositivevalueforrv.Allthecurveswithr00.Weshallrefertothispointastheturningpointofther=r0curve.Theturningpointisdierentfordierentr0curves.Bydenition,theapparenthorizonhastopassthroughtheturningpointsofallthether=constantcurves. FortheSchwarzschildcase(Fig. 5-1 ),weseethatallther=constantcurvesasymptotestoau=constantline(eventhorizon).SuchasituationdoesnotreallyhappeninFig. 5-2 .Thisisbecause,wehavenotidentiedtheeventhorizonhere.AsmentionedinChapter4,wecannotevolvetheEinsteinequationsbeyondthecriticalradiusp 89

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5-1 ,wehavetabulatedthecoordinatepositionuapandtheradiusoftheapparenthorizonwhenitrstappears.Wehavealsogiventhevaluesofucutoffforvariousvaluesof.Thesizeofthebulgeregionisucuoffuap. Weshouldnotethattheplotonlyshowstheregionexteriortotheshellatv=0.Allther=constantcurvesintheinterioroftheshell(whichisatspace)aretimelike. Table5-1. CoordinatepositionofBulge rap0 4.579 -9.158 -8.569 4 4.469 -8.938 -8.514 3 4.356 -8.713 -8.381 2 4.241 -8.482 -8.252 1 4.122 -8.245 -8.125 0.5 4.062 -8.123 -8.062 0.25 4.031 -8.062 0.125 4.015 -8.031 3{6 )givesTvv==2 4r2M r3+3 2M2 90

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AplotshowingtheApparenthorizonandtheregionwhereWeakenergyconditionisviolatedforanevaporatinggeometrywith=5.ThisplotisjustarescaledversionofFig. 5-2 conditionisviolated.Thisisasituationwherethebackreactionofthequantumstresstensoronthegeometryisneglected.Letusnowlookfortheviolationofweakenergyconditioninthegeometrywherethebackreactionisincluded. InFig. 5-3 ,whichisjustarescaledversionofFig 5-2 (theevaporatinggeometrywith=5),theorangecurveistheapparenthorizonandthebrowncurveistheboundaryoftheregionwhereTvvisnegative.AsseenfromtheanalysisofBergmannandRoman[ 30 ],weseethatthereexistsaregionsurroundingtheapparenthorizonwhereTvvisnegative,implyingtheviolationofweakenergycondition. 91

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5-4 5-5 5-6 5-7 5-8 and 5-9 .Therearetwocurvesineachofthesegraphs,theredonerepresentsthemasslossinferredfromthequasi-staticapproximationandtheblueonerepresentsthemasslossinferredfromthefullnumericalcalculation. Ineachofthesegraphs,weplacethreemarkerstoidentifyandcomparethedierentstagesofevaporation.Table 5-2 givesthepositionofthesethreemarkers. Table5-2. Positionofmarkers LateTimes DivergingPoint ApparentHorizon MLT 1.77 91.28 1.68 72 1.489 79.2 4 1.802 99.39 1.72 74 1.567 81 3 1.83 112.89 1.75 77 1.65 83.43 2 1.90 139.86 1.79 81 1.744 86.77 1 1.94 220.8 1.82 90 1.85 92.63 0.5 1.97 382.6 1.85 124 1.915 97.9 Therstmarkercorrespondstothelatetimesregimeofevaporation.RecallthatinChapter3,Section3.3.3,wecategorizedthequasi-staticevaporationintothreestages,whereinthelaststage,whichwecalledLatetimes,hadthepropertythat _M Foreachvalueof,theevaporationentersthisregimeatadierentinstancesandtherstmarkerMLTrepresentsthemassoftheblackholeatthisinstance.endcorrespondstotheendofevaporationobtainedfromthequasi-staticapproximation. Thegraphsclearlyshowthatthemasslossfromthequasi-staticapproximationandthecompletenumericalcomputationmatchescloselyduringtheinitialstages.Thenthereisaninstantwherethetwocurvesbegintodivergefromeachother.Fordierentvaluesof,thishappensatdierentinstances.ThesecondmarkerDPcorrespondstothisinstance 92

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Comparisonofmasslosscurveswithrespecttoanobserveratr=30,for=5.Theredcurveindicatesthequasi-staticmasslossandthebluecurvecorrespondstothemasslossfromtheexactnumericalcalculations.ThedottedlineistheMLTmarker,thedashedlineistheMDPmarkerandthedotteddashlineistheMAHmarker. wherethemasslosscurvesofthequasi-staticapproximationandthecompletenumericalcalculationdivergeawayandMDPcorrespondstothemassatthattime. FromTable 5-2 ,weknowthatforvariousvaluesof,theapparenthorizonstartsformingatdierentvaluesofu.Theapparenthorizonishiddenfromtheobserveruntilhereachesu=uap.ThethirdmarkerAHrepresentstheinstantwhentheobserverstartsseeingtheapparenthorizonandMAHrepresentsthemassoftheblackholeatthatinstant.Inotherwords,beyondthismomenttheobserverwillstartseeingtheeectsofthe\Bulge"intheevaporationprocess. ThereareseveralimportantpointswewouldliketoobservefromthegraphsandtheTable 5-2 93

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Comparisonofmasslosscurveswithrespecttoanobserveratr=30,for=4.Theredcurveindicatesthequasi-staticmasslossandthebluecurvecorrespondstothemasslossfromtheexactnumericalcalculations.ThedottedlineistheMLTmarker,thedashedlineistheMDPmarkerandthedotteddashlineistheMAHmarker. Toquantitativelycharacterizethedeviationofthequasi-staticmasslosscurvefromtheexactmasslosscurve,weshalltthedatatoapowerlawcurve.Thatis,wetthedata 94

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Comparisonofmasslosscurveswithrespecttoanobserveratr=30,for=3.Theredcurveindicatesthequasi-staticmasslossandthebluecurvecorrespondstothemasslossfromtheexactnumericalcalculations.ThedottedlineistheMLTmarker,thedashedlineistheMDPmarkerandthedotteddashlineistheMAHmarker. to _MS whereSandPdependon.Foranidealobserververyfaraway,r0M.Butthiscannotbepracticallyachievedinthenumericalcalculation,becauseevolvingtheequationsouttolargedistanceswouldaccumulatealargeerror.Foraniter0,theaboveequationcanbesolvedbyTaylorexpandingp r01 2(P+3)M r025 8(P+4)M r03+:::#=S 95

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Comparisonofmasslosscurveswithrespecttoanobserveratr=30,for=2.Theredcurveindicatesthequasi-staticmasslossandthebluecurvecorrespondstothemasslossfromtheexactnumericalcalculations.ThedottedlineistheMLTmarker,thedashedlineistheMDPmarkerandthedotteddashlineistheMAHmarker. Iftheexactcurvefollowsquasi-staticevaporation,thenthecorrecttshouldcorrespondtoP=2andS=1.FromhavingcheckedtheconvergenceofthedataM()forvariousstepsizes,weexpectM()tohaveanerroroflessthan104.Foreachvalueof;weattempttotvariousvaluesofPandS.Wechoosethebestttobetheonewhichgivestheleastresidualerror.Ifthemaximumresidualerrorinttingislessthan104,thenwecanconsiderthetas\good".Table 5-3 showsthebesttvaluesofPandS.Now,wecanplotthesevalueswithrespectofasshowninthegraphsFig. 5-10 andFig. 5-11 Ifweextrapolatetheseresultstoarealisticblackholebytakingthelimit!0,weseethatP!2andS!1.Thistellsusthatquasi-staticapproximationisvalidwheneverisverysmall. 96

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Comparisonofmasslosscurveswithrespecttoanobserveratr=30,for=0.5.Theredcurveindicatesthequasi-staticmasslossandthebluecurvecorrespondstothemasslossfromtheexactnumericalcalculations.ThedottedlineistheMLTmarker,thedashedlineistheMDPmarkerandthedotteddashlineistheMAHmarker. toparameterizetheevolution,theBondimassMBwouldbethecorrectmeasureofmass.Letusnowconsideraninstant,whentheobserver'svcoordinatepositionisv2.Theapparenthorizonvisibletohimisatthecoordinatepositionv=v1.Hence,theapparenthorizonmasswithrespecttotheobserverisMap(v1).LetussaythathemeasuresaBondimassMB.WeseethattherateofchangeofMapisslowerthantherateofchangeofMB.ThismeansthatMBandMap(v1)arenotcomparablequantities.ButtheredoesexistarelationbetweenMBandMap(v2),asshownbyBardeen. Bardeen[ 1 ]hasshownthatforverysmallvaluesof,dMB=d'dMap(v2)=dv.Thatis,therateofdecreaseofapparenthorizonmassisequaltotheluminosityoftheblackhole(whichispreciselytherateofdecreaseoftheBondimass).ThisinuencedPiranandParentini[ 26 ]tomeasuretheapparenthorizonmassasafunctionofvintheirnumerical 98

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P S 5 1.48 0.651 0.002 4 1.62 0.72 0.001 3 1.78 0.798 0.0002 2 1.87 0.867 0.0002 1 1.93 0.928 0.0001 0.5 1.975 0.973 0.00005 evolution.Theirclaimisthat,whenthemassMapisstilllargecomparedtothecriticalmass(p AnimportantassumptionthatgoesintoBardeen'sanalysisisthattheobserveratr0iswellinsidethequasi-staticregime.InFig. 5-12 ,weseethatwhentheobserverisatv=v1,theblackholehasnotevenformed,soobviouslyhecannotbeinthequasi-staticregime.Fromourcalculations,wecanseethatuntiltheobserverreacheslargevalueofv,dMB=disnotcomparablewithdMap(v2)=dv.Thatistosay,donotgetintothe"quasi-staticregime"asdescribedbyBardeenuntiltheobserverreacheslargevaluesofv.Hence,ifwearetouseMapasthemeasureofthemassanddMap=dvasameasureoftheluminosityoftheblackhole,wehavetogotoverylargevaluesofv.Buttracingtheapparenthorizontoaverylargevalueofvisnotnumericallypossiblebecausetheapparenthorizonwouldstriketheucutoatanitevalueofv.Hence,weclaimthatourmethodofcalculatingtheBondimassbyfollowinganobserveratconstantr0isamuchmorereliableandaccuratemeasureofthemass.Moreover,thiscanbedirectlycomparedtothemasscurveobtainedfromquasi-staticapproximation(whichispreciselywhatwedidinthegraphs). 99

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BesttforP(). Figure5-11. BesttforS(). 100

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DiagramshowingthedierencebetweentheBondimassandtheApparenthorizonmassasmeasuredbyanobserveratlarger. 101

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2{25 .ThisformforthestressenergytensorisinspiredfromathetwodimensionalblackholemodelgivenbyUnruh,FullingandDavies. Inthissimplemodel,wherethestresstensorisexplicitlyexpressedintermsofthemetricofthegeometry,itispossibletonumericallysolvethesemi-classicalEinsteinequations(Chapter4).TheinitialconditionsforthenumericalevolutionaretakentobeSchwarzschildinitialconditionsonthecollapsingshell.Withrespecttothismodel,wecanexplicitlyworkoutthequasi-staticprediction(Chapter3).WeseefromFig. 3-2 thatthequasi-staticallyapproximatedgeometrydoesnotcontainaBulgeregionthatexistsintheexactgeometryFig. 1-4 Theevaporationstrengthinourmodelisxedbythedimensionlessparameter=M2,whereMistheinitialmassoftheblackhole.Foranastronomicalsizeblackholeandasinglequantumeld=M21050.Sinceitisnotpossibletonumericallyevolvetheequationsforsuchlowparameters,wedothecomputationsfornitevaluesoftheparameter.Fromthenumericalevolutionoftheexactgeometry,weconrmthattheredoesexistabulgeregionandthesizeofthebulgeregiondependsonthevalueoftheparameter=M2. WecalculatetheBondimassoftheblackholeasseenbyafarawayobserver.Weevaluatethiswithrespecttotheexactmetricandwithrespecttothequasi-staticmetricandthencomparethetwo.Weseethatforsmallvaluesof=M2,mostofthemassoftheblackholeisevaporatedaftertheformationoftheapparenthorizon.Thatis,mostoftheradiationappearstocomeoutoftheBulgeregion.Thisisinperfectagreementwiththeexpectation.ComparisonoftheBondimasscurvesfromtheexactmetricandthequasi-staticmetricrevealsthatthereisasignicantdierencebetweenthemwhen=M2islarge,butthedierenceappearsinconsequentialwhen=M21.Foranyvalueof=M2,thelatetimesbehaviorofthequasi-staticmasscurvetstheequationdM dt=

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Itwillbeinterestingtoseeifitispossiblegeneralizeourmodeltoincludethebackscatteringeects.Thatis,ifitispossibletodoasimilarcalculationwiththequantumstresstensorgivenbythecompletes-waveapproximation([ 11 ]).Thiswouldmakethecalculationmuchmorerealisticthanwhatwehave. 105

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whereeachofthesefunctionsF;G;F1;G1;:::areseventhorderpolynomialsinx.Letusdenexj+1xjandt(xxj)=.NoweachofthesefunctionsF;G;F1;G1;:::isaseventhorderpolynomialint. Wecancalculatethederivativeswithrespecttot,andthendividebytoobtainthederivativeswithrespecttox. 107

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Weshallnowimposetheeightconditionsinthefollowingway.Firstly,forthefunctionF(t),whichisaseventhorderpolynomialint. ForthefunctionG(t), 108

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Sincethesefunctionshavetherstthreederivativeszeroatt=0,itiseasytoinferthatthecoecientsgnarenonzeroonlyforn>3.Takingderivativeswithrespecttotyields, Inthesameway,wecandierentiatethefunctionsG1(t),G2(t),G3(t).Weshallnowusetheconditions( A{7 )att=1toobtaing4+g5+g6+g7=1;4g4+5g5+6g6+7g7=0;12g4+20g5+30g6+42g7=0;24g4+60g5+120g6+210g7=0:

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21 6843971 2703413 21 2201021 6377777775:(A{16) Wenowwritetheaboveequationsobtainedfromtheconditions( A{7 )as [M]266666664g4g5g6g7377777775=2666666641000377777775;(A{17) Similarly,wecanwritetheequationsobtainedfromtheconditions( A{9 ),( A{11 ),( A{13 )as [M]266666664g41g51g61g71377777775=266666664000377777775;[M]266666664g42g52g62g72377777775=2666666640020377777775;[M]266666664g43g53g63g73377777775=2666666640003377777775:(A{18) Solvingtheequations( A{17 )and( A{18 ),weobtaing4=35;g5=84;g6=70;g7=20;g41=15;g51=39;g61=34;g71=10; g42=5 22;g52=72;g62=13 22;g72=22;(A{19) 63;g53=1 23;g63=1 23;g73=1 63:(A{20) LetusnowsolveforthefunctionsF;F1;F2;F3.Fromtheconditions( A{6 ),( A{8 ),( A{10 ),( A{12 ),weseethatallthesefunctionsandtheirderivativesvanishatt=1. 110

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Sincethesefunctionshavetherstthreederivativeszeroatt=1,itiseasytoinferthatthecoecientsfnarenonzeroonlyforn>3.Takingderivativeswithrespecttotyields, Inthesameway,wecandierentiatethefunctionsF1(t),F2(t),F3(t).Wecansolveforthecoecientsfnsbyapplyingtheconditions( A{6 ),( A{8 ),( A{10 ),( A{12 )on( A{21 )att=0.Wecannowsolvetheseequationsinexactlythesamewayaswesolvedforthecoecientsgn,exceptforthefactthattakingaderivativewithrespecttotisequivalenttotakingaderivativewithrespectto(1t)withaminussign.Hencethecoecientsfnandgnshouldbeverycloselyrelated.Carefulanalysisyields,f4=g4;f5=g5;f6=g6;f7=g7;f41=g41;f51=g51;f61=g61;f71=g71;f42=g42;f52=g52;f62=g62;f72=g72; f43=g43;f53=g53;f63=g63;f73=g73:(A{24) WenowknowthefunctionsF;G;F1;G1;F2;G2;F3;G3andtheirderivativesfromEqs. A{21 A{23 A{14 A{15 A{24 and A{20 .So,wecanuseEqs. A{2 A{3 A{4 and A{5 tondthevaluesofy,y0,y00,y000atanyxinbetweenxjandxj+1(ortinbetween0and1). Inparticular,sinceweintendtoaddanewgridpointrightinbetweenxjandxj+1,thatisat(xj+xj+1)=2,weneedthevaluesofy,y0,y00,y000att=1=2. 111

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2;G0=F0=35 16;G00=F00=0;G000=F000=105 2;G1=F1=11 64;G01=F01=19 32;G010=F010=15 8;G0100=F0100=105 4;G2=F2=3 1282;G02=F02=1 162;G020=F020=7 162;G0200=F0200=15 42;G3=F3=1 7683;G03=F03=1 3843;G030=F030=1 323;G0300=F0300=3 163: A{2 ,wehave 2(yj+1+yj)11 64(y0j+1y0j)+3 128(y00j+1+y0j0)21 768(y000j+1y0j00)3;(A{25) FromEq. A{3 ,wehave 16(yj+1yj) 19 32(y0j+1+y0j)+1 16(y00j+1y0j0)1 384(y000j+1y0j00)2;(A{26) FromEq. A{4 ,wehave 8(y0j+1y0j) 7 16(y00j+1+y0j0)+1 32(y000j+1y0j00);(A{27) FromEq. A{5 ,wehave 2(yj+1yj) 3+105 4(y0j+1+y0j) 215 4(y00j+1y0j0) +3 16(y000j+1+y0j00):(A{28) Thususingtheabovefourequations,wecanndthevaluesofy,y0,y00,y000attheinterpolatedpoint.Thethirdderivativescalculatedattheinterpolatedpointaresometimesnotaccurate,hence,weuseadierentinterpolationschemetocalculatethethirdderivativeattheinterpolatedpoint. Letssaythatweneedthethirdderivativeatapointinbetweenxjandxj+1.Whatwedois,weusethevalueofy000atxj1,xjandxj+1calculateitsvalueatthemidpointofxjandxj+1byttingaquadraticfunction.Weshallwritethequadraticfunctionas,y000=Axxj

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Hence,usingtheEqs. A{25 A{26 A{27 and A{29 ,wecanobtaintheinterpolatedvaluesofyanditsderivativesat(xj+xj+1)=2. 113

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4e2f+rurv Wecancalculatethehigherderivativesofthefunctionsrandfwithrespecttouandvwiththeknowledgeofthelowerderivatives.Notethat,ifQ=[Numer] [r=r];P=[Numer] [r=r];P0=[Numer]0+Pr0 B{1 ,wenowobtain, 2e2ffu+ru2rv+ruvru+ruvru(1+=r2)(B{2) 2e2ffv+ruvrv+rurv2+ruvrv(1+=r2)(B{3) 2e2ffuv+ru2rv22ruvrurv=r3 114

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2e2ffv2+ruv2rv(3+2=r2)+ruvrv2(3+=r2)+rv3ru2ruvr2v=r3 2e2ffu2+ru2vru(3+2=r2)+ruvru2(3+=r2)+ru3rv2ruvr2u=r3 2fuv2+ru2rv3+ru2vrv2+6ruvrur2v=r4+ru2v2rv(1+=r2) (B{12) 2fu2v+ru3rv2+ruv2ru2+6ruvrvr2u=r4+ru2v2ru(1+=r2) (B{13) 115

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2fu2v2 116

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f=f0+cf1+c22f2+::: 4e2f+rurv 4e2(f0+cf1)+(r0u+cr1u)(r0v+cr1v) (r0+cr1=r0)r0uv+cr1uv=1 4e2f0(1+2cf1)+r0ur0v+c(r1ur0v+r1vr0u) 4e2f0+r0ur0v 2e2f0f1+r1ur0v+r1vr0u 2e2f0f1+r1ur0v+r1vr0u Similarlyforthefunctionf(u;v),f0uv+cf1uv=r0uv+cr1uv WecanfurtherdierentiateEqs. B{19 and B{20 toobtainthehigherderivatives. 117

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2e2f0f1+r1ur0v+r1vr0ur0u=r201 2e2f0f1u+e2f0f0uf1+r1u2r0v+r1ur0uv+r1uvr0u+r1vr0u2=r0: Dierentiatingwithrespecttovgives, 2e2f0f1+r1ur0v+r1vr0ur0v=r201 2e2f0f1v+e2f0f0vf1+r1v2r0u+r1vr0uv+r1uvr0v+r1ur0v2=r0: Dierentiatingoncemorewithrespecttougives, 2e2f0f1+r1ur0v+r1vr0ur0uv=r202r0ur0v=r30+1 2e2f0f1u+e2f0f1f0u+r1u2r0v+r1ur0uv+r1uvr0u+r1vr0u2r0v=r20+1 2e2f0f1v+e2f0f0vf1+r1v2r0u+r1vr0uv+r1uvr0v+r1ur0v2r0u=r201 2e2f0f1uv+e2f0f1vf0u+e2f0(f0uvf1+f0vf1u+2f0uf0vf1)=r0(r1uv2r0u+r1v2r0u2+r1uvr0uv+r1vr0u2v)=r0(r1u2vr0v+r1uvr0uv+r1u2r0v2+r1ur0uv2)=r0 WecannowdierentiatethefunctionsEq. B{20 toobtain 118

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Similarly, Anotherdierentiationwithrespecttovyields, U0 wheretheconstant(1U0=4M)isalargenumber(sayL)whichdependsontheinitialconditions.evu 4MvU0 4MvU0 119

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r"1+vU0 4M12M r"vU0 r"1+u 2ruv="1+vU0 r2ru(B{30) Letsnowcalculatethesecondderivativeswithrespecttouandv.ru 4MLvU0 Similarly,rv 4MLu 2r 120

PAGE 121

B{30 2ruv2=2ruv SimilarlydierentiatingEq. B{30 withrespecttou,yields2ru2v="1+vU0 Dierentiatingthepreviousequationoncemorewithrespecttov,gives Letusnowcomputethederivativesofthefunctionf(u;v). ru Dierentiationyields2e2ffu=(1+2ru) 4M2M ru r1 4M2M r2u 4M+1 Similarly, 2e2ffv=(12rv) 4M2M rvU0 r1 4M2M r2vU0 4M+1 121

PAGE 122

2fuv=2ruv 2M+1 2fu2=2ru2 2fv2=2rv2 2M+1 2M+1 2M+1 2M+1 2M+1 2M+1

PAGE 123

[1] J.M.Bardeen,BlackHolesdoEvaporateThermally.Phys.Rev.Lett.46,382(1981). [2] C.Barrabes,W.IsraelThinshellsingeneralrelativityandcosmology:Thelightlikelimit.Phys.Rev.D43,1129(1991). [3] J.D.Bekenstein,BlackHolesandEntropy.Phys.Rev.D7,2333(1973). [4] N.D.Birrel,P.C.W.Davies,QuantumeldsinCurvedspace.CambridgeUniversityPress,Cambridge,1982. [5] P.Candelas,K.W.Howard,Vacuum<2>inSchwarzschildspacetime.Phys.Rev.D29,1618(1984). [6] P.CandelasVacuumpolarizationinSchwarzschildspacetime.Phys.Rev.D21,2185(1980). [7] S.M.Christensen,S.A.Fulling,TraceanamoliesandtheHawkingeect.Phys.Rev.D15,2088(1977). [8] S.M.Christensen,Regularization,renormalization,andcovariantgeodesicpointseparation.Phys.Rev.D17,946(1977). [9] P.C.W.Davies,S.A.Fulling,W.G.Unruh,Energy-momentumtensornearanevaporatingblackhole.Phys.Rev.D13,2720(1976). [10] T.Elster,Vacuumpolarizationnearablackholecreatingparticles.Phys.Lett.A94,205(1983). [11] A.Fabbri,J.Navarro-Salas,ModelingBlackHoleEvaporation.ImperialCollegePress,London,2005. [12] K.Fredenhagan,R.Haag,OnthederivationofHawkingRadiationAssociatedwiththeformationofaBlackhole.Commun.Math.Phys.127,273(1990). [13] S.A.Fulling,AspectsofQuantumFieldTheoryinCurvedSpace-Time.CambridgeUniversitypress,Cambridge,1989. [14] P.Hajicek,W.Israel,What,NoBlackHoleEvaporation?Phys.Lett.80A,382(1980). [15] P.Hajicek,OriginofHawkingradiation.Phys.Rev.D36,1065(1987). [16] S.W.Hawking,ParticlecreationbyBlackholes.Commun.Math.Phys.25,152(1975). [17] S.W.Hawking,Breakdownofpredictabilityingravitationalcollapse.Phys.Rev.D14,2460(1976). 123

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R.M.Wald,QuantumFieldTheoryinCurvedSpacetimeandBlackHoleThermody-namics.TheUniversityofChicagopress,ChicagoandLondon,1994. [36] J.W.York,Jr.,BlackHoleinthermalequilibriumwithascalareld:Theback-reaction.Phys.Rev.D31,775(1985). [37] J.W.York,Jr.,Dynamicaloriginofblack-holeradiance.Phys.Rev.D28,2929(1983). 125

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Full Text

BLACK( HOLE EVAPORATION:
VALIDITY OF QUASI-STATIC APPROXIMATION

By
K(ARTHIK( SHANK(AR

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007

S2007 K~arthik Shankar

To the entire physics community

ACKENOWLED GMENTS

I would like to thank my advisor Prof. Bernard Whiting for his valuable guidance

and insightful comments that helped me complete this work. I would like to thank all

my teachers, in particular all the professors at UF physics who contributed to my physics

education. In this respect, I would specifically like to thank Prof. John K~lauder, Prof.

James Fry and Prof. Pierre Ramond. I am extremely grateful to Prof. Steve Detweiler

and Prof. Richard Woodard for their time and availability to help me clarify most

of the fundamental concepts in physics. I would also like to acknowledge my friends

and colleagues Dr. Aparna Baskaran, Mr. Anand Balaraman, Mr. Ian Vega and Dr.

K~yongchul K~ong, for the time they spent with me in stimulating physics discussions. Most

of all, I would like to thank my parents for being supportive of my education throughout

my life.

page

ACKNOWLEDGMENTS

LIST OF TABLES.

LIST OF FIGURES

ABSTRACT

1 INTRODUCTION

1.1 Classical Black Holes.
1.1.1 Gravitational Collapse
1.1.2 Theorems
1.1.3 Black Hole Dynamics.
1.2.1 Particle Creation
1.2.2 Back Scattering.
1.2.3 Quantum Stress Tensor.
1.3 Back Reaction.
1.3.1 Quasi-static (Qs) Approximation.
1.3.2 Motivation For The Problem: Is The
1.:3.3 Violation Of WEC
1.3.4 Method Of Approach.

Approximation

Valid?

2 MODEL.

2.1 Classical Metric
2.1.1 Schwarzschild Exterior
2.1.1.1 Jump conditions on the Null shell
2.1.1.2 Gaugfe choice
2.1.2 General Exterior .
2.2 Quantum Field
2.2.1 (T,,,) In Two Dimensions.
2.2.2 4D-Model
2.2.3 Stress Tensor In Schwarzschild Exterior..

:3 QUASI-STATIC APPROXIMATION.

Constructing The Metric
Computing The Energy Flux .
Analysis.
The Complete Quasi-Static Geometry

CHAPTER

4 NITAERICAL EVOLUTION

4.1 Algorithm.
4.1.1 Initial Data
4.1.2 Constraint Equations.
4.1.3 Algorithm
4.2 Testing The Code.
4.2.1 Accumulated Error
4.2.2 Constraint Violation
4.2.3 Correspondence Cl...1I : With
4.2.4 Validity Of Gauge Clun...I ~ At
4.3 Retrieving Output
4.3.1 Trace A Constant r Surface
4.3.2 Apparent Horizon
4.3.3 Negative Energy Density.
4.3.4 Energy Flux.

Schwarzschild Geometry
The Surface S2

5 RESITLTS AND DISCUSSION

Position Of Apparent Horizon
Violation Of WEC
Comparison Of Mass Loss
Apparent Horizon mass.

6 SITAINARY AND CONCLUSION .......... ...

6.1 Future Directions.

APPENDIX

A INTERPOLATION .......... .......

B DERIVATIVE TERMS ......... .. .

B.1 Nonlinear Equations ..........
B.2 Linearized Equations ..........
B.:3 Schwarzschild Derivatives ..........

REFERENCES ........_. ....._.

114

114
116i
119

BIOGRAPHICAL SK(ETCH.. .

. .. 126

LIST OF TABLES

Table page

5-1 Position of the Bulge ........ . .. 90

5-2 Position of markers ........ . .. 92

5-3 P(a~) and S(a~) ........ .. .. 99

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11

5-12

Constant r curves

Constant r curves

AH and Tvy .

Mass loss a~=5 .

Mass loss a~=4 .

Mass loss a~=3 .

Mass loss a~=2 .

Mass loss a~=1

Mass loss a~=0.5 .

P(a~) plot ....

S(a~) plot ....

Bondi mass ...

of Schwarzschild

of evaporating BH

. .

. .

. .

. .

. .

. .

. .

.

.

. .

LIST OF FIGURES

re

G

Pc

Figu:

1-1

1-2

1-3

1-4

1-5

1-6

3-1

3-2

4-1

page

12

13

23

29

32

34

58

67

74

77

88

89

91

93

94

95

96i

97

98

100

100

101

ravitational collapse ...

enrose diagram for a collapse geometry

Geometric optics approximation .

Evaporation geometry .....

Quasistatic geometry 1 ......

Collapse and Evaporation without

Quasi-static geometry 2 .....

Quasi-static geometry 3 .....

Algorithm for evolution .....

Igularity

. .

.

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

BLACK( HOLE EVAPORATION:
VALIDITY OF QUASI-STATIC APPROXIMATION

By

K~arthik Shankar

May 2007

Cl.! ny~: Bernard F. Whiting
Major: Physics

Hawking's discovery that a black hole quantum mechanically radiates energy like a

black body -II_e-r- -is that its mass should decrease, leading to a process known as black

hole evaporation. Solving for the evaporating black hole geometry (that is, its metric)

exactly doesn't seem possible because it involves many complications. The most serious

complication is that we do not have an analytic functional form for the quantum stress

energy tensor in terms of the unknown metric. One approach to solving this problem

is to use a Quasi-static approximation, which assumes that the evaporating black hole

at every instant can be approximated by a stationary black hole. Effectively, it assumes

that the luminosity of the black hole at any instant goes as 1/M~2, Where M~ is the mass

of the black hole at that instant. In this dissertation, the validity of this approximation

is examined in the context of a simple model where exact numerical calculations can be

performed. In this model, we assume an analytic form for the quantum stress energy

tensor in terms of the unknown metric. This model is a four dimensional extension of the

two dimensional black hole geometry originally investigated by Unruh, Fulling and Davies.

We explicitly compare the results obtained from the quasi-static approximation and the

exact numerical calculation performed in this model. We observe that there is a significant

difference between the quasi-static approximation and the exact calculation whenever the

quantum effects are large. When the quantum effects are very small, as in .I-1 in thli-- scal

black holes, the quasi-static approximation matches the exact calculations very closely.

CHAPTER 1
INTRODUCTION

1.1 Classical Black Holes

General relativity views gravity as a manifestation of curvature in the geometry. The

spacetime is not flat. There exists a metric on the spacetime manifold and test particles

move on geodesics with respect to this metric. Gravity, which is now defined by the metric

on the manifold is in turn produced by the matter. In Newtonian gravity, the sources were

just masses (single component). In General relativity, the source is a 10 component object

called the stress-Energy Tensor. The equation which relates the stress energy tensor to the

curvature of the metric is called Einstein's field equation.

8xGC

Here, G,,, is the Einstein tensor which can he calculated from the metric and T,,, is

the Stress energy tensor. In this entire thesis, we will work in a system of units called

geometrized units, where G = c = 1. With respect to this system, length, mass and time,

all have the same dimensions. The Einstein equations are then G,,, = 8xT,, .'

There are some very interesting solutions to Einstein's field equations known as

black holes. There is a variety of such solutions. But all these solutions carry a common

property, namely, a section of the spacetime manifold ( 11,- BH) is causally disconnected

from the rest of the manifold ( 11,- Ext), that is, no information from the section BH

can leak out to the other section Ext. In other words, no timelike or null curve can

originate inside BH and enter Ext, but there exist timelike and null curves that originated

in the Ext and enter BH. The boundary between Ext and BH is commonly known as the

Event horizon. Another common feature of these black hole solutions is the existence of

singularities. A point in the spacetime is supposed to be singular if there exists a geodesic

which cannot he continued through that point. In most situations, this happens at a point

where the curvature tensor diverges.

The mathematical existence of such black hole solutions does not tell us how these

black holes are physically formed. To understand that, in section 1.1.1, we will briefly

go over the key points involved in a gravitational collapse. In this respect, we will also

introduce the notion of Penrose diagrams. Representing the causal structure of the

spacetime geometry using Penrose diagrams is very useful in interpreting many of the

physical properties of the geometry, and they will be frequently used in this work.

To better understand the singularities, black holes and the collapse process that gives

rise to black holes, there is a number of theorems we need to be familiar with. We will

spend Section 1.1.2 and Section 1.1.3 doing that. In Section 1.1.2, we will briefly discuss

the relevant theorems which categorized the various black hole solutions, and in Section

1.1.3, we will go over the classical laws of black hole mechanics which resemble the laws of

thermodynamics.

1.1.1 Gravitational Collapse

The physical picture of a star collapsing under its own gravity to form a black hole is

fairly simple. If the star is so massive that the matter pressure in it is not strong enough

to overcome gravity, the star will collapse until it becomes a single point with infinite

density. Let us see what General relativity has to ;?i about such a collapse. To keep

things simple, let us restrict ourselves to spherical collapse.

There is a simple theorem by Birkhoff which is very useful in the context of collapse.

It states that the metric on the manifold is static and uniquely determined if we assume

that the spacetime is spherically symmetric and that there is no matter (T,,, = 0). In

particular, when a spherical star collapses symmetrically, its exterior geometry has to be

static (independent of time) and the metric outside the surface of the star is given by the

Schwarschild solution.

d~s2 ___ 2 d2I 2 ~2~22
r. (1 12

Figure 1-1. Gravitational collapse of spherical star which results in formation of a black
hole.

A thorough description of the Schwarzschild geometry and the various coordinate

systems generally used to describe it can be found in any standard general relativity text

book (for eg. [23], [34]). Hence let us not discuss these details. Nevertheless, we would

like to explicitly mention one of the important properties of Schwarzschild geometry. The

r = constant surfaces are timelike surfaces outside the event horizon (r = 2M~) and they are

spacelike surfaces inside the event horizon. This implies that any timelike curve inside the

horizon should reach the singularity at r = 0. Hence any r = constant surface inside the

horizon is a Trapped surface, basically because even light cannot move out in the direction

of increasing r. In a general geometry, a Trapped surface is defined as a closed spacelike

2-surface on which all null geodesic congruences have a positive convergence.

Coming back to the collapse of a spherical star, Birkhoff's theorem tells us that the

geometry outside the surface of the star is the Schwarzschild geometry. This means, once

the surface of the star shrinks to a size less than its gravitational radius (r = 2M~), an

event horizon forms and outgoing light rays from the surface of the star will move in the

direction of decreasing r. In Fig. 1-1, we trace the outgoing light rays from the point r = 0

r=o

Figure 1-2. Penrose diagram representing a gravitational collapse which results in
formation of a black hole.

at various stages of collapse. The x-axis is the r coordinate and the y-axis corresponds to

some time coordinate such that each of the t = constant surfaces are spacelike surfaces.

We can clearly see that the rays that come out of r = 0 after a particular time, never get

out to infinity. They reach a maximum value of r, then they turn back and move in the

direction of decreasing r. The null ray which cr li- at a constant value of r after leaving

the surface of the star is the Event horizon. The formation of a singularity is shown in the

figure by thickening the r = 0 line after a particular time.

Let us now look at the behavior of r = constant surfaces in the collapse geometry.

Let us first draw the conformal diagram (Penrose diagram) for the spherically symmetric

collapse geometry. Look at Fig. 1-2. At every point, the null cones are 450 lines. The

surface of the star R(t) is shown in red. The various r = constant curves are shown in

the diagram. The boundary of the diagram is the future and past null infinities, I+ and

I- respectively. The point where r=0 line becomes horizontal is the beginning of the

singularity. The event horizon, denoted by H, coincides with the curve r = 2M~ in the

exterior of the star surface. The radius of the star's surface decreases from infinity and

crosses the event horizon. Observe that the r= constant curves outside the surface of

star and inside the event horizon are spacelike. So, once the surface of the star crosses

its gravitational radius, it has no choice but to hit the singularity. So, classical general

relativity predicts the fate of a sufficiently massive star to be a singularity.

1.1.2 Theorems

To have a reasonably complete discussion, it is necessary to mention some of the

milestone theorems proved with respect to collapse and classical black holes. A thorough

description of these theorems can he found in [18].

Israel(Uniqueness theorem): A static black hole in vacuum spacetinle necessarily has

to be spherically syninetric, that is, it must correspond to the Schwarzschild solution.

Hawking: A Stationary black hole has to be either Static or Axially syninetric.

Carter: A stationary and axially syninetric black hole in vacuum has a unique form

given by the K~err Solution.

No hair theorem: A stationary black hole in vacuum is uniquely determined by its

mass, charge and angular montentunt. An external observer cannot find anything about

the black hole other than these three parameters. That is, an observer at future infinity

has no way of finding whether the black hole was formed out of collapse of Television

sets or collapse of cars. Hence, with respect to an external observer, there is a huge

information loss when black holes are formed.

Price: During a realistic gravitational collapse, all modes of perturbations are

radiated away as gravitational waves and the spacetinle will eventually settle down to a

stationary geometry.

All the above theorems lead to a wonderful implication: The final result of any

gravitational collapse is a stationary black hole characterized by its mass, charge and

angular montentunt. In particular, if the initial angular montentunt of the star is zero,

and if the star is uncharged, then the final black hole geometry is just the Schwarzschild

geometry.

Weak energy condition (WEC) is the requirement that the local energy density

measured by any observer must he positive. mathematically this amounts to asking for

T,,W"W^TT" > 0 for all possible timelike and null IT'^ vectors.

Singularity theorems : There are a few of these theorems. The first of these is

given by Penrose [28]. It states that a singularity will necessarily form if the following

three conditions are satisfied, (i) Weak Energy condition is satisfied, (ii) There exists

a non-compact Co 111v: surface in the space-time manifold, (iii) there exists a trapped

surface in the manifold.

This is a very powerful theorem in the context of collapse, for it states that the

moment the first trapped surface forms within a collapsing star, its fate is determined to

be singular.

Hawking: Two-Surface area of the event horizon never decreases, if Weak Energy

condition is satisfied. This theorem is very general and it does not require the black hole

to be stationary, nor does it require it to be a vacuum solution.

1.1.3 Black Hole Dynamics

Before we discuss the consequences of quantum effects on black holes, let us first

briefly go over some important properties of classical black holes. Four laws were derived

according to which the black holes behave. These had an excellent mathematical analogy

with the laws of thermodynamics. A good description of these laws can he found in [29].

They are briefly summarized below.

1. The surface gravity n of a stationary black hole 1 is uniform over the event
horizon. {n2 -~ e, For Schwarzschild black hole, a 1/4M}). The

analogous statement in thermodynamics is that a system in equilibrium has uniform
temperature. Thus, we make a connection between surface gravity and temperature.

1 The geometry should have a killing vector (0, which is timelike outside the event
horizon.

2. If the charge Q, mass Af, angular momentum .7, of a stationary black hole changes
infinitesimally due to some process of external interference, then the area of the
event horizon changes by

SM=6,4 + RHb 6 -
87r

This equation is analogous to the first law of thermodynamics wherein Af is
proportional to energy, A is proportional to entropy, a is proportional to temperature,
and the last two terms of the equation correspond to the work done in the
quasi-static limit.

3. If null energy condition is satisfied, then the surface area of the black hole never
decreases 64 '> 0. This is just Hawking's Area theorem. This is analogous to the
second law of thermodynamics, where a system's entropy ahr-l- .- increases if it
absorbs positive energy from the surroundings. It can also be shown that the total
entropy of the black hole and the external universe together should ah-- .1-< increase.
This is called the Generalized Second Law (GSL).

4. The a of the black hole cannot he reduced to zero within a finite advanced time.
The thermodynamic analogue of this statement is that no system can he brought
down to zero temperature in finite time.

Bekenstein [3] showed that these laws hear more than just mathematical resemblance

with thermodynamics. He argued that information content is thermodynamically nothing

but entropy, and the information content in a black hole can he evaluated by considering

the bits of information lost into it during collapse. With this argument, he finds that the

entropy of a blacks hole is ~(In 2)4/h..

The analogy between black holes and thermodynamics is further strengthened when

Hawking showed that black holes would radiate as though it were a black hody with

a temnperatur of s~. If we assume that this quantity is the correct expression for Ithe

temperature of the black hole, then by law 2, the correct expression for entropy should be

A/4h, which is very close to the result of Bekenstein's argument. So, it appears that there

should be some deeper connection between thermodynamics and black holes. With these

four powerful laws we can ask the question : How does the event horizon behave after the

collapse? Is it possible to extract energy from it?

By analyzing the above laws, one can come to the conclusion that energy can he

extracted from a charged rotating black hole until the black hole stops rotating, O = 0

and the black hole becomes neutral, Q = 0. The processes which leave the entropy (Area)

unchanged are called reversible processes. These processes can of course change the mass,

charge and angular momentum, but with a constraint that

Q2
4.1T 41T-'

where -T T = (T{i)1/2. This constraints is derived fr~om the second lawv. By dlefinition, If

is invariant under a reversible process. An irreversible process is one which increases if .

It is clear that when the maximum possible energy is extracted from the black hole, if i,

would be its final mass. Hence, as a consequence of the second law, no energy can he

removed from a Schwarzschild black hole.

Let us now take a small digression to emphasize that there exist at least two well

known processes by which energy can he extracted out of a K~err black hole until it looses

all its angular momentum and becomes a Schwarzschild black hole. For details, refer to

[23].

Penrose process : Let a particle get into the ergosphere of a rotating black hole

and break into two pieces (;?i a homb explodes). Let one piece have a positive energy

with respect to the observer at infinity and let the other piece have a negative energy

with respect to infinity. The important point to note is that, inside the ergosphere, there

can exist a timelike four velocity for which the energy is negative (which is not possible

outside the ergosphere). Now, when the positive energy particle comes out to infinity, by

conservation of energy, the particle has more energy than the initial particle. So, we have

succeeded in extracting energy from a black hole which has an ergosphere (R / 0). Note

that, there is no such ergosphere in a Schwarzschild black hole.

Superradiance (Misner effect) : Consider a classical wave incident on a rotating

black hole. It can been shown that, if the wave has the same direction of angular

momentum as the black hole, then it get scattered off with an increased amplitude

and in the process reducing the angular momentum of the black hole. Thus, again we have

retrieved energy from a rotating black hole.

The above discussions were all about classical processes. One main assumption that

goes into our conclusion that no energy can be extracted from a Schwarzschild black hole

is that the WEC is satisfied. All known classical matter satisfy this assumption. But

quantum fields violate the WEC. Hence our conclusion is not justified when quantum

fields are present in the black hole geometry. In fact, in 1975, Hawking showed that even

Schwarzschild black holes would radiate energy quantum mechanically [16]. The result of

this process is referred to as Hawking radiation.

1.2.1 Particle Creation

Hawking's original derivation of black hole radiation [16] was followed by mathematically

more precise derivations [12] that confirmed his predictions on black hole radiation. In this

Section, we shall go over the key steps involved in Hawking's original derivation. Let us

start with a brief discussion on the process of particle creation in curved spacetime. In flat

spacetime, the concept of particles arise from second quantization of fields. We shall first

generalize the concept of 1. rI~; 1. to curved spacetime. A detailed description of field

theory in curved space, can be found in [35], [13] and [4]. To keep things simple, we will

restrict ourselves to the simplest of the field theories, a real, massless, scalar field.

1. We first solve the classical field equations V,VM@(x) 0 We then define the
K~lein-Gordon inner product of two solutions cp(x), (x) as

The field equations guarantee that the above integral when evaluated over any
Cauchy surface E gives the same result. We define H to be the Hilbert space of
smooth solutions to the field equations which vanish sufficiently rapidly at spatial
infinity and have a finite K~lein-Gordon norm. We then obtain a orthogonal basis
to H. Let us call the individual elements of the basis set mode functions. These
mode functions are split into two sets, positive frequency modes {ui} and negative

frequency~^ mods {* such that (ui, uj) = 6ij, (uf* up = Tad(s p=0 n
flat spacetime, these modes actually correspond to plane waves exp(ik.x' iwt). The
modes with w > 0 are the positive frequency modes and the modes with w < 0 are
the negative frequency modes. Moreover, in flat spacetime, the splitting into positive
and negative frequency modes is natural, in the sense that, with respect to the global
Minkowski time coordinate t, the positive frequency solutions have the property that

In an arbitrary space time, which does not have a natural time axis, there is no such
natural splitting.

2. We then expand the field operator #(x) with respect to the mode functions in terms
of creation and annihilation operators as

3. Now, the canonical quantization procedure leads to the following commutation
relations betweePn the creation and1 annihilation operators. [as a l] b e, and [ai, aj]
r~[a al] 0. Note that, the quantization procedure very much depends upon our
choice of the mode functions.

4. Let us now construct the Fock space, which is defined as the space of all possible
states of the quantum field.
The vacuum state |vac) or the zero particle state is the one which is annihilated by
all the annihilation operators, that is,

alvac) = 0.

If we index the set of mode functions by the momentum associated with that mode,
then the single particle state, characterized by its momentum p, is defined as the
state created by th~e creation operator a) when? it acts on? th~e vacuum? state |vac)j.
Extending this definition, the multi particle states are defined by a set of creation
operators acting on the vacuum state. For example, a three particle state with
m~omenta py, p2 93 is defined as t ~a~a 3 |Ivac).(Since these particles are bosons, we
symmetrize with respect to pl, p2,p3). The set of all possible multi particle states
forms the basis of the Fock space.2

2 The basis states can be normalized by a multiplicative factor.

To calculate the number of particles in a given state, let us define the number

operator as NV = Eafai, so that in any given state | ), the expectation value of the total

number of particles can be obtained as ( |N| ~).
Let us now choose another set of mode functions, ;?i {vyj,'* v]} andcll the creation

and annihilation operators with respect to these mode functions as by and b!, so that, the

field expansion is,

#(x) =i E(byvy.i(x) + b~lv]*(x)).

Quantization with respect to any set of mode functions would ultimately give rise

to the same Fock space, but the basis of the Fock space constructed in terms of multi

particle states would differ depending on our choice of mode functions. In other words, the

definition of "Particle" depends on our choice of mode functions. The vacuum state |vac)

with respect to the old set of mode functions {ui, of }, is no longer a zero particle state

with respect to these new mode functions {vj, vj}. To calculate the expectation value of

the number of particles, we make use of the Bogoluboy transformations [11], [4]. It turns

out that,

(v~ac|Ey3b3byuvac) = Egy|Ay|2 ~

where pij is defined as the inner product of the two sets of mode functions, Pij = (I uj*

This is essentially the concept of la Irticle creation".

Given a background geometry and two sets of mode functions, we now know the

procedure to calculate the particle creation. At the first glance, the concept of particle

creation seems to be a purely mathematical artifact, because it ultimately just depends on

the choice of the mode functions, and there is no natural choice of mode functions except

in flat spacetime. But that is not true, there are some situations where the concept of

particle creation does have a physical meaning. To understand this, we should first note

an important property of the mode functions: if the mode functions are fixed on any one

Cauchy surface, the field equations determine them everywhere in spacetime. Let us now

ask how we should choose the mode functions on a surface so that corresponding states

have the correct physical meaning with respect to the relevant observers? If this surface is

from a region in spacetime which is almost flat, then we know the answer to that question:

We make the natural choice of mode functions as in flat spacetime. Otherwise, we do not

Let us now consider a spacetime which is almost flat at early times (t = -oo) and late

times (t = co). Consider two spacelike Cauchy surfaces at these two times C_, and E.

respectively. The natural choice of mode functions on these two surfaces may not match,

because the spacetime is not flat everywhere. That is, when one set of mode functions

is propagated through time from one C 111 hv: surface to the other, they need not match

the other set of mode functions. This might lead to a nonzero Bugoluboy coefficient Pij.

Hence, the vacuum state with respect to an observer at early times on the surface E_,

would be seen as a nonzero particle state by an observer at late times on the surface Em.

A variety of examples of particle production has been worked out. Particle creation

effects on different cosmological backgrounds have been calculated [4]. Unruh calculated

the particle production in flat spacetime with respect to an accelerated observer [32]. It

turns out that the accelerated observer would see a thermal bath of particles. Hawking

applied the concept of particle creation to the gravitational collapse geometry, and he

found that an observer at late times would observe particles in accordance to a thermal

spectrum [16], [17]. Let us now briefly discuss Hawking's result.

In a real collapse geometry as in Fig. 1-2, we have two .I-imptotically flat surfaces

{I-} and {I+}, and hence two sets of inertial flat space observers. {I-} is a Cauchy

surface by itself, and {I+, H} together form a Cauchy surface. The observer at I- will

choose his natural set of mode functions form such that

fwim ~ e-'^E(0, ~) on I-.

The observer at I+ will choose a natural set of mode functions pwim such that

and panz = 0 on H+. For brevity, we will suppress the angular dependence in the modes,

but we should keep in mind that all of the following arguments apply to all the (1, m)

modes. We see that f, forms a complete basis, but p, does not form a complete basis hv

itself. To specify the basis completely, we have to specify some in falling modes q.. with

nonzero behavior on the horizon. The form of q, is not needed to compute the particle

creation at I+. We should just make sure that whatever he the q, we choose, it should

have zero data at I+.

Our aim is to find the number of particles observed at I+ as a function of the

retarded time u. Since the mode functions p., do not have a compact support at I+, when

we calculate the particle creation, the particles will be delocalized everywhere on I+. To

calculate the number of particles observed at I+ as a function of time, the mode functions

p, should not he used. We have to construct a complete set of wave packets out of these

mode functions. Let us use P"\' to denote a wave packet of characteristic frequency wo and

peaked around uo with a width of Auo. We will not get into construction of these wave

packets, a detailed description can he found in [11].

Let us consider the field to be in a vacuum state with respect to the observer at I-.

To compute the number of particles emitted at I+ in a time interval Auo around the time

uo, with frequency wo, we need to calculate the Bugolubov coefficient P,",. = _(~R p ),

which involves evaluating the inner product integral on a Cauchy surface. We will evaluate

the inner product on I-. For this, we first need to know the behavior of the modes POo at

I-.

The way to obtain the functional form of POo at I- is to back propagate it from I'

using the classical field equations. When we back propagate the modes front late times uo,

these modes would travel very close to the the event horizon r ~ 2M~, and are the highly

blue-shifted. We now invoke Geometric optics approximation [23], [34], which ;7 a that,

if the effective wavelength of the wave is very short compared to the curvature scale of

the geometry, the surfaces of constant phase of a wave can he approxiniated by null rays.

Figure 1-3. Diagram representing Geometric optics approximation by which modes at I+
are traced back to I-.

Hence, we can back propagate the P,"O modes using null rays. This is pictorially shown in

the Fig. 1-3.

1.2.2 Back Scattering

The classical K~lein-Gordon field equation in Schwarzschild geometry can be viewed as

a free field equation (as in flat spacetime) with a potential term. The potential term peaks

at r = 3M~ and vanishes at r = 2M~ and r = 00. It depends on the 1 value of the wave

mode. The potential is larger for a higher value of 1.

If we neglect the potential, then we can freely propagate the wave through spacetime

to get the functional form of P,"o at I- (that is, we do not even need the help of

Geometric optics approximation). By evaluating the inner product integral at I-, we

can compute the Bugoluboy coefficient P,",O, and hence the number of particles emitted. It

turns out that at sufficiently late times (large uo), the flux of particles (number of particles

emitted per unit time) at I+ is a constant (independent of the time uo).

8,= 8rMw J

The flux is similar to a thermal flux, in the sense that it obeys Planck distribution with a

temperature

TH

Since we neglected the effects of the potential term which depends on the 1 value of the

modes, we will find that every 1 mode contributes to the energy flux equally. The energy

flux or the luminosity of the black hole in each I mode is

21 +1 oo he-0 (21 + 1)h
LI = dw =
2x o e8rr^^ 1 768x.11-

If we sunt over all the I modes, we see that the total luminosity of the black hole is

infinite. This happens because we neglected the effects of the potential.

Let us now include the effects of the potential for each I mode. Since, the potential

term is larger for a higher value of 1, the higher 1 modes will be back scattered more. It

turns out that including the effect of back scattering modifies the results of particle flux

and luminosity by just a niultiplicative factor T,I.

Tl (21 + 1) 8
n, = LI= TI.
es"r^n/r 1 768;r11-

This can he physically understood in the following way. A fraction RI of the

back propagated mode from I+ is reflected by the potential back to I- with the same

frequency. This piece of back propagated mode does not contribute to particle creation

when we calculate the Bugolubov coefficient integral at I-. The other fraction(T,I)

propagates along null rays (geometric optics approximation) close to the horizon and gets

reflected at r = 0 back to I-. It is this piece of the back propagated mode that contributes

to particle creation when we calculate the Bugolubov coefficient integral at I-. Hence, it is

understandable how the transmission coefficient (TI) enters the results.

For each mode w, 1, we can numerically compute the transmission coefficient by

solving the K~lien-Gordon wave equations with appropriate boundary conditions. For a

given value of w, TI 0 as I becomes large. If we compute the total luminosity of the

black hole by summing over all the I modes, it now turns out to be finite [25]. Explicit

numerical computations have been performed [10] and it turns out that the luminosity of

the 1 = 0 mode is

1.625
Clo =7680x.11- '

and the total luminosity of the black hole is

L= 1.795
7680;i~l-

It is interesting to note that more than 90 percent of the black hole luminosity is from

the s-wave modes (1 = 0). Another point to note is the importance of backscattering.

Without back cl Ir1 r1 ),! the s-wave luminosity would be around five to ten times 1

than what it should be. Yet another point we would like to note is that, even though the

magnitude of luminosity is strongly affected if we neglect back scattering, the functional

form of luminosity (L ~ &/M~2) is nevertheless preserved. So, in a sense, neglecting back

scattering is acceptable if we are interested in studying just the qualitative aspects of

1.2.3 Quantum Stress Tensor

So far, we used the concept of 1' 'I'ticle creation" to discuss black hole radiation. But

as we discussed in Section 1.2.1, the concept of II 'I'ticle" has a physical meaning only

with respect to inertial observers at I- and I+. Let us now discuss a more rigorous way to

approach the problem of black hole radiation.

In General relativity, the covariant stress energy tensor of the matter field at any

point in space time contains all the physically important information about the energy

fluxes and momentum fluxes. The quantum field which apparently contributes to an

energy flux at I+ (as seen in Section 1.2.2), should have a corresponding stress energy

tensor. Fr-om the Lagrangian of the K~lein-Gordon field, we see that classically, it has

followingf stress tensor.
I,, dd,~- ,gdd"1

When we quantize the field, the stress energy tensor becomes an operator. Quantum

theory then tells us that the physically measured stress energy tensor is given by the

expectation value of this operator ( ,).

Unfortunately, calculation of the expectation value of the stress energy tensor with

respect to any state in curved space time is rather complicated. Even in flat space, when

the expectation value of stress tensor operator is calculated, we get an infinite result.

This is because when we sunt over all the modes, the zero point energy of each of them

add up to give infinity. No matter which state we calculate the expectation value of the

stress tensor, this infinite zero point energy is ah-- .--4 a problem. To rectify the problem,

we use the argument that, the absolute energy of a state is really of no consequence, it

is the difference in energy between various states that is physically measurable. So, it is

justified to ignore the infinite zero point energy. Since in flat space, an inertial observer is

supposed to measure zero energy in the vacuum state (ground state with zero particles),

the physically measured energy in any other state can he computed by calculating the

expectation value of the stress energy tensor in that state and then subtracting out the

infinite zero point energy which appears while calculating the expectation value of the

stress tensor in the vacuum state. This process of subtracting out "infinity" is called

Renornialization. 1\athentatically, this renornialization is done by 1...)~! II.. orderingt the

stress tensor, which amounts to rearranging the creation and annihilation operators which

appear in the field expansion in such a way that all the creation operators are to the right

of all the annihilation operators. We represent the normal ordered stress tensor as : T, :.

The expectation value of the normal ordered stress tensor with respect to any state |4)

is then (4| : T,,, : |4). Unfortunately, this process of normal ordering is not a covariant

procedure. Hence the resulting expectation value of the normal ordered stress tensor will

not he a tensor. A correct procedure for renornialization has to be adapted to obtain a

covariant stress energy tensor. There are various methods of doing this : dimensional

regularization, covariant geodesic point split regularization [8], ... A detailed description of

all these methods are given [4], [13]. Without getting into the details of these procedures,

we shall go over the results relevant to black hole radiation.

To calculate (I,,), we need to know the state of the quantum field and the background

geometry completely. We do not know the collapse geometry completely, all we know is

that the exterior geometry is Schwarzschild. It turns out that is possible compute ( ,,)

in the exterior Schwarzschild geometry. The expression for (I,,) contains messy integrals.

Using a Gaussian path integral approximation, an approximate analytic expression [24] for

it can he derived. But a complete computation can he performed only numerically [5], [6],

[21].

Let us now briefly discuss the properties of ( ,,) in the exterior section of Schwarzschild

geometry for various states of the quantum field. The region under consideration is

bounded by H-, I- (the past horizon and the past .I-i-inju..l~e null infinity)and H+, I+

(the future horizon and the future .I-i-midal'tic null infinity). The state of the field will be

described with respect to inertial observers on these surfaces.

Boulware vacuum: This state corresponds to zero particle state with respect to

observers at I- and I+. When (I,,) is evaluated with respect to this state, it turns out

that it vanishes at I- and I+ as expected. But, near the horizon, when evaluated in a

locally inertial coordinate system, (I,,) blows up. This tells us that Boulware vacuum

state is physically unstable.

Hartle-Hawking vacuum: On the horizon H-, the K~ruskal U coordinate is a

locally inertial coordinate and on the horizon H+, the K~ruskal V coordinate is a locally

inertial coordinate. Hartle-Hawking state is the vacuum state with respect to the standard

mode functions exp(iwUi) on H- and exp(iwV/) on H+. Note that, even though U and V

are locally inertial coordinates on the horizon, this state does not physically correspond

to a zero particle state at the horizon, because the geometry near the horizon is not flat.

In fact when ( ,,) is calculated, it turns out to be nonzero and negative near the horizon.

At I+ and I-, the energy density turns out to be a constant. There is an outgoing energy

flux at I+ matched exactly by an ingoing energy flux at I-. Hence, this state represents a

black hole in thermal equilibrium.

Unruh vacuum: This state corresponds to vacuum state with respect to locally

inertial observers at I- and H-. The stress tensor ( ,,) is regular everywhere on the

horizon. Moreover, we find a positive energy flux at I+, and negative energy flux at H*.

Note that, this state closely resembles the "in" vacuum state of the collapse geometry (in

a collapse geometry, we do not have a H-). Hence, we expect this state to reproduce the

radiation seen in the collapse geometry. In fact, the energy flux at I+ has the same form

as what we obtained using the concept of pI Ioticle". This -II__- -r- that Hawking's original

treatment of the problem is indeed valid.

Further discussion on the Quantum stress tensor given in C'!. Ilter 2.

1.3 Back Reaction

Fr-om Section 1.2, we learn that when we apply quantum field theory on a fixed

black hole geometry, we see that the black hole radiates with a constant luminosity at

late times. This implies that the total energy radiated out is infinite. This obviously is

~~l~rn because conservation of energy tells us that the energy radiated out of the black

hole should somehow be compensated by the decrease in mass of the black hole. Where

did we go wrong? As discussed in Section 1.2.3, calculation of the quantum stress energy

tensor in the Unruh vacuum state shows a negative energy flux across the horizon. This

ingoing negative energy should decrease the mass of the black hole. This process is called

Black hole evaporation. A genuine way to treat this problem within general relativity is

to consider the back reaction effect of the quantum stress tensor on the geometry. That

is, the geometry outside the collapsing star is not Schwarzschild, but is determined by the

quantum stress tensor as dictated by the semi-classical Einstein field equations.

Figure 1-4. Penrose diagram representing a gravitational collapse, formation of a black
hole and evaporation of the black hole.

Since, as mentioned in Section 1.2.3, we do not have a generic expression for the

quantum stress tensor in terms of the metric, we cannot solve the above equation to get

a solution to the evaporating black hole geometry. Before considering owsi~ to tackle this

problem, let us look at the conformal diagram of the evaporating black hole geometry.

We shall assume that the black hole evaporates completely so that its singularity

finally disappears and that after all the radiation has escaped, the geometry is flat.

The conformal diagram representing such a situation is shown in Fig. 1-4. r = constant

curves of this geometry are shown in black. The large r curves are everywhere timelike as

expected. For small values of r, including r = 0, the r = constant curves are initially time

like, then become spacelike, and once again become timelike. The point where r = 0 curve

turns spacelike indicates the formation of singularity. The point where the r = 0 curve

turns back to a timelike curve indicates the completion of evaporation. Event horizon

(EH) is the outgoing null ray (in red) that starts from the initial timelike piece of the r=0

curve and ends at the point where it evaporation ends. The region where the r = constant

curves are spacelike is the trapped region. That is, any timelike or null curve will have to

move in the direction of decreasing r. The boundary of trapped region AH is called the

apparent horizon.

A crucial point to note is that the apparent horizon is not confined within the event

horizon. That is to ?-w that the trapped region extends beyond the event horizon. We

shall call this region the Bulge. Note that, in a collapse geometry without evaporation as

in Fig. 1-2, there exists no such bulge.

1.3.1 Quasi-static (Qs) Approximation

Let us now get to the problem of solving for the completely back reacting geometry.

We already noted that solving this problem completely is not possible since we do not

have the form of the quantum stress tensor in an arbitrary metric. The problem with

calculating the energy radiated out of a fixed background is that the conservation of

energy is violated. A commonly accepted method to fix this problem is the Quasi-static

approximation. It wei~ that, since the temperature of macroscopic black holes is very low,

the process of evaporation is quasi-static. That is, the radiation from the black hole at

any instant can be viewed as thermal radiation from a static black hole of mass M~ with a

temperature T ~ 1/M~. Then, the conservation of energy requires that the rate of decrease

in mass of the black hole is equal to the luminosity of the black hole. Since the luminosity

of the black hole goes as h/M~2, We have

dM &

This tells us that the total life time of the black hole revap ~ M3~

In terms of physical units, the temperature of a black hole is roughly 10 8(M,/M,!~oK

and the life time is roughly 10n1(M~/M1,)3yeasS, Where il. is the mass of our sun. As the

evaporation proceeds, the mass decreases, causing the temperature to increase, and hence

the rate of evaporation increases. Since the evaporation is very slow, at any instant, it

might he possible to find a time scale ATr much bigger than the dynamical time scale of

the geometry (A-r > Af), such that (dlf/dt)a-r < Af. If there exist such a time scale,

then it seems reasonable to neglect the back reaction of the quantum field on the geometry

in that period of time A-r and approximate the radiation as that emitted by a static

black hole. Thus, the quasi-static approximation seems to be justified if there exist such

a time scale A-r. The two conditions on the time scale (dlf/dt)aTr < Af and aTr > At

together imply that dlf/dt < 1. In the units where G=c=1, we then have h/Af2 ~i

This so__~-r-;- that the quasi-static approximation should hold good until the late stages

of evaporation, that is, until the black hole mass goes down to Planck scale. During the

late stages of evaporation, when dM/ldt becomes large, the quasi-static approximation does

not make sense. During this stage of evaporation, the black hole is very small and very hot

and the curvature at the horizon is very high (Planck scale). At this stage there are bigger

problems to worry about than validity of quasi-static approximation, the semi classical

Einstein equations are themselves not valid, quantum gravity effects must he included.

1.3.2 Motivation For The Problem: Is The Qs Approximation Valid?

Let us now understand the implications of the quasi-static approximation. If we

note carefully, the basic assumption is that the fully back reacting evaporating black hole

geometry can he considered as a sequence of snapshots of non back reacting black hole

geometries. This approximation implies that, at every instant during evaporation, the

geometry outside the event horizon(in particular, the region responsible for radiation)

can he mapped to a region of Schwarzschild exterior of a given mass. Let us pictorially

represent this in Fig. 1-5. We see that, at any instant (a specific snapshot), the apparent

horizon and the event horizon coincide outside the surface of the star and the radiation

coming out to an observer far away is generated from a Schwarzschild exterior like region.

Let us take a look at the fully back reacting geometry. We see that there is a region

where the apparent horizon bulges out of the event horizon. Various arguments [33],

Figure 1-5. Representation of quasi-static geometry as a sequence of snapshots of
Schwarzschild geometries.

[37], [15] can he found in the literature advocating for the point of view that most the

radiation coming out to an observer far away would be generated from the bulge. That

is, the observer who sees a 1 in r~ chunk of radiation also sees the trapped region. But

in the quasi-static geometry there is no such bulge, so the observer does not see the

radiation coming from the trapped region. This generates suspicion on the validity of the

quasi-static approximation.

So, the point is: Since most of the radiation is generated from bulge, and the

quasi-static geometry does not have a bulge, there are strong reasons to suspect the

quasi-static approximation.

Examining the validity of quasi-static approximation is the aim of this dissertation.

Before, we describe the strategy we are going to follow to address the problem, let us

now briefly discuss the validity of Weak energy condition in the completely back reacting

geometry.

1.3.3 Violation Of WEC

It turns out that we can make substantial conclusions with respect to the validity of

weak energy conditions just by looking at the conformal diagram. For details, refer to [30].

Let us express the spherically symmetric metric in terms of the null coordinates,

u being the ingoing (retarded) null coordinate and t' heing the outgoing (advanced) null

coordinate.

In this coordinate system, the Weak energy condition can he expressed as

From the Einstein equations, we have

-r""v +r,,. f,,. Tn -rear' 2rs, few
r r r r

In order to analyze the weak energy conditions in our evaporating geometry, we first

note that at any point, where an r =constant curve is timelike, we have dr/du < 0 and

dr/dt' > 0. In the trapped region, we have dr/du < 0 and dr/dt' < 0. On the apparent

horizon, we have dr/dt' = 0. Also, note that there is one section of the apparent horizon

where d2T it,2 > 0.

We can then conclude that there exists a region around that section of the apparent

horizon (where d2T it,2 > 0), wherein the weak energy condition should be violated. In

particular, we will have T,,. < 0.

What does it physically mean to have a WEC violation? It means that (i) there

exists inertial observers for whom the local energy density is negative and (ii) any inertial

observer in these regions will measure a nonzero energy flux, in this case, a negative

energy flux goes towards the center.

The violation of weak energy condition in this region prompts us to consider an

interesting alternative, where there is actually no singularity or event horizon because the

singularity theorems are not applicable. The conformal diagram for such a situation is

shown in Fig. 1-6. The trapped region forms and disappears in a finite amount of time

and the weak energy condition is violated in a neighborhood of the apparent horizon.

Even though there is no event horizon, an external observer will measure same kinds of

radiation and even the classical gravitational effects would be the same as a conventional

I

Figure 1-6. Diagram representing collapse without singularity. The circle represents the
apparent horizon.

black hole. So, observationally there will not be any difference until final stages of

evaporation. An interesting model of formation and evaporation of such a singularity

free black hole is given by Hayward [19].

1.3.4 Method Of Approach

Researchers have addressed the problem of including the back reaction of quantum

fields on the background geometry in various owsi~. York [36] calculated the back reaction

effect of the quantum field in Hartle-Hawkingf state on the Schwarzschild background.

He considered the geometry to be a perturbation about the Schwarzschild geometry

and used linearized Einstein equations to solve for the metric perturbations. He used

an analytic approximation [24] to the quantum stress tensor to be the source for the

linearized Einstein equations. He found that the metric perturbations diverge at larget

value distances. That is, there is regime beyond which the linear perturbation theory fails.

He finally concluded that a black hole can be in thermal equilibrium with the quantum

fields only if we apply reasonable boundary conditions at some finite large distance, and

this turns out to be equivalent to placing the black hole in a thermal box of appropriate

temperature.

Let us now focus on the situation where the black hole forms and evaporates.

Tipler [31] addressed the question of whether spherically symmetric black holes really

evaporate quasistatically. By analyzing the null geodesics very close to the event horizon,

he concluded that the event horizon is so unstable that even a solar mass black hole

would evaporate away in 1 sec. Israel and Hajicek [14] used a model to show that there

exists evaporating black hole solutions with a stable horizon, and hence disproved Tipler's

arguments. Later Bardeen [1] generalized their analysis and pointed out exactly where

Tipler went wrong. Bardeen showed that if there is an outgoing flux from the black

hole at large distances ;?-, LH and if the stress energy tensor on the apparent horizon

is regular, then the rate at which the black hole looses its mass is given by LH, that

is dM/ldt = -LH, where M~ is the apparent horizon mass at any instant. For further

discussion on Bardeen's analysis, see ChI Ilpter 5. We know that the stress tensor on

the horizon is regular when no back reaction is considered (as seen in Unruh vacuum).

For Bardeen's analysis to be applicable to a realistic situation, we need to know if the

stress tensor on the horizon is regular even when the back reaction on the geometry is

considered. This was shown to be true by 1\assar [22]. Thus Bardeen's description of

black hole evaporation is justified.

Bardeen's analysis is applicable for any kind of LH. We know that, if the evaporation

is quasi-static, then LH N 112. The point of this thesis is to examine if LH N 112. The

way in which we address this problem is the following:

We consider a null shell collapsing to form a black hole. The quantum fields are

in the "in" vacuum state. The presence of quantum fields changes the geometry from

Schwarzschild. Since we do not have an explicit expression for the expectation value of

the quantum stress tensor, we use a simple model for the quantum stress tensor which is

inspired from the two dimensional model of Unruh, Fulling and Davies [9]. We explain this

model in detail in Chapter 2. Using this model, we construct the quasi-static geometry

and write the evolution equations in ('! .pter 3. In Chapter 4, we use the Einstein

equations to numerically evolve the initial conditions. This gives us the exact geometry.

We have the quasi-static geometry from ('!! Ilpter 3, and the exact geometry (numerical)

from ('!! Ilpter 4. We compute the energy flux with respect to an observer in both the

geometries and we compare the results in C'! Ilpter 5. It turns out that, whenever the

quantum effects are small (as in the case of astronomical black holes) the quasi-static

approximation holds good. Even though our results are based on calculations with respect

to this simple model, in C'!s Ilter 2 and Chapter 6, we give reasons for why we expect these

results to hold in the general case.

Numerical calculations on this model has previously been performed by Piran and

Parentini [26], to identify the bulge and to evaluate the apparent horizon mass. In ('!! I pter

5, we give arguments for why the apparent horizon mass loss is not a very good quantity

to rely upon in deciding whether the black hole evaporation is quasi-static or not. We also

explain the reason for why the method we choose for the mass loss calculation is more

appropriate show that it yields readily comparable results with quasi-static approximation.

CHAPTER 2
1\ODEL

In this C'!s Ilter, we will construct a model which describes a spherically symmetric

space time containing a radially in falling null shell and a scalar quantum field. Ignoring

the presence of the quantum field would lead to a classical metric which is flat inside the

shell and Schwarzschild outside the shell. But the presence of the quantum field changes

the geometry. The interior remains flat, while the exterior changes to something other

than Schwarzschild. In the Section 2.1, we shall construct the coordinate system in which

the Einstein equations will be evolved. In the Section 2.2 we describe a model for the

stress energy tensor of the quantum field which will later he used as the source terms for

the evolving Einstein equations.

2.1 Classical Metric

The most general spherically symmetric geometry can he described in terms of

two arbitrary functions. We can write the metric in various v- .--s depending on the

coordinate choice. While constructing the coordinate system, it is useful to keep in

mind that the only geometrically invariant quantity at any point in the space-time is

the circumferential radius. 1 Let us first define the circumferential radius. The meaning

of spherical symmetry is that, at any point in space-time, there exists a unique closed,

spacelike two-surface with the property that, the Lie derivative of the metric vanishes

along the tangential directions to the surface (that is, the metric is exactly the same at all

points on that surface). The simplest coordinate system to represent a closed two-surface

is the standard (0, 4) coordinates. Let us label each of these surfaces by a variable r such

that its surface area is 47r T2. This r is the circumferential radius. In many cases, it would

be very convenient to use r itself as one of the coordinates. For example, we can generalize

1 Of course, the proper time of an observer is an invariant quantity, but it would not he
useful to construct a global coordinate system.

the ingoing Eddington-Finkelstein coordinate system so that the metric takes the following

form.

dS2 _62 JU2 2edvdr +. r.'dG (2-1)

The angular part of the metric d82 Sin2 8d2 is represented as dR. The quantities and

m are functions of r and v. In the above metric, v = coast is an ingoing radial null ray

and the outgoing null ray is given by

dr eL 1 2m
dv 2 r

If we take = 0 and m = const, then we get the Schwarzschild black hole. A simple

coordinate transformation will get us back to the usual Schwarzschild coordinates (t, r).

v: = t+ ~rs where lr = r + 2nln 1)

2m 2m

One main drawback of the usual Schwarzschild coordinate system (t, r) is the coordinate

singularity that occurs at r = 2m (event horizon), meaning these coordinates cannot be

continuously extended across the horizon. Hence, we have to use two different patches

of coordinates, one inside the horizon and the other outside. But no such problem exists

with the coordinates of (2-1). Hence, this coordinate system (2-1) will be very good to

deal with situations where a star collapses to form a black hole. Bardeen's use of this

coordinate system [1] in analyzing the change in properties of the null geodesics close

to the horizon due to the back reaction of a stress tensor on the geometry is a classic

example.

The Einstein equations for this geometry (2-1) can be written out in an extremely

simple form,

8m 8m 80
= 4xrr2T~ r __ 2T~ v __ 2Tr (2-3)

All other components of the Einstein equations can be obtained from the above equations

by using the conservation of stress-energy tensor and the Bianchi identities.

For the model we are going to construct, we will not be using r as a coordinate. We

will use a double null coordinate system or a light cone system (u, v), a is the advanced

time coordinate and v is the retarded time coordinate. In this coordinate system, we can

write the most general spherically symmetric metric in terms of two independent functions

r (u, v) and f (u, v).

ds2 .t I' .>)dudv T 2(u, v)dR. (2-4)

The null cones in this coordinate system are u = coast (outgoing) and v = coast ingoingg).

Observe that, any coordinate re-definition U(u) or V(v) is just a gauge choice, meaning

the new coordinates (U, V) would also carry the interpretation of null coordinates. So, any

such transformation can be done for convenience.

The Einstein tensor G,,, of (2-4) can be calculated to be

Sd2 82p ii~

1 Br iir 82r

2 82r ifiir
G = -2
r 8v2 U
si21 t4 8 2 d 2f

Let us now consider the physical situation of collapse of a spherical null shell of mass

M~ in flat space at v = vo. The metric inside the shell (v < vo) is flat, hence,

ds2 __ _8880v T2 u, U)dR Where r =(2-5)

If the region outside the shell(v > i,,) is vacuum, then Birkhoff's theorem tells us that the

exterior region is nothing but a Schwarzschild geometry But, the presence of a quantum

stress tensor changes this. In any case, let us initially ignore the quantum effects and just

study the Schwarzschild exterior.

2.1.1 Schwarzschild Exterior

We can express the Schwarzschild geometry (2-2) in terms of the standard null

coordinates used in the literature,

ds2 = --(1 2M/lr)dadv + r2 8, U)dR Where r,= (2-6)

u and v are defined with respect to the coordinates (t, r) in (2-2) as

U = t rs v = t + rs.

To stitch the interior and exterior geometries at the shell, we shall avail the jump

conditions formulated by Israel and Barrabes [2].

2.1.1.1 Jump conditions on the Null shell

Since the shell has a finite mass M~ and zero thickness, it is obvious that it should

have infinite density. Let us represent the stress tensor of the shell in terms of a delta

function,

T,, = S,,6(v vo), (2-7)

where S"" is interpreted as the surface stress energy tensor. The question now is, can the

interior Eq. 2-5 and the exterior Eq. 2-6 geometries be stitched smoothly so that the

Einstein equations G,, = 8xrT,, are valid across the shell? Clearly, the LHS and RHS

of the Einstein equations diverge on the shell because of the delta function. But, that

is alright. What we need to know is if it is possible to start from one side of the shell

with appropriate initial conditions and integrate the Einstein equations with the given

stress energy tensor Eq. 2-7 and reach the other side of the shell to obtain the correct

geometry. This question is well posed because the delta function in the stress tensor is

well behaved under integration. To answer this question we have to first construct a single

patch of continuous coordinate system that exists on either sides of the shell, and then

express the metric gy,,, the Einstein tensor G,,, and the stress tensor T,,, in terms of these

coordinates. In these coordinates, the metric must be continuous everywhere. Only then

can integration with respect to these coordinates have a physical meaning. (To perform

an integration, we need a measure. The metric serves as the measure, hence we require it

to be finite and continuous everywhere.) And, only in these coordinates can the Einstein

equations be integrated. From the analysis of Israel and Barrabes [2](a clearer description

is available in [29]), it turns out that, for the Einstein equations to consistently hold up

across the shell, the following two conditions should be satisfied.

1. The induced metric on the shell from either sides of it (interior and the exterior)
should be the same.

2. With respect to the coordinate system of Eq. 2-5 and Eq. 2-6, the surface stress
tensor is given by

On the shell v = vo, the induced metric from the interior is

dS HELL = 2( Ild2

and the induced metric from the exterior is

de HELL r2 8, 00)d ,

The first condition would then require that r(u, i n) = r(u, I n). Since we have

r(u, vo) = r,(u, vo) =* r=T+ 2Mln( -1 1) Y (28)
2 2 2

we can rewrite the equation r(u, i n) = r(u, I,,), to obtain a one to one relation between the

coordinates n (exterior) and u (interior),

a = 4M In 1 .-- (2-9)

The event horizon that appears at u = co, appears at u = vo 4M~. Differentiating the

above equation gives,
dil4M~ 2M~
= 1 -1 ,(2-10)

where ro(u) is the radius of the shell when it is at the point (u, vo).

We can now express the exterior metric(2-6) in terms of the coordinate u instead of

1~ -ud 2M/r (uB) v)uu, ~R

dS2 Jdvr T2'(ur(il),)dl. (2-11)

By comparing the exterior metric in the above form Eq. 2-11 with the interior metric

Eq. 2-5, we see that the metric is continuous across the shell. Hence, we observe that

the coordinate system (u, v) is a continuous coordinate system, so integrating a function

defined on the manifold is meaningful with respect to this coordinate system. Note

that the coordinate v is the standard flat space retarded time coordinate in the interior

(v < vo) and it is the standard Schwarzschild retarded time coordinate in the exterior

(v > I,,). Any other coordinate system (U(u), V(v)) will also be a continuous coordinate

system if the first derivatives of U(u) and V(v) are continuous across the shell.

Fr-om the second condition, we see that the only non vanishing component of the

stress energy tensor for the shell with respect to (u, v) coordinate system is

T,, = ~-6(v vo).

If we change the coordinate from v to V, the stress tensor would change as

dv 2 M dv 2 M 6(V Vo) do~; 2'
Tyv = T,, d ro2u 6(v(V) vo) 4,

Mdv

Note that this stress tensor is meaningful only in a continuous coordinate system.

2.1.1.2 Gauge choice

We are going to choose a coordinate system (U, V) such that, r = (V U)/2 on the

surfaces S1 and S2, Where S1 is an ingoing null surface at shell (V = Vo) and S2 is an

outgoing null surface at U = Uo

Vo U V Uo
r (U, Vo) = r (Uo, V) =
2 2

The gauge condition imposed on S1 implies that U(u) = u given by the Eq. 2-9.

A crucial reason for choosing the coordinate u instead of a is that u is valid on either

sides of the event horizon, unlike a which goes to infinity at the horizon. There is another

important reason behind this choice. The form of stress tensor of the quantum field is

markedly simplified when expressed in terms of u. We shall go over this in Section 2.2.

Let us now an~ lli. .. the implications of the gauge condition imposed on S2. The

surface S2 has two regions, one in the interior of the shell and other in the exterior. The

condition imposed on S2 has different implications for these two regions. For the interior,

it implies V(v) = v (the standard flat space retarded time coordinate), and for the

exterior, it implies V(v) = v(v), where

+ 2M~1n 1 o'U-U (2-12)
2 4M~ 2

In here, v is the standard Schwarzschild retarded time coordinate and no is just u(uo)

given by Eq. 2-9. Differentiating the above equation gives

1 (2-13)
dv v uo

With respect to the continuous coordinate system (u, v) constructed in Section 2.1.1.1, we

see that the there is a discontinuity in the derivative of V(v) at the shell (v = I,,).

dV 1vu v dv1 v > vo

This implies, our choice of the coordinate system (U, V), is not a continuous coordinate

system. Hence, we cannot integrate the Einstein equations across the shell.

If we wish to evolve the Einstein equations in this coordinate system in the region

exterior to the shell, we cannot start with initial conditions inside the shell, we will have

to start with initial conditions on or outside the shell. In such a case, the simplest place to

lay the initial conditions are on the surfaces S1 and S2-

Let us now rewrite the exterior Schwarzschild metric(2-6) in terms of our new

coordinates (U, V), equivalently (u, v).

ds2 __ _6~' .)udv T 2(u, v)dR. (2-14)

The functional dependence of r on the coordinates u and v can be inferred the following

way. The dependence of r on the coordinates u and v are already known.

r, = ->e 4M 2M.( 15
v-2 2M, I 25

We can rewrite Eq. 2-9 and Eq. 2-12, to obtain

e 4M = e 4M 1) (2-16)
4M '

e 4M e 4M -1 (2-17)

Using the above three Eqs. 2-16, 2-17 and 2-15, we have

e 4M e 4M = 4M e 4M Ug- -

(1IL) I1(vo--" 1) (4Mv-uo 1) 28
-T- ezM = e 4M= e 4M 4M / 4M .(218

Solving the above equation gives us r(u, v). Comparing the metric (2-14) with (2-6), we

see that
du dv
e-~ ~ ~ d d 1- Mr

Using the equations (2-10) and (2-13), we can see that

4M~ 4M~
e =(1 2M/r) 1U 1gI I--] (2-19)

This equation appears to have an indeterminate form at u = vo 4M~. But we can use Eq.

2-18 to rewrite it so that it is well defined everywhere.

2M (nro- -oou
e-" ') 4M r 4 M .-- (2-20)
e~ 4M /M

One can easily see from Eq. 2-19 that, on the surfaces S1 and S2,

e-I '0 = 1 (2-21)

is a constant. One important reason for choosing this gauge is that the metric(2-14) takes

a very simple form on the surfaces S1 and S2, T iS given by (v u)/2 and f is a constant

given by the above equation. Hence, specifying initial conditions on these surfaces in order

to evolve the Einstein equations becomes very simple.

If we are to evolve the Einstein equations from the initial data on a spacelike

hypersurface S in a specified coordinate system(gauge), the initial data should comprise

of the metric and the normal derivatives of the metric on S. The result of the evolution

would then give us the geometry (effectively the metric, since the coordinate system(gauge)

is already fixed) everywhere in the domain of dependence of S. But if we are to evolve the

Einstein equations from the initial data on a light cone defined by an ingoing null surface

S1 and an outgoing null surface S2, We Only need the metric on the initial surfaces.

In the appendix, we calculate all the relevant derivatives of the functions r(u, v)

and f (u, v). These will be used in the algorithm for numerically evolving the Einstein

equations in Chapter 4.

2.1.2 General Exterior

As we have mentioned earlier, because of the presence of quantum fields, the exterior

geometry is going to be different from the Schwarzschild geometry.

1. A reasonable assumption is that the quantum fields do not have any significant
effect on the geometry at very early times of collapse, then we can assume that on a
surface S2 ClOSe to I-, the exterior geometry is approximately Schwarzschild.

2. Another reasonable assumption is that the metric on the shell S1 is Schwarzschild.

By choosing uo to be a large negative number, S2 can be taken close to I-. We will use

the same gauge as in Section 2.1.1.2. The exterior metric is again given by

ds2 __ _6~' .ddv T2(u, v)dR. (2-22)

The functions f(G, v) and r(u, v) are not given by Eq. 2-20 and Eq. 2-18 anymore. They

take those values only on the surfaces S1 and S2-

If the quantum stress tensor is known in this coordinate system, then we have all the

initial data required to evolve the semi-classical Einstein equations

G,,, = 8 x~ (T,,,) .

The result of evolving the Einstein equations would be the knowledge of the functions

f (6, v) and r (u, v) everywhere.

2.2 Quantum Field

Let us now consider a massless (K~lien-Gordon) scalar quantum field # on a spacetime

which classically looks like the geometry discussed in Section 2.1 (a null shell collapsing to

form a Schwarzschild black hole exterior). The presence of the quantum field tells us that

the geometry outside the shell is not Schwarzschild.

The stress tensor of this field is the following operator

I,,= d~d~ -~gdd". (21

Classically, the field takes a value zero everywhere, hence the classical stress tensor

of the field is zero. But the quantum fluctuations at any point in spacetime would result

in a nonzero stress energy tensor proportional to 5. If we know the state of the quantum

field, then the measured stress tensor can be found from the expectation value of the stress

tensor operator with respect to that state. The most relevant state corresponding to black

hole evaporation is the so called "In v. ..I1I1n, state. Long before the onset of collapse of

the shell, the space time is Minkowskian. Hence, an observer at past .I-i-inidllicl~ infinity,

Z-, should see a zero particle state and thus measure the stress tensor due to the quantum

field to be zero. This is precisely the definition of the "In v.. 11 InI~ state. Let us represent

it as |in). Since in a collapse geometry, past .I-i-inisllicl~ infinity Z- is a cauchy surface, the

|in) state is uniquely defined.

As discussed in the C'!s Ilter 1, calculation of the expectation value stress energy

tensor with respect to any state in curved space time is rather complicated. A detailed

description of the procedures used for this calculation is given in [4] and [13]. We will not

get into the details of these procedures, but we will just adopt their results for our model.

But before that, let us first briefly discuss some properties which we expect the stress

tensor to satisfy [7].

Just from considering the requirement that the expectation value of the renormalized

stress tensor in anyv state should obey the conservation laws, that is V,, (T") =0,

C!~! -I. Is-- is and Fulling [7] showed that in the exterior Schwarzschild metric, (T,,) can be

expressed in terms of two integration constants and two arbitrary functions. One of the

integration constants corresponds to the magnitude of (Tt,) at infinity and the other one

is determined if we assume that the physically normalized components of (T,,) are finite

every where on the future horizon. One of these arbitrary functions corresponds to the

trace (T,,) g"". If the matter field theory is conformally invariant, then at the classical

level, the trace of the stress tensor has to vanish. But the quantized stress tensor (T,,)

does not necessarily have a vanishing trace. The trace should be proportional to the Weyl

Scalar of the geometry (which is 48M~2r-6 for the Schwarzschild geometry). This is called

the Trace anomaly. When the same argument is applied to analyze the stress tensor in a

two dimensional geometry, the conclusion is that the stress tensor is entirely determined

by its trace. It turns out that this trace has to be proportional to the Ricci curvature

scalar. To obtain any further information about (T,,,), explicit renormalization has to be

performed.

The procedures for renormalization are very tedious and in most of the cases they do

not give an analytic expression for the renomalized stress tensor. That is, for a general

metric gy,,, there is no generic expression for (T,,,(gy,,)), except when there is a high degree

of symmetry in the matter field # and the space time gp,,. This typically happens in

conformally flat spacetimes and for matter fields for which the classical action is invariant

under conformal transformations.

This is a bad news for us because, in general a spherically symmetric geometry as in

Eq. 2-4 is not conformally flat. Not even the specific geometry of Schwarzschild exterior

(2-14) is conformally flat. But things get better when we go down to two dimensions.

2.2.1 (Tp,,) In Two Dimensions

Any two dimensional geometry is conformally flat and minimally coupled scalar

matter theories in two dimensions are conformally invariant. Because of this symmetry,

it turns out that there exists an analytic expression for the renormalized stress tensor

(T,,,(gy,,)), in an arbitrary metric gy,,. This was first calculated by Unruh, Fulling and

Davies [9] using covariant geodesic Point split regularization method [7]. The results of

their calculation can be summarized in the following way.

Consider the two dimensional conformally flat geometry given in terms of the

null(conformal) coordinates.

The scalar field # is quantized with respect to the choice of the positive frequency modes

u, = (4xu~o)-1/26-iw n Vw = (qiTUo)-1/26-w

In most of the physical applications, where the coordinate system is bounded by

a physical condition such as r(u, v) > 0, these modes are not all linearly independent.

In such cases, we can obtain the modes u, from the other set of modes v, by reflecting

them on the r (u, v) = 0 surface. So, we will define a set of linearly independent modes on

some Cauchy surface, and then use the classical field equations to evolve these functions

to every point in space time. Generally (at least in collapse geometries), I-(u = -oo)

is a cauchy surface. The natural set of linearly independent modes at I- is v,. In two

dimensions, the classical solution to the field equations ensures that there is no back

cl Ir1 r1 ;), that is, the mode v, has the same functional form through out the space

time until it hits the r = 0 surface. After that the ingfoingf mode turns into an outgoing

mode u,. If the coordinate v is a locally inertial coordinate at I-, then quantization with

respect to these modes actually corresponds to the "In v.. 11 InI~ state |9). Calculation of

the expectation value of the quantum stress tensor in the state gives |9) gives,

12xr Biu2 v

& 82f
12x r 8v2 Ui

is state dependent. To do this, we first need to identify the positive frequency modes with

It then turns out that,

where the the curly brackets indicates the Schwarzian derivative. The Schwarzian

derivative of a function f(x) with respect to x is defined as

d3 /X 3 2 2 22
{ f (), x}-
df/dx 2 df/dx

The state on which we want the expectation value to be calculated is the |in) vacuum

state. For our model, we start with initial data on the Cauchy surface S1US2. The |in)

vacuum state corresponds to a zero particle state with respect to any inertial observer at

I-. This state should also correspond to zero particle state with respect to any inertial

observer in the interior of the shell. So, by continuity, it corresponds to zero particle

state with respect to an inertial observer at the shell. On the surface S1, fi is an inertial

coordinate, and if we choose the surface S2 far eHOugh (near I-), then v is an inertial

coordinate on S2. Hence, if we use the coordinates constructed in Section 2.1 (il, v),

we can directly employ Eq. 2-24 with a replaced by fi and v replaced by v to get the

expectation value of the quantum stress tensor in the |in) vacuum state.
2.2.2 4D-Model

Since an elegant analytic expression such as Eq. 2-24 cannot be obtained in 4

dimensions, we create a model for the quantum stress tensor in 4 dimensions by

multiplying a factor 1/4xrr2 to the stress tensor obtained from the two dimensional

model Eq. 2-24, so as to make physical sense in four dimensions. According to this model,

(2-25)

The constant a~ determines the strength of the quantum field. A single quantum field

would correspond to a~ = &/12xr. One can easily verify that in a generic spherically

symmetric geometry (2-4), the expectation value of the above stress tensor is conserved,

V" ( ,,) = 0. Hence, it is physically consistent to solve the semi-classical Einstein

equations G,,, = 8xr (T,,,). An important point to keep in mind is that the stress tensor is

of this form Eq. 2-25 only with respect to the coordinate system (u, v) we constructed in

the Section 2.1.

This model gives us an analytic form of the stress tensor in any spherically symmetric

geometry. But fact that the model gives (leo) = 0 -II__- -0 that this is not a very realistic

model in four dimensions, because explicit calculations of the stress energy tensor in four

dimensional Schwarzschild geometry performed by Candelas and Howard shows that (leo)

is non zero and is determined by the trace anomaly. One can attempt to generalize the

stress tensor Eq. 2-25, so as to include a nonzero function (lo). But it turns out that any

nonzero (leo) leads to a stress tensor that is not conserved. So, let us stick to our model

with (ieo) = 0.

Furthermore, there are some basic differences in the behavior of the fields in two

dimensions and four dimensions, and that contributes to the reason for why this model

cannot he very realistic.

1. Unlike two dimensions, in four dimensions, each mode (;?i u,) is further split into
angular modes. We have to include the contribution from all these modes.

2. Unlike two dimensions, in four dimensions, the wave modes do backscatter. Hence,
we have to include the grey body factors (transmission probability).

To have an estimate of how close this model depicts the realistic case which incorporates

the above two criteria, let us briefly compare the results of Hawking's original calculation

in 4 dimensions when performed with and without incorporating the above two criteria.

We know that more than CII' of the radiation is via the 1 = 0 mode. Hence, an s-wave

approximation is a fairly good approximation to the black hole radiation problem. But

the model we have constructed is not an s-wave approximation, because we have not

considered the backscattering of the waves on the potential.

When the back scattering effects are neglected, the s-wave approximation luminosity

of the black hole is

76b.11-

If we include the back scattering effects and the other 1 modes, then numerical calculations

show that
L=1.795
76801.11-

Without the back scattering effects, we see that the radiation coming from the black hole

is about 5 to 10 times larger. So, considering the back scattering effects is important.

Nevertheless, we shall accept this model because it appears that even though

neglecting the backscattering effects changes the luminosity by a factor of 5 or so, its

qualitative behavior, namely L ~ &/M~2 is Still retained.

2.2.3 Stress Tensor In Schwarzschild Exterior.

Now that we have a model for the quantum stress tensor in a general spherically

symmetric geometry Eq. 2-25, let us compute the quantum Stress tensor in Schwarzschild

background by pline-_~r~in in the functional form of flu, v) from Eq. 2-20. Since the

expectation value of the quantum stress tensor is a tensor quantity, we have

(du 2 dv2 ,,.,, i d
du d du d

Let us now express the stress tensor in terms of the standard Schwarzschild coordinates

(u, v). The relationship between fi and a is given by Eq. 2-10 and v and v is given by Eq.
2-13.
a~ M~ 3 M2 8M ~ 24M\2 26
(T,,) = +(-6
8KTr2 2 r4 _U 1)3 4

(T,) 2 312 4 (2-27)
8xrr r. 2 r4

(T,,) = 8;r2 3 2-8

We will use this form of the stress tensor in ChI Ilpter 3 to construct the energy flux in the

quasi-static approximation.

CHAPTER 3
QUASI-STATIC APPROXIMATION

Fr-om Hawking's calculation in 1975, it is evident that a quantum field in the vacuum

state with respect to a locally inertial observer at past infinity (before the star collapses

to form a black hole) evolves into a state with a thermal spectrum of outgoing particles at

late times at future infinity. The temperature of this spectrum turns out to be the surface

gravity n, of the black hole, which for a Schwarzschild black hole is 1/4M~. At late times,

the total number of particles emitted per unit time can be calculated by evaluating the

expectation value of the number operator. It turns out that this quantity is a constant.

This implies that the energy flux emitted by the black hole is a constant at late times. As

discussed in OsI Ilpter 1, the luminosity of the black hole turns out to be

L= 1.795
7680x) / l-

A more rigorous way of computing the luminosity of the black hole is given by

evaluating the expectation value of the stress energy tensor in the appropriate field

configuration (namely vacuum state at past infinity). Unfortunately, the expectation value

of the Stress tensor diverges and hence a tedious procedure of renormalization has to be

performed in order to obtain a finite meaningful result. Various methods for doing this

exist and is explained in detail in [4] and [13]. By early 1980's, it was well established

that all these methods yield the same prediction for the luminosity of the black hole as

given by Hawking's calculation. The advantage of this more rigorous method compared

to Hawking's calculation is that, in here we can calculate the energy density and energy

flux at any point in the space time, unlike Hawking's calculation which can only yield the

energy flux at future infinity.

If we use this energy flux (which is a constant at late times) to compute the total

energy emitted by the black hole, we are obviously going to get a diverging result,

meaning the total energy radiated away by the black hole is infinity. Clearly, this is a

physically wrong interpretation. A finite mass black hole cannot radiate out infinite

energy. Where did we go wrong?

Right in the first step. The assumption that the black hole geometry is fixed and is

not dependent on the quantum field is wrong. The original Hawking's calculations and the

later calculations of the stress energy tensor are all performed on a fixed Schwarzschild

background. We need to consider the fact that the stress energy tensor of the quantum

field will serve as source terms for the Einstein equations. Thus the spacetime geometry

will be different from the geometry of the pure black hole which Hawking used in his

calculations. Physically, the black hole should lose mass as it emits radiation. In fact,

calculation of the renormalized stress energy tensor shows that the positive energy flux

going out to infinity is exactly balanced out by a negative energy flux going into the black

hole horizon. This negative energy flux going into the black hole should decrease the mass

of the black hole. In other words, the physically correct thing to do would be to somehow

include the back reaction of the quantum field on the background geometry. How do we do

this? The simplest way of fixing this problem is through Quasi-Static approximation.

Since the luminosity of the black hole is very small for large black holes, according

to this approximation scheme, the radiation from the black hole at any instant can be

thought of as being described very accurately by thermal radiation with temperature

depending on the mass as 1/4M~. Then, we can explicitly write down the conservation of

energy equation by requiring the total energy radiated to be equal to the total mass lost

by the black hole,

dM 1.795
dt 7680x) / -

We can determine the life time of an evaporating black hole by solving the above equation.

7680x Mf~,
1.795 3 '

where il. is the initial mass of the black hole.

The Quasi-static approximation does succeed in including the back reaction effects of

the quantum field in such a way that the conservation of energy is not violated. But the

question is whether conservation of energy alone is sufficient to justify the validity of this

approximation. Let us now focus on the geometrical implication this approximation leads

us to. Geometrically, this approximation effectively oils down to treating the complete

geometry of the evaporating black hole (which includes the back reaction of the quantum

field) as a sequence of pure Schwarzschild geometries of varying mass. The entire point

of this work is to examine whether this approximation is really valid. We have already

discussed in ('! .pter 1, the motivation and reasons for doubting the validity of this

approximation. Let us anyway go over the reasoning briefly.

The complete evaporating black hole geometry has an apparent horizon which bulges

out of the event horizon. The radiation emitted out at late times is causally in contact

with this bulge. Hence we are justified in expecting that this bulge would change the

characteristics of evaporation when compared to evaporation taking place in a geometry

where there is no such bulge, as in the quasi-static geometry.

In this C'!s Ilter, we shall apply the quasi-static approximation to the model

constructed in C'!s Ilter 2. In Section :3.1, we will construct the quasi-static metric for

the model. In Section :3.2, we will compute the energy flux going out and then use it to

write down the conservation of energy equation. We shall see that solving the Quasi-static

problem oils down to solving two nonlinear coupled ODEs. An exact solution can he

obtained only numerically, and we defer it to ('! .pter 5. In Section :3.3, we shall discuss

some analytic properties of the solution without having to solve it exactly. In Section :3.4,

we show that the quasi-static metric constructed in Section :3.1 can smoothly be extended

into flat space at the end of evaporation.

3.1 Constructing The Metric

Recall that our model consists of a null shell (collapsing to form a black hole) and

a quantum field (which is in vacuum state at past infinity) whose stress energy tensor is

analytically known at every point of space time. The metric inside the shell is flat

ds2 = -dudv

and the metric outside the shell is piecewise Schwarzschild, that is, with a changing mass

to compensate for the energy loss that can be calculated from the known stress energy

tensor of the quantum field. We shall use an outgoing Vaidya type metric to describe the

exterior geometry.

ds2 _2M~(u)),,,, 2 2dudr +r2

where M~(u) is the Bondi mass. Note that the only non vanishing component of the

Einstein tensor for the above metric is

dM (u)

This geometry is not static and it does not possess a timelike killing vector. Nevertheless,

we can choose a time like coordinate t such that the metric takes the following form.

ds2 ____2M(u) )rl 2 2M~(u)), dT2 2

For this, we would need

a = t r*, where r, = r + 2(u)In( 1 .(

Our aim here, is to follow an observer on an r = (mini- I10 li-ro) curve and calculate

the energy flux crossing a detector of unit area. We shall calculate this energy flux

from the stress energy tensor of the quantum field. As we follow the observer, we can

parameterize the coordinate positions of the observer in terms of proper time, namely

n,(-r) and t,(-r). Also, let us reparameterize the mass function M~(u) in terms of the proper

time of the observer M~(r).

Let us now construct the quasi-static metric. The quasi-static metric should be

parameterized by the proper time 7r, such that any instant (characterized by the proper

Figure 3-1. Sequence of snapshots of Schwarzschild geometries with varying mass.

time -r of the observer), the metric should be a pure Schwarzschild metric of mass M~(r).

Hence we shall erect an instantaneous null coordinate coordinate system (u, v) and an

instantaneous timelike coordinate t, satisfying the following conditions.

The shell collapse occurs at v = (vo = 0),

the proper time elapsed after the observer crossed the collapsing shell is -r,

the geometry exterior to the shell is given by Schwarzschild metric with mass M~(r),

The coordinates (u, v) and (t, r) are related by n = t r, and v = t + r,, as in

Schwaarschild geometry. Observe that here r, =r + 2Ml(r) In (i 1
Fig. 3-1 represents the construction of the quasi-static metric. At various moments, the
observer sees a Schwarzschild exterior of different mass.

At any instant -r, the quasi-static metric would then take the form

'!2 ___ 2M~(r) 2 _2M~(r) dT2 2dR, (3-2)

ds2 ____2M~(r)),,, dudy +

With respect to these null coordinates, let us denote the position of the observer as

(u,, v,). To evaluate (u,, v,), we need to find the coordinate value of t at the observer,

that is t,. Since the shell collapse occurs at v = vo = 0, the coordinate value of t at the

instant the shell crossed the observer is given by to = -ro, + i,, The proper time elapsed

after the shell crossed the observer is given by

7 = 1 2M(t)( to).

We can thus calculate

Up = t ro, = 71 2(-ro 2ro, + in, (3-3)

v, = t+ro*= ( 1(-ro +1,,

We have now obtained the position of the observer (U,, v,) in a quasi-static metric

wherein the collapse occurs at v = vo = 0.

3.2 Computing The Energy Flux

The four-velocity of our observer U has only one component with respect to the

coordinate system (t, rl, 8,) : Ut = 1/ 1 2M/

We shall now invoke the quasi-static approximation to calculate the rate of change of

mass of the black hole as seen by the observer. The energy flux crossing the observer can

be calculated from the stress energy tensor of the quantum field. The instantaneous energy

flux measured by the observer across the surface r = ro is given by T/U". We know that

the energy crossing the observer gets redshifted as it goes out to infinity by a factor z/.t

Hence, the change in mass of the geometry is given by this redshifted energy. That is,

-(- Mr) = -4xrr2E' /t,
d-r

where E" = -T"UM and r is the locally inertial coordinate at the position of the observer:

dr" = z/gdr. This implies,

-E'=-gir Uter,= Tr9r 1-2/r Ter,,

From the above two equations, we can write the quasi-static equation as

M~r) = 4r2 Ttr

Recall from C'!s Ilter 2 that the Stress energy tensor calculated for the model in a fixed

Schwarzschild background is

M 3 M~2 8M24M\2 i(5
(in |Tu,| in) = +(35
8KTr2 2 r4 _U 1)3 4

(in |Tv,,|in) = +,(36
8;,T2 732 r4
a~ 2M~ M
(i s |i)=1-- ,(3-7)

where the coordinates u and v are the standard Schwarzschild coordinates and the

coordinate fi is the standard flat space advanced time coordinate. The relation between u

and fi is given by

fi M In1 -> = 1 .(3-8)
u ~4 u in fi1 U

Since we have constructed the quasi-static metric and the instantaneous null coordinate

system (Section 3.1) in such a way that it matches with a Schwarzschild spacetime of

fixed mass M~(r), we shall adapt the same form of stress energy tensor for the quasi-static

metric with M~ replaced by M~(r) in the Eqs. 3-5, 3-6 and 3-7.

Since the null coordinates are defined at every instant in such a way that n = t r,

and v = t + r,, we have

Bu 8v 8v Bu 1
= land.
8it iit dr dr 1-2/

Now, a coordinate transformation directly yields

Bu du 8v 8v v 8vB
Ttr = T,, + T,, +? Ts, + ,v d u
8t dr 8t dr 8t dr 8t r

ca/4xr2 4 12M\2 (9
Ter=+. 39
(1 M/r)(i vo)3 ( 'U0 4

In particular, we need Tt, to be evaluated at the observer's position, that is at r = ro
and a = n,(u = u,). We should note that the dependence of Tt, on the proper time -r
is through the dependence of M~ and a on -r. The quasi-static approximation gives us the
following equation for the rate of change of black hole mass with respect to the observer,

a4M~(r) 12M~2(
M(-r) = + (3-10)

The functional form of u,(-r) can be calculated implicitly from our knowledge of n,(-r), Eq.
3-3 and u(u), Eq. 3-8

2M~(r)r z

ni(T) =l T, +71

2 [ro + 2M~(r) In(ro/2M(r) 1) ,

(3-11)

(3-12)

n,3(-r) = U?3(r) l-j 4(r) In( 1pl~~ .

Differentiating the above expressions with respect to the proper time gives

ro ro ro)3 MI~r:M I

o 2M), (3-13)

a, = n 4M n 1 +" (3-14)

Note that we have suppressed the proper time dependence of the functions M~(r), U,(-r)
and U,(-r) in the above equations for brevity. Eq. 3-14 can be rewritten as

it, + 4M In(-G,/4M 1) +

u, = (1 + 4M~/u,)(u, + 4M~ In(-u,/4M~ 1)) 4M~,

(3-15)

where u, is given by Eq. 3-13.

1'[ ]

Solving Eq. 3-10 together with Eq. 3-15 is tantamount to solving the quasi-static

problem. This involves solving two first order nonlinear coupled differential equations

involving M~(r) and u,(-r). We need two initial conditions to solve these equations.

The mass of the shell when the observer crossed it, M~(r = 0) = Mn,

The fact that u is the flat space advanced time null coordinate in the interior

region of the shell determines that the radius of the shell at any instant is given by

R = (vo u)/2. In particular, u,(-r = 0) = -2ro.

It is not possible to solve these equations completely analytically. Hence we solve these

equations numerically using the mathematical software Maple.

These solutions are given in C'! Ilpter 5 and are also compared with the solutions from

complete numerical evolution of Einstein's equations, which are discussed in ('! .pter 4.

3.3 Analysis

Even though it is not possible to completely solve the quasi-static equations Eqs.

3-10 and 3-15 analytically, various limits and analytic properties of the solution can be

discussed.

Since the observer at any instant -r, is ahr-l- .- outside the event horizon with respect

to the instantaneously defined metric, we have u,(-r) vo < -4M~(r). We can hence define

a positive quantity E(Tr) Such that

Up(r) Vo = -4M~(r) E(Tr).

For convenience, let us also define x(-r) as

x ()--(7 1. (3-16)
4M~ 4M~(r)

Note that the quantity x(-r) is ah-- .1-< positive and initial value of x at -r = 0 is given by

x(-r = 0) = ro/2Mn, 1.

Eq. 3-10 can now be rewritten in terms of x as

a4M~ 12M~2
'iI1 2M/r (1 + E/4M~)3 (4M~)3 (1 E/4M~)4 (4M~)4

a~ 1 3
M -2!) = (1 +i.1 x)- -4](37

Then we can replace u, in Eq. 3-15 by x to obtain,

-4M~(1+ x) 4MAx= u, = ( )(u, 4~~) M
1+x

-.w 4MA = 6, )+ 4M Inr) + 4M (3-18)

Fr-om Eq. 3-17, observe that M~(r) < 0, which just means that the mass of the black

hole monotonically decreases.

Fr-om Eq. 3-13, we see that for reasonably large ro, up(-r) > 0. The reason for such

a conclusion is the following : If at all u,(-r) should become negative, it can happen only

when -r ~ ro/M~. But the evaporation would have reached a completion long before this

time because, -evap ~ M~/M is the timescale for evaporation. For large ro, we can clearly

conclude that for the relevant evaporation scenario, we ah-- .1-< have u,(-r) > 0. More

precisely, by retaining only the dominant terms in Eq. 3-13, we have

6,(r ~ 1- 4Mn~ro2M -1).(3-19)

Fr-om Eq. 3-18, it is possible to see that the right hand side is ahr-l- .- positive, which

means x is ah-- .1-< negative. To see how this works out, we shall categorize the time scale

into three categories, namely

1. Early times, x(-r) > 1. The mass loss due to evaporation is negligible during this
period. Fr-om Eq. 3-17, it is clear that

M~- a

which implies that the terms which are negative in the RHS of Eq. 3-18, namely
4M~x, scales aS Z-2. Hence the term u, dominates the RHS of Eq. 3-18, making it
positive. Thus we conclude that, during early times x is negative.

2. Intermediate times, x(-r) ~ 1. The mass loss due to evaporation is small but is not
necessarily negligible. The strength of evaporation, a~, determines whether the mass
loss in this stage is negligible or not. To check if the RHS of Eq. 3-18 is positive
during this time, we can plug in the approximate form of u, from Eq. 3-19. The
RHS of Eq. 3-18 takes the form

1+x 1x

We can clearly see that for large enough ro, the above term is abor-l- positive. Thus
we conclude that, during the intermediate times x is negative.

3. Late times, x(-r) stage, Eq. 3-17 can be Taylor expanded to give

M1-') = a 1 (1 3x + 62" + ...) + 6.1 4x + 10Z2 .)
1i-r 2M/r 10. 1 64.11
a~ -1 1
M~) =+ O(Z23_20)

It is straight forward to see that the right hand side of Eq. 3-18 is positive in this
stage of evaporation. This is because, for small x, In(x) is a big negative number, u,
is still a positive number and M~ is a negative number. Thus during the late times
stage of evaporation x is negative. One can also neglect x in Eq. 3-20 and solve for
M~(r). With a suitable definition of Q(-r), it turns out that the late times solution to
Eq. 3-20 can be expressed as

2 2M~ (r) M 3M~2(r) M3 3c)
Q~r)- 1 1 -1 +- +- 15 -_-r (3-21)
35 rot 2 64

We have so far seen that during all stages of evaporation, M~(r) and x(-r) are both

negative, which implies that they are both monotonically decreasing functions. Let us ;?i

that the evaporation comes to an end at a time Tr = -rend, that is M~(-end) = 0. From the

definition of x in Eq. 3-16, we know that x is positive that x f 0 as long as M~ 0 But

we do not know the limit x reaches as M~ goes to zero, that is we do not know the value of

X(Tend). Ifen~d) is nonzero, then the evaporation would be over before it reaches the late

times (stage 3). Let us now assume that the evaporation reaches the late times stage and

solve Eq. 3-18 to find x(-rend). We shall first rewrite Eq. 3-18 into the following form.

-4 M = 6+Mnx

For large ro, we can plug in Eq. 3-19 for u, in the above equation. This gives us

d~ x 2MI Mllli(,I (
d-r 1+xro T

Since we consider M/lro
only the dominant terms, we have

d M

Using Eq. 3-20 for M~, we have

d -a ~ Mxr~o(32
d-r 6i4M\3

Expanding Eq. 3-21 in terms of M/lro and considering terms only upto the leading order,

we get an expression for M~(r).

3a~
[1 + O(M~/ro)] M~3(~ = (end 7). (3-23)
6i4

We can now rewrite Eq. 3-22 in such a way that the quantity M~x gets grouped together.

d M
Mx11 = Mx (In( )/(ed-).(3-24)

Solving this equation yields,

Mx 1
In In( nTed-7
ro 3

In(Mx) = A(Tend 7 -1/3

Since we know that as (Ten~rd T) gOeS to zero, M1 goes to zero and In(""~) goes to -ono.

Hence, the constant A should be negative. Now, we can get a solution for x as a function

of M~, by using Eq. 3-23

Mx~J = oA-n-)13 0e .M (3-25)

This clearly shows that as M~ goes to zero, x also goes to zero. Thus we conclude that

X(Tend) = 0.

From Eq. 3-3 and Eq. 3-8, we have the

n, 7 2ro +'n~= p-4Mln(~ 2"-1).

Since we know that aS -r i end, M~ 0 and x 0 from the definition of x in Eq. 3-16,

we see that u, i ,, Then, the above equation takes the form,

Tend 2ro + vo = vo 4M1~n x = vo 4M ~ :o' In.

We can now find A in terms of Tend, r0, i,, This gives us,

A (3)1/3ll = 2ro0 7end. (326i)

3.4 The Complete Quasi-Static Geometry

Now that we have an analytic solution near the end of evaporation, we can analyze

the complete structure of quasi-static geometry. Let us assume that the spacetime

becomes flat after the black hole evaporates completely We have to match the Vaidya

geometry Eq. 3-1 to the flat space geometry at the surface u = upend. See Fig. 3-2.

Applying the junction conditions [2] on this surface, we see that the geometries can be

smoothly mapped only if dM~(u)/du is zero. Let us now show that dM~(u)/du is indeed

zero.

Since we know that x and M~ are ah-- .1-< negative, from the definition of x Eq. 3-16,

we can conclude that u, should ahr-l- .- be positive.

u, = -4M~(x + 1) -> u, = -4 (x M
d-r

Figure 3-2. Penrose diagram representing a quasistatically evaporating black hole.

More over, near the end of evaporation, by differentiating Eq. 3-25, we obtain

dIx A (3a?)1/3 n
d-r 4.1/-

Since M~ ~ -1/M~2, We have

d x
(M~x) = 0 aS i Tend-
dr M

This tells us that, toward the end of evaporation, u, = -4M~. Note that from the

definition of x in Eq. 3-16 and the relationship between u and u, we have

du 4 1 x
= 1 1 = 0, aS- i Tend
du, 'I,,- U x+l 1 x+-I1

We had previously observed that the quantity u, is .ll.k- li-- positive, so u, monotonically

increases tO Upend = end 2ro + in, as -r goes tO Tend. Using Eq. 3-13, we can see that at

Tendl M end 8)(/)Tn /
Upend = 1 + + Mn2/o + In ~3~(/)~n 00.1/ )
To to4ro

Front the above equations, we can conclude that towards the end of evaporation, ~lps,

0. This implies dM~(u)/du = 0 toward the end of evaporation. Hence, we have a Vaidya

spacetinle with mass Af(u) with the property that Af(u) monotonically decreases to zero

as a goes to Upendt Since Af(u) goes to zero in a smooth enough manner, we can readily

stitch this Vaidya spacetinle to a flat spacetinle along the null ray u = unenet.

CHAPTER 4
NUMERICAL EVOLUTION

Let us first recapitulate the highlights of the model constructed in C'!s Ilter 2. We

consider a null shell collapsing to form a black hole. For convenience, we will denote the

fi, v coordinate constructed in C'!s Ilter 2 as n, v. The metric outside the shell is written as

The coordinates n, v are fixed by the gauge choice that on the surface Sl(v = 0) the value

of the function r is -u/2 and on S2( 0 U), the value of the function r is (v Uo)/2. In

this coordinate system, the stress energy tensor quantized with respect to "In v., 1n InI~ is

taken as

The constant a~ determines the strength of the quantum field. For a single quantum field

a~ = &/12xr.We can verify that the expectation value of this stress tensor is conserved,

The Einstein tensor G,,, of the above geometry is

G, =- [i -2 d r
-2af ar(4-2)

The field equations, G,,, = 8xrT,,,, then give us four differential equations.

-2f f r, rs,, = --0f, 2 _

2'f,r, s,= a [~f,, ff] (4 5)

fuvr + rs, = 0.

(4-6)

From Eq. 4-4 and Eq. 4-6, we can obtain

le2f + uy

In this C'!s Ilter, we will describe the procedure use to solve the Einstein equations

with appropriate initial conditions, so as to obtain the geometry everywhere. That is,

we will numerically solve the differential equations Eqs. 4-3, 4-4, 4-5 and 4-6 to get

functions f and r everywhere. Then we will calculate some physically important quantities

like apparent horizon, energy flux, etc... on this geometry, so as to compare with the

corresponding results obtained from the quasi-static approximation we discussed in

Terminology: At any given point (u, v) the functions f and r and their derivatives

will be referred to in the following way. A subscript a or v to a function denotes the

corresponding derivative. For higher derivatives are represented by superscripts to u and

v. We shall use the term pure u-derivatives to refer to the set

{rs, r,2 rg~3 f U2 f3) )

and pure v-derivatives to refer to the set

{rs, r,2 r,~3 ) i)i2 f3 .

We shall use the term mixed derivatives to refer to the set

(Two fu~v Fu3v, Ty2,~U3~2 Ty3 T22T32T2, 3 T3,

fU1) fU21)) fU31)) fU12, )U3, ) 21,2, ) 3,U2, ) 2,3, ) 3U3) .

The knowledge of the functions f and r at every point implies the knowledge of all the

derivatives. For convenience, we shall use a representative function A(u, v) to denote the

set of all known functions including f, r, all their derivatives, the components of Einstein

tensor and Stress energy Tensor and any other function constructed from the known

functions .

4.1 Algorithm

4.1.1 Initial Data

To evolve the Einstein equations with initial data on the surfaces S1 and S2, We need

to specify the functions f and r on S1 and S2. The gauge conditions already specify r

on these two surfaces. The value of f on these two surfaces are taken to be a constant fo

given by

afo~~o (8

This choice basically means that the geometry on the surfaces S1 and S2 1S Schwarzschild

geometry. In C'!s Ilter 2, Section 2.1.2, we justify this choice. All the derivatives of f and r

vanish on these two surfaces except the following, r, = -1/2 on S1 and r, = 1/2 on S2- It

is now possible to evolve these equations numerically inside the future light cone.

To solve the differential equations numerically, we have to convert the equations to a

finite difference equation. To do this, let us start by laying down a two dimensional grid

with a step size An in the u-direction and Av in the v-direction.

4.1.2 Constraint Equations

Observe that Eq. 4-3 is a constraint equation on the initial data on the surface

St (v = I n) and the Eq. 4-5 is a constraint equation on the initial data on the surface

S2( = u0). The initial data r(u, v) and flu, v) should satisfy these constraints on the

initial data surface (the chosen lightcone) to have a meaningful evolution. From the initial

conditions, one can clearly check that these constraint equations are satisfied.

Let us now evolve the functions flu, v) and r(u, v) from the surface v = vo to a

surface v = in + Av by using the dynamical equations Eq. 4-4 and Eq. 4-6, or effectively

Eq. 4-7 and check if the constraint equation Eq. 4-3 holds on this evolved surface

v = vo + av. That is, we want to check if

(r+ av) 2 ( f + av) (r+ av)
882 Uv du dU du U

I~(f + av) (f + Av)
By!Byivu'
Retaining terms up to first order in Av, we have

(rs, 2 for,) + [r,,, 2 for,, 2r, f ] av

(.f~cU- f~)+-fr-ffr(,-f/]
T T

From Eq. 4-3, the terms independent of Av cancel each other. Let us check if the terms

linear in Av satisfy the equation.

[TUe,,,,- 2.fiLrs, 2rlLfsU]= [few~,1L- 2LfiLfeU r,(fe,1- .f)/r]

By pline-_rine in Eq. 4-7 for r,, and f,,, and then using Eq. 4-3 to eliminate f,,, we can

verify the above equation after a bit of tedious algebra.

Now, let us evolve the functions f (u, v) and r (u, v) from the surface n = uo to a

surface n = uo + Au by using the dynamical equations Eq. 4-4 and Eq. 4-6 (or effectively

Eq. 4-7) and check if the constraint equation Eq. 4-5 holds on this evolved surface

u = uo + au. That is, we want to check if

(r+ au) 2 ( f + au) (r+ au)
8v2 du dU dt Uv d

(f + afu) ( f + A)
r. + An 8v itcr U 8Y'I
Retaining terms up to first order in Au, we have

(rs, 2 far,) + [r,,, 2 far,, 2r, f, ] au

(f, ro ) [f ,,, -i 2f ,f s, rs(f ,, -f~)T 2)l
T T

From Eq. 4-5, the terms independent of Au cancel each other. Let us check if the terms

linear in As satisfy the equation.

By pline-_rine in Eq. 4-7 for r,, and fu,,, and then using Eq. 4-5 to eliminate f,,, we can

verify the above equation after a bit of tedious algebra.

Thus, we have shown that, if the constraint equations Eq. 4-3 and Eq. 4-5 are

satisfied on the initial surface, then we can use Eq. 4-7 to evolve the functions f (u, v)and

r(u, v), and automatically the constraint equations will be satisfied everywhere.

4.1.3 Algorithm

We will now describe explicitly how Eq. 4-7 is used to evolve the functions and their

derivatives from one point on the grid to the next point.

Our strategy is to start on a v = constant surface, and evaluate all the derivatives of

functions at every grid point by moving in the u-direction. Once we have the the entire

solution on this surface, we jump to a parallel surface given by v + Av = constant. We shall

call the first surface S1 and the next parallel surface S1.

We start with the assumption that on the surface S1, all the a derivatives are known,

and at the point p, where this surface meets with the initial data surface S2, both u

and v derivatives are known. At any point p, if we know all the pure derivatives of the

functions, then all the mixed derivatives can be calculated from the evolution equations.

Explicit calculation of these mixed derivatives are performed in the appendix. These

mixed derivatives will be used to compute the pure derivatives at a neighboring point of p.

We shall explain this with the help of Fig. 4-1.

1. We first compute the v derivatives of the functions f and r on the surface S1. Since
these values are known at the point p, we will Taylor expand these functions to
obtain the values at the immediate neighboring point q on S ,

1 1

St S2

Figure 4-1. Diagram representing the u-v grid on which the functions r and f are evolved.

1 1
rT 2 = T 2 + T ,2 (au) + 22 (U2 6 3 U2 (U3
1 1
v3 3 32 2 2 6 3
Similarly for the function f,

1 1

1 1
ff," = f",z + f", (Au) +f~ 2 u2 6" 3 2 3
1 1

2. Since we already have the u-derivatives of the functions at the point q, and now we
also know the v-derivatives of the functions at the point q, we can compute all the
mixed derivatives at the point q. This is explicitly worked out in the appendix. Once
we know all the derivatives at q, we automatically know the Einstein tensor G,, and
the stress tensor T,, from Eqs. 4-1 and 4-2.

3. With the knowledge of all the derivatives at q, we can proceed to compute all the
derivatives at the next point on S1 by repeating the steps 1 and 2. In the same way,
we can continue to find all the derivatives at every point on the surface S1.

4. Our next step is to compute the functional values and the u-derivatives on the
surface S1. Since all the derivatives at the points p, q, ... on S1 are known, we can
Taylor expand the functions about each of these points to obtain the functions
and the u-derivatives at the points P, Q, ... respectively. To illustrate this, let us

explicitly write out the functions and their u-derivatives at the point P.

r"= r" + r (a v) + r", (av)2 -T Up 3a)

r = r"- + r" ,(Av) + r"~, (av)2 3 3~,aj
r =r" + 7r:2 (Av1) + r"2 ,(Av)2 2 3~? ()3

fP = f"P + f (av) + f~ (av)2 3 ~(a)
f~ = f" + ff (av) + f 7,(Av)2 +;3 a~3
ff = f", + f", (av)j + f ~i,(av)2 2 3 3,(~j

This step gives a prescription for computing the functions and the u-derivatives on
the surface S1.

5. Since we know the functions and the u-derivatives everywhere on the surface S1, we
can use the steps 1 and 2 to compute all the other derivatives on S1. We can then
move from S1 to the next parallel surface, and similarly proceed moving from one
v = constant surface to the next.

The above five steps prescribe the algorithm for evolving the initial conditions on the

null surfaces S1 and S2 into the entire future light cone.

It seems like, the above algorithm can be used to numerically evaluate the functions

on each parallel surface Sl(at some fixed v) starting from the initial point u = Uo and

moving from one grid point to the next until a point is reached where r = 0 But this is

not true. Take a look at the evolution equation Eq. 4-7. We have

~e2f + uy

There is a singularity at r = re = 2. We shall refer to this re as the critical radius

[26]. The denominator goes to zero as r goes to re. If the numerator also goes to zero

sufficiently fast so that the r,, is finite in the limit r re, then we can jump over this

point in the evolution. But unfortunately, this is not the case. It is easy to check this on

the initial surface S1, where f = fo and r, = -1/2. In order to have the numerator vanish

at r = r we need r,. at that point to be positive. But it turns out that, on the surface

S1, r,. hecones negative before reaching the critical radius. That is, the critical radius lies

within the apparent horizon. This tells us that r,,, blows up to infinity when r = r Hence

on each surface S1, we have to stop before we reach the critical radius.

It is obvious that the accuracy of the numerical algorithm (used to generate the

functional values at all points on the grid) depends very much on our choice of the

distance between the grid points, which we refer to as the stepsize. An obvious place

where the numerical recipe would loose accuracy is where the higher derivatives of the

functions become large. To maintain accuracy, we need to reduce the stepsize, because

when we reduce the stepsize, the contribution of the higher derivatives in the algorithm is

significantly reduced. This brings in the concept of Adaptive 1\eshing. That is, we need

to add more grid points in between the already existing grid points whenever the higher

derivatives become large. Adding in a new grid point is very straightforward. What is

not straightforward is assigning functional values and derivative values to that added grid

point. This process is called Interpolation. There are various schemes of interpolation

people use. Here, we use a Polynomial Interpolation, that is, we fit a polynomial between

two grid points such that it takes the correct functional values and derivative values

at the two grid points. We then choose the midpoint of the two grid points to be a

new grid point and the functional values and derivative values at this new grid point is

obtained front the interpolated function. The polynomial interpolation scheme used in this

algorithm is derived in the appendix.

When do we choose to interpolate? We interpolate when we suspect that the errors

are beyond some tolerance level. Since the algorithm is third order, that is, it includes

third order terms in the Taylor expansion, a justifiable criterion is that, we should

interpolate whenever the contribution front the third derivatives in any function is

significantly comparable to the second derivative contribution. In the algorithm, we

Figure 4-2. The change in grid structure due to adaptive nieshing.

interpolate whenever the third derivative of r contributes to more than 1 of the second

derivative contribution. That is, we interpolate in the u-direction if r,,:(Au) > (0.01)r,,2.

Similarly, we interpolate in the v-direction if r,3:(Au) > (0.01)r,.s. When we use this

criterion for adaptive nieshing, the grid structure at the end of the evolution looks like Fig.

4-2.

4.2 Testing The Code

Fr-on the evolution, we have the functional values and the derivatives on all the grid

points inside the future light cone bounded by the surfaces S1 and S2. To verify if these

values are correct, we have to perform a few tests.

4.2.1 Accumulated Error

Let us take a look at the steps and 4 of the algorithm. We use the functional values

and the pure derivatives at the point p to evaluate the derivatives at the point P and q.

Since we include terms up to (Au)" and (Av)" in the Taylor expansion, the error in the

functional values and their derivatives at the point P and q due to this expansion is of the

order O(Au)4 and O(Av)4 Tespectively.

In the evolution, if it took No, steps in the a direction and NV, steps in the v direction

to reach any point (u, v) on the grid, then the accumulated error at that point is going to

be NV,0(Au)4 + #,0 U)4. To verify this, we consider three different step sizes and look

at the convergence of a representative function A(u, v). In practice, we use the components

of G,,, and T,,, for A. Let the exact value of the function at the grid point (uo, i n) be

denoted by Ao. Then,

A = Ao + Nv,0(au)4 + #,0 U)4. _9)

The number of steps needed to reach the point (u, v) is inversely proportional to the step

size. That is, NV,- ~ 1(Au) and NV, ~ 1/(Av) Hence, we have

A = Ao + c,(Au)3 C/ Ua)3, _10)

where c, and c, are numbers that do not depend on the step sizes An and Av. The order

of convergence of an algorithm with respect to a stepsize a, can be calculated in the

following way
An Ana/
=2", (4-11)
Ana/ AA/4
where n is the order of convergence. With respect to this definition, we expect a third

order convergence in both u-direction and v-direction. We verify this on an arbitrarily

chosen ingoing null line and on an arbitrarily chosen outgoing null line. We do observe

that n = 3 in both u and v directions. Moreover, we can calculate the coefficients c, and

c,, and it turns out that c, is more than 100 times smaller than c,. Hence, for an effective

error control, having a small step size in a direction An is much more important than

having a small step size in v direction, Av.

4.2.2 Constraint Violation

In Section 4.1.2, we discussed how constraint equations are automatically satisfied on

every surface when the evolution equation Eq. 4-7 are used. But the error accumulated

in the functions will contribute to violation of the constraints. We have to keep the

constraint violation under check. At every grid point, we have to check for the constraint

violations.

To compare the LHS and RHS of the constraint equations, we define a quantity called

constraint violating percentage

~=l G vT G 8;TT, 42
C,, G, 0 in; T' C. 100 .(-2

From the Section 4.2.1, we know that each of the quantities Gazz, T,,, G,.,, T,,. shows a

third order convergence. Hence, these constraint violating percentages should also show a

third order convergence. An appropriate step size can now he chosen with the criteria that

both C,, and C,. at every grid point is lesser than ;?i 0.1

4.2.3 Correspondence Check With Schwarzschild Geometry

Having a low constraint violating percentage is certainly a requirement for the

numerical values to correspond to a valid solution to Einstein's equations. But is that

sufficient for us to believe that the numerical solution is indeed correct? It is ahr-7-a

advisable to perform this evolution for a known solution, so that we can verify the validity

of the numerical recipe.

We know that if we turn off the evaporation and then evolve the equations, that

is with a~ = 0, we should get the Schwarzschild geometry. At any given point in our

coordinate system, we can explicitly calculate the functions f, r and all their derivatives

for the pure Schwarzschild exterior geometry. This is explicitly calculated in the appendix.

Now, we can compare the numerically evolved solutions ,4 to the exact solution ,do.

Specifically, we choose an outgoing null ray, not very close to the event horizon, and we

compare A and 240 for various step sizes to see if Eq. 4-9 is satisfied.

This correspondence check works fine, hence we have confidence in our numerical

recipe.

4.2.4 Validity Of Gauge Choice At The Surface S2

Recall that choosing the surface S2 at u 0 U iS actually a gauge choice. The

evolution should yield the same physical results for different choice of the surface

S2(different Uo). For different choices of Uo, the functions A(u, v) will certainly have

different numerical values at any given coordinate position (u, v). This happens even in

pure Schwarzschild geometry. So, how do we confirm that the evolution using different

values of Uo actually corresponds to the same physical geometry? We need to find gauge

invariant quantities that can be compared. What we do is the following.

From C'!. Ilter 2, Section 1.1.1, we see that the choice of Uo does not affect the a

coordinate choice, it only affects the v coordinate choice. Hence, any outgoing null surface

S2 can be characterized by its a coordinate. Also, since the surface S2 hits the null shell

S1 at v = 0, and since the radius of the shell at this surface is a gauge invariant quantity,

we can use this radius to characterize the surface S2. But on the shell, we have r = u/2,

which means that the coordinate a can also be effectively used characterize the surface S2-

For different choices of Uo, the surface S2, Which corresponds to ;?i (u = uo) is

the same physical surface. On this surface, any function A(no, v) which is parameterized

by the coordinate v, can be reparameterized in terms of r : A(r). But, this function is

not gauge invariant, so we have to construct a gauge invariant quantities from A(r). In

practice, the gauge invariant quantity which we construct is the local energy flux crossing

an observer at r = coast surface. We explain the construction of this gauge invariant

quantity E(r) in Section 4.3.3.

4.3 Retrieving Output

As a result of evolving the Einstein equations, we have the numerical solution

everywhere. In this Section, we will obtain some physically useful quantities constructed

from which will give us a physical picture of the behavior of this geometry.

4.3.1 Trace A Constant r Surface

We evolve the Einstein equations by parallel moving the v= constant surfaces. To

trace a surface r = ro, we do the following.

At every step of evolution, that is, at every v coordinate, we find the two successive

grid points uj and uj 1, such that the value ro lies in between r(uj) and r(ujp1). The

exact value of a at which r (u, v) = ro can be calculated by solving a quadratic equation

o = r(uj) + (u uj)r,(uj) + ,(y.(4-13)

Thus, for every value of v, we can find the value of a such that r (u, v) = ro.

4.3.2 Apparent Horizon

The same way as we traced the r = ro surface, we can also trace the apparent horizon.

The criteria is that r, vanishes on the apparent horizon. So, for every v, we can find the

two successive grid points uj and uj 1, such that r,(uj) > 0 and r,(ujp1) is negative. The

exact value of a at which r,(u, v) = 0 can be calculated by solving the quadratic equation.

O = rv(Uj) + (U Uj)ruv(Uj) + r,2,(uy).. (4-14)

Thus, for every value of v, we can find the value of a at which the apparent horizon is

located. The radius of the apparent horizon can now be found as a function of v.

4.3.3 Negative Energy Density

Recall that in ChI Ilpter 1, we discussed about the region around around the apparent

horizon where we expect the weak energy condition to be violated. We saw that the T,,

component of the stress tensor should become negative in this region. This physically

means that there are locally inertial observers who would see a negative energy density.

We can verify if this happens in our numerical solution. At each step of evolution (each

v), we can find the grid points uj and uj 1 between which the quantity T,, changes sign.

Using this data, we will be able to trace out the region where the weak energy condition is

violated. As expected, we find that the region around the apparent horizon has negative

T,,. In fact, we find that T,, is negative everywhere in the exterior.

4.3.4 Energy Flux

We shall now construct the gauge invariant energy flux measured by an observer

moving on the surface r = ro. The four velocity of this observer in (u, v) coordinates is

dr d 2 d-r 'd

The normalization condition U"U, = -1, implies that

d-r d-r

Since the observer moves on a constant r surface,

dv r, du
redu + redy = 0 +

From the above two equations we can conclude that

= e-'"'"), = -' "") ,(4-16)

4 U" =(,-r).(4-17)

The energy flux with respect to our observer is given by E" = -T,"U". Specifically,

we are interested in calculating the energy flux across a constant r surface. Let us first

calculate the quantity Er, and later understand how exactly this quantity is connected to

the energy flux across a constant r surface.

Er = E"r, + E"rs,

where

E" = -T,"U" TU"U = 2e-2fTUy + 2e-2fT vU"

E' = -T,"U" TU"U = 2e-2fTUn + 2e-2fTUy -

This gives us
2e-3f
Er Ta -T, (4 18)

How is the quantity Er related to the energy flux through the r = ro surface?

Let us first construct a locally inertial reference frame at the position of the observer.

el el
t= (U+U) X= (vU- U),
2 2

v = e-f (t + x) a = e-f (t x). (4-19)

The metric takes the form,

ds2 = -62fdudv t 2d -d2 d2 t 2d2

The stress tensor when expressed in terms of (t, x) coordinates become,

Tu, = e-2 f (Tun onT + 2T, ) ,

Tz = e-2f (Tun ovT 2T,, ),

T,, = e -2f _T, Tu) -

The interpretation of the stress tensor in a locally flat coordinate system is very

simple. For an observer at rest with respect to this locally inertial coordinate system, Ttt

is the energy density, Texis the energy flux across a constant x surface, and T"z is the

momentum flux across a constant x surface.

Our observer is not at rest with respect to this coordinate system (t, x). Since the

observer moves on a constant r surface, we have to boost the coordinate system (t, x) to

(t, x) using the appropriate four velocity U". The metric is still locally Minkowskian

ds2 __ _dt 2 d2 d2 __ _d" + d2 + 2d

with the coordinate transformation for the boost given by,

iit 8ix 8ix iit
=Ut -

(4-20)

The energy flux across a constant x surface measured by an observer at rest with

respect to the boosted coordinate system is given by Ttz fT,. From the definition

E"- T "U), it, is clear that Ez Tpe, is the energy flux measured by a locally inertial

observer at rest with respect to the coordinate system (t, x).

88 8t x ax tx x

Ti t (( Tu ) + TUz)2 + (Tt:Z + t)U

The components of the four velocity of our observer is obtained via a coordinate

transformation.

iit iit el
Ust = TU" + U'U = (U'U + UU),
Biu 8iv 2'
8ix 8ix el
Uz = TU" + U'U = (U'U UU)
Biu 8iv 2

Plugging in the expressions for Tut, Tz,Ttz, we get

Ez = Ty, = Ts, (U")~ T,, ( U")

How do we compare Eq. 4-21 with Eq. 4-18? For this, we first find the re

between r and the coordinate x, that is, express the function r(u, v) in terms o

inertial coordinates (t, x).

Fr-om the coordinate transformations Eq. 4-20 and Eq. 4-19, we find that

(4-21)

nation

f the locally

Bu dt
iit 81

8iv iit

Bu 8x
8ix 81

8iv 8ix
+i~ d

e-f (U" Uz)

e-f (Ut + U")

U",

U",

Bu du dt Bu 8x
+= e-f (U" Ut) = -U",

8iv 8iv iit 8iv 8ix
+= e-f(U" + Ut) = U".

We can now represent the surface r(u, v) = ro in the (t, x) coordinates. Using the

above equations and Eq. 4-17, we get

8 Bu 8iv
v (t, X) = ru r = re U" + r, U' = 0,

8 Bu 8v e-f
r(t, 2) ru dr, = ~ -r~,UU + r~,U' = (-2rar,) = 2e-f /r~r.

The fact that Br/8t = 0, clearly demonstrates that in the neighborhood of any point

the surface r = ro, is equivalent to the surface x = 0. Hence, both Er and E" are eligible

to physically represent the energy flux through the given physical area. Since we would

like to measure the energy flux with respect to the proper time of the observer, E" is the

quantity we are looking for. From the above equations, we can verify that Er and E" have

the following relation

Hence we conclude that the energy flux crossing the observer is given by 4xrr2E", where

E" is given by Eq. 4-21. After every step of evolution, this quantity is also calculated

and stored. We shall refer to this function as E(u, v). This function E(u, v) is a positive

function which means that there is a net energy flux in the outward direction.

Mass loss

Since an observer at large ro sees a net energy flux, the inferred mass loss of the black

hole can be calculated in the following way. At every step in the evolution, that is, for

every v step,

Follow the observer at r = ro as in Section 4.3.1.

Calculate the energy flux E.

*Calculate the proper time of the observer from Eq. 4-16. We will set -r = 0 at the
instant the observer crosses the shell (v = 0).

The differential equation Eq. 4-16 can be converted to finite difference equation and

solved numerically as the evolution proceeds.

Using conservation of energy, we can write an equation for mass loss as in CI Ilpter 3.

We know that the energy flux measured by the observer at finite ro would be redshifted

when it reaches infinity. We shall assume that for ro large enough, this redshift factor is

the same as the redshift factor in Schwrarzschild geometry J1- 2M/ro Recall that we

used this fact while writing down the quasi-static equations in OsI Ilpter 3. This assumption

might not be valid, but since our aim is to compare these results with the quasi-static case

where this assumption holds true, this assumption is justified. At any instant of time 7r,

the mass lost can then be written as,

aM~ = 4r 1, M/o(r)ATr, (4-22)

and hence the mass at any instant is

CHAPTER 5
RESULTS AND DISCUSSION

In this C'!s Ilter, we will discuss the results obtained from the numerical computation

described in ('! .pter 4 and compare the mass loss due to evaporation of the black hole

obtained from the exact numerical computations to the mass loss of the black hole

obtained from the quasi-static approximation scheme described in ('! .pter 3.

The stress energy tensor of our model (Eq. 2-25) has a parameter a~ in it. For a single

quantum field, a~ is proportional to R. To consider the presence of more than one field, we

have to multiply a~ by an appropriate number. In a sense, a~ determines the strength of

the quantum field. In terms of geometrized units (which we have been using throughout),

mass and time has the same dimension as length and a~ has the dimension of length. The

dimensionless quantity that characterizes the evaporation of a black hole of initial mass

Af, is then C~/Af2. To have a perspective on the size of this parameter, it is useful to note

that

For a solar mass black hole (1028K~g) and a single quantum field c~/Af2 ~ 10-74

For a primordial black hole of mass (1012K~g) and a single quantum field c~/Af2
10-5o. The primordial black holes of this mass range created during the Big hang are
expected to have evaporated completely by now.

For a Planck mass black hole (10-sEKg) and a single quantum field c~/Af2 -

Since this quantity is very small for realistic black hole, it would be impossible to carry

out numerical computations. We perform the numerical computations in the regime where

Ca/Af2 ~ 1. Let us not think of this as a Planck scale black hole evaporation problem

because our approach of using semi-classical Einstein equations might not he valid in this

regime, because important quantum gravity effects (if they exist) will have to be included.

Instead, we will think of the large value of a~ as an outcome of a very large number of

quantum fields working together to increase the strength of evaporation.

All data that will be shown in this C'!s Ilter will correspond to a choice of At = 2.

0 5 10 15 20 25 30 35

-14-

Figure 5-1. A plot of r= constant curves of Schwarzschild geometry. The six curves
correspond to r=8, r=6, r=4.5, r=4, r=3.5, r=3

5.1 Position Of Apparent Horizon

Let us first consider Fig. 5-1 which shows the r = constant (;?i ro) curves obtained

from the Schwarzschild geometry. We see that the curve ro = (2M~ = 4) is a straight line

at constant n = -8. All the curves with ro > 4 start at v = 0 with a s value less than

-8 and as v increases, a monotonically increases to .I-i-upha' tically reach the n = -8

line. This shows that all these curves are timelike. Another way to see this is, on all these

curves the quantity r, is positive. The curves with ro < 4 start at v = 0 with a s value

greater than -8 and as v increases, a monotonically decreases to .l-i-mptotically reach the

u = -8 line. This shows that all these curves are spacelike. Another way to see this is, on

all these curves the quantity r, is negative.

Let us now consider Fig. 5-2 which shows the r = constant (ro) curves obtained from

the evaporating geometry for a~ = 5. A rescaled version is shown in Fig. 5-3. There exists

an ri'such that, for ro > r, these curves are ahr-l- timelike (r, > 0). The curve r = r,

0 5 10 15 20 25 30 35

-14-

Figure 5-2. A plot of r= constant curves of evaporating geometry with a~ = 5. The six
curves correspond to r=8, r=6, r=4.5, r=4, r=3.5, r=3

starts out at v = 0 with r, = 0, and then attains a positive value for r,. All the curves

with ro < r," start out at v = 0 with a negative r,, and as v increases, the value of r,

on these curves increases to become positive. This means that, for some value of v, r,

becomes zero. Up to this value of v, the curve is spacelike, and beyond this value of v, the

curve will become timelike with r, > 0. We shall refer to this point as the turning point of

the r = ro curve. The turning point is different for different ro curves. By definition, the

apparent horizon has to pass through the turning points of all the the r = constant curves.

For the Schwarzschild case (Fig. 5-1), we see that all the r = constant curves

.I-i-uspini.'1 to a n =constant line (event horizon). Such a situation does not really happen

in Fig. 5-2. This is because, we have not identified the event horizon here. As mentioned

in C'!s Ilter 4, we cannot evolve the Einstein equations beyond the critical radius 2. This

actually sets a cutoff value cuto,ff beyond which the equations cannot be evolved, and the

event horizon lies beyond this value of autonf. Hence, it turns out that every r = constant

surface intersects the last n = constant (ucatonf) surface, and cannot be evolved further. In

Table 5-1, we have tabulated the coordinate position uap and the radius of the apparent

horizon when it first appears. We have also given the values of autor y for various values of

a~. The size of the bulge region is amoyf Uap.

We should note that the plot only shows the region exterior to the shell at v = 0. All

the r = constant curves in the interior of the shell (which is flat space) are timelike.

Table 5-1. Coordinate position of Bulge
0~ Mp cUtoff
5 4.579 -9.158 -8.569
4 4.469 -8.938 -8.514
3 4.356 -8.713 -8.381
2 4.241 -8.482 -8.252
1 4. 122 -8.245 -8.125
0.5 4.062 -8.123 -8.062
0.25 4.031 -8.062
0.125 4.015 -8.031

5.2 Violation Of WEC

As mentioned earlier, when the metric is expressed in terms of the null coordinates

the weak energy condition takes the form,

T,, > 0 T,, > 0 T,, > ( Ts,T,,) 1/ 2

From our model, the stress energy tensor of the quantum field when evaluated on the

Schwarzschild background (Eq. 3-6) gives

4xr2~ I 3 \~2 r4

We see that T,, < 0 if r > 2M~/3. This means that the weak energy condition is violated

by the quantum stress tensor in the region r > 2M~/3. Specifically, there is a finite region

surrounding the event horizon of the Schwarzschild geometry where the weak energy

0 5 10 15 20 25

-11-

Figure 5-:3. A plot showing the Apparent horizon and the region where Weak energy
condition is violated for an evaporating geometry with a~ = 5. This plot is just
a rescaled version of Fig. 5-2

condition is violated. This is a situation where the back reaction of the quantum stress

tensor on the geometry is neglected. Let us now look for the violation of weak energy

condition in the geometry where the back reaction is included.

In Fig. 5-3, which is just a rescaled version of Fig5-2 (the evaporating geometry with

a~ = 5), the orange curve is the apparent horizon and the brown curve is the boundary of

the region where T,., is negative. As seen from the analysis of Bergmann and Roman [:30],

we see that there exists a region surrounding the apparent horizon where T,,. is negative,

implying the violation of weak energy condition.

5.3 Comparison Of Mass Loss

In ('!, Ilter :3, we used quasi-static approximation on the model to calculate the mass

of the black hole as a function of proper time of the observer. In ('! .pter 4, we use a

numerical algorithm to evolve the completely back reacting geometry from which we can

compute the mass loss of the black hole as a function of proper time of the observer. Now,

we shall compare the two results. For various values of a~ ranging from 5 to 0.5, we have

graphically expressed the function M~(r) in the figures 5-4, 5-5,5-6, 5-7, 5-8 and 5-9. There

are two curves in each of these graphs, the red one represents the mass loss inferred from

the quasi-static approximation and the blue one represents the mass loss inferred from the

full numerical calculation.

In each of these graphs, we place three markers to identify and compare the different

stages of evaporation. Table 5-2 gives the position of these three markers.

Table 5-2. Position of markers
Late Times Diverging Point Apparent Horizon
a~ MET Tend M~DP TDP M~AH TAH
5 1.77 91.28 1.68 72 1.489 79.2
4 1.802 99.39 1.72 74 1.567 81
3 1.83 112.89 1.75 77 1.65 83.43
2 1.90 139.86 1.79 81 1.744 86.77
1 1.94 220.8 1.82 90 1.85 92.63
0.5 1.97 382.63 1.85 124 1.915 97.9

The first marker corresponds to the late times regime of evaporation. Recall that in

OsI Ilpter 3, Section 3.3.3, we categorized the quasi-static evaporation into three stages,

wherein the last stage, which we called Late times, had the property that

Mi ~ ~ (5-1)

For each value of a~, the evaporation enters this regime at a different instances and the first

marker M~LT represents the mass of the black hole at this instance. -rend COTTOSponds to the

end of evaporation obtained from the quasi-static approximation.

The graphs clearly show that the mass loss from the quasi-static approximation and

the complete numerical computation matches closely during the initial stages. Then there

is an instant where the two curves begin to diverge from each other. For different values of

a~, this happens at different instances. The second marker -rDP COTTOSponds to this instance

0.8-

0.4-

0 20 40 60 80 100

Figure 5-4. Comparison of mass loss curves with respect to an observer at r=30, for
a~=5. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the M~LT marker, the dashed line is the M~DP marker and the
dotted dash line is the M~AH marker.

where the mass loss curves of the quasi-static approximation and the complete numerical

calculation diverge away and M~DP COTTOSponds to the mass at that time.

Fr-om Table 5-2, we know that for various values of a~, the apparent horizon starts

formingf at different values of u. The apparent horizon is hidden from the observer until

he reaches n = uap. The third marker TAH represents the instant when the observer

starts seeing the apparent horizon and M~AH represents the mass of the black hole at that

instant. In other words, beyond this moment the observer will start seeing the effects of

the "B- lg, in the evaporation process.

There are several important points we would like to observe from the graphs and the

Table 5-2

0.8-

0.4-

0 20 40 60 80 100

Figure 5-5. Comparison of mass loss curves with respect to an observer at r=30, for
a~=4. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the Afer marker, the dashed line is the AfDP marker and the
dotted dash line is the IAlsH marker.

For smaller values of c0, we see that IAGH is larger. This means that, for small c0,
most of the evaporation takes place after the observer sees the bulge.

For any value of c0, we see that Afur is greater than IAGH and AfDp. This means that
the evaporation reaches the Late time regime before the apparent horizon forms and
before the quasi-static curve departs from the exact curve.

For larger values of c0, we see that IAGH is lesser than AfDp, while for smaller
values of c0, IAGH is greater than AfDP. This tells us that for small values of c0,
the evaporation appears to be quasi-static even after the apparent horizon is
formed. This leads us to an interesting conclusion: Even though most of the
evaporation takes place after the apparent horizon is formed, the evaporation
resembles quasi-static evaporation for small values of a~.

To quantitatively characterize the deviation of the quasi-static mass loss curve from the

exact mass loss curve, we shall fit the data to a power law curve. That is, we fit the data

0.8-

0.4-

0 20 40 60 80 100 120

Figure 5-6. Comparison of mass loss curves with respect to an observer at r=30, for
a~=3. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the M~LT marker, the dashed line is the M~DP marker and the
dotted dash line is the M~AH marker,

to
So~ 1 1
M ~ (5-2)
64 1 2M~/r o MEP '
where S and P depend on a~. For an ideal observer very far away, ro > M~. But this

cannot be practically achieved in the numerical calculation, because evolving the equations

out to large distances would accumulate a large error. For a finite ro, the above equation

canl be solved by TIaylor expanlding -21 2M/ro With a suitable definition for Q, we halve

1 1 M1 M2 5 M3S
Q M .. (Tend -T\
P +1 P + 2 ro 2(P + 3) ro8(P + 4) ro 64.S
(5-3)

0.8-

0.4-

0 20 40 60 80 100 120 140

Figure 5-7. Comparison of mass loss curves with respect to an observer at r=30, for
a~=2. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the M~LT marker, the dashed line is the M~DP marker and the
dotted dash line is the M~AH marker.

If the exact curve follows quasi-static evaporation, then the correct fit should correspond

to P = 2 and S = 1. Fr-om having checked the convergence of the data M~(r) for various

step sizes, we expect M~(r) to have an error of less than 10-4. For each value of a~, we

attempt to fit various values of P and S. We choose the best fit to be the one which gives

the least residual error. If the maximum residual error in fitting is less than 10-4, then we

can consider the fit as t;ood". Table 5-3 shows the best fit values of P and S. Now, we

can plot these values with respect of a~ as shown in the graphs Fig. 5-10 and Fig. 5-11.

If we extrapolate these results to a realistic black hole by taking the limit a~ 0 we

see that P 2 and S 1 This tells us that quasi-static approximation is valid whenever

a~ is very small.

0.8-

0.4-

0 50 100 150 200

Figure 5-8. Comparison of mass loss curves with respect to an observer at r=30, for
a~=1. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the MarT marker, the dashed line is the AfDP marker and the
dotted dash line is the IAlsH marker.

5.4 Apparent Horizon mass

Fr-om Fig. 5-3, we see that the apparent horizon (AH) is a timelike curve everywhere

outside the shell and radius of the apparent horizon ro;, decreases as time progresses. As

the black hole evaporates, the apparent horizon shrinks representing a mass loss. Following

Bardeen [1], we define the apparent horizon mass ifl,,. at any instant as roy, = 2.11,,. The

mass Af(-r) as observed by the far away observer at r = ro, is the Bondi mass Afe which

is different from the apparent horizon mass i T, We shall use Fig. 5-12 to illustrate this

difference.

We know that as time progresses, both i T,,, and Afe decreases. Bardeen considered

i T,,, as a measure of the mass of evaporating black hole because he used the advanced

time coordinate v to parameterize the evolution. If we use the retarded time coordinate u

1.8-

1.6-

1.4-

1.2-

0 50 100 150 200 250 300

Figure 5-9. Comparison of mass loss curves with respect to an observer at r=30, for
a~=0.5. The red curve indicates the quasi-static mass loss and the blue curve
corresponds to the mass loss from the exact numerical calculations. The
dotted line is the M~LT marker, the dashed line is the M~DP marker and the
dotted dash line is the M~AH marker,

to parameterize the evolution, the Bondi mass MsB would be the correct measure of mass.

Let us now consider an instant -r, when the observer's v coordinate position is v2. The

apparent horizon visible to him is at the coordinate position v = vl. Hence, the apparent

horizon mass with respect to the observer is MI,l,(vl). Let us ;?i that he measures a Bondi

mass Ms. We see that the rate of change of Ml,l, is slower than the rate of change of MsB.

This means that MsB and MI,l,(vl) are not comparable quantities. But there does exist a

relation between MsB and Ml,l,(v2), aS Shown by Bardeen.

Bardeen [1] has shown that for very small values of a~, dl~B/dvr ~ iM,, (v2)/dy. That

is, the rate of decrease of apparent horizon mass is equal to the luminosity of the black

hole (which is precisely the rate of decrease of the Bondi mass). This influenced Piran and

Parentini [26] to measure the apparent horizon mass as a function of v in their numerical

Table 5-:3. P(a~) and S(ca)
a~ P S Alax. Res. Error
5 1.48 0.651 0.002
4 1.62 0.72 0.001
:3 1.78 0.798 0.0002
2 1.87 0.867 0.0002
1 1.9:3 0.928 0.0001
0.5 1.975 0.97:3 0.00005

evolution. Their claim is that, when the mass

mass (~ )~,

ifl,,, is still large compared to the critical

When we repeat this calculation, we find that the coefficient in the above equation is c0/64

An important assumption that goes into Bardeen's analysis is that the observer at

ro is well inside the quasi-static regime. In Fig. 5-12, we see that when the observer is at

t' = vl, the black hole has not even formed, so obviously he cannot he in the quasi-static

regime. From our calculations, we can see that until the observer reaches large value

of to dl~B/dv is not comparable with .11.1[, (<'2) it. That is to ;?i, do not get into the

"quasi-static regime" as described by Bardeen until the observer reaches large values of

to Hence, if we are to use ifl,,. as the measure of the mass and 1.11.,1 /dt' as a measure

of the luminosity of the black hole, we have to go to very large values of v. But tracing

the apparent horizon to a very large value of t' is not numerically possible because the

apparent horizon would strike the auton at a finite value of v. Hence, we claim that our

method of calculating the Bondi mass by following an observer at constant ro is a much

more reliable and accurate measure of the mass. Moreover, this can he directly compared

to the mass curve obtained from quasi-static approximation (which is precisely what we

did in the graphs).

2.1-

2-

1.9

1.8-

1.7-

1.6-

1.5-

0 1 2 3 4 5

Figure 5-10. Best fit for P(a~).

1.1-

1-

0.9-
S

0.8-

0.7-

12 3 4 5

Figure 5-11. Best fit for S(~).

100