Hawking radiation and masses in generalized dilaton theories

gr-qc/9605044

Hawkingradiationdilatonandtheories

massesingeneralized

H.Liebl

1

Institutf¨urTheoretischePhysikTechnischeUniversit¨atWien

WiednerHauptstr.8-10,A-1040Wien

Austria

D.V.Vassilevich

2

andS.Alexandrov

DepartmentofTheoreticalPhysics

St.PetersburgUniversity1904St.Petersburg

Russia

Abstract

Ageneralizeddilatonactionisconsideredofwhichthestandarddilatonblackholeandsphericallyreducedgravityareparticularcases.TheArnowitt-Deser-Misner(ADM)andtheBondi-Sachs(BS)massarecalculated.Specialattentionispaidtoboththeasymptoticcon-ditionsforthemetricaswellasforthereferencespace-time.Forthelatteronewesuggestamodifiedexpressiontherebyobtaininganewdefinitionofenergy.DependingontheparametersofthemodeltheHawkingradiationbehaveslikeapositiveornegativepowerofthemass.

1

1Introduction

Overthelastfewyears1+1dimensionaldilatontheorieshavebeenstudiedextensivelyintheirstringinspired(CGHS)version[1]aswellasinmoregen-eralforms[2].Oneofthemainmotivationstostudysuchmodelsarisesfromthehopethatwithinthesimplifiedsettingof2Dmodelsonecangaininsightintophysicalpropertiesof4Dgravity.TheyactuallyallowforexampleblackholesolutionsandHawkingradiationandaremoreamenabletoquantumtreatmentsthantheir4Dcounterparts.Thetwomostfrequentlyconsideredtheories,thestringinspiredCGHSaswellassphericallyreducedgravity(SRG)differdrasticallyinsomeoftheirphysicalproperties.Forexampledifferenceswereobservedwithrespecttothecompletenessofnullgeodesicsforthosetwomodels[3].Thesedifferencesdirectlyleadonetoinvestigatephysicalpropertiesofageneralizedmodelofwhichthetwoprominentexam-plesaresimplyparticularcases.

Importantclassicalquantitiesaretheenergyatspatialandnullinfinity,thesocalledArnowitt-Deser-Misner(ADM)andtheBondi-Sachs(BS)mass,respectively(providedtheglobalstructureofthespace-timeisthesameasforSRG).Forgeneralmodels,however,thereexistingeneralnoflatspacetimesolutions[3]andthereforeaproperreferencespace-timehastobechosen.Asecondimportantpointistorealizethatthese“masses”dependontheasymptoticbehaviorofthemetricandarethereforeonlydefinedwithrespecttoaparticularobserver.Ourapproachisbasedonthesecondorderformalism.Significantinsightcanalsobegainedfromafirstorderformalism[4],[5].

AnimportantfeatureinsemiclassicalconsiderationsisthebehaviorofHawkingradiation.IntheCGHSmodelitisjustproportionaltothecos-mologicalconstantwhereasthedependenceinSRGisinversetoitsmass,whichimpliesanacceleratedevaporationtowardstheendofitslifetime.AswewillshowageneralizedtheorywillexhibitHawkingradiationwhichisproportionaltotheblackholemassintermsofpositiveornegativepowersoftheblackholemass,dependingontheparametersofthemodel.

Insection2werepeatsomerelevantresultsof[3]forthemetricandtheglobalstructureoftheclassicalsolutionofageneralizeddilatonLagrangianwhich,foracertainrangeofparametersposessesasingularitystructurecoincidingwiththeoneoftheSchwarzschildblackhole.TheADMandBSmassarecalculatedinsection3.Furthermorewewillpresentadefinitionforenergy,differingfromtheoneproposedbyHawking[6],bytakinga

2

modifiedreferencespace-time.Thepathintegralmeasureandtheproblemofinterpretationofvariousenergydefinitionswillbediscussedinsection4beforewefinallydemonstratethedynamicalformationofblackholesintheframeworkofconformalgauge,withspecialemphasisonthenecessaryboundaryconditions.

2ClassicalSolution

AmongthenumerousdifferentgeneralizationsoftheCGHSmodel[1]weconsidertheaction

L=󰀁

d2x

√

2

,b=−1

8

8

C

˜+󰀂

C−

2B

4Bφ󰀄

,C˜=C−2Bln2.(5)

Thetransformation

󰀂

a=1:u=

e−2(1−a)φ

with

b=−1:a=1:l(u)=Ba

1|ua−1

,

a=1:l(u)=1

|

b+1

e−2(b+1)u󰀇

,b=−1:a=1:l(u)=

1

a−1

C+

2B

8

󰀂

C

˜+4Bu󰀄

󰀅

,wheretheconstantsaregivenby

BC

1=

a−1

BB

2=

a−1

.Thescalarcurvatureofthemetric(2)hastheform

b=−1R=1

b+1

(b+1−a)e−2(1+b)φ

󰀉

b=−1

R=

1

a−1

+B′a+b−12uBb

2

2

e2u−

a−1

󰀈

aC

a−1

b=−1,a=1:R=

1

l(u)

du2,

isobtainedbymeansofthetransformation

dv=dt−

du

3

(9)(10)

(12)

(13)

(17)

󰀉

(19)

4

SincebelowwewillrepeatedlymakeuseofthespecialcasesofasymptoticallyMinkowski,RindleranddeSitterspace-timeswelistthemforcompleteness.Afterasuitablerescaling(seeeq.(34)below)theyare

AsymptoticallyMinkowskispace:(b=a−1)

(ds)2=(MU

a

a−1AsymptoticallyRindlerspace:(b=−0)

1)−1dU2(ds)2=(MUa22a

2U)dt2

−12Sitterspace:−(MU2U)dUAsymptoticallyde(b=1−a)

(ds)2

=(MU

a

a−1

−B2U2)−1dU2

2−a

withM=B1B

(21)(22)(23)

5

gravity[7][15]andwilllateronapplythemtosolutionsofourmodel.Wereviewwhatkindofslicingwehavetoworkwithandwillthenoutlinethemainstepsonhowtoobtaintheexpressionforthetotalenergy.

Consideraone-dimensionalspacelikesliceΣdrawninourspacetime(seeFig.1).AssumethatΣhasaboundarypointB=∂Σ.Lettheunit,outward-pointing,spacelikenormalofthepointBasembeddedinΣbenµ.IfΣisasurfaceofconstanttwithmetricΛ2dr2thenthespace-timemetricnearΣcanbewritteninADMform[16]

ds2=Λ2dr2−N2(dt+Λtdr)2.

(24)

B(t)NuΣ

NrBnµFigure1:Spacetimefoliation:ΛtdenotingtheradialshiftandΛtheradiallapse

Toobtaintheexpressionforthetotalenergyonehastocasttheactionintohamiltonianform.Boundarytermshavetobeaddedtotheactiontoensurethatitsassociatedvariationalprinciplefixestheinducedmetricandthedilatonontheboundary.Asshownin[7],theformofasuitableHamiltonianwithboundarytermsatBisthefollowing:

H=

󰀁

Σ

dr(NH+NrHr)+N(Eql+NrΛPΛ)|B,

(25)

wherePΛistheADMmomentumconjugatetoΛandHandHraretheHamiltonianandthemomentumconstraintrespectively.SincetheexpressionfortheHamiltonian(25)divergesingeneral,areferenceHamiltonianH0hastobesubtractedtoobtainthephysicalHamiltonian.E,whichdefinesthequasilocalenergyisgivenby

Eql=e−2φ(n[φ]−n[φ]0)

(26)

6

wherethesecondtermindicatesthatthevalueisreferencedtoabackground.WeshallonlyconsiderthecasewhenNr=0atB,i.etheon-shellvalueofHamiltonianwillbeassociatedonlywithtimetranslations(Hdoesnotgeneratedisplacementsnormaltotheboundary).Definingthetotalenergyasthevalueofthephysicalhamiltonianontheboundarywefinallyget

E≡H|B=NEql=4Ne−2φ(n[φ]−n[φ]0)|B

(27)

OnlyforN=1thiscoincideswiththeADMmass,whereastheadditionalfactorNin(27)givestheproperdefinitionforthetotalenergyforspace-timeswhoselapsedoesn’tgotooneasymptotically[17,6].However,asweshallseeattheendofthissection,theformof(27)isnotunique.Toapplythisresulttoouraction(1)wetakethelineelement

(ds)2=g(φ)l(φ)dt2−

g(φ)

Λ

󰀈

∂

l(φ)

weobtain

EADM=4e

−2φ

N2=−g(φ)l(φ)

(30)

󰀅

l(φ)−

󰀃

7

wehavenoa-dependenceonEfortheaction(1).CalculatingthisquantityatspacelikeinfinitycorrespondstotheADMmass.Inserting(4)into(31)withC=0forl0thedivergingtermscanceleachotherwhereasthenextordertermsgiveafinitecontribution.Weobtain

E=

C

+O(e−(b+1)φ)

for

(b+1)φ→+∞.

(33)

2

ThiscoincideswiththeexpectationthattheparameterCisproportionaltothemass.Noticethatfortheseexpressionsthereferencespace-time(C=0)itselfcontainsalreadyacurvaturesingularity[3]exceptforthespecialcasesconsideredbelow.Forsphericallyreducedgravity(a=−b=1

2

dv

dU=K−

1

K

(35)

andatthesametimeweareabletoobtaindimensionsoflengthforUandV.SimultaneouslyEADMischanged.WeseethatanunambiguousdefinitionofEADMincludestwoimportantingredients.Thefirstoneisthereferencepointforenergywhichisspecifiedbythechoiceofgroundstatesolution.WedefinethegroundstateconfigurationsbyC=0.FortheimportantparticularcasesofasymptoticallyMinkowski,RindleranddeSittermodelsthischoicegiveszeroorconstantcurvaturegroundstatesolutions.Thesecondingredientistheasymptoticconditionforthemetric.ItcorrespondstothechoiceofanobserverwhomeasuresenergyandHawkingradiationattheasymptoticregion.Thisambiguityinthechoiceoftimefortheasymptoticobserverisexpressedin(35).Inthespiritoftheseremarks,thevalue(32)and(33)oftheADMmassshouldbespecifiedas”theADMmasswithrespectto

8

C=0solutionmeasuredbyanasymptoticobserverwithtimeandlengthscalesdefinedbymetric(2)”.Thisvalueisunambiguouslydefinedforallvaluesofaandb.Fortheimportantparticularcasesmentionedaboveotherdefinitionsaremorerelevant:

•AsymptoticallyMinkowskiandRindlerspaces:

Asmentionedabove,eq.(34)canbeusedtoobtaindimensionsoflengthforUandV.ThishappenstobethecaseifKhasdimensionsofenergysquared.

m1−m

ThereforeacombinationofB1andB2,e.g.B1B2,orequivalentlyofCandB,isanaturalchoicesinceallofthesequantitieshavedimensionofenergysquared.However,becauseourvacuumisdefinedasC=0,i.e.B1=0,onlyB2shouldcontributetoK.Ifwerequireinadditionthatfortheminkowskiancaseourmetricapproachesasymptoticallytheunitoneηµν=diag(-1,1),wehavetouse

K=B2

whichinturngives

2−a

a−1−b

a−1

(36)

L(U)=B1B

With

n=

1∂U

=

−B

∂

a−1

.(37)

√Λg(φ)

2

󰀈

b+1

2

(2(a−1))

b−a+1

2

󰀅

a

2

andEADM=

C

2

,(41)

respectively.

•AsymptoticallydeSitterspaces:

9

Arescalingof(8)asin(34)doesnotchangetheasymptoticbehaviorof(19),whichisgivenby

(ds)2=−B2u2dt2+(B2u2)−1du2,

(42)

butthecoefficientoftheasymptoticallyvanishingtermischangedascanbeseenfrom(37).Thereforeweget

EADM=

C

B

󰀇1

2

,(45)

whichnowdoesnotonlyholdfor|φ|→∞butonthewholespace-time.Thisvaluedependsagaininthesamemannerasaboveontheasymptoticconditionforthemetric.NotethatforasymptoticallyMinkowski,RindlerordeSittersolutionsthetwoexpressions(33)and(45)forEagree.Ontheotherhand,forarbitraryasymptoticbehaviorthetwovaluesofEcorrespondtotwodifferentdefinitionsoftheasymptoticobserverinreferencespace-time.Inarecentpaper[18]Mannalsoconsideredamodifiedreferencespace-timefortheenergytoobtainthesocalledquasilocalmass,whichalsodivergesfromHawking’sproposal.Notethatthesamerescalingasin(34)couldcertainlybeappliedto(44)therebygivingadditionalcoefficientsofBandtheparameters.Asimilarresultwasalsoobtainedin[19]byapplyingaRegge-Teitelboimargument.

10

3.2Bondi-SachsMass

InthefirstpartofthischapterweobtainedthevalueofthetotalenergywhichcorrespondstotheADMmassatspatialinfinity.HerewewillshowthatEonI+,thesocalledBSmassEBS,equalsEADManddoesnotdependonthevalueoftheretardedtimecoordinatev.However,theproceduretoobtainthisquantityislogicallydifferent.ThemaindifferencewithrespecttothecalculationoftheADMmassaboveisthatwedonottakethelimitφ→∞alongasingleslicebutinsteadwewillconsiderthelimitofexpression(27)alongaparticularnullhypersurfaceNassociatedwithdifferentslicesalongalineofconstantretardedtime[20](seeFig.2).ForconveniencewewillworkagainwiththeEF-metric(8).Firstwedefineafuturedirectedlightlike

lim

φ−>

8N

a) ADMb) Bondi

Figure2:ConceptualdifferencesintheconstructionoftheADMandtheBondimass

vectorkµandpickanotherlightlikevectorhµsuchthathµkµ=1.Theyare

kµ∂/∂xµ=

󰀆

hµ∂/∂xµ=−

󰀆

2

∂/∂u2∂/∂u+

󰀆

(46)

l∂/∂v.

(47)

11

Next,alongthenullhypersurfaceNwelet

u˜µ∂/∂xµ:=

1

(kµ+hµ2)(48)=

1

l

∂/∂v(49)

definethetimelikenormaltotheΣslicesspanningthepointsofN.Similarlywedefineitsspacelikenormal

n˜µ∂/∂xµ:=

1

(kµ−hµ)(50)

=

√

2√∂/∂v=N−1∂/∂t,whichisobtainedbyusing(20).Therefore,

withN=

󰀃

2

l(u)l0(u)

󰀇

∂φ

∂u

.(53)

Using(52)in(27)andrememberingthatg(φ)dφ=duandl(u)=l(φ)g(φ)

wefinallyobtain

EBS=4e

−2φ

󰀅

l(φ)−

󰀃

12

whichisexactlythesameasexpression(31)andthereforetheADMmassandtheBondi-Sachsmassturnouttohavethesamevalue.ThiscanbereadilyunderstoodbyrecallingthatthedifferencebetweentheBSandtheADMmassistheintegralofastressenergyfluxwhichvanishesatthispurelyclassicallevel.Similarly(53)exactlyreproducestheresultof(44)therebyshowingthatalsothemodifieddefinitionofmassonI+agreeswiththevalueatspatialinfinity.

4

HawkingradiationofgeneralizedSchwarzschildblackholes

ThereareanumberofwaysofcalculatingtheHawkingradiation[21].Oneofthemconsistsincomparingvacuabeforeandaftertheformationofablackhole.Inthecaseofgeneralizeddilatongravitythiswayistechnicallyratherinvolved.Wepreferasimplerapproachbasedonananalysisofstaticblackholesolutions.

ConsiderageneralizedSchwarzschildblackholegivenby

ds2=−L(U)dτ2+L(U)−1dU2,

whereL(U)hasafixedbehaviorattheasymptoticregionI+:

L(U)→L0(U)

(56)(55)

withL0(U)correspondingtothegroundstatesolution.AtthehorizonwehaveL(Uh)=0.WecancalculatethegeometricHawkingtemperatureasthenormalderivativeofthenormoftheKillingvector∂/∂τatthe(nonde-generate)horizon[22]

TH=|

1

2

lnL.(59)

13

Inconformalcoordinatesthestressenergytensorlookslike[21]

T−−=−

1

2

∂z,∂zρ=

1

2

L′′L,

()

whereprimedenotesdifferentiationofLwithrespecttoU.Bysubstituting()into(63)weobtaintheHawkingflux

T−−|asymp=

12

L′(U))2|hor.

(65)

Itiseasytodemonstratethatourresultisindependentofaparticularchoiceofconformalcoordinatesprovidedthebehaviorattheasymptoticregionisfixed.Withoutdestroyingtheconformalgaugeonecanchangecoordinatessothat

ρ→ρ+h+(x+)+h−(x−)

(66)

14

withtwoarbitraryfunctionsh±.SincethebehaviorofρatI+isfixed,h−=0.Thetransformationwitharbitraryh+doesnotchangeTtoapply(65)tosomespecificspace-timesweremark−−.

BeforewestartthatasinthecaseoftheADMandBondimassthevaluewillonceagaindependonourchoiceoftheasymptoticbehaviorofthemetric.WewillusethesameformofthemetricasforthecalculationoftheADMmass,equations(35)and(37),whichgivethemetric(55)afteratransformationlike(20).Forthemostinterestingspace-timesofouraction(1)wewillexplicitlypresentthesolutions.

•AsmentionedAsymptoticaboveMinkowskithiscorrespondsspacetimes:

tob=a−1.From(37)wegetfora=1

2−a

L(U)=B1B

a−1

−1.(67)

ThehorizondeterminedbyL(Uh)=0islocatedat

U1−a

h=B

2

2a

(68)

andthereforeweget

Ta2

−−|asymp=

a󰀅

2B

a

.(69)

Forthespecialcasea=1,theCGHSmodel,wehave

L(U)=

e

√

2B

−1(70)

whichresultsin

Tλ2

−−|asymp=

2(a−1)

2

U

a

15

andgetforthevalueofHawkingradiation

T−−|asymp=

2−a

2

2

󰀇

1

2

2(a−1)

U

a

2

2

U(74)

andtheHawkingradiationreads

T−−|asymp=

B(a−1)

.

(75)

Notice,thatifwehadnotusedtherescalingcoefficientasin(37),butifinsteadwehadset

K=

B

48

whichisjusttheresultgivenin[11].

,(77)

5

Pathintegralmeasure,asymptoticcondi-tionsandtheproblemofinterpretation

Itiswellknown(seee.g.[26])thatthereisauniqueultralocalpathintegralmeasureforascalarfieldfyieldingacovariantlyconservedstressenergytensor.Thismeasureisdefinedbytheequation

󰀁

Dfexpi

󰀅󰀁

√

16

Asaconsequence,HawkingradiationandADMmassdependonachoiceofasymptoticconditionsandpathintegralmeasure.

IndilatongravitywehaveanewentitywhichisabsentinEinsteingrav-ity,thedilatonfield.Oneistemptedtousearescaledmetricatsomestepsofthecalculations.Inthepresentpapersucharescalingistrivialeverywhere.Thismeansthatweareusingjustthemetricgµνwhichappearsintheaction(1)todefinethepathintegralmeasure,stressenergytensoretc.Ofcourse,thisisjustamatterofinterpretation.Onecanclaimthattherescaledmet-ricΦ(φ)gµνdescribesthegeometryofspace-timeandthusshouldbeusedinthedefinitionsoftheabovementionedquantities.However,therescaledmetricshouldbeusedeverywhere.Otherwise,thequantumtheorybecomesinconsistent.Forexample,apartofdiffeomorphisminvariancecanbelost.WebelievethatifonewishestokeepcontacttofourdimensionalEinsteingravity,whichisdiffeomorphisminvariant,oneshouldretaingeneralcoordi-nateinvariance3.Inthecontextofdilatongravitymodelsatransitionfrom

gµνtogΦ

µν=Φ(φ)gµνmeansreplacementofoneparticulardilatoninterac-tionbyanother.SuchareplacementshouldingeneralchangetheHawkingradiationbecausethedefinitionsofthestressenergytensorandpathin-tegralmeasureandtheasymptoticbehaviorofmetricarenotconformallyinvariant.Atleast,conformalinvarianceoftheHawkingradiationwasneverproved.SuchconformalequivalencewasconjecturedinarecentpaperbyCadoni[29]whousedconformaltransformationingeneraldilatontheorytoremovekineticterm(∇φ)2ofthedilatonfield.Healsousedaφ-dependentpathintegralmeasureandφ-dependentstressenergytensortodefinetheHawkingradiation.AcomparisonshowsthathisresultsfortheHawkingtemperaturedifferbyanaandbdependentfactorfromours,whichwereob-taineddirectlywithoutuseofconformaltransformation.WeconcludethatconformallyequivalenttheoriesdoreallygivedifferentresultsfortheHawk-ingradiation.Thisstatementcanbeillustratedbyanexamplefrom.[30],wheretheCGHSblackholewastransformedtotheflatRindlerspace-time.DuetotheabsenceofblackholecurvaturesingularityanyradiationinthelatterspaceshouldbeconsideredastheUnruhradiationratherthantheHawkingone.

WehaveseenthatbothADMmassandHawkingradiationdependontheasymptoticbehaviorofthemetricandonthesubtractionprocedureofrefer-

17

encespace-timecontribution.Thisdependenceisquitenaturalfromaphys-icalpointofview.Sinceenergyandenergyfluxarenotcoordinateinvariant,inadditiontoazeroenergystateoneshouldalsodefinetwoobservers.Oneofthemcorrespondstoaparticularsolution,andtheothertothereferencezeropointconfiguration.ForasymptoticallyMinkowski,RindlerordeSittersolu-tionthereisachoicewhichisclearlypreferable.WiththischoiceourresultsforADMmassandHawkingradiationareindependentofcoordinatetrans-formationswhichvanishrapidlyenoughatasymptoticregion.InthecaseoffourdimensionalEinsteingravityappropriateasymptoticconditionswereformulatedbyFaddeev[27].FortheCGHSmodelwithPolyakov–Liouvilletermthesuitableasymptoticconditionswerefoundrecently[14].

6

6.1

Dynamicalformationofblackholes

Generalsolution

Considerascalarmatterfieldfminimallycoupledtogravity.Weaddthefollowingtermtotheaction(1)

Lm=−

1

−ggµν∇µf∇νf.B

e2φTµν=0

(79)

Inthepresenceofthismatterfieldtheequationsofmotiontaketheform

gµν((4−2a)(∇φ)2−2∇2φ−

2

R−4a(∇φ)2+4a∇2φ+B(a+b)e2(1−a−b)φ=0

∇2f=0(80)

andinconformalgaugetheseequationsare

2∂+∂−ρ+4a∂+φ∂−φ−4a∂+∂−φ+

B

e2(ρ+(1−a−b)φ)=0

1

∂+∂−f=0

(82)(83)

4

4(a−1)∂±φ∂±φ+2∂±∂±φ−4∂±ρ∂±φ=

18

Thematterequationofmotion(83)canbesolvedbysettingf=f(x+).Thenρcanbedeterminedfromthe(−,−)componentof(84):

∂−ρ=(a−1)∂−φ+

1

2

ln|∂−φ|+ξ(x+)

(86)

Thearbitraryfunctionξ(x+)canberemovedbyusingresidualgaugefreedomoftheconformalgauge.Inwhatfollowswesetξ=0.ρcanberemovedfromtheequationsofmotionwhicharereducedtothefollowingthreeindependentequations:

1

∂+∂−(φ−

1

B

8

8(f′)2

(87)(88)

|∂−φ|e−2bφ

−2∂+∂−φ+4∂+φ∂−φ+

8(b+1)

e−2φ(b+1)=−2η(x+)e−2φ+λ(x+)

(90)

withtwoarbitraryfunctionsηandλ.Fromeq.()oneobtains

∂+e−2φ∓

B

16(b+1)

e

−2bφ

−

1

2

(f′)2

(93)

19

Thetwoequations(92)and(93)completelydefineφforagivenmatterdistribution.Letustakethematterfieldintheformofshockwave

1

Bbt

2bln|

b+1>0

φ=

1

4(b+1)

|,for

B

±

B

Bbx−

2b

ln|

±

B

20

intheasymptoticregionsbeforeandaftertheformationofablackholethemetricofthesolutions(96)and(95)behavesas

e2ρ→|B/16a|.

(99)

ThustoobtainablackholesurroundedbyMinkowskispacewithunitmetricweneedaproperrescalingofcoordinates.

Letusmodifytheprocedureoftheprevioussubsectioninordertoobtaintherescaledsolution.Firstofall,oneshouldtakethefunctionξin(86)intheform

ξ=

1

B

󰀊󰀊󰀊.󰀊󰀊󰀊a󰀊󰀊

(100)

Withthischoiceequation(86)nowreads

ρ=(a−1)φ+

1

4

󰀅

a

2

.(102)

ThisconstantisproportionaltotheenergyoftheshockwavemeasuredbyaMinkowskispaceobserver.Thisconfirmsourpreviousresult(41)fortheADMmassofblackholeinasymptoticallyminkowskianspace-time.

ItistemptingtocalculatetheHawkingradiationbycomparingnormalorderingprescriptionsandGreenfunctionsintwoasymptoticregions,beforeandafterformationofablackhole[31].Ifsuchamethodcouldbeappliedinthecoordinatesystems(x+,x−)and(x+,h(x−)),thestressenergytensoroftheHawkingradiationwouldbecalculatedbydifferentiatingthefunctionh(x−).Thisappeared,however,nottobethecase.Thefullgroundstatesolutioniscoveredbypositive(ornegative)valuesofxort.HencetheGreenfunctiondiffersfromthestandardexpressionvalidforinfiniterangeofallcoordinates.

21

7Conclusions

InthepresentpaperwecalculatedtheADMandBSmassandHawkingradiationoftheblackholesolutionsingeneralizeddilatongravitiesdescribedbytheaction(1).Specialattentionwaspaidtoasymptoticconditionsforthemetricfield.Itturnedoutthatbothmassandstressenergytensordependontheseconditions.Fromaphysicalpointofviewthismeansthattheyareinfluencedbythechoiceofanasymptoticobserver.Forgenericaandbthischoiceisnotunique.Itratherdependsontheinterpretationofparticulartwodimensionalmodelanditsrelationtofourdimensionalgravity.FortheselectedvaluesofaandbcorrespondingtoasymptoticallyMinkowski,RindlerordeSittersolutionsthereexistsapreferredcoordinateframeatinfinity.Thisfactallowssomephysicalconclusions.Wecanseethat,dependingontheparametersofthedilatonicaction,theHawkingtemperaturecanberelatedtotheblackholemasswithpositiveornegativepower.Inafullyquantizedtheorytheparametersaandbshouldbecomerunningcouplingsbeingfunctionsofascaleparameterwhichcouldbetheblackholemass.Thusitcouldhappenthatablackholewhichstartsitsevolutionnearthepointa=−b=1

22

[1]C.G.Callan,S.B.Giddings,J.A.Harvey,andA.Strominger,Phys.

Rev.D,45(1992)1005;[2]G.Mandal,A.SenguptaandS.R.Wadia,Mod.Phys.Lett.A6(1991),

1685(preprintIASSNS-HEP-91/10);E.Witten,Phys.Rev.D44(1991)314;V.P.Frolov,Phys.Rev.D46(1992)5383;J.G.RussoandA.A.Tseytlin,Nucl.Phys.B382(1992)259;J.Russo,L.Susskind,andL.Thorlacius,Phys.Lett.B292(1992)13;T.Banks,A.Dabholkar,M.Douglas,andM.O’Laughlin,Phys.Rev.D45(1992)3607;S.P.deAlwis,Phys.Lett.B2(1992)278;E.Elizalde,P.Fosalba-Vela,S.NaftulinandS.D.Odintsov,Phys.Lett.B352,(1995),235[3]M.O.Katanaev,W.KummerandH.Liebl,“Onthecompletenessof

theBlackHoleSingularityin2dDilatonTheories”,Techn.Univ.Wienprep.TUW95-24(gr-qc/9602040)[4]T.Kl¨oschandT.Strobl,Class.QuantumGrav.13(1996)965,(gr-qc/9508020);T.Kl¨oschandT.Strobl,ClassicalandQuantumGravity

in1+1Dimensions:PartII:Theuniversalcoverings,(gr-qc/9511081)tobepublishedinClassicalandQuantumGravity;[5]W.KummerandS.R.Lau(inpreparation)

[6]S.W.HawkingandG.T.Horowitz,TheGravitationalHamiltonian,Ac-tion,EntropyandSurfaceTerms,(gr-qc/9501014)[7]S.R.Lau”Onthecanonicalreductionofsphericallysymmetricgrav-ity”,Techn.Univ.Wienprep.TUW95-21(gr-qc/9508028),toappearinClass.QuantumGrav.[8]C.Teitelboim,Phys.Lett.126B(1983)41;R.Jackiw,1984Quantum

TheoryofGravity,edS.Christensen(Bristol:Hilger)p.403[9]J.S.LemosandP.M.Sa,Phys.Rev.D49(1994)27[10]S.Mignemi,Phys.Rev.D50,R4733(1994)

[11]A.FabbriandJ.G.Russo,’Solublemodelsin2ddilatongravity’,prep.

CERNTH/95-267,(hep-th/9510109),tobepublishedinPhys.Rev.D

23

[12]A.S.Eddington,AComparisonofWhitehead’sandEinstein’sFormulas,

Nature/113,(1924),192;D.FinkelsteinPast-FutureAsymetryoftheGravitationalFieldofaPointParticle,Phys.Rev.110(1958),965-967[13]M.Iofa,EnergyinDilatonGravityinCanonicalApproach,(hep-th/9602023)[14]W.T.KimandJ.Lee,Int.J.Mod.Phys.A11no.3(1996)553[15]J.D.BrownandJ.W.York,Phys.Rev.D47,1407,(1993)

[16]R.Arnowitt,S.Deser,andC.W.Misner,inGravitation:AnIntroduc-tiontoCurrentResearch,ed.L.Witten,Wiley,NewYork,(1962)[17]L.AbbottandS.Deser,Nucl.Phys.B195(1982)76

[18]K.C.K.Chan,J.D.E.CreightonandR.B.Mann,Conservedmassesin

GHSEinsteinandstringblackholesandconsistentthermodynamics,(gr-qc/9604055)[19]W.KummerandP.Widerin,Phys.Rev.D51(1995)4265

[20]S.R.Lau,”Energyofisolatedsystemsatretardedtimesasthenulllimit

ofthequasilocalenergy”,personalnotes01/96[21]S.B.Giddings,e.g.N.D.BirrellandP.C.W.Davies,Quantumfieldsin

curvedspace,CambridgeUniversityPress,Cambridge1982;R.Brout,S.Massar,R.Parentani,Ph.Spindel,Phys.Rep.260(1995)329[22]Cf.e.g.R.M.Wald,“GeneralRelativity”,UniversityofChicagoPress,

1984[23]C.W.Misner,K.S.ThorneandJ.A.Wheeler,Gravitation(Freeman,San

Francisco,1973)[24]S.M.ChristensenandS.A.Fulling,Phys.Rev.D15(1977)2083.[25]R.M.Wald,Commun.Math.Phys.45(1975)9.

[26]K.Fujikawa,U.Lindstrom,N.K.RocekandP.vanNieuwenhuizen,

Phys.Rev.D37(1988)391.[27]L.D.Faddeev,UspekhiFiz.Nauk[Sov.Phys.Uspekhi]136(1982)435.

24

[28]R.Jackiw,Anotherviewonmasslessmatter-gravityfieldsintwodimen-sions,(hep-th/9501016);

D.Amelino-Camelia,D.BackandD.Seminara,Phys.Lett.B354(1995)213;

H.S.La,Areapreservingdiffeomorphismsand2Dgravity,MIT-CTP-2462,(hep-th/9510147)

J.CruzandJ.Navarro-Salas,Solvablemodelsforradiatingblackholesandarea-preservingdiffeomorphisms,ValenciapreprintFTUV/95-31(1995),(hep-th/9512187)[29]M.Cadoni,Phys.Rev.D53,(1996),4413

[30]M.CadoniandS.Mignemi,Phys.Lett.B358(1995)217.

[31]A.Strominger,“LesHouchesLecturesonBlackHoles”(hep-th/9501071);L.Thorlacius,“BlackHoleEvolution”(hep-th/9411020);S.B.Giddings,“QuantumMechanicsofBlackHoles”,(hep-th/9412138);