The Cardy-Verlinde formula and entropy of Topological Reissn

a r X i v :h e p -t h /0210187v 1 20 O c t 2002The Caedy-Verlinde formula and entropy of Topological Reissner-Nordstr¨o m black holes in de Sitter spaces M.R.Setare ?Department of Science,Physics group,Kordestan University,Sanandeg,Iran and Institute for Theoretical Physics and Mathematics,Tehran,Iran and Department of Physics,Sharif University of Technology,Tehran,Iran February 1,2008Abstract In this paper we discuss the question of whether the entropy of cosmological horizon in Topological Reissner-Nordstr¨o m-de Sitter spaces can be described by the Cardy-Verlinde formula,which is supposed to be an entropy formula of conformal ?eld theory in any dimension.Furthermore,we ?nd that the entropy of black hole horizon can also be rewritten in terms of the Cardy-Verlinde formula for these black holes in de Sitter spaces,if we use the de?nition due to Abbott and Deser for conserved charges in asymptotically de Sitter spaces.Our result is in favour of the dS/CFT correspondence.

1Introduction

Holography is believed to be one of the fundamental principles of the true quantum the-

ory of gravity[1,2].An explicitly calculable example of holography is the much–studied AdS/CFT correspondence.Unfortunately,it seems that we live in a universe with a pos-itive cosmological constant which will look like de Sitter space–time in the far future. Therefore,we should try to understand quantum gravity or string theory in de Sitter space preferably in a holographic way..Of course,physics in de Sitter space is interesting even without its connection to the real world;de Sitter entropy and temperature have always been mysterious aspects of quantum gravity[3].

While string theory successfully has addressed the problem of entropy for black holes,dS entropy remains a mystery.One reason is that the?nite entropy seems to suggest that the Hilbert space of quantum gravity for asymptotically de Sitter space is?nite dimensional, [4,5].Another,related,reason is that the horizon and entropy in de Sitter space have an obvious observer dependence.For a black hole in?at space(or even in AdS)we can take

the point of view of an outside observer who can assign a unique entropy to the black hole.

The problem of what an observer venturing inside the black hole experiences,is much more tricky and has not been given a satisfactory answer within string theory.While the idea of black hole complementarity provides useful clues,[6],rigorous calculations are still limited to the perspective of the outside observer.In de Sitter space there is no way to escape the problem of the observer dependent entropy.This contributes to the di?culty

of de Sitter space.

More recently,it has been proposed that de?ned in a manner analogous to the AdS/CFT correspondence,quantum gravity in a de Sitter(dS)space is dual to a certain Euclidean CFT living on a spacelike boundary of the dS space[7](see also earlier works[8]-[10]). Following the proposal,some investigations on the dS space have been carried out re-cently[9]-[27].According to the dS/CFT correspondence,it might be expected that as

the case of AdS black holes[28],the thermodynamics of cosmological horizon in asymp-totically dS spaces can be identi?ed with that of a certain Euclidean CFT residing on a spacelike boundary of the asymptotically dS spaces.

In this paper,we will show that the entropy of cosmological horizon in the Topologi-

cal Reissner-Nordstr¨o m-de Sitter spaces(TRNdS)can also be rewritten in the form of Cardy-Verlinde formula.We then show that if one uses the Abbott and Deser(AD)pre-scription[29],the entropy of black hole horizons in dS spaces can also be expressed by

the Cardy-Verlinde formula.

2Topological Reissner-Nordstr¨o m-de Sitter Black Holes We start with an(n+2)-dimensional TRNdS black hole solution,whose metric is

ds2=?f(r)dt2+f(r)?1dr2+r2γij dx i dx j,

f(r)=k?ωn M

8(n?1)r2n?2

?

r2

n Vol(Σ)

,(2) 2

whereγij denotes the line element of an n?dimensional hypersurfaceΣwith constant cur-vature n(n?1)k and volume V ol(Σ),G n+2is the(n+2)?dimensional Newtonian gravity constant,M is an integration constant,Q is the electric/magnetic charge of Maxwell?eld. When k=1,the metric Eq.(1)is just the Reissner-Nordstr¨o m-de Sitter solution.For gen-eral M and Q,the equation f(r)=0may have four real roots.Three of them are real, the largest on is the cosmological horizon r c,the smallest is the inner(Cauchy)horizon of black hole,the middle one is the outer horizon r+of the black hole.And the fourth is negative and has no physical meaning.The case M=Q=0reduces to the de Sitter space with a cosmological horizon r c=l.

When k=0or k<0,there is only one positive real root of f(r),and this locates the position of cosmological horizon r c.

In the case of k=0,γij dx i dx j is an n?dimensional Ricci?at hypersurface,when M=Q=0the solution Eq.(1)goes to pure de Sitter space

ds2=r2

r2

dr2+r2dx2n(3)

in which r becomes a timelike coordinate.

When Q=0,and M→?M the metric Eq.(1)is the TdS(Topological de Sitter)solution [32,33],which have a cosmological horizon and a naked singularity,for this type of solution,the Cardy-Verlinde formula also work well.

In the BBM prescription[22],the gravitational mass,subtracted the anomalous Casimir energy,of the TRNdS solution is

E=?M=?r n?1

c

l2

+

nω2n Q2

4πr c ?(n?1)k+(n+1)r2c8r2n?2c ,

S=r n

c

Vol(σ)

4(n?1)

ωn Q

16πG

,(6)

when k=0,the Casimir energy vanishes,as the case of asymptotically AdS space.When k=±1,we see from Eq.(6)that the sign of energy is just contrast to the case of TRNAdS space[30].

Thus we can see that the entropy Eq.(5)of the cosmological horizon can be rewritten as

S=2πl

|

E c

where

E q=1

8(n?1)

ωn Q2

4πr+ (n?1)?(n+1)r2+8r2n?2+ ,

?S=r n+Vol(σ)

4(n?1)

ωn Q

ωn 1?r2+8(n?1)r2n?2+ .(10)

In this case,the Casimir energy,de?ned as?E c=(n+1)?E?n?T?S?n?φQ,is

?E c =

2nr n?1

+

Vol(σ)

n

2

?φQ=nωn Q2

horizon entropy and cosmological horizon entropy.If one uses the BBM mass of the asymptotically dS spaces,the black hole horizon entropy cannot be expressed by a form like the Cardy-Verlinde formula[32].In this paper,we have found that if one uses the AD prescription to calculate conserved charges of asymptotically dS spaces,the TRNdS black hole horizon entropy can also be rewritten in a form of Cardy-Verlinde formula, which indicates that the thermodynamics of black hole horizon in dS spaces can be also described by a certain CFT.Our result is also reminiscent of the Carlip’s claim[34]that for black holes in any dimension the Bekenstein-Hawking entropy can be reproduced using the Cardy formula[35].

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