Asymptotically nonflat Einstein-Born-Infeld-dilaton black holes with Liouville-type potential

A. Sheykhi, N. Riazi,* and M. H. MahzoonPhysics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran

(Received 25 February 2006; published 21 August 2006)

We construct some classes of electrically charged, static and spherically symmetric black hole solutionsof the four-dimensional Einstein-Born-Infeld-dilaton gravity in the absence and presence of Liouville-type potential for the dilaton field and investigate their properties. These solutions are neither asymptoti-cally flat nor (anti)-de Sitter. We show that in the presence of the Liouville-type potential, there exist twoclasses of solutions. We also compute temperature, entropy, charge and mass of the black hole solutions,and find that these quantities satisfy the first law of thermodynamics. We find that in order to fully satisfyall the field equations consistently, there must be a relation between the electric charge and otherparameters of the system.

DOI: 10.1103/PhysRevD.74.044025 PACS numbers: 04.70.�s, 04.30.�w

I. INTRODUCTION

It is quite possible that gravity is not given by theEinstein action, at least at sufficiently high energies. Instring theory, gravity becomes scalar-tensor in nature. Thelow-energy limit of the string theory leads to the Einsteingravity, coupled nonminimally to a scalar dilaton field [1].When a dilaton is coupled to Einstein-Maxwell theory, ithas profound consequences for the black hole solutions.Some efforts have been done to construct exact solutions ofEinstein-Maxwell-dilaton gravity. For example exactcharged dilaton black hole solutions of EMd gravity inthe absence of a dilaton potential have been constructed bymany authors [2,3]. The dilaton changes the casual struc-ture of the spacetime and leads to curvature singularities atfinite radii. These black holes are asymptotically flat. Inrecent years, nonasymptotically flat black hole spacetimesare attracting much interest in connection with the so-called AdS/CFT correspondence. Black hole spacetimeswhich are neither asymptotically flat nor dS/AdS have beenfound and investigated by many authors. The unchargedsolutions have been found in [4], while the charged solu-tions have been considered in [5]. In the presence ofLiouville-type potential, static charged black hole solu-tions have also been discovered with a positive constantcurvature event horizons and zero or negative constantcurvature horizons [6,7]. The extension to the dyonic blackhole solutions in four-dimensional and higher-dimensionalEMd gravity with one and two Liouville-type potentialshave also been done in [8]. These solutions possess bothelectric and magnetic charge and they are neither asymp-totically flat nor dS/AdS.

The idea of the nonlinear electromagnetism was firstintroduced in 1934 by Born and Infeld in order to obtaina finite value for the self-energy of pointlike charges [9].Although it became less popular with the introduction ofQED, in recent years, the Born-Infeld action has beenoccurring repeatedly with the development of superstring

theory, where the dynamics of D-branes is governed by theBorn-Infeld action [10,11]. For various motivations, ex-tending the Reissner-Nordstrom black hole solutions inEM theory to the charged black hole solutions in EBItheory has attracted some attention in recent years [12].For example, exact solutions of spherically symmetricEinstein-Born-Infeld black holes in (A)dS spacetime withcosmological horizon in arbitrary dimensions has beenconstructed in [13]. The extension to the case where blackhole horizon (cosmological horizon) is a positive, zero ornegative constant curvature surface have also been studied[14]. Unfortunately, exact solutions to the Einstein-Born-Infeld equation coupled to matter fields are too compli-cated to find except in a limited number of cases. Indeed,exact solutions to the Einstein Born-Infeld dilaton (EBId)gravity are known only in three dimensions [15].Numerical studies of the EBId system in four-dimensionalstatic and spherically symmetric spacetime have been donein [16]. In the absence of a dilaton potential a class ofsolution to the four-dimensional EBId gravity with mag-netic charge has been constructed [17]. Our aim in thispaper is to generalize these solutions to the case of one andtwo Liouville type potential and investigate how the prop-erties of the solutions will be changed in the presence ofpotential for the scalar field. In addition, in each case, wecompute, the mass, electric charge, temperature and en-tropy of the system. We will consider three special cases:(a) V��� � 0, (b) V��� � 2�e2�� and(c) V��� � 2�1e

2�1� � 2�2e2�2�. The first case corre-

sponds to the action considered in [18]. When � � 1, itreduces to the four-dimensional low-energy action ob-tained from string theory in terms of Einstein metric.Case (b) corresponds to a Liouville-type potential. Thiskind of potential appears when one applies a conformaltransformation on the low-energy limit of the string treelevel effective action for massless boson sector and writethe action in the Einstein frame [19,20]. This potential hasbeen considered previously by a number of authors[6,8,21]. One may refer to � as the cosmological constant,since in the absence of the dilaton field the action reduces*Electronic address: riazi@physics.susc.ac.ir

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to the action of EBI gravity with cosmological constant[13,14]. The potential in case (c) was previously investi-gated by a number of authors both in the context of FRWscalar field cosmologies [22] and EMd black holes [6,8].This kind of potential function can be obtained when ahigher-dimensional theory is compactified to four-dimensional spacetime, including various supergravityand string models.

The organization of this paper is as follows: Section II isdevoted to a brief review of the field equations and generalequations of motion. In Sec. III we consider EBId blackholes without potential. In Sec. IV, we present two classesof solutions with a Liouville type potential and generaldilaton coupling. In Sec. V, we extend these solutions to thecase of two Liouville potentials. We finish our paper withsome concluding remarks.

II. FIELD EQUATIONS

We consider the four-dimensional action in which grav-ity is coupled to dilaton and Born Infeld fields with anaction

S �Zd4x

��������gp

�R� 2�r��2 � V��� � L�F;��� (1)

where R is the Ricci scalar curvature,� is the dilaton fieldand V��� is a potential for�. The Born-Infeld L�F;�� partof the action is given by

L�F;�� � 4�e�2���1�

��������������������������1�

F��F��2�

s �: (2)

Here, � is the dilaton coupling constant and � is called theBorn-Infeld parameter with dimension of mass. In the limit�! 1, L�F;�� reduces to the standard Maxwell fieldcoupled to a dilaton field

L�F;�� � �e�2��F��F��: (3)

On the other hand, L�F;�� ! 0 as �! 0. It is convenientto set

L�F;�� � 4�e�2��L�Y�: (4)

where

L �Y� � 1��������������1� Yp

; (5)

Y �F2

2�: (6)

where F2 � F��F��. The equations of motion can beobtained by varying the action (1) with respect to thegravitational field g��, the dilaton field � and the gaugefield A� which yields the following field equations

R �� � 2@��@���1

2g��V���

� 4e�2��@YL�Y�F��F��

� 2�e�2���2Y@YL�Y� �L�Y��g��; (7)

r2� �1

4

@V@�� 2��e�2��L�Y�; (8)

r��e�2��@YL�Y�F

��� � 0: (9)

We wish to find static and spherically symmetric solu-tions of the above field equations. The most general suchmetric can be written in the form

ds2 � �U�r�dt2 �dr2

U�r�� R2�r��d�2 � sin2�d’2�:

(10)

The Maxwell equation can be integrated immediately,where all the components of F�� are zero except Frt:

Frt �qe2��

R2�r���������������������1� q2e4��

�R4�r�

r (11)

Here q is the electric charge, defined through the integral

q �1

4

Zs2e�2�� � Fd�; (12)

where � is the Hodge dual and s2 is any two-sphere definedat spatial infinity, which its volume element denoted byd�. We note that the electric field is finite at r � 0. This isexpected in Born-Infeld theories. It is interesting to con-sider three limits of (11). First, for large � (where the BIaction reduces to Maxwell case) we have Frt � qe2��

R2�r� as

presented in [6]. On the other hand, if �! 0 we get Frt �0, finally in the case of �! 0 it reduces to the case ofEinstein Born Infeld theory without dilaton field [13,14].With the metric (10) and Maxwell field (11), the fieldEqs. (7) and (8) reduce to the following system of coupledordinary differential equations

1

R2

ddr

�UdR2

dr

��

2

R2 � V���

� 4�e�2���2Y@YL�Y� �L�Y��; (13)

1

R2

ddr

�R2U

d�dr

��

1

4

dVd�� 2��e�2��L�Y�; (14)

1

R

d2R

dr2 �

�d�dr

�2� 0: (15)

In particular, in the case of the linear electrodynamicswith L�Y� � � 1

2Y, the system of Eqs. (7)–(9) and (13)–

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(15) reduce to the well-known equations of EMd gravity[6].

To solve these equations, we make the ansatz

R�r� � e���r�; (16)

By this ansatz, Eq. (6) becomes Y � � q2

q2��. In addition,

we introduce the constant A:

A � ��2Y@YL�Y� �L�Y�� �������������������������q2 � ��

q� �: (17)

Using (16) in Eq. (15), immediately gives

��r� ��

1� �2 ln�br� c�; (18)

where b and c are integration constants. For later conve-nience, without loss of generality, we set b � 1 and c � 0.

III. SOLUTIONS WITH V��� � 0

Let us begin by looking for the solutions withoutLiouville potential (V��� � 0).

A. String coupling case � � 1

We first consider the string coupling case � � 1 withV��� � 0. In this case we find the following solution

U�r� � 2r�1� 2A�

r0

2r

�; (19)

with A is a constant related to the electric charge q as onecan see from Eq. (17) and r0 > 0 is an integration constantrelated to the mass of the system. In order to fully satisfythe system of equations, there must be a relation betweenelectric charge q and � parameter

q2 �1�

��������������������16�2 � 1

p8�

: (20)

Now we compute the mass of the system. In order todefine the mass, we use the so-called quasilocal formalism[23]. The quasilocal mass is given by

M �1

2

dR2�r�dr

U1=2�r��U1=20 �r� �U

1=2�r��; (21)

where U0�r� is an arbitrary non-negative function whichdetermines the zero of the energy for a background space-time and r is the radius of the spacelike hypersurfaceboundary. If no cosmological horizon is present, the larger limit of (21) determines the asymptotic mass M. For thesolution under consideration, there is no cosmologicalhorizon and the natural choice for the background isU0�r� � 2�1� 2A�r. The large r limit of (21) gives themass of the solution

M �r0

4: (22)

The metric corresponding to (19) and the other metricthat we will present in this paper are neither asymptoticallyflat nor (anti)-de Sitter. The solution has several properties.First, there is an event horizon at rh � 2M=�1� 2A�. Inorder to study the general structure of these solutions, wefirst look for the curvature singularities in the presence ofdilaton gravity. It is easy to show that the Kretschmannscalar R���R��� diverges at r � 0, it is finite for r � 0and goes to zero as r! 1. Also, it is notable to mentionthat the Ricci scaler is finite every where except at r � 0,and goes to zero as r! 1. Therefore r � rh is a regularhorizon and we have an essential singularity located at r �0. This can be seen from the explicit expression of K andR:

K �1

4r4 �4r2�3� 4A� 12A2� � 4rr0�1� 2A� � 3r2

0�;

(23)

R �2r�6A� 1� � r0

2r2 : (24)

Second, even though ��r� diverges at r! 1, but sincethe mass, charge and curvature all remain finite, the solu-tion is well behaved at infinity. Note that the dilaton field isregular on the horizon, too. The spacial infinity is confor-mally null and the solution describes a black hole with thesame causal structure as the Schwarzschild spacetime.Black hole entropy typically satisfies the so-called arealaw of the entropy [24], which states that the entropy is aquarter of the event horizon area. It is easy to see that thetemperature and entropy of the black hole can be written as

T �1

4dUdr�rh� �

1� 2A2

; (25)

S � rh �2M

1� 2A: (26)

Note that the temperature depends on A and is independentof M. In the limit A! 1=2, the temperature goes to zero,while the entropy S becomes infinite. Since the tempera-ture is always non-negative quantity thus A 1=2.

Finally we investigate the first law of thermodynamics.The black hole solution we found here have mass andcharge, thus in general all thermodynamic quantities arefunctions of M and q. Since the electric charge is fixed, thefirst law of thermodynamics may be written as

dM � TdS: (27)

Then it is easy to see that thermodynamics quantitiesobtained above satisfy the first law (27). The solutionwith zero mass (M � 0) is singular with a null singularityat r � 0. The case M< 0 corresponds to naked timelikesingularity located at r � 0.

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B. General dilaton coupling �

It is straightforward to generalize the solution (19) toarbitrary dilaton coupling constant �. In this case, we findthe following solution

U�r� � r2�2N��1� 2A�=N �

r0

r

�; (28)

withN � �2=�1� �2�. Here r0 > 0 is again an integrationconstant related to the mass of the black hole. The con-sistency of all field equations forces that the electric chargeq satisfies the following equation:

����������������� q2

q�2���2 � 1� � 1� � 2

�����p�q2 � ���2 � 1�� � 0:

(29)

Note that the solution is ill defined for � � 0. In theparticular case � � 1, the solution reduces to (19). There isno cosmological horizon and the mass can be computedfrom (21). For the background function U0�r� �r2�2N�1� 2A�=N, the mass is found to be

M �Nr0

2: (30)

On the other hand, if � and � go to infinity, we will haveN � 1 and A � 0, respectively. Thus the metric reduces tothe Schwarzschild black hole. There is an event horizon atrh � 2M=�1� 2A� which is regular only for A< 1=2. TheKretschmann invariant and Ricci scalar are finite at r � rh,diverge at r � 0, and both of them vanish at r! 1, there-fore r � rh is a regular horizon and we have an essentialsingularity located at r � 0. Note that the dilaton field isregular on the horizon, too. The temperature and the en-tropy of the black hole on the event horizon are

T �1

4dUdr�rh� �

1� 2A4N

r1�2Nh ; (31)

S � r2Nh : (32)

Finally we investigate the first law of thermodynamics.Again, since the electric charge is fixed, thus the first law ofthermodynamics can be written as

dM � TdS: (33)

The solution with zero mass (M � 0) is singular with anull singularity at r � 0.

IV. SOLUTION WITH A LIOUVILLE TYPEPOTENTIAL

In this section, we consider the action (1) with aLiouville type potential,

V��� � 2�e2��; (34)

where � and � are constants. One may refer to � as thecosmological constant, since in the absence of the dilaton

field the action reduces to the action of EBI gravity withcosmological constant [13,14].

A. Solution with � � 1

At first, we consider the case � � 1. We have found thefollowing solution

U�r� � 2r�1� 2A���

r0

2r

�; (35)

with � � �1 and r0 � 4M, where M is the mass of thesystem defined via the Eq. (21). The electric charge, isgiven by Eq. (20). For the background function U0�r� ��1� 2A���r. There is an event horizon at rh �

2M1�2A��

which is regular only for �< 1� 2A. The Kretschmanninvariant and Ricci scalar, are regular except for r � 0,where they diverge. Note that the dilaton field is regular onthe horizon, too. As an illustration we present the Ricciscalar R:

R �2r�6A� 1� 3�� � r0

2r2 ; (36)

The temperature and the entropy of the black hole on theevent horizon are

T �1� 2A��

2; (37)

S �2M

1� 2A��: (38)

which satisfy the first law. Note that the temperature de-pends on A and � and is independent of M. In the limit�! �1� 2A�, the temperature goes to zero, while theentropy S become infinite. It is interesting to see that oursolutions are well behaved in the limit �! 0. In otherwords all of our results presented in this section reduce tothe ones presented in section (III A), in this limit.

B. Solutions with general coupling �

In this section, we present exact black hole solutions ofEBId with an arbitrary dilaton coupling � and Liouvillepotential V��� � 2�e2��. In this case we can distinguishtwo classes of solutions which satisfy all the field equationsdepending on a suitable choice of the � parameter.

I. � � ��. In this case, using (16) and (18), one caneasily show that Eqs. (13) and (14) have solution of theform

U�r� �r2�2N

N

�1� 2A���

2Mr

�; (39)

where M is the mass of the system defined via Eq. (21).Again, the consistency of all field equations forces that theelectric charge q satisfies the following equation:

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����������������� q2

q��2������2 � 1� � 1�

� 2�����p�q2 � ���2 � 1�� � 0:

(40)

For the background function U0�r� � r2�2N�1� 2A���=N. There is an event horizon at rh �

2M1���2A which is

regular only for �< �1� 2A�. The Kretschmann invariantand Ricci scalar, are regular except for r � 0, where theydiverge. Again the dilaton field is regular on the horizon.The temperature and the entropy of the black hole on theevent horizon are

T �1

4dUdr�rh� �

1��� 2A4N

r1�2Nh ; (41)

S � r2Nh �

2M1��� 2A

r2N�1h : (42)

which satisfy the first law (27). In the limit �! �1� 2A�,the temperature goes to zero, while the entropy S becomesinfinite. One may note that the solution is ill defined for� � 0. In the particular case � � 1, the solution reduces tothe (35), while in the absence of Liouville potential (� �0), the above solutions reduce to (28).

II. � � �1=�. In this case, using (16) and (18), one caneasily show that Eqs. (13) and (14) have solution of theform

U�r� �r2�2N

N

�1� 2A�

2Mr�

��1� �2�

1� 3�2 r2�2N�1�

�;

(43)

In order to have consistency of all the field equations, theelectric charge must satisfy Eq. (29). The Kretschmanninvariant and Ricci scalar, diverge at r � 0, and both ofthem vanish as r goes to infinity, so there is a singularitylocated at r � 0. Note that the solution is ill defined for�2 � 1=3. In the limit �2 ! 1 it reduces to the solution ofsection (IVA), and in the limit �! 0 the solution reduceto that with V��� � 0.

On the other hand, if � and � go to infinity, the solutionbecomes

U�r� � 1�2Mr�

�

3r2; (44)

which is the Schwarzschild ds/Ads black hole, dependingon the sign of �. In order to investigate the causal structureof the solution, we must investigate the zeros of the metricfunction U�r�. In fact, for 0< r<1 the zeros of U�r� aregoverned by the function

f�r� � 1� 2A�2Mr�

��1� �2�

�1� 3�2�r2�2N�1�: (45)

We investigate the function g�r� � rf�r�, for simplicity.The cases with �2 > 1=3 and �2 < 1=3 should be consid-ered separately. We should also consider the sign of theparameter � in each case.

In the first case where �2 < 1=3 and �< 0 we may haveone horizon since dg

dr > 0. But the more interesting casehappens for �> 0 where we only obtain one local mini-mum at r � rmin where

rmin �

�1� 2A

�

���2�1�=2��2�1�

: (46)

The function g�r� possesses horizon if g�rmin�< 0. Thereare two zeros for g�rmin�< 0 and one degenerate zero forg�rmin� � 0 which corresponds to an extremal black hole.The condition g�rmin�< 0 gives

M>�1� 2A��1� �2�

1� 3�2

�1� 2A

�

���2�1�=2��2�1�

: (47)

In the second case for �2 > 1=3 and �< 0, the functiong�r� increases monotonically, thus we can conclude thatthere is one point where g�r� � 0 which is the black holehorizon. For �> 0 we find local extremum for the func-tion. The sign of d2g�r�

d2r determines whether we have localmaximum or minimum. For �2 > 1 we have local maxi-mum and g�rmax� should be positive in order to have anyhorizon. The latter condition gives

M<�1� 2A��1� �2�

1� 3�2

�1� 2A

�

���2�1�=2��2�1�

: (48)

If we have 13 <�2 < 1, then we would have local minimum

and in case of any horizon existing g�rmin� Eq. (46) shouldbe negative which implies Eq. (47).

The above considerations show that the solutions de-scribe black holes with two horizons or an extremal blackhole hiding a singularity at the origin r � 0, when the masssatisfies (47) and (48). The radius of inner and outerhorizons cannot be expressed in a closed analytical formexcept for the extremal case. The radius of the extremalsolution rext, coincides with rmin:

rext �

�1� 2A

�

���2�1�=2��2�1�

��1� 3�2�Mext

�1� 2A��1� �2�: (49)

Unfortunately, because of the nature of the exponents ofr in (45), the event horizon determined by f�r� � 0 cannotbe expressed in a closed analytical form for arbitrary �. Asan example, we consider the special case � �

���3p

. For thisvalue of �, and large � limit, the action (1) is simply theKaluza-Klein action which is obtained by dimensionallyreducing the five dimensional vacuum Einstein action. Fordetails, see [25–27].

In this case, there are two zeros of f�r� at r:

r �1

��1� 2A

���������������������������������������1� 2A�2 � 4�M

q�: (50)

The extremal solution corresponds to Mext ��1�2A�2

4� . In

this case f�r� has only one root at rext �1�2A

� . When �<�1�2A�2

4M we have two horizons located at r � r. For �>�1�2A�2

4M , there is a naked singularity at r � 0.

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The temperature and the entropy of the black hole on thehorizons are

T �r�3=2

12�4M� 2�1� 2A�r � 3�r2

�; (51)

S � r3=2 : (52)

which satisfy in the first law of thermodynamics.

V. SOLUTIONS WITH A GENERAL COUPLINGPARAMETER AND TWO LIOUVILLE

POTENTIALS

In this section, we present exact solutions to the EBIdgravity equations with an arbitrary dilaton coupling pa-rameter � and dilaton potential

V��� � 2�1e2�1� � 2�2e2�2�: (53)

Where �1, and �2, �1 and �2 are constants. This kind ofpotential was previously investigated by a number of au-thors both in the context of FRW scalar field cosmologies[22] and EMd black holes [6,8]. This generalizes furtherthe potential (34). If �1 � �2, then (53) reduces to (34), sowe will not repeat these solutions. Requiring �1 � �2,ones obtains

U�r� �r2�2N

N

�1� 2A��1 �

2Mr

��2�1� �

2�

1� 3�2 r2�2N�1�

�: (54)

In order to fully satisfy the system of equations, the �1 and�2 parameters must satisfy �1 � 1=�2 � ��, and q pa-rameter should satisfy Eq. (40), by replacing �! �1.Note that the solution is ill defined for �2 � 1=3. In theparticular case �2 � 0, this solution reduces to (39) andwhen �1 � 0, it reduces to (43). The Kretschmann invari-ant and Ricci scalar, diverge at r � 0, and both of themvanish as r goes to infinity, so there is a singularity locatedat r � 0. Another solution with the same spacetime metricis generated via the discrete transformation �1 $ �2 and�1 $ �2.

In order to investigate the causal structure of the solutionand subsequently find the horizons (similar to what wasdone in the previous section) we find the zeros of thefunction

f�r� � �1� 2A��1� �2Mr�

�2�1� �2�

1� 3�2 r4N�2; (55)

Again, we investigate the function g�r� � rf�r�, for sim-plicity. The cases with �2 > 1=3 and �2 < 1=3 should beconsidered separately. We should also consider the sign ofthe parameter �1 and �2.

For the first case, where �2 > 1=3, we certainly governextremum if �1 > 1=2 (�1 < 1=2) and �2 < 0 (�2 > 0).The sign of second derivative will show whether we havelocal minimum or maximum. Here, for 1=3<�2 < 1

(�2 > 1) and �2 > 0 (�2 < 0) the function f�r� wouldhave local minimum and in opposite, for 1=3<�2 < 1(�2 > 1) and �2 < 0 (�2 > 0) the function g�r� will havelocal extrema at

rmin�max� �

�1� 2A��1

�2

���2�1�=2��2�1�

: (56)

The value of the function g�r� at its extremum is

g�rext� � �2M� 2�1� 2A��1��1� �

2�

1� 3�2

�

�1� 2A��1

�2

���2�1�=2��2�1�

: (57)

In order to have any horizon, g�rmin��g�rmax�� should belarger(less) than or equal to zero in order to possess anylocal extremum and subsequently to have any horizon forthe black hole. The case g�rmin��g�rmax�� � 0 correspondsto an extremal black hole. The condition g�rmin�< 0 gives

M>�1� 2A��1��1� �2�

1� 3�2

�

�1� 2A��1

�2

���2�1�=2��2�1�

: (58)

and we obtain the following inequality for the conditiong�rmax�> 0

M<�1� 2A��1��1� �2�

1� 3�2

�

�1� 2A��1

�2

���2�1�=2��2�1�

: (59)

For the second case where �2 < 1=3 the function g�r�possesses local minimum for �2 < 0 and local maximumfor �2 > 0. In this case, the function diverges both at r � 0and at infinity. The local minimum(maximum) happens at(56) and the value of the function g�rmin� is given by (57).Since in this case we have both a local minimum andmaximum, condition (59), should hold for both cases.

We see that in both cases we obtain horizons for anygiven value of the parameter �. Here we express the radiusof the extremal solution like the preceding section

rext �

�1� 2A��1

�2

���2�1�=2��2�1�

��1� 3�2�Mext

�1� �2��1� 2A��1�: (60)

Unfortunately, because of the nature of the exponents of rin (45), the event horizon determined by g�r� � 0 cannotbe expressed in a closed analytical form for arbitrary �.

VI. CONCLUSION

Born-Infeld theory and dilaton gravity are well-motivated and extensively studied theories, not only sepa-

A. SHEYKHI, N. RIAZI, AND M. H. MAHZOON PHYSICAL REVIEW D 74, 044025 (2006)

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rately, and also coupled to each other. In this paper, wederived some classes of exact, electrically charged, staticand spherically symmetric black hole solutions to four-dimensional Einstein-Born-Infeld-dilaton gravity withoutpotential or with one or two Liouville type potentials. Theblack hole solutions have unusual asymptotics. They areneither asymptotically flat nor asymptotically (anti-) deSitter. In particular, in the case of the linear electrodynam-ics with L�Y� � � 1

2Y the method presented here gives thewell-known asymptotically nonflat and non-(A)dS blackhole solutions of the EMd gravity [6]. We showed that inthe presence of dilaton field, both Kretschmann invariantand Ricci scalar diverge at r � 0, they remain finite for r �

0 and tend to zero as r! 1. Thus we have an essentialsingularity located at r � 0. We found that in the presenceof Liouville type potentials for dilaton field, there exist twoclasses of solutions which satisfy all the field equationsdepending on suitable choice of the � parameter. We alsocomputed—for each case—temperature, entropy, chargeand mass of the black hole solutions, and find that these

quantities satisfy the first law of thermodynamics. Wefound that in order to fully satisfy all the field equationsconsistently, there must be a relation between the electriccharge and other parameters of the system. In general, theelectric charge depends on the three parameters �, � and�. We found that in the large � and � limit, our solutionsreduce to Schwarzschild, and Schwarzschild ds/Ads blackholes, depending on the sign of �.

Finally, it should be noted that the solutions were basedon an ansatz and consistency checks demanded a relation-ship between the parameters of the theory. An attempt forfinding exact solutions of EBId gravity by relaxing theansatz (16), is under investigation. Note that the four-dimensional EBId black hole solutions obtained here arestatic. Therefore, it would be interesting if one could con-struct charged rotating solutions of EBId gravity in fourdimensions. One can also attempt to construct static androtating solutions of the EBId gravity with both flat andcurved horizons in various dimensions.

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