Thermodynamic instability of charged dilaton black holes in AdS spaces

A. Sheykhi,1,2,* M.H. Dehghani,3,† and S.H. Hendi4,‡

1Department of Physics, Shahid Bahonar University, P.O. Box 76175-132, Kerman, Iran2Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran

3Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran4Physics Department, College of Sciences, Yasouj University, Yasouj 75914, Iran

(Received 8 December 2009; published 20 April 2010)

We study thermodynamic instability of a class of (nþ 1)-dimensional charged dilatonic spherically

symmetric black holes in the background of the anti–de Sitter universe. We calculate the quasilocal mass

of the anti–de Sitter dilaton black hole through the use of the subtraction method of Brown and York. We

find a Smarr-type formula and perform a stability analysis in the canonical ensemble and disclose the

effect of the dilaton field on the thermal stability of the solutions. Our study shows that the solutions are

thermally stable for small �, while for large � the system has an unstable phase, where � is a coupling

constant between the dilaton and matter field.

DOI: 10.1103/PhysRevD.81.084040 PACS numbers: 04.70.Dy

I. INTRODUCTION

Strong motivation for studying thermodynamics ofblack holes in anti–de Sitter (AdS) spaces arises from thecorrespondence between the gravitating fields in AdSspacetime and conformal field theory (CFT) living on theboundary of the AdS spacetime [1]. It was argued that thethermodynamics of black holes in AdS spaces can beidentified with that of a certain dual CFT in the hightemperature limit [2]. With the AdS/CFT correspondence,one can gain some insights into thermodynamic propertiesand phase structures of strong ’t Hooft coupling CFTs bystudying thermodynamics of AdS black holes. It is wellknown that the Schwarzschild black hole in AdS space isthermodynamically stable for a large horizon radius, whileit becomes unstable for a small horizon radius. That is,there is a phase transition, named the Hawking-Page phasetransition, between the high temperature black hole phaseand the low temperature thermal AdS space [3]. It has beenexplained by Witten [2] that the Hawking-Page phasetransition of Schwarzschild black holes in AdS spacescan be identified with a confinement/deconfinement tran-sition of the Yang-Mills theory in the AdS/CFT correspon-dence. It is important to note that for the (locally) AdSblack holes with a zero or negative constant curvaturehorizon the Hawking-Page phase transition does not appearand these black holes are always locally stable [4–6].

On the other side, there has been a renewed interest instudying scalar coupled solutions of general relativity eversince new black hole solutions have been found in thecontext of string theory. The low energy limit of the stringtheory leads to the Einstein gravity, coupled nonminimallyto a scalar dilaton field [7]. The dilaton field couples in anontrivial way to other fields such as gauge fields and

results in interesting solutions for the background space-time [8–10]. These solutions [8–10], however, are allasymptotically flat. It has been shown that with the excep-tion of a pure cosmological constant, no dilaton de Sitter oranti–de Sitter black hole solution exists with the presenceof only one Liouville-type dilaton potential [11]. In thepresence of one or two Liouville-type dilaton potential,black hole spacetimes that are neither asymptotically flatnor (anti)–de Sitter [(A)dS] have been studied in differentsetups (see e.g. [12–17]). With the combination of threeLiouville-type dilaton potentials, charged dilaton blackhole/string solutions in the background of (A)dS spacetimein four- [18] and higher-dimensional spacetime [19,20]have been explored. Such potential may arise from thecompactification of a higher-dimensional supergravitymodel [21], which originates from the low energy limitof a background string theory.In this paper, we would like to study thermodynamic

instability of asymptotically AdS dilaton black holes in allhigher dimensions. In particular, we shall disclose theeffect of the dilaton field on the thermal stability of thesolutions. The motivation for studying higher-dimensionalsolutions of Einstein gravity originates from string theory,which is believed to be the most promising approach toquantum theory of gravity in higher dimensions. In fact,the first successful statistical counting of black hole en-tropy in string theory was performed for a five-dimensionalblack hole [22]. This example provides the best laboratoryfor the microscopic string theory of black holes. Besides,recently it has been realized that there is a way to make theextra dimensions relatively large and still be unobservable.This is if we live on a three-dimensional surface (brane)embedded in a higher-dimensional spacetime (bulk)[23,24]. In such a scenario, all gravitational objects suchas black holes are higher dimensional. Furthermore, asmathematical objects, black hole spacetimes are amongthe most important Lorentzian Ricci-flat manifolds in any

*sheykhi@mail.uk.ac.ir†mhd@shirazu.ac.ir‡hendi@mail.yu.ac.ir

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dimension. One striking feature of the Einstein equationsin more than four dimensions is that many uniquenessproperties holding in four dimensions are lost. For in-stance, four-dimensional black holes are known to possessa number of remarkable features, such as uniqueness,spherical topology, dynamical stability, and the laws ofblack hole mechanics. One would like to know which ofthese are peculiar to four dimensions and which hold moregenerally. For a recent review on higher-dimensional blackholes see [25]. In the light of all mentioned above, itbecomes obvious that further study on the thermodynamicsof higher-dimensional dilaton black holes in AdS spaces isof great interest.

This paper is outlined as follows. In Sec. II, we presentthe (nþ 1)-dimensional black hole solutions of Einstein-Maxwell–dilaton theory in AdS background. In Sec. III,we obtain the conserved and thermodynamic quantities ofthe solutions and verify the validity of the first law of blackhole thermodynamics. We perform a stability analysis inthe canonical ensemble and disclose the effect of thedilaton field on the thermal stability of the solutions inSec. IV. We summarize our results in Sec. V.

II. DILATON BLACK HOLES IN ADS SPACES

The action of (nþ 1)-dimensional (n � 3) Einstein-Maxwell–dilaton gravity can be written

S ¼ � 1

16�

Zdnþ1x

ffiffiffiffiffiffiffi�gp �

R� 4

n� 1ðr�Þ2 � Vð�Þ

� e�4��=ðn�1ÞF��F��

�; (1)

where R is the scalar curvature, Vð�Þ is a potential for thedilaton field �, � is an arbitrary constant governing thestrength of the coupling between the dilaton and theMaxwell field, F�� ¼ @�A� � @�A� is the electromag-

netic field tensor, and A� is the electromagnetic potential.

While � ¼ 0 corresponds to the usual Einstein-Maxwell–scalar theory, � ¼ 1 indicates the dilaton-electromagneticcoupling that appears in the low energy string action inEinstein’s frame. Varying action (1) with respect to thegravitational field g��, the dilaton field �, and the gauge

field A� yields the following field equations:

R�� ¼ 4

n� 1

�@��@��þ 1

4g��Vð�Þ

�

þ 2e�4��=ðn�1Þ�F��F�

�

� 1

2ðn� 1Þg��F��F��

�; (2)

r2� ¼ n� 1

8

@V

@�� �

2e�4��=ðn�1ÞF��F

��; (3)

r�ðe�4��=ðn�1ÞF��Þ ¼ 0: (4)

We assume the (nþ 1)-dimensional spherically symmetricmetric has the following form:

ds2 ¼ �N2ð�Þf2ð�Þdt2 þ d�2

f2ð�Þ þ �2R2ð�Þd�2n�1; (5)

where d�2n�1 denotes the metric of a unit (n� 1) sphere

and Nð�Þ, fð�Þ, and Rð�Þ are functions of �, which shouldbe determined. First of all, the Maxwell equations (4) canbe integrated immediately, where, for the spherically sym-metric spacetime (5), all the components of F�� are zero

except Ft�:

Ft� ¼ Nð�Þqe4��=ðn�1Þ

½�Rð�Þ�n�1; (6)

where q, an integration constant, is the charge parameter ofthe black hole. Our aim here is to construct exact, (nþ 1)-dimensional black hole solutions of Eqs. (2)–(4) for anarbitrary dilaton coupling parameter �. The dilaton poten-tial leading to AdS-like solutions of dilaton gravity hasbeen found recently [19]. For an arbitrary value of � inAdS spaces the form of the dilaton potential Vð�Þ in (nþ1) dimensions is chosen as

Vð�Þ ¼ 2�

nðn� 2þ �2Þ2 f��2½ðnþ 1Þ2 � ðnþ 1Þ�2

� 6ðnþ 1Þ þ �2 þ 9�e�4ðn�2Þ�=½ðn�1Þ��

þ ðn� 2Þ2ðn� �2Þe4��=ðn�1Þ

þ 4�2ðn� 1Þðn� 2Þe�2�ðn�2��2Þ=½ðn�1Þ��g: (7)

Here� is the cosmological constant. For later conveniencewe redefine� ¼ �nðn� 1Þ=2l2, where l is the AdS radiusof spacetime. It is clear the cosmological constant iscoupled to the dilaton in a very nontrivial way. This typeof dilaton potential can be obtained when a higher-dimensional theory is compactified to four dimensions,including various supergravity models [21]. In the absenceof the dilaton field action (1) reduces to the action ofEinstein-Maxwell gravity with cosmological constant.Using metric (5) and the Maxwell field (6), one can showthat the system of equations (2) and (3) have solutions ofthe form

N2ð�Þ ¼�1�

�b

�

�n�2

���ðn�3Þ; (8)

f2ð�Þ ¼��1�

�c

�

�n�2

��1�

�b

�

�n�2

�1��ðn�2Þ

� 2�

nðn� 1Þ�2

�1�

�b

�

�n�2

���

��1�

�b

�

�n�2

��ðn�3Þ

; (9)

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�ð�Þ ¼ n� 1

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð2þ 2�� n�Þ

qln

�1�

�b

�

�n�2

�; (10)

R2ð�Þ ¼�1�

�b

�

�n�2

��: (11)

Here c and b are integration constants and the constant � is

� ¼ 2�2

ðn� 2Þðn� 2þ �2Þ : (12)

The charge parameter q is related to b and c by

q2 ¼ ðn� 1Þðn� 2Þ22ðn� 2þ �2Þ cn�2bn�2: (13)

According to the Gauss theorem, the electric charge of theblack hole is

Q ¼ 1

4�

Z�!1

Ft�

ffiffiffiffiffiffiffi�gp

dn�1x ¼ �n�1

4�q; (14)

where �n�1 is the volume of the unit (n� 1) sphere. For� � 0 the solutions become imaginary for 0< �< b, andtherefore we should exclude this region from the space-time. For this purpose we introduce the new radial coor-dinate r as

r2 ¼ �2 � b2 ) d�2 ¼ r2

r2 þ b2dr2:

With this new coordinate, the above metric becomes

ds2 ¼ �N2ðrÞf2ðrÞdt2 þ r2dr2

ðr2 þ b2Þf2ðrÞ þ ðr2 þ b2Þ

� R2ðrÞd�2n�1; (15)

where the coordinates r assume the values 0 � r <1, andNðrÞ, fðrÞ, �ðrÞ, and RðrÞ are given by Eqs. (8)–(11) with

replacement � ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ b2

p.

The Kretschmann invariant R���R��� and the Ricci

scalar R diverge at r ¼ 0 (� ¼ b). Thus, r ¼ 0 is a curva-ture singularity. It is worthwhile to note that the scalar field�ð�Þ and the electromagnetic field Ft� become zero as

� ! 1. It is also notable to mention that these solutionsare valid for all values of �. When (� ¼ 0 ¼ �), thesesolutions describe the (nþ 1)-dimensional asymptoticallyAdS Reissner-Nordstrom black holes. One should note thatthe singularity for � � 0 is null, while it is timelike for� ¼ 0.

The quasilocal mass of the dilaton AdS black hole canbe calculated through the use of the subtraction method ofBrown and York (BY) [26]. Such a procedure causes theresulting physical quantities to depend on the choice ofreference background. In order to use the BY method, themetric should have the form

ds2 ¼ �WðRÞdt2 þ dR2

VðRÞ þR2d�2: (16)

Thus, we should write the metric (5) in the above form. Todo this, we perform the following transformation:

R ¼ �

�1�

�b

�

�n�2

��=2

:

It is a matter of calculations to show that the metric (5) maybe written as (16) with the following W and V:

WðRÞ ¼ N2ð�ðRÞÞf2ð�ðRÞÞ;

VðRÞ ¼ f2ð�ðRÞÞ�dRd�

�2

¼�1þ 1

2ð�ðn� 2Þ � 2Þ

�b

�

�n�2

�2

��1�

�b

�

�n�2

�ð��2Þf2ð�ðRÞÞ:

The background metric is chosen to be the metric (16) with

W0ðRÞ ¼ V0ðRÞ ¼ f20ð�ðRÞÞ

¼

8>><>>:

1þ �2

l2� 2�2b�

l2ð1þ�2Þ þ �4b2

l2ð1þ�2Þ2 for n ¼ 3

1þ �2

l2� �2b2

l2ð2þ�2Þ for n ¼ 4

1þ �2

l2for n � 5:

(17)

As you see from the above equation, the solutions for n ¼3 and n ¼ 4 have not ‘‘exact’’ asymptotic AdS behavior.Because of this point, we cannot use the AdS/CFT corre-spondence to compute the mass. Indeed, for n � 5 themetric is exactly asymptotically AdS, while for n ¼ 3, 4it is approximately asymptotically AdS. This is due to thefact that if one computes the Ricci scalar, then it is notequal to �nðnþ 1Þ=l2. It is well known that the Ricciscalar for AdS spacetime should have this value (see e.g.[27]). Also, the metrics with f20ð�Þ given by Eq. (17) for

n ¼ 3 and n ¼ 4 do not satisfy the Einstein equation withthe cosmological constant, while an AdS spacetime shouldsatisfy the Einstein equation with a cosmological constant,and an asymptotical AdS should satisfy at infinity. On theother side, at large �, the metric behaves as �2, and there-fore we used the word approximately asymptotically AdS.To compute the conserved mass of the spacetime, we

choose a timelike Killing vector field on the boundarysurface B of the spacetime (16). Then the quasilocalconserved mass can be written as

M ¼ 1

8�

ZBd2’

ffiffiffiffi�

p fðKab � KhabÞ� ðK0

ab � K0h0abÞgnab; (18)

where � is the determinant of the metric of the boundaryB, K0

ab is the extrinsic curvature of the background metric,

and na is the timelike unit normal vector to the boundaryB. In the context of the counterterm method, the limit inwhich the boundaryB becomes infinite (B1) is taken, and

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the counterterm prescription ensures that the action andconserved charges are finite. Although the explicit functionfð�ðRÞÞ cannot be obtained, at large R this can be done.Thus, one can calculate the mass through the use of theabove modified Brown and York formalism as

M ¼ �n�1

16�ðn� 1Þ

�cn�2 þ n� 2� �2

n� 2þ �2bn�2

�: (19)

In the absence of a nontrivial dilaton field (� ¼ 0), thisexpression for the mass reduces to the mass of the (nþ 1)-dimensional asymptotically AdS black hole.

III. THERMODYNAMICS OF ADS DILATONBLACK HOLE

In this section we intend to study thermodynamics ofdilaton black holes in the background of AdS spaces. Theentropy of the dilaton black hole typically satisfies the so-called area law of the entropy, which states that the entropyof the black hole is a quarter of the event horizon area [28].This near universal law applies to almost all kinds of blackholes, including dilaton black holes, in Einstein gravity[29]. It is a matter of calculation to show that the entropy ofthe dilaton black hole is

S ¼ �n�1bn�1��ðn�1Þ=2

þ4ð1� �þÞðn�1Þ=ðn�2Þ ; (20)

where

� ¼ 1��

b2

r2 þ b2

�ðn�2Þ=2;

and �þ ¼ �ðr ¼ rþÞ in which rþ is the outer horizon andis related to the parameters b, c, �, and �. The Hawkingtemperature of the dilaton black hole on the outer horizonrþ can be calculated using the relation

T ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ b2

p ðN2f2Þ04�Nr

��������r¼rþ; (21)

where a prime stands for the derivative with respect to r.One can easily show that

T ¼ �bðn� 2Þ�1��ðn�1Þ=2þ

2nðn� 1Þ�ð1� �þÞ1=ðn�2Þ

�nðn� 1Þð1� �þÞ2=ðn�2Þ

2�b2

� n��ðn�1Þ�1þðn� 2Þ � ½�ðn� 1Þ � 1�ð1� �þÞ

�2��ðn�1Þþ

�: (22)

We have shown the behavior of T versus �þ in variousdimensions in Figs. 1 and 2. From these figures we find outthat, independent of the spacetime dimensions, for smallvalues of � and �þ, the temperature may be negative (T <0). In this case we encounter a naked singularity. On theother hand, for an extremal black hole the temperature iszero and the horizon is degenerate. In this case rext is thepositive root of the following equation:

3�2��ðn�1Þext ð1� �extÞð4�nÞ=ðn�2Þ � 6�

nðn� 1Þ b2

��

n�ext

ðn� 2Þð1� �extÞ þ �ðn� 1Þ � 1

�¼ 0;

where

�ext ¼ 1��

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2ext þ b2

p�n�2

: (23)

Finally, for large values of � it is always positive provided�þ > b. Substituting solutions (8)–(11) in Eq. (6), theelectromagnetic field can be simplified as

Ftr ¼ q

ðr2 þ b2Þðn�1Þ=2 ; (24)

while the corresponding gauge potential At may be ob-tained as

At ¼ � q

ðn� 2Þ�n�2: (25)

The electric potential U, measured at infinity with respectto the horizon, is defined by [30]

U ¼ A���jr!1 � A��

�jr¼rþ ; (26)

–1

0

1

2

3

0.4 0.6 0.8 1 1.2

FIG. 1. T versus �þ for b ¼ 0:2, l ¼ 1, and � ¼ 0:1. n ¼ 4(solid line), n ¼ 5 (bold line), and n ¼ 6 (dashed line).

–0.4

–0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

FIG. 2. T versus �þ for b ¼ 0:2, l ¼ 1, and n ¼ 5. � ¼ 0:1(solid line), � ¼ 1 (bold line), and � ¼ 2 (dashed line).

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where � ¼ @t is the null generator of the horizon.Therefore, the electric potential may be obtained as

U ¼ q

ðn� 2Þ�n�2þ; (27)

where �2þ ¼ r2þ þ b2. Finally, we check the first law ofthermodynamics for the black hole. In order to do this, weobtain the mass M as a function of extensive quantities Sand Q. Using the expression for the charge, the mass, andthe entropy given in Eqs. (14), (19), and (20), we can obtaina Smarr-type formula per unit volume as

MðS;QÞ ¼ ðn� 1Þ16�

�32�2ðn� 2þ �2ÞQ2b2�n

ðn� 1Þðn� 2Þ2

þ n� 2� �2

n� 2þ �2bn�2

�; (28)

where b ¼ bðQ; SÞ. One may then regard the parameters Sand Q as a complete set of extensive parameters for themass MðS;QÞ and define the intensive parameters conju-gate to S and Q. These quantities are the temperature andthe electric potential

T ¼�@M

@S

�Q¼ ð@M@b ÞQð @b@rþ

ÞQð @S@rþ

ÞQ þ ð@S@bÞQð @b@rþÞQ

; (29)

U ¼�@M

@Q

�Sþ

�@M

@b

�S

�@b

@Q

�S; (30)

where

�@b

@rþ

�Q¼ �ð @Z@rþ

ÞQð@Z@bÞQ

; (31)

Z ¼�1� 32�2ðn� 2þ �2ÞQ2

ðn� 1Þðn� 2Þ2r2n�4þ

�rþb

�n�2

�

��1�

�b

rþ

�n�2

�1��ðn�2Þ

� 2

nðn� 1Þ�r2þ�1�

�b

rþ

�n�2

��: (32)

Straightforward calculations show that the intensive quan-tities calculated by Eqs. (29) and (30) coincide withEqs. (22) and (27). Thus, these thermodynamics quantitiessatisfy the first law of black hole thermodynamics,

dM ¼ TdSþUdQ: (33)

IV. STABILITY IN THE CANONICAL ENSEMBLE

Finally, we study the thermal stability of the solutions inthe canonical ensemble. In particular, we will see that thescalar dilaton field makes the solution unstable. The stabil-ity of a thermodynamic system with respect to smallvariations of the thermodynamic coordinates is usuallyperformed by analyzing the behavior of the entropy

SðM;QÞ around the equilibrium. The local stability inany ensemble requires that SðM;QÞ be a concave functionof the intensive variables. The stability can also be studiedby the behavior of the energy MðS;QÞ, which should be aconvex function of its extensive variable. Thus, the localstability can in principle be carried out by finding thedeterminant of the Hessian matrix of MðS;QÞ with respectto its extensive variables Xi, HM

XiXj¼ ½@2M=@Xi@Xj�

[30,31]. In our case the mass M is a function of entropyand charge. The number of thermodynamic variables de-pends on the ensemble that is used. In the canonicalensemble, the charge is a fixed parameter and thereforethe positivity of ð@2M=@S2ÞQ is sufficient to ensure local

stability. In Fig. 3 we show the behavior of ð@2M=@S2ÞQ as

a function of the coupling constant parameter � for adifferent value of n. This figure shows that there existsan upper limit for �, named �max, for which ð@2M=@S2ÞQis negative provided�> �max and positive otherwise. Thatis the black hole solutions are unstable for large values of�. It is important to note that �max depends on the parame-ters b, rþ, and the dimensionality of spacetime (see Fig. 4).On the other hand, Figs. 5 and 6 show the behavior ofð@2M=@S2ÞQ as a function of �þ for different values of

–200

0

200

400

600

800

1000

1200

1400

2 4 6 8 10 12 14 16 18 20

FIG. 3. ð@2M=@S2ÞQ versus � for b ¼ 0:2, l ¼ 1, and �þ ¼0:4. n ¼ 5 (solid line), n ¼ 6 (bold line), and n ¼ 7 (dashedline).

–20

–10

0

10

20

30

40

1 2 3 4 5 6 7 8

FIG. 4. ð@2M=@S2ÞQ versus � for b ¼ 0:2, l ¼ 1, and n ¼ 5.�þ ¼ 0:5 (solid line), �þ ¼ 0:55 (bold line), and �þ ¼ 0:6(dashed line).

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coupling constant parameter � and different values of n.These figures show that for a fixed value of � the solutionis thermally stable for �þ > b in five, six, and sevendimensions provided �< �max. It is also easy to plot thesefigures for arbitrary n and generalize the conclusions forhigher dimensions.

In comparison with the asymptotically AdS black holesof Einstein gravity, which have a small unstable phase, thestability phase structure of the black holes in Einstein-Maxwell–dilaton gravity shows that the dilaton field cru-cially changes the stability phase structure.

V. SUMMARYAND DISCUSSION

Thermodynamics of black holes in AdS spaces havebeen the subject of much recent interest. This is primarilydue to their relevance for the AdS/CFT correspondence. Itwas argued that the thermodynamics of black holes in AdSspaces can be identified with that of a certain dual CFT inthe high temperature limit. In this paper, we consideredasymptotically AdS black holes in (nþ 1)-dimensionalEinstein-Maxwell–dilaton gravity. We computed thecharge, mass, temperature, entropy, and electric potentialof the AdS dilaton black holes and found that these quan-tities satisfy the first law of black hole thermodynamics.We also obtained a Smarr-type formula, MðS;QÞ, andperformed a stability analysis in the canonical ensembleby considering ð@2M=@S2ÞQ for the charged black hole

solutions in (nþ 1) dimensions and showed that there isno Hawking-Page phase transition in spite of the charge ofthe black holes for small �, while the solutions have anunstable phase for large values of �. Indeed, for fixedvalues of the metric parameters, we found that there existsa maximum value of � for which the solutions are ther-mally unstable if �> �max, where �max depends on thedimensionality of the spacetime and the metric parametersb and rþ. This phase behavior shows that although there isno Hawking-Page transition for black object whose hori-zon is diffeomorphic to Rp for small � and therefore thesystem is always in the high temperature phase [2], in thepresence of dilaton with �> �max the black hole solutionshave some unstable phases.Finally, we would like to mention that although charged

AdS black holes in dilaton gravity are thermodynamicallyunstable for large values of dilaton coupling constant �, itis worthwhile to examine the dynamical (gravitational)instability of these dilaton black holes. This is due to thefact that there are black holes in Einstein gravity that arethermodynamically unstable, while they are dynamicallystable [32]. However, there may be some correlationsbetween the dynamic and thermodynamic instability ofblack hole solutions in dilaton gravity [31].

ACKNOWLEDGMENTS

We thank the anonymous referees for constructive com-ments. This work has been supported by Research Institutefor Astronomy and Astrophysics of Maragha, Iran.

[1] E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998); J.M.Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).

[2] E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998).[3] S.W. Hawking and D.N. Page, Commun. Math. Phys. 87,

577 (1983).

[4] D. Birmingham, Classical Quantum Gravity 16, 1197(1999).

[5] D. R. Brill, J. Louko, and P. Peldan, Phys. Rev. D 56, 3600(1997); A. Chamblin, R. Emparan, C. V. Johnson, andR. C. Myers, Phys. Rev. D 60, 064018 (1999).

–10

0

10

20

30

40

50

0.5 1 1.5 2 2.5 3 3.5 4

FIG. 6. ð@2M=@S2ÞQ versus �þ for b ¼ 0:2, l ¼ 1, and � ¼0:5. n ¼ 5 (solid line), n ¼ 6 (bold line), and n ¼ 7 (dashedline).

–100

–50

0

50

100

150

200

0.4 0.6 0.8 1 1.2

FIG. 5. ð@2M=@S2ÞQ versus �þ for b ¼ 0:2, l ¼ 1, and n ¼ 5.� ¼ 0:5 (solid line), � ¼ 2 (bold line), and � ¼ 5 (dashed line).

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