Comments on Entropy of 2D Black Holes from Counting Microstates

Commentson“Entropyof2DBlackHolesfromCounting

Microstates”

by

Mu-InPark1andJaeHyungYee

DepartmentofPhysics,YonseiUniversity,Seoul120-749,Korea

arXiv:hep-th/9910213v1 27 Oct 1999ABSTRACT

WepointoutthatarecentanalysisbyCadoniandMignemionthestatisticalentropyof2DblackholeshasaseriouserrorinidentifyingtheVirasoroalgebrawhichinvalidatesitsprincipalclaims.

PACSNos:04.70.Dy,11.25.H,11.30.-j,04.20.Fy9September1999

RecentlyCadoniandMignemi[1]presentedamicroscopicalderivationoftheentropyofthetwo-dimensional(2D)blackholes.Theyhaveshownthatthecanonicalalgebraoftheasymptoticsymmetryof2DAnti-deSitter(ADS)spaceincludedtheVirasoroalgebra.UsingthisalgebraandtheCardy’sformulatheyobtainedthestatisticalentropywhichagreed,upto

√

afactorof

Butthereareseriousmismatches[10]withourpurpose:a)Becausethealgebraisdefinedonlyatone(boundary)pointthereisnoroomfortheinfinitetowerofsymmetrygeneratorsthroughtheFourier-seriesexpansionofthealgebraasrequiredintheBHSprocedure.b)J[χ]istime-dependentingeneralwhichmeansthatJ[χ]isnotconservedquantity.ThisiscontrasttotheusualfactthatJ[χ]isa(boundarypartof)conservedNoethercharge[5].Moreover,thealgebraandthecentraltermc(χ,ω)arenowtime-dependentandeventually“time-dependent󰀅entropy”wouldbeexpectedS∼

2π

󰀄

2π/λ

0

dtJ[χ](3)

withtimeperiodof2π/λandtheyobtainedaVirasoroalgebraintheasymptoticsymmetryalgebraof2Dblackholes.Butthisprocedureisanerroneousone.Letusexplainourargumentindetail.

WestartourargumentbyconsideringEq.(2)intheirsuggestingframe.Itistruethatthe(overall)timeintegrationof(2)hasadefinitemeaningas

ˆ[[χ,ω]]+cδω󰀃J[χ]=Jˆ(χ,ω)

(4)

withthecentralcharge(23)ofRef.[1]andthislookslikeaVirasoroalgebra.Buttheproblem

ofthismethodisthattheleft-handsideof(4)cannotbewrittenas

ˆ[χ],Jˆ[ω]}DB,{J

(5)

whichisessentialfortheinterpretationofthetimeintegrated(1)asaVirasoroalgebra:Let

usconsider

{J[χ],H[ω]}DB={J[χ],J[ω]}DB=δωJ[χ]

whichimpliesthatH[ω]isthecorrect(asymptotic)symmetrygenerator,i.e.,

{φ(x),H[ω]}DB=δωφ(x).

(7)(6)

forthesymmetrytransformationδωφofthefieldφ.[ThisDiracbracketaswellasthePoissonbracketinEq.(19)ofRef.[1]canbewell-definedcontrasttotheclaimsofCadoniandMignemifollowingtheworkofRef.[11].]HereitisanimportantfactthattheDiracbracketsarealldefinedatequaltimesbecausetheconstitutingPoissonbracketsareequaltimesinnatureforthelocalfieldtheory:

{A,B}=

󰀄

dx

󰀁

δA

δπk(x,t)3

−

δA

δφk(x,t)

󰀂

forthecanonicalconjugatespairsφk(x,t),πk(x,t).Ifweperformthe(overall)timeintegrationover(3)theleft-handsidebecomes

λ

2π

󰀂2󰀄

2π/λ

0

dt

′

󰀄

2π/λ

0

dt

2π

δ(t−t′)intheintegrand.

Thisisadirectconsequenceof(6)whichisawell-knownfactrelatedtotheNoethertheoreminthefieldtheory;iftheirclaimwhichequating(4)and(5)wascorrect,thetime-integrated

ˆ[χ](≈Jˆ[χ])shouldbetreatedasthesymmetrygeneratornecessarilycontrasttoquantityH

(7).

Inconclusion,theymadeaseriousmistakebyidentifying(4)and(5)andhencetheirwayofapplyingtheCardy’sformulatoobtaintheblackholeentropy(25)ofRef.[1]withthemisidentifiedc(centralcharge)andl0(lowesteigenvalueoftheVirasorogenerator)cannotbejustified.

MIPwouldliketothankProf.StevenCarlip,Drs.Gung-wonKangandHyuk-jaeLeeforseveralhelpfuldiscussions.MIPwassupportedinpartbyapostdoctoralgrantfromtheNaturalScienceResearchInstitute,YonseiUniversityintheyear1999andinpartbytheKoreaResearchFoundationunder98-015-D00061.

λ

References

[1]M.CadoniandS.Mignemi,Phys.Rev.D59,081501(1999).[2]M.CadoniandS.Mignemi,Phys.Rev.D51,4319(1995).

[3]J.D.BrownandM.Henneaux,Comm.Math.Phys.104,207(1986).

[4]A.Strominger,J.HighEnergyPhys.02,009(1998);D.Birmingham,I.SachsandS.Sen,

Phys.Lett.B424,275(1998).[5]ThishasbeenrecentlystudiedalsointheChern-Simonstheories,M.Ba˜nados,Phys.Rev.

D52,5816(1996),M.Ba˜nados,T.Brotz,andM.Ortiz,Nucl.Phys.B545,340(1999);M.-I.ParkandP.Oh,Mod.Phys.Lett.A14,231(1999);M.-I.Park,Nucl.Phys.B544,377(1999);P.Oh,e-printhep-th/9906197.

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[6]S.Carlip,Class.Quant.Grav.15,3609(1998).

[7]R.Jackiw,in“QuantumTheoryofGravity”editedbyS.M.Christensen(Hilger,Bristol,

UK,1984);C.Teitelboim,ibid.[8]T.ReggeandC.Teitelboim,Ann.Phys.88,286(1974).

[9]P.A.M.Dirac,Lecturesonquantummechanics(YeshivaUniversityPress,NewYork,

19).[10]Similarsituationoccursalsointhehigherdimensionblackholesystems,S.Carlip,Phys.

Rev.Lett.82,2828(1999);e-printgr-qc/9906126;M.-I.ParkandJ.Ho,Phys.Rev.Lett.(inpress),Alberta-Thy-09-99,hep-th/9910158.[11]M.-I.Park,quotedinRef.[5].

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