Slowly rotating dilaton black hole in anti–de Sitter spacetime

Tanwi Ghosh* and Soumitra SenGupta†

Department of Theoretical Physics, Indian Association for the Cultivation of Science, Jadavpur, Calcutta–700 032, India(Received 13 December 2006; revised manuscript received 23 August 2007; published 24 October 2007)

A rotating dilaton black hole solution for asymptotically anti-de Sitter spacetime is obtained in thesmall angular momentum limit with an appropriate combination of three Liouville-type dilaton potentials.The angular momentum, magnetic dipole moment, and the gyromagnetic ratio of such a black hole aredetermined for arbitrary values of the dilaton-electromagnetic coupling parameter.

DOI: 10.1103/PhysRevD.76.087504 PACS numbers: 04.50.+h, 04.20.Ha, 04.20.Jb, 04.70.Bw

I. INTRODUCTION

Scalar coupled black hole solutions with differentasymptotic spacetime structure has been a subject of inter-est for a long time. There has been a renewed interest insuch studies ever since new black hole solutions have beenfound in the context of string theory. The low energyeffective action of string theory contains two masslessscalars, namely, dilaton and axion. The dilaton field cou-ples in a nontrivial way to other fields such as gauge fieldsand results in interesting solutions for the backgroundspacetime [1]. It is found that the dilaton changes thecausal structure of spacetime and leads to curvature singu-larities at finite radii. Subsequently various other propertieshave also been explored for such scalar coupled blackholes [2–9]. Most of these scalar coupled black holesolutions, however, were asymptotically flat. Gao andZhang [10] obtained the asymptotically nonflat chargeddilaton black hole solutions in anti-de Sitter and de Sittermetric in Schwarzschild coordinate using the combinationof three Liouville-type dilaton potentials. Such potentialmay arise from the compactification of a higher dimen-sional supergravity model [11,12], which originates fromthe low energy limit of a background string theory.

At this time, there was also a growing interest to studythe rotating black hole solutions in the presence of a dilatoncoupled electromagnetic field in the background. An exactsolution for a rotating black hole with a special dilatoncoupling was derived using the inverse scattering method[13]. In an alternative approach, Horne and Horowitzdeveloped a simpler perturbative technique to find such arotating black hole solution where a small angular momen-tum is introduced as a perturbation in a nonrotating systemin the presence of a general dilaton-electromagnetic cou-pling. They studied the properties of asymptotically flatcharged dilaton black holes in the limit of infinitesimallysmall angular momentum [14,15]. A similar work thenfollowed by introducing an infinitesimal small charge tothe black hole [16]. Following the method adopted in [14],asymptotically nonflat rotating dilaton black holes are

obtained in [17] for small values of the angular momentum.In a different context, M. H. Dehghani et al. obtainedcharged rotating dilaton black string solutions [18]. How-ever, charged rotating dilaton black hole solutions for anarbitrary dilaton-electromagnetic coupling parameter in ade Sitter or anti-de Sitter spacetime is yet to be found.

The interest for studying such dilaton black holes withnonvanishing cosmological constant has several reasons.Gauged supergravity theories in various dimensions areobtained with negative cosmological constant in a super-symmetric theory. In such a scenario, anti-de Sitter space-time constitutes the vacuum state, and the black holesolution in such a spacetime becomes an important areato study [19–24]. Apart from the general interest as aclassical solution of field equations in the framework ofgeneral relativity, there are also cosmological implicationsof exploring the features of anti-de Sitter and de Sitterspacetime [25–29]. The behavior of galaxies away fromtheir centers suggests the presence of a very large amountof dark matter with 1

r2 distribution of energy. One canimagine this dark matter in the form of a scalar field suchas a dilaton. Asymptotically flat behavior is not consistentwith the observed nondecaying velocity profile and there-fore the AdS or dS solutions are of considerable interest inthis context. Moreover, the observed rotational motion ingalaxies further suggests the need to study the rotatingmetric in such a spacetime.

In the backdrop of the scenarios described so far, it istherefore worthwhile to study the rotating black hole solu-tions in a spacetime with nonzero cosmological constant inthe presence of dilaton-electromagnetic coupling. Ex-ponential potentials for scalar fields have been used bymany workers in this context, which closely resemblesGao-Zhang’s model. We here study various features of arotating charged dilatonic black hole solution in a spacewith nonzero cosmological constant. In particular we con-centrate on the anti-de Sitter spacetime as considered in[10]. Introducing a small angular momentum, followingHorne and Horowitz, we aim here to obtain a rotating blackhole solution in anti-de Sitter spacetime by generalizingGao and Zhang’s work. We then determine the expressionsfor angular momentum, magnetic dipole moment, and thegyromagnetic ratio for such a black hole.

*tanwi.ghosh@yahoo.co.in†tpssg@iacs.res.in

PHYSICAL REVIEW D 76, 087504 (2007)

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II. FIELD EQUATIONS AND SOLUTIONS

We begin with an action of the form

S �Zd4x

��������gp

�R� 2@�’@�’� V�’� � e�2�’F2�

(1)

where R is the scalar curvature, F2 � F��F��, F�� beingthe usual Maxwell field tensor, V�’� is the potential for thedilaton field’, and�measures the strength of the couplingbetween the electromagnetic field and ’. While � � 0corresponds to the usual Einstein-Maxwell-scalar theory,� � 1 indicates the dilaton-electromagnetic coupling thatappears in the low energy string action in Einstein’s frame.Varying the action with respect to the metric, the Maxwellfield, and dilaton field, respectively, we obtain the equa-tions of motion as

Rab � 2@a’@b’�12gabV

� 2e�2�’�FacFcb �12gabFcdF

cd� (2)

@a���������gp

�e�2�’Fab� � 0 (3)

r2’ �1

4

@V@’��2e�2�’FcdF

cd: (4)

For arbitrary value of � in anti-de Sitter space, the form ofthe dilaton potential is chosen as [10],

V�’� �2

3�

1

��2 � 1�2��2�3�2 � 1�e�2’=�

� �3� �2�e2�’ � 8�2e�’�’=�� (5)

with the metric of the form [10],

ds2 � �

��1�

r�r

��1�

r�r

��1��2�=�1��2�

�1

3�r2

�1�

r�r

�2�2=��2�1�

�dt2

�

��1�

r�r

��1�

r�r

��1��2�=�1��2�

�1

3�r2

�1�

r�r

�2�2=��2�1�

��1dr2

� r2

�1�

r�r

�2�2=��2�1�

d�2 � 2af�r�sin2�dtd�:

(6)

Here r� and r� are, respectively, the event horizon andCauchy horizon of the black hole. The parameter ‘‘a’’measures the angular momentum, which we shall chooseto be small. In other words it can be viewed as a smallaxisymmetric perturbation in an anti-de Sitter background.We now determine the form of f�r� for such an anti-deSitter metric and look for the possible black hole structurein that spacetime.

Integrating the Maxwell’s equations we obtain the firstderivative of the scalar potential as

� A00 � F01 �Qe2�’

h; (7)

where Q, the integration constant, is related to the physicalelectric charge as will be shown later, and h for the anti-deSitter metric is given by

h � r2

�1�

r�r

�2�2=��2�1�

: (8)

It may be noted that the only term in the metric thatchanges to the order of the angular momentum parametera is gt�. It is easy to see that the dilaton solution does notchange to the order of a and the solution for the � compo-nent of the vector potential in the presence of rotation isgiven by

A� � �aQc�r�sin2� (9)

where c�r� satisfies

Q�uc0e�2�’�0 �2Qce�2�’

h� �Q

�fh

�0

(10)

with

u ���

1�r�r

��1�

r�r

��1��2�=�1��2�

�1

3�r2

�1�

r�r

�2�2=��2�1�

�

For anti-de Sitter spacetime the solution for the dilatonfield is [10]

e2�’ � e2�’0

�1�

r�r

�2�2=��2�1�

: (11)

The remaining equation of motion which is needed to besatisfied for small a is

f00

f�h00

h�

4Q2c0

fh: (12)

For small value of a, using the ansatz c�r� � 1r [17,30], the

solution of f�r� from Eqs. (12) and (15) turns out to be

f�r� �r2�1� �2�2e�2�’0

�1� �2��1� 3�2�r2�

�1�

r�r

�2�2=��2�1�

(13)

� e�2�’0

�1�

r�r

��1��2�=�1��2�

�

�1�

r�r��1� �2�

�1� �2�

rr��

r2�1� �2�2

�1� �2��1� 3�2�r2�

�

��3e�2�’0r2

�1�

r�r

�2�2=��2�1�

(14)

It is straightforward to verify that in our case for � � 0, theexpression for f�r� agrees with the corresponding expres-

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sions for a slowly rotating dilaton black hole solution in aflat spacetime as obtained in [14,15]. For a slowly rotatingdilaton black hole, we learn from [14] that the surfacegravity and the area of the event horizon do not changeto the order of the angular momentum parameter a. In thelinear approximation in rotation parameter a we obtain theexpression for angular momentum for anti-de Sitter caseusing Komar approach [31]. For rotating Ads black hole,angular momentum and mass are defined by

M0 � �1

8�

Id��̂�t�� (15)

and

J0 �1

16�

Id��̂��; (16)

where * denotes the Hodge dual and the Killing oneform. (�̂) represents the difference between the Killingisometries of the spacetime and its background spacetime.Using the above two integrands in the asymptotic limit, weobtain

�t;r�t� �

r�2r2 �

�1� �2

1� �2

�r�2r2 �O

�1

r2

�(17)

and

�t;r��� � e�2�’0

r�r2 � e

�2�’0r�r2

�3� �2�

3��2 � 1��O

�1

r2

�:

(18)

Integrating (19) and (20) over two sphere in r! 1 limitwe get

M0 �r�2�

�1� �2

1� �2

�r�2

(19)

and

J0 �ae�2�’0

2

�r� �

�3� �2�

3�1� �2�r�

�: (20)

For the anti-de Sitter dilatonic black hole the expression foractual mass can be written as [31]

Mactual � M0 � a�3J0

� M0 � a2 �3

e�2�’0

2

�r� �

�3� �2�

3�1� �2�r�

�� M0:

(21)

Here for a slowly rotating black hole we have ignored theterm of the order of a2.

It is interesting to note that the expression for J is sameas that obtained in [14]. Using the expression for h [fromEq. (8)] and the expression for e2�� [from Eq. (11)] inEqs. (7) and (9), one obtains the following expressions forthe radial fields [30]:

Er �Qe2�’0

r2 (22)

and

Br �aQsin2�

r2 : (23)

The Gaussian flux of electric field gives the value ofelectric charge as follows:

Q0 �1

4�

IF � Qe2�’0 : (24)

Magnetic dipole moment for this slowly rotating dilatonblack hole in anti-de Sitter spacetime can be defined as

�0 � Q0a � gQ0J0

2M0: (25)

Substituting M0, J0 and Q0 from Eqs. (17)–(19) and (22), gcan be obtained as

g � 2�4�2r�

��3� �2�r� � 3�1� �2�r��: (26)

It may be noted that while using Eq. (20) to estimate gabove, we have ignored the term quadratic in the rotationparameter a. As a result, in the linear approximation in athe above expression for g turns out to be same as thatfound in [14]. Moreover, it is easy to check that for � � 0,we retrieve g � 2 and in the limit �! 1 the value of g liesbetween 2 and 3=2.

III. CONCLUDING REMARKS

Starting from the nonrotating charged dilaton black holesolutions in anti-de Sitter spacetime, we have obtained thesolution for the rotating charged dilaton black hole byintroducing a small angular momentum following the per-turbative method suggested in [14]. Since it is well knownthat in the presence of one Liouville-type dilaton potentialno de Sitter or anti-de Sitter dilaton black hole exists evenin the absence of rotation, therefore considering the pro-posal in [10] of three Liouville-type dilaton potential, weshow that such potential leads to slowly rotating chargeddilaton black holes solutions in an anti-de Sitter spacetime.As expected, our solutions f�r� reduce to [14] for � � 0.The forms of angular momentum J and gyromagnetic ratiog are obtained. Their values to the linear order in theangular momentum parameter a turn out to be same asthat obtained in [14]. It is interesting to note that the dilatonfield modifies the value of g from 2 through the couplingparameter �, which measures the strength of the dilaton-electromagnetic coupling.

As discussed earlier, the presence of such anti-de Sitterdilatonic charged rotating black hole is inevitably associ-ated with an accompanying scalar field with appropriateLioville-type potential. Their study therefore may lead to abetter understanding of the origin of the dark matter in the

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Universe. This work brings out the corresponding space-time solution along with the associated black hole structurein the limit of small angular momentum.

ACKNOWLEDGMENTS

T. G. wishes to thank CSIR (India) for financial support.

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