Gauss-Bonnet black holes in dS spaces

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PHYSICAL REVIEW D 69, 104025 ~2004!

Gauss-Bonnet black holes in dS spaces

Rong-Gen Cai*Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China

and Interdisciplinary Center for Theoretical Study, University of Science and Technology of China, Hefei, Anhui 230026, Chi

Qi Guo†

Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, China~Received 5 November 2003; published 26 May 2004!

We study the thermodynamic properties associated with the black hole horizon and cosmological horizon forthe Gauss-Bonnet solution in de Sitter space. When the Gauss-Bonnet coefficient is positive, a locally stablesmall black hole appears in the case of spacetime dimensiond55, the stable small black hole disappears, andthe Gauss-Bonnet black hole is always unstable quantum mechanically whend>6. On the other hand, thecosmological horizon is found to be always locally stable independent of the spacetime dimension. But thesolution is not globally preferred; instead, the pure de Sitter space is globally preferred. When the Gauss-Bonnet coefficient is negative, there is a constraint on the value of the coefficient, beyond which the gravitytheory is not well defined. As a result, there is not only an upper bound on the size of black hole horizon radiusat which the black hole horizon and cosmological horizon coincide with each other, but also a lower bounddepending on the Gauss-Bonnet coefficient and spacetime dimension. Within the physical phase space, theblack hole horizon is always thermodynamically unstable and the cosmological horizon is always stable;furthermore, as in the case of the positive coefficient, the pure de Sitter space is still globally preferred. Thisresult is consistent with the argument that the pure de Sitter space corresponds to an UV fixed point of dualfield theory.

DOI: 10.1103/PhysRevD.69.104025 PACS number~s!: 04.70.Dy, 04.20.Jb

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I. INTRODUCTION

Higher-derivative curvature terms naturally occur in maoccasions, such as in the quantum field theory in curspace@1# and in the effective low-energy action of strintheories. In the latter case, due to the AdS and conforfield theory ~CFT! correspondence@2#, these terms can bviewed as the corrections of the largeN expansion of bound-ary CFTs on the side of dual field theory. On the sidegravity, however, because of the nonlinearity of Einstequations, it is quite difficult to find nontrivially exact analytical solutions of the Einstein equations with these highderivative terms. In most cases, one has to adopt someproximation methods or find solutions numerically.

Up to quadratic curvature terms, there is a special coposition

LGB5RmngdRmngd24RmnRmn1R2, ~1.1!

which is often called the Gauss-Bonnet term. Einstein grity with the Gauss-Bonnet term has some remarkabletures in some sense. For instance, the resulting equationmotion have no more than second derivatives of metricthe theory has been shown to be free of ghosts whenexpanded about flat space, evading any problems withtarity @3#. Further, it has been argued that the Gauss-Bonterm appears as the leading correction@4# to the effectivelow-energy action of the heterotic string theory. In additio

*Email address: [email protected]†Email address: [email protected]

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it has already been found that exact analytical solutions wspherical symmetry can be obtained in this gravity the@3,5–7#.

The thermodynamics and geometric structure ofGauss-Bonnet black hole in asymptotically flat space hbeen analyzed in Refs.@8,9#. In a previous paper@7#,1 westudied the thermodynamics and phase structure of topolcal black holes in Einstein gravity with the Gauss-Bonnterm and a negative cosmological constant. Those topolcal black holes are asymptotically anti–de Sitter~AdS! andtheir event horizon can be a hypersurface with positive, zeor negative constant curvature. In the present paper, westudy the properties of Gauss-Bonnet black holes in asytotically de Sitter ~dS! space. Studying the Gauss-Bonnblack hole in dS space is of interest in its own right. On tother hand, we hope to gain some insights into the dual fitheory in the sense of the dS/CFT correspondence@12#.

It is well known that unlike the cases of asymptoticaflat space and asymptotically AdS space, it is not an ematter to calculate conserved charges associated with aymptotically dS space because of the absence of spatiafinity and a globally timelike Killing vector in such a spacetime. On the other hand, there is a cosmological evhorizon, except for the black hole horizon, for the spacetiof black holes in dS space. Like the black hole horizon, this also a thermodynamic feature for the cosmological horiz

1The thermodynamics and phase structure of black hole solutperturbed by quadratic curvature terms in asymptotically AdS sphave also been discussed in Ref.@10#; see also@6# for the case ofblack holes in dimensionally continued gravity.

©2004 The American Physical Society25-1

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R.-G. CAI AND Q. GUO PHYSICAL REVIEW D69, 104025 ~2004!

@13#. In general the Hawking temperatures associated wthe black hole horizon and cosmological horizon, resptively, are not equal; therefore the spacetime for a black hin dS space is unstable quantum mechanically.

In this paper we will discuss separately the thermodynaics of the black hole horizon and cosmological horizoNamely, we view the black hole horizon and cosmologihorizon as two thermodynamic systems. For the case ofblack hole horizon, we calculate the black hole mass witdefinition due to Abbott and Deser~AD! @11# by consideringthe deviation of metric from the pure dS space being defias the vacuum~lowest-energy state!.2 In terms of this defini-tion, the gravitational mass of asymptotically dS spacealways positive and coincides with the Arnowitt-DeseMisner ~ADM ! mass in asymptotically flat space, when tcosmological constant goes to zero. For the case of themological horizon, we will adopt the prescription dueBalasubramanian, de Boer, and Minic~BBM! @16#. In thisprescription, except for a constant, which depends oncosmological constant and space dimension and can bgarded as the Casimir energy of the dual field theory inspirit of the dS/CFT correspondence, the gravitational mis just the AD mass, but with an opposite sign@16–18,20#.The BBM mass is measured at the far past (I 2) or far fur-ther (I 1) boundary of dS space, which is outside the cmological horizon. With these definitions, thermodynamquantities associated with the black hole horizon and coslogical horizon obey the first law of thermodynamics, resptively. In Ref. @19# we have also shown that they satisfrespectively, the Cardy-Verlinde formula this way. In particlar, it was argued@21# that for the Euclidean black hole–dSitter geometry which is closely related to the horizon thmodynamics, when one deals with the thermodynamicsone of two horizons, one should view the other as the bouary. In this way, one has well-defined Hamiltonians assoated with the black hole horizon and cosmological horizrespectively. Therefore the point of viewing the black hohorizon and cosmological horizon as two thermodynamsystems should be reasonable.

The organization of the paper is as follows. In the nesection we present a solution of the Gauss-Bonnet blackin dS space. In Secs. III and IV we discuss the thermonamics and phase structure of the black hole horizoncosmological horizon, respectively. The paper is endedSec. V with some conclusions and a discussion.

II. GAUSS-BONNET BLACK HOLE SOLUTIONIN de SITTER SPACE

We start with the Einstein-Hilbert action with the GausBonnet term~1.1! and a positive cosmological constantL5(d21)(d22)/2l 2 in d dimensions:

2In Ref. @11# the authors consider Einstein gravity with a cosmlogical constant. When higher-derivative curvature termspresent, a similar mass definition of the gravitational field has bdiscussed in Ref.@14# ~see also discussions for Gauss-Bonnet grity @15#!.

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16pGE ddxA2gS R2~d21!~d22!

l 21aLGBD ,

~2.1!

whereG is the Newton constant anda is the Gauss-Bonnecoefficient with dimension (length)2. From this action weobtain the equations of motion

Rmn21

2gmnR52

~d21!~d22!

2l 2gmn

1aS 1

2gmn~RgdlsRgdls24RgdRgd1R2!

22RRmn14RmgR ng 14RgdR m n

g d

22RmgdlRngdlD . ~2.2!

For the metric we adopt the following ansatz of spherisymmetry:

ds252e2ndt21e2ldr21r 2dVd222 , ~2.3!

wheren andl are functions ofr only, anddVd222 represents

the line element of a (d22)-dimensional unit sphere withvolume Vd2252p (d21)/2/G@(d21)/2#. To find a solutionwith metric ~2.3!, there is a simple method@3#: substitutingthe metric ansatz~2.3! into the action~2.1! yields

S5~d22!Vd22

16pG E dtdren1lF r d21w~11aw!2r d21

l 2 G 8,~2.4!

where the prime denotes a derivative with respect tor, a5a(d23)(d24), andw5r 22(12e22l). From the actionone has

en1l51,

w~11aw!21

l 25

16pGM

~d22!Vd22r d21. ~2.5!

Then one obtains the exact solution

e2n5e22l511r 2

2aS 17A11

64pGaM

~d22!Vd22r d211

4a

l 2 D ,

~2.6!

where M is an integration constant, which is just the Amass of the solution. This exact solution was first foundBoulware and Deser in Ref.@3#. In @7# we extended thissolution to the case where the unit spheredVd22 is replacedby a hypersurface with positive, zero, or negative constcurvature.

Note that the solution~2.6! has a singularity atr 50 ifa.0. When a,0 there is an additional singularity at thplace where the square root vanishes in Eq.~2.6!. In addition

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GAUSS-BONNET BLACK HOLES IN dS SPACES PHYSICAL REVIEW D69, 104025 ~2004!

note that there are two branches in the solution~2.6! with‘‘ 2 ’’ and ‘‘ 1’’ signs, respectively. When the integration costantM vanishes, the solution reduces to

e2n5e22l511r 2

2aS 17A11

4a

l 2 D . ~2.7!

This is a dS or AdS solution depending on the effective cvature radius:

1

l eff2

521

2aS 17A11

4a

l 2 D . ~2.8!

When a.0, one hasl eff2 .0 for the branch with the ‘‘2 ’’

sign, while l eff2 ,0 for the ‘‘1’’ sign. Therefore, in this case

the solution is asymptotically dS for the branch with t‘‘ 2 ’’ sign and asymptotically AdS for the sign ‘‘1,’’ al-though the cosmological constantL in the action~2.1! ispositive. On the other hand, whena,0, one hasl eff

2 .0 forboth branches, which means the solution is always asytotically dS. But in that case, one can see from Eq.~2.8! thatthe Gauss-Bonnet parameter has to satisfy

a/ l 2>21/4 ~2.9!

for the branch with the ‘‘2 ’’ sign. Otherwise, the theory isnot well defined. Here it should be stressed that the cstraint ~2.9! is obtained from the vacuum solution of ththeory~2.1!. To avoid a naked singularity, the more stringeconstraint will be Eq.~3.19!, as discussed below.

That the solution~2.6! has two branches implies that ththeory has two different vacua~2.7!. In Ref. @3# Boulwareand Deser have shown that the branch with the ‘‘1’’ sign isunstable; the graviton propagating on the background inbranch is a ghost, while the branch with the ‘‘2 ’’ sign isstable and the graviton is free of ghosts. The branch with‘‘ 1’’ sign is of less physical interest. Therefore we will ndiscuss this branch and focus on the branch with the ‘‘2 ’’sign in what follows.

III. THERMODYNAMICS OF THE BLACK HOLEHORIZON

On can see from Eq.~2.7! that whenM50, there is acosmological horizon atr c5 l eff . When M increases fromzero, like the case of the Schwarzschild-dS solution, a blhole horizon appears in the solution~2.6! and the cosmologi-cal horizon shrinks. That is, in general there are two posireal roots for the equatione2n50. The large one is the cosmological horizonr c and the small one is the black hohorizon r 1 . In this section we first discuss the thermodnamics associated with the black hole horizon.

In terms of the black hole horizon, the mass of the GauBonnet black hole—namely, the AD mass of the solutioncan be expressed as

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M5~d22!Vd22r 1

d23

16pG S 11a

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2r 1

2

l 2 D . ~3.1!

Obviously, whena50, this quantity reduces to the massthe Schwarzschild-dS black hole ind dimensions. TheHawking temperature associated with the black hole horican easily be obtained by the requirement of the absenceconical singularity at the black hole horizon in the Euclidesector of the Gauss-Bonnet black hole solution in dS spaIt turns out that

T5~d25!a1~d23!r 1

2 2~d21!r 14 / l 2

4pr 1~r 12 12a !

. ~3.2!

Another important thermodynamic quantity is the entropythe black hole horizon. In Einstein gravity, the entropy ofblack hole satisfies the so-called area formula@22#. Namely,the entropy is equal to one-quarter of the horizon area. Whigher-derivative curvature terms are present, however,statement no longer holds. Wald has shown that the entrof a black hole in any gravity theory is always a functionthe horizon geometry@23#. From Refs.@7,8# we can read theentropy of the Gauss-Bonnet black hole in dS space,

S5Vd22r 1

d22

4G S 112~d22!a

~d24!r 12 D , ~3.3!

since the cosmological constant does not explicitly occuthis expression. Indeed we can show that three thermonamic quantities~3.1!, ~3.2!, and ~3.3! obey the first law ofthermodynamics,dM5TdS.

The quantity indicating the local stability of the blachole is the heat capacity. For the Gauss-Bonnet black holdS space, it is

C[S ]M

]T D5S ]M

]r 1D S ]r 1

]T D , ~3.4!

where

]M

]r 15

~d22!Vd22

4Gr 1

d25~r 12 12a !T,

]T

]r 15

1

4p l 2r 12 ~r 1

2 12a !2@2~d21!r 1

6 2~d23!

3 l 2r 14 26~d21!ar 1

4 12~d23!a l 2r 12

23~d25!a l 2r 12 22~d25!a2l 2#. ~3.5!

By definition,F5M2TS, the free energy of the black holis

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F5Vd22r 1

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16pG~d24!l 2~r 12 12a !

@~d24!r 16 1~d24!l 2r 1

4

16~d22!ar 14 1~d28!a l 2r 1

2 12~d22!a2l 2#.

~3.6!

Now we are in a position to discuss the thermodynamstability and phase structure of the black hole.

~1! Let us first consider the case ofa.0, which is thecase of the heterotic string theory@4#. When the Hawkingtemperature~3.2! vanishes, we obtain

r 12 5r 1,2

2 5~d23!l 2

2~d21! S 16A11~d21!~d25!

~d23!2

4a

l 2 D ,

~3.7!

from which we see that whend55 only, there are two reapositive roots: one isr 15r 250, and the other isr 1

2 5r 12

5 l 2/2. The large oner 1 is the horizon radius of the maximablack hole in the solution~2.6!, beyond which the singularitybehind the black hole horizon becomes naked. The maxiblack hole with radiusr 1 in Eq. ~3.7! is therefore the counterpart of the Nariai black hole in the Gauss-Bonnet gravthere, the black hole horizon and cosmological horizon cocide with each other and therefore the Hawking temperais zero.

Whend55, the inverse temperature starts from infinityr 150, reaches a minimal value at some place, and then gto infinity again at the maximal black hole horizon radiuThis behavior can be seen from the heat capacity~3.5!. Whenthe black hole horizon satisfies

0,r 12 ,r 0

25l 2

4 S 1112a

l 2 D FA1116a

l 2 S 1112a

l 2 D 22

21G ,

~3.8!

the heat capacity is positive and it becomes negative for 02

,r 12 , l 2/2. Here l /A2 is the maximal black hole horizo

radius in the case of five dimensions. This behavior is qudifferent from the case without the Gauss-Bonnet tethere, the heat capacity of the black hole horizon is alwnegative. Therefore the small Gauss-Bonnet black holeisfying Eq. ~3.8! is thermodynamically stable. Here thereno restriction on the Gauss-Bonnet coefficient, except forpositivity of the coefficient.

Whend>6, the equationT50 has only one real positiveroot r 1 in Eq. ~3.7!; the other is negative, without any physcal meaning. One can see from Eq.~3.2! that the inversetemperature always starts from zero atr 150 and goes toinfinity monotonically at the maximal horizon radiusr 1 inEq. ~3.7!, which implies that the heat capacity is alwanegative in this case. This indicates the instability of tGauss-Bonnet black hole. Whend>6, therefore, the thermodynamic properties of the Gauss-Bonnet black hole inspace are qualitatively similar to those of thSchwarzschild-dS black hole, the black hole in dS sp

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without the Gauss-Bonnet term. Thus the thermodynamicthe black hole horizon becomes remarkably related tospacetime dimension. In Fig. 1 we plot the inverse tempeture versus the radius of the black hole horizon.

Checking the free energy~3.6!, however, we find that it isalways positive whatever the spacetime dimension andGauss-Bonnet coefficient are. In Fig. 2 the free energy ofive-dimensional Gauss-Bonnet black hole is plotted verthe horizon radius and the Gauss-Bonnet coefficient. In F3 we plot the free energy versus the horizon radius aspacetime dimension with a fixed Gauss-Bonnet coefficieThe positivity of the free energy implies that the black hosolution is not globally preferred; instead, the dS space~2.7!is globally preferred since we have taken the dS space asvacuum state.

~2! When a,0, from the solution~2.6! we find that theblack hole horizon must satisfy

r 12 >22a. ~3.9!

However, from the entropy formula~3.3! one can see when

FIG. 1. The inverse temperature of the Gauss-Bonnet blholes in dS space. The curves from up to bottom correspond to

case of (d55, a/ l 250.2), (d55, a/ l 250), and (d56, a/ l 2

50.2), respectively. Note that the region with negative temperashould be ruled out in physical phase space.

FIG. 2. The free energy of the five-dimensional Gauss-Bonblack hole versus the Gauss-Bonnet coefficient and horizon rad

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22a<r 12 ,2

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the entropy of the black hole horizon is negative, whishould be ruled out in physical phase space since a negentropy is meaningless. Therefore we obtain a constrainthe minimal horizon radius of the Gauss-Bonnet black ho

r 12 >22a

d22

d24, ~3.11!

whena,0. As the above equation holds, the black hole hvanishing entropy.

In this case, whend55, from the temperature~3.2!, wefind that the horizon radius falls into the range

26a<r 12 < l 2/2. ~3.12!

As in the case ofa.0, herer 15 l /A2 is the horizon radiusof the maximal black hole; there, both the black hole horizand cosmological horizon coincide with each other andHawking temperature vanishes. From Eq.~3.12! we see thatthere is a more stringent constraint than the one~2.9!:

a/ l 2>21/12. ~3.13!

Further, one can see from Eq.~3.5! that the heat capacitychanges its behavior at the place

r 12 5

l 2

4 S 1112a

l 2 D F216A1116a

l 2 S 1112a

l 2 D 22G .

~3.14!

In order for Eq.~3.14! to have a real root, one has to hav

a/ l 2<21/12, ~3.15!

FIG. 3. The free energy of the Gauss-Bonnet black holes vethe horizon radius and spacetime dimension with a fixed Ga

Bonnet coefficienta/ l 250.2.

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which contradicts the condition~3.13!. This means that theinverse temperature always starts monotonically from a finvalue at the minimal radius given by Eq.~3.12! to infinity atthe maximal radiusl /A2 given in Eq.~3.12!. This indicatesthat whena,0, the five-dimensional Gauss-Bonnet blahole in dS space has a negative heat capacity and thenunstable as the case without the Gauss-Bonnet term.inverse temperature of the black hole horizon, plotted in F4, shows this fact.

Whend>6, the condition that the Hawking temperatu~3.2! vanish is

r 12 5r 3,4

2 5~d23!l 2

2~d21! S 16A11~d21!~d25!

~d23!2

4a

l 2 D .

~3.16!

To have two positive real roots, one has

a

l 2.2

~d23!2

4~d21!~d25!. ~3.17!

The large root corresponds to the maximal Gauss-Bonblack hole in dS space. But the small one is outsideconstraint~3.11!. Therefore, the behavior of the inverse temperature is similar to the case ofd55: it starts from a finitevalue at the minimal black hole horizon given by Eq.~3.11!and goes to infinity at the maximal radius given by Eq.~3.16!monotonically. As a result, thed>6 Gauss-Bonnet blackhole in dS space is also unstable whena,0, like the casewithout the Gauss-Bonnet term. In Fig. 5 we plot the invetemperature of the black hole in seven dimensions witfixed Gauss-Bonnet coefficient. Note that the value ofright-hand side in Eq.~3.17! is smaller than21/4 given inEq. ~2.9!. Therefore it seems that the true constraint oncoefficienta is Eq. ~2.9!, rather than Eq.~3.17!. It turns outthis is not correct. The reason is that the horizon radius finto the range

22ad22

d24<r 1

2 <r 32 , ~3.18!

uss-

FIG. 4. The inverse temperature of the five-dimensional Gau

Bonnet black holes with21/12,a/ l 2,0.

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which gives us a more stringent constraint

a

l 2>2

~d22d28!~d24!

4~d21!~d22!2. ~3.19!

We have checked numerically that within the ranges~3.18!and ~3.19!, the heat capacity is always negative, while tfree energy of the black hole horizon is always positivethe case ofa.0.3

IV. THERMODYNAMICS OF THE COSMOLOGICALHORIZON

For the cosmological horizon denoted byr c , the associ-ated Hawking temperatureTc is

Tc52~d25!a2~d23!r c

21~d21!r c4/ l 2

4pr c~r c212a !

~4.1!

and entropy

Sc5Vd22r c

d22

4G S 112~d22!a

~d24!r c2 D . ~4.2!

The thermodynamic energy associated with the cosmologhorizon can be calculated using the BBM prescription@16#~namely, the surface counterterm approach!. In this prescrip-tion, it has been found that the BBM mass for black holesdS spaces in Einstein theory is just the negative AD m~see, for example, Refs.@16–18,21#!, except for a constantwhich is not relevant to the present discussion. ForGauss-Bonnet black holes in dS space, the BBM prescripis also applicable. As in the case of Einstein gravity, it tu

3In the range22a,r 12 ,22a(d22)/(d24), a stable small

black hole with positive heat capacity may appear, but it hanegative entropy. As a result, it should be ruled out in physphase space; the true constraint on the horizon radius is giveEq. ~3.18!.

FIG. 5. The inverse temperature of the seven-dimensio

Gauss-Bonnet black holes with217/100,a/ l 2,0.

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out that the thermodynamic energy of the cosmologicalrizon is the negative AD mass~see also@20,10#!:

E52M52~d22!Vd22r c

d23

16pG S 11a

r c2

2r c

2

l 2 D . ~4.3!

A self-consistency check is that these three thermodynaquantities obey the first law of thermodynamics,dE5TcdSc . To see the thermodynamic stability, we calculathe heat capacity of the cosmological horizon:

Cc[S ]E

]TcD5S ]E

]r cD S ]r c

]TcD , ~4.4!

where

S ]E

]r cD5

~d22!Vd22

4Gr c

d25~r c212a !Tc ,

]Tc

]r c5

1

4p l 2r c2~r c

212a !2@~d21!r c

61~d23!

3 l 2r c416~d21!ar c

422~d23!a l 2r c2

13~d25!a l 2r c212~d25!a2l 2#. ~4.5!

And the free energyFc5E2TcSc is

Fc5Vd22r c

d25

16pG~d24!l 2~r c212a !

@2~d24!r c62~d24!l 2r c

4

26~d22!ar c42~d28!a l 2r c

222~d22!a2l 2#. ~4.6!

The cosmological horizon radius has a range in size:minimal value is just the maximal black hole horizonr 3 inEq. ~3.16! discussed in the previous section, while the mamal radius isl eff given in Eq. ~2.8!; there, the integrationconstantM vanishes. Namely, the cosmological horizon isthe region

r 32<r c

2< l eff2 . ~4.7!

Within this region, it is easy to show that the heat capac~4.5! is positive and therefore the inverse temperature alwstarts from infinity, where the cosmological horizon coicides with the black hole horizon, and monotonically goesa finite value, which corresponds to the inverse temperaof the vacuum dS space~2.7!. This implies that the thermodynamics of the cosmological horizon is locally stable. FroEq. ~4.6! we see that the free energy is always negative.this does not mean that the solution is globally prefersince, when we calculate the gravitational mass, the purespace~2.7! is regarded as the vacuum. This vacuum has zAD mass, but has nonzero Hawking temperature and entrassociated with the cosmological horizon:

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Tcvac5

1

2p l eff, Sc

vac5Vd22l eff

d22

4G S 112~d22!a

~d24!l eff2 D .

~4.8!

And the corresponding free energy is

Fcvac52

Vd22l effd23

8pG S 112~d22!a

~d24!l eff2 D . ~4.9!

To see whether or not the solution with nonvanishingM isglobally preferred, we have to compare the two free energ~4.6! and ~4.9!:

DF5Fc2Fcvac. ~4.10!

If DF.0, the solution with nonvanishingM is not globallypreferred; otherwise, it is preferred. It seems difficult to alytically prove DF.0, but we have checked numericalthat indeedDF.0 within the range~4.7!. Therefore the puredS space~2.7! is globally preferred. Namely, although ththermodynamics of the cosmological horizon is locastable, it will decay to the pure dS space. In Fig. 6 we pthe difference of the two free energies,DF, versus theGauss-Bonnet coefficient and the cosmological horizondius in the case of five dimensions. The case of ten dimsions is plotted in Fig. 7.

When a,0, the coefficient has to satisfy the constra~3.19!, again. Within the horizon range~4.7! and the con-straint ~3.19!, we have numerically checked that the diffeence of the free energies associated with the cosmologhorizon is always positive as in the casea.0. ~As an ex-ample, we plot the free energy difference associated withcosmological horizon in the case of five dimensions in F

FIG. 6. The differenceDF of two free energies associated withe cosmological horizon for the case of five dimensions.

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8.! As a result, the pure de Sitter space~2.7! is globallypreferred again.

V. CONCLUSION AND DISCUSSION

In summary we have discussed the thermodynamic prerties and phase structures associated with the blackhorizon and cosmological horizon for the Gauss-Bonblack hole–de Sitter spacetime. The black hole horizoncosmological horizon are viewed as two thermodynamic stems. When the Gauss-Bonnet coefficient is positive, whis the case for the effective low-energy action of the heterostring theory, a locally stable small black hole whose radsatisfying Eq.~3.8! appears ind55 dimensions, which isabsent in the case without the Gauss-Bonnet term. Whenspacetime dimensiond>6, the stable small black hole disappears; the black hole is always unstable thermodynacally as in the case without the Gauss-Bonnet term. Contto the black hole horizon, the cosmological horizon is alwathermodynamically stable with positive heat capacity. Bthe Gauss-Bonnet black hole solution in de Sitter space isglobally preferred; instead, the pure de Sitter space~2.7! is

FIG. 7. The differenceDF of two free energies for the case oten dimensions.

FIG. 8. The differenceDF of two free energies in the case o

five dimensions with21/12,a/ l 2,0.

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globally preferred, which has lower free energy than the cwith nonvanishingM.

On the other hand, when the Gauss-Bonnet coefficiennegative, there is a bound on the coefficient given by~2.9!; otherwise, the gravity theory is not well defined.@Notethat the constraint~2.9! is derived from the vacuum solutioof the theory; it does not warrant that a naked singularityoccur in this case. In fact, a true constraint is Eq.~3.19!,under which a black hole solution is meaningful.# In thiscase, the horizon radius of the Gauss-Bonnet black holenot only an upper boundr 3 given by Eq.~3.16!; there, theblack hole horizon coincides with the cosmological horizobut also a lower bound. From the solution~2.6!, the lowerbound seems to be22a given by Eq.~3.9!. Checking theentropy~3.3! of the black hole horizon tells us that within thrange~3.10!, the entropy associated with the black hole hrizon is negative. As a result this range~3.10! should be ruledout in physical phase space. Therefore the true lower boof the horizon radius is given by Eq.~3.11!. Further it givesa more stringent constraint on the value of the Gauss-Bocoefficient~3.19!. Within the coefficient~3.19! and the hori-zon range~3.18!, the black hole horizon becomes alwathermodynamically unstable and the cosmological horizostill thermodynamically stable; that is, in this case the stasmall black hole in five dimensions disappears. Checkingfree energies associated with the black hole horizon andmological horizon, respectively, reveals that pure de Sispace is still globally preferred.

Therefore, both thermodynamic discussions of black hhorizon and cosmological horizon lead to the same consion: that a pure de Sitter space is globally preferred. Tresult is consistent with the argument that a pure de Sspace corresponds to an UV fixed point of the renormaltion group flow of the dual field theory in the dS/CFT corrspondence@16,24#.

Finally we would like to stress that as argued in thetroduction, a black hole–de Sitter spacetime is unsta

d

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quantum mechanically because two Hawking temperatuassociated with the black hole horizon and cosmologicalrizon are in general not equal, except for the Nariai solutor its generalizations, where the two temperatures are eto each other. So it is not an easy matter to study the tmodynamic properties of spacetime for a black hole inspace in its entirety. In particular, Teitelboim recently argu@21# that for the Euclidean black hole–de Sitter geomewhich is closely related to the horizon thermodynamiwhich deals with the thermodynamics of one of two hozons, one should view the other as the boundary. In this wone has well-defined Hamiltonians associated with the blhole horizon and cosmological horizon, respectively. In tpaper we just followed this spirit to discuss the thermodnamic properties associated with the black hole horizoncosmological horizon, respectively, and to obtain the concsion that pure de Sitter space is globally preferred and ithe end point of decay. In addition, the local stability analyof black hole horizon and cosmological horizon mightless motivated just due to different temperatures. Howewhen two horizons separate with a very large distance,effect of the Hawking evaporation of one horizon couldnegligible on the other horizon. In this sense it might besome interest and be of meaning to discuss the local stabof two horizons, respectively. We wish that the presentvestigation together with a lot of existing literature conceing black hole–de Sitter spacetimes is in the way to copletely understand the classical and quantum propertieasymptotically de Sitter spaces.

ACKNOWLEDGMENTS

This work was initiated while one of authors~R.G.C.! wasvisiting the ICTS at USTC, whose hospitality is gratefulacknowledged. The research was supported in part by a gfrom the Chinese Academy of Sciences, Grant No. 10325from NSFC, and by the Ministry of Science and Technoloof China under Grant No. TG1999075401.

s.

. D

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Gauss-Bonnet black holes in dS spaces