Charged rotating Kaluza-Klein black holes in dilaton gravity

Masoud Allahverdizadeh,1 Ken Matsuno,2,* and Ahmad Sheykhi3,†

1Institut fur Physik, Universitat Oldenburg, D-26111 Oldenburg, Germany2Department of Mathematics and Physics, Graduate School of Science, Osaka City University,

3-3-138 Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan3Department of Physics, Shahid Bahonar University, Kerman 76175, Iran

(Received 19 August 2009; revised manuscript received 20 December 2009; published 1 February 2010)

We obtain a class of slowly rotating charged Kaluza-Klein black hole solutions of the five-dimensional

Einstein-Maxwell-dilaton theory with arbitrary dilaton coupling constant. At infinity, the spacetime is

effectively four dimensional. In the absence of the squashing function, our solution reduces to the five-

dimensional asymptotically flat slowly rotating charged dilaton black hole solution with two equal angular

momenta. We calculate the mass, the angular momentum, and the gyromagnetic ratio of these rotating

Kaluza-Klein dilaton black holes. It is shown that the dilaton field and the nontrivial asymptotic structure

of the solutions modify the gyromagnetic ratio of the black holes. We also find that the gyromagnetic ratio

crucially depends on the dilaton coupling constant, �, and decreases with increasing � for any size of the

compact extra dimension.

DOI: 10.1103/PhysRevD.81.044001 PACS numbers: 04.20.Ha, 04.50.�h, 04.70.Bw

I. INTRODUCTION

The study of black holes in more than four spacetimedimensions is motivated by several reasons. Strong moti-vation comes from developments in string/M theory, whichis believed to be the most consistent approach to quantumtheory of gravity in higher dimensions. In fact, the firstsuccessful statistical counting of black hole entropy instring theory was performed for a five-dimensional blackhole [1]. This example provides the best laboratory for themicroscopic string theory of black holes. Besides, theproduction of higher-dimensional black holes in futurecolliders becomes a conceivable possibility in scenariosinvolving large extra dimensions and TeV-scale gravity.Furthermore, as mathematical objects, black hole space-times are among the most important Lorentzian Ricci-flatmanifolds in any dimension. While the nonrotating blackhole solution to the higher-dimensional Einstein-Maxwellgravity was found several decades ago [2], the counterpartof the Kerr-Newman solution in higher dimensions, that is,the charged generalization of the Myers-Perry solution [3]in higher-dimensional Einstein-Maxwell theory, still re-mains to be found analytically. Indeed, the case of chargedrotating black holes in higher dimensions has been dis-cussed in the framework of supergravity theories and stringtheory [4–6]. Recently, charged rotating black hole solu-tions in higher dimensions with a single rotation parameterin the limit of slow rotation have been constructed in [7](see also [8–10]).

On the other hand, a scalar field called a dilaton appearsin the low energy limit of string theory. The presence of thedilaton field has important consequences on the causal

structure and the thermodynamic properties of black holes.Thus, much interest has been focused on the study of thedilaton black holes in recent years. While exact dilatonblack hole solutions of Einstein-Maxwell-dilaton (EMd)gravity have been constructed by many authors (see e.g.[11–17]), exact rotating dilaton black hole solutions havebeen obtained only for some limited values of the dilatoncoupling constant [18–20]. For the general dilaton cou-pling constant, the properties of charged rotating dilatonblack holes only with infinitesimally small charge [21] orsmall angular momentum in four [22–24] and five dimen-sions have been investigated [25]. Recently, chargedslowly rotating dilaton black hole solutions in the back-ground of anti–de Sitter spaces have also been constructedin arbitrary dimensions [26–29].Most authors have considered mainly asymptotically flat

and stationary higher-dimensional black hole solutionssince they would be idealized models if such black holesare small enough for us to neglect the tension of a brane oreffects of compactness of extra dimensions. However, ifnot so, we should consider the higher-dimensional space-times which have another asymptotic structure. Therefore,it is also important to study black hole solutions with awide class of asymptotic structures. Recently, the blackobject solutions with nontrivial asymptotic structures havebeen studied by various authors. For example, squashedKaluza-Klein black hole solutions [30–48] asymptote tothe locally flat spacetime, i.e., a twisted S1 fiber bundleover the four-dimensional Minkowski spacetime. In otherwords, the spacetime is effectively four dimensional atinfinity. The black ring solutions with the same asymptoticstructures were also found [49–53]. As far as we know,charged Kaluza-Klein dilaton black holes with asymptoti-cally locally flat structures have been constructed in thestatic case only [38]. In the present work we generalize

*matsuno@sci.osaka-cu.ac.jp†sheykhi@mail.uk.ac.ir

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such static dilaton black holes to the rotating ones. We alsoinvestigate the properties of these rotating black holesrelated to the presence of the dilaton field and the differ-ence of the asymptotic structures. Especially, we want toconstruct a new class of charged rotating squashed Kaluza-Klein black hole solutions in five-dimensional EMd grav-ity. We also study the properties of the solutions in thevarious limits. Finally, we investigate the effects of thedilaton field and the twisted compact extra dimension onthe angular momentum and the gyromagnetic ratio of theserotating black holes.

II. ROTATING DILATON BLACK HOLES WITHSQUASHED HORIZONS

We consider five-dimensional EMd theory with action

S ¼ 1

16�

ZM

d5xffiffiffiffiffiffiffi�g

p ðR� 2@��@��� e�2��F��F��Þ

� 1

8�

Z@M

d4xffiffiffiffiffiffiffi�h

p�ðhÞ; (1)

where R is the scalar curvature, � is the dilaton field,F�� ¼ @�A� � @�A� is the electromagnetic field tensor,

and A� is the electromagnetic potential. � is an arbitrary

constant governing the strength of the coupling betweenthe dilaton and theMaxwell field. The last term in Eq. (1) isthe Gibbons-Hawking boundary term which is chosen suchthat the variational principle is well defined. The manifoldM has metric g�� and covariant derivative r�. � is the

trace of the extrinsic curvature �ab of any boundary @M

of the manifold M, with induced metric hab. The equa-tions of motion can be obtained by varying the action (1)with respect to the gravitational field g��, the dilaton field

�, and the gauge field A�, which yields the following field

equations:

R�� ¼ 2@��@��þ 2e�2��ðF��F�� � 1

6g��F��F��Þ;(2)

r2� ¼ ��

2e�2��F��F

��; (3)

@�ð ffiffiffiffiffiffiffi�gp

e�2��F��Þ ¼ 0: (4)

We would like to find rotating solutions of the above fieldequations. For small rotation, we can solve Eqs. (2)–(4) tothe first order in the angular momentum parameter a.Inspection of the slowly rotating black hole solutions[29] shows that the only terms in the metric that changeto the first order of the angular momentum parameter a aregt� and gtc . Similarly, the dilaton field does not change to

OðaÞ and, A� and Ac are the only components of the vector

potential that change. Therefore, for an infinitesimal angu-lar momentum, we assume the metric, the gauge potential,and the dilaton field being of the following forms:

ds2 ¼ �uðrÞdt2 þ hðrÞ�wðrÞdr2 þ r2

4fkðrÞd�2

S2þ �2

3

þ 2afðrÞdt�3g�; (5)

A ¼ffiffiffi3

psinhð#Þ coshð#Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4þ 3�2p r21r2þðr2þ � r21Þ

ðr2þðr21 � r2Þsinh2ð#Þ þ ðr21 � r2þÞr2Þðr2þcosh2ð#Þ � r21Þ�dt� a

2�3

�; (6)

�ðrÞ ¼ �3�

4þ 3�2ln

�1þ r2þ

r21 � r2þ

r21 � r2

r2sinh2ð#Þ

�;

(7)

where �3 ¼ dc þ cosd�, d�2S2

¼ d2 þ sin2d�2 de-notes the metric of the unit two-sphere, and rþ and r1 areconstants. The functions uðrÞ, hðrÞ, wðrÞ, kðrÞ, and fðrÞshould be determined. In the particular case, a ¼ 0, thismetric (5) reduces to the static Kaluza-Klein dilaton blackhole solutions [38].

Here, we are looking for the asymptotically locally flatsolutions in the case a � 0. Our strategy for obtaining thesolution is the perturbative method proposed by Horne andHorowitz [22]. For small a, we can expect to have solutionswith uðrÞ, hðrÞ, wðrÞ, and kðrÞ still functions of r alone.Inserting metric (5), the gauge potential (6), and the dilatonfield (7) into the field equations (2)–(4), one can show thatthe static part of the metric leads to the following solutions

[38]:

uðrÞ ¼ 1� r2þr2

ð1� r2þr21Þh2ðrÞ

; (8)

hðrÞ ¼�1þ r2þ

r21 � r2þ

r21 � r2

r2sinh2ð#Þ

�4=ð4þ3�2Þ

; (9)

wðrÞ ¼ ðr21 � r2þÞ2r41ðr21 � r2Þ4ð1� r2þ

r2Þ; (10)

kðrÞ ¼ ðr21 � r2þÞr21ðr21 � r2Þ2 ; (11)

while the rotating part of the metric admits a solution

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fðrÞ ¼ 1

h3ðrÞr4ð2� 3�2Þðr21 � r2þÞðr2þ � r21 þ r2þsinh2ð#ÞÞr2þsinh2ð#Þ½ðð6�2 � 4Þr21r2þ þ 12r21r2 � ð4þ 3�2Þ

� ðr41 þ r4ÞÞr4þsinh4ð#Þ � ðr21 � r2þÞðð6�2 � 4Þr21r2þ þ 12r21r2 � ð8þ 6�2Þr4Þr2þsinh2ð#Þ� ð4þ 3�2Þðr21 � r2þÞ2r4�: (12)

The coordinates ðt; r; ; �; c Þ run the ranges of�1< t <1, 0< r < r1, 0 � � �, 0 � � � 2�, and 0 � c �4�, respectively. The spacetime (5) has the timelikeKilling vector field, @=@t, and the spacelike Killing vectorfields with closed orbits, @=@� and @=@c . To avoid theexistence of naked singularities and closed timelike curveson and outside the black hole horizon, we choose theparameters such that 0< rþ < r1.

The black hole horizon is located at r ¼ rþ. The in-duced metric on the three-dimensional spatial cross sectionof the black hole horizon with the time slice is obtained as

ds2jr¼rþ;t¼const ¼ ½coshð#Þ�8=ð4þ3�2Þ r2þ4½kðrþÞd�2

S2þ�2

3�;(13)

which implies the shape of horizon is the squashed S3, atwisted S1 fiber bundle over an S2 base space with thedifferent sizes. We see that the function kðrÞ causes thedeformation of the black hole horizon. We also note thatthe rotation parameter a has no contribution to the shape ofthe horizon, in contrast to the rotating squashed Kaluza-Klein black hole solutions in [30,37,40].

III. ASYMPTOTIC STRUCTURE OF THESOLUTIONS

In the coordinate system ðt; r; ; �; c Þ, the metric (5)diverges at r ¼ r1, but we see that this is an apparentsingularity and corresponds to the spatial infinity. In orderto see this, we introduce the new radial coordinate givenby

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 � r2þ

q2

r2

r21 � r2: (14)

We also define the parameters

þ ¼ ðrþÞ ¼ r2þ

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 � r2þ

q ; (15)

0 ¼ 12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 � r2þ

q: (16)

In terms of the coordinate , the metric, the gauge poten-tial, and the dilaton field can be written as

ds2 ¼ �UðÞdt2 þ VðÞd2 þHðÞ�ðþ 0Þd�2

S2

þ ð0 þ þÞ0

þ 0

�23 þ aFðÞdt�3

�; (17)

A ¼ffiffiffi3

psinhð#Þ coshð#Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4þ 3�2p

� þðþ 0Þð0 � þsinh2ð#ÞÞðþ þsinh2ð#ÞÞ

��dt� a

2�3

�; (18)

�ðÞ ¼ �3�

4þ 3�2ln

�1þ þ

sinh2ð#Þ

�; (19)

where the functions UðÞ, VðÞ, HðÞ, and FðÞ are,respectively, given by

UðÞ ¼�1� þ

�1

H2ðÞ ; (20)

VðÞ ¼ HðÞ�þ 0

� þ

�; (21)

HðÞ ¼�1þ þ

sinh2ð#Þ

�4=ð4þ3�2Þ

; (22)

FðÞ ¼ 1

2ð3�2 � 2Þþðsinh2ð#Þþ � 0Þðþ 0ÞH3ðÞ� ½sinh2ð#Þðð4� 6�2Þþ ð8� 3�2Þ0Þ3þþ ð2ð3�2 � 2Þ2sinh2ð#Þ þ ð6�2 þ ð3�2 þ 4Þ� sinh2ð#Þ � 4Þ2

0 þ ð9�2 þ ð3�2 � 2Þ� coshð2#Þ þ 6Þ0Þ2þ þ 0ðð�6�2

þ ð3�2 þ 4Þcsch2ð#Þ þ 4Þ þ 120Þþþ ð3�2 þ 4Þ2csch2ð#Þ2

0�: (23)

The new radial coordinate runs from 0 to 1. In the limit ! 1 (i.e. r ! r1), the metric (17) reduces to

ds2 ¼ �dt2 þ d2 þ 2d�2S2þ 0ð0 þ þÞ�2

3

þ aF1dt�3; (24)

where

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F1 ¼ Fð1Þ ¼ ð3�2 þ 4Þ0ðþ þ 0Þcsch2ð#Þ þ ð3�2 � 2Þþ½þ coshð2#Þ � þ � 20�2ð3�2 � 2Þþðsinh2ð#Þþ � 0Þ

: (25)

Next, in order to transform the asymptotic frame into therest frame, we define the coordinate c � given by

c � ¼ c þ aF120ð0 þ þÞ t: (26)

Then, the metric takes the following asymptotic form:

ds2 ¼ �dt2 þ d2 þ 2d�2S2þ L2��3

2; (27)

where

��3 ¼ dc � þ cosd�; (28)

and the size of the extra dimension L is given by L2 ¼0ð0 þ þÞ ¼ r21=4. We see that the spacetime is asymp-totically locally flat, i.e., the asymptotic form of the metricis a twisted S1 bundle over the four-dimensionalMinkowski spacetime.

IV. VARIOUS LIMITS

A. � ! 0

One may note that in the absence of a nontrivial dilaton,� ¼ 0, solution (5) reduces to the slowly rotating chargedsquashed Kaluza-Klein black hole solutions of the five-dimensional Einstein-Maxwell theory with two equal an-gular momenta. To see this, we introduce a coordinate ~rsuch that

~r 2 ¼ r2 þ r2þr21 � r2þ

ðr21 � r2Þsinh2ð#Þ; (29)

and the new parameters

~rþ ¼ rþ coshð#Þ; (30)

~r� ¼ rþr1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir21 � r2þ

q sinhð#Þ; (31)

~r1 ¼ r1: (32)

Therefore, the metric and the gauge potential reduce to

ds2 ¼ � �wð~rÞ�wð~r1Þdt

2 þ �kð~rÞ2 d~r2

�wð~rÞ þ~r2

4½ �kð~rÞd�2

S2þ �2

3

þ 2a �fð~rÞdt�3�; (33)

�A ¼ffiffiffi3

p~rþ~r�

2~r2ffiffiffiffiffiffiffiffiffiffiffiffiffi�wð~r1Þ

p�dt� a

2�3

�; (34)

where the functions �wð~rÞ, �kð~rÞ, and �fð~rÞ are now given by

�wð~rÞ ¼ ð~r2 � ~r2þÞð~r2 � ~r2�Þ~r4

; (35)

�kð~rÞ ¼ ð~r21 � ~r2þÞð~r21 � ~r2�Þð~r21 � ~r2Þ2 ; (36)

�fð~rÞ ¼ 2~r41ð~r2þ~r2� � ð~r2þ þ ~r2�Þ~r2Þð~r21 � ~r2þÞð~r21 � ~r2�Þ~r6

: (37)

To avoid the existence of the naked singularities and closedtimelike curves on and outside the black hole horizon, wechoose the parameters such that 0< ~r� < ~rþ < ~r1. For~r1 ! 1, the solution (33) is just the five-dimensionalslowly rotating Kerr-Newman black hole with two equalangular momenta [7]. When the rotation parameter van-ishes, a ! 0, the solution (33) coincides with the five-dimensional charged static Kaluza-Klein black hole withsquashed horizons [35]. For ~r� ! 0, the solution (33)reduces to the vacuum rotating squashed Kaluza-Kleinblack hole in the limit of slow rotation [30,37].

B. r1 ! 1For r1 ! 1, solution (5) reduces to the five-

dimensional slowly rotating charged dilaton black holesolution with two equal angular momenta [29]. To showthis, we define the new parameters

~� ¼ffiffi32

q�; (38)

~a ¼ a

2; (39)

then, the metric reduces to

ds2 ¼ � ~Uð~rÞdt2 þ d~r2

~Wð~rÞ þ ~r2 ~R2ð~rÞd�2S3þ ~a ~Fð~rÞdt�3;

(40)

where d�2S3

¼ 14 ðd�2

S2þ �2

3Þ denotes the metric of the

unit three-sphere, and the functions ~Uð~rÞ, ~Wð~rÞ, ~Rð~rÞ, and~Fð~rÞ are now given by

~Uð~rÞ ¼�1� ~r2þ

~r2

��1� ~r2�

~r2

�ð2�~�2Þ=ð2þ~�2Þ; (41)

~Wð~rÞ ¼�1� ~r2þ

~r2

��1� ~r2�

~r2

�2=ð2þ~�2Þ

; (42)

~Rð~rÞ ¼�1� ~r2�

~r2

�~�2=½2ð2þ~�2Þ�

; (43)

~Fð~rÞ ¼�1� ~r2�

~r2

�ð2�~�2Þ=ð2þ~�2Þ�2þ ~�2

1� ~�2

~r2

~r2�

��1�

�1� ~r2�

~r2

�½2ð~�2�1Þ�=ð2þ~�2Þ�þ 2

�1� ~r2þ

~r2

��:

(44)

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The spacetime (40) asymptotes to the five-dimensionalMinkowski spacetime at infinity. In the absence of a dilatonfield, ~� ¼ 0, the metric (40) reduces to Eq. (33) with ~r1 !1 [7]. When ~a ! 0, the solution (40) coincides with thefive-dimensional charged static dilaton black hole [12].

C. �0 ! 0

We consider the limit 0 ! 0 with þ finite. We intro-duce the coordinates ðt0; c 0Þ and the parameter a0 definedas

t0 ¼ t� a

2c ; c 0 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ð0 þ þÞ

qc ;

a0 ¼ affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ð0 þ þÞ

p :(45)

Then, the metric (17) reduces to

ds2 ¼�H�2ðÞ�1�þ

�dt02

þHðÞ��

1�þ

��1d2 þ2d�2

S2þ dc 02

�; (46)

where the function HðÞ is given by Eq. (22). This metric(46) coincides with that of charged static dilaton blackstrings [54].

V. PHYSICAL QUANTITIES

In this section we calculate the mass, the angular mo-mentum, and the gyromagnetic ratio of these rotatingKaluza-Klein dilaton black holes. Starting with (17), aftera few calculations, the Komar mass M associated with thetimelike Killing vector field @=@t at infinity and the Komarangular momentums J� and Jc associated with the space-

like Killing vector fields @=@� and @=@c � at infinity canbe obtained as

M ¼ 3�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið0 þ þÞ0

p ð3�2 þ 4 coshð2#ÞÞ8þ 6�2

; (47)

J� ¼ 0; (48)

Jc ¼ a�þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0ð0 þ þÞ

q �ð5� 3�2Þþ � 9�2þ þ ðð�8þ 3�2Þþ � ð16þ 3�2Þ0Þ coshð2#Þ þ 3þ coshð4#Þ2ð4þ 3�2Þð�þ � 20 þ þ coshð2#ÞÞ

�:

(49)

We see that the spacetime (17) has only one angularmomentum in the direction of the extra dimension. Forr1 ! 1, in terms of the parameters ~a, ~r�, and ~�, the mass(47) and the angular momentum (49) reduce to

M ¼ 3�½ð2þ ~�2Þ~r2þ þ ð2� ~�2Þ~r2��8ð2þ ~�2Þ ; (50)

J ¼ �~a½2ð2þ ~�2Þ~r2þ þ ð4� ~�2Þ~r2��8ð2þ ~�2Þ ; (51)

which are the mass and the angular momentum of the five-dimensional charged slowly rotating dilaton black holewith equal rotation parameters [29].

Next, we calculate the gyromagnetic ratio g of slowlyrotating charged Kaluza-Klein dilaton black holes. Thegyromagnetic ratio is an important characteristic ofcharged rotating black holes. Indeed, one of the remarkablefacts about a Kerr-Newman black hole in four-dimensionalasymptotically flat spacetime is that it can be assigned agyromagnetic ratio g ¼ 2, just as an electron in the Dirac

theory. It should be noted that, unlike four dimensions, thevalue of the gyromagnetic ratio is not universal in higherdimensions [8]. Besides, scalar fields, such as the dilaton,modify the value of the gyromagnetic ratio of the blackhole and consequently it does not possess the gyromagneticratio g ¼ 2 of the Kerr-Newman black hole [22]. In oursolution, we also expect the modification of the gyromag-netic ratio by the nontrivial Kaluza-Klein asymptotic struc-ture, which is related to the parameter r1. The magneticdipole moment can be defined as

� ¼ Qa; (52)

where Q denotes the electric charge of the black hole. Thegyromagnetic ratio is defined as a constant of proportion-ality in the equation for the magnetic dipole moment

� ¼ gQJ

2M: (53)

Substituting M and J from Eqs. (47) and (49), the gyro-magnetic ratio can be obtained as

g ¼ 6ð3�2 þ 4 coshð2#ÞÞð�þ � 20 þ þ coshð2#ÞÞþð5� 3�2Þ � 9�20 þ ðð�8þ 3�2Þþ � ð16þ 3�2Þ0Þ coshð2#Þ þ 3þ coshð4#Þ : (54)

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In terms of ~�, ~r�, and ~r1, the gyromagnetic ratio is givenby

g ¼ 6½ðð2þ ~�2Þ~r21 � 4~r2�Þ~r2þ þ ð2� ~�2Þ~r2�~r21�2ðð2þ ~�2Þ~r21 � 3~r2�Þ~r2þ þ ð4� ~�2Þ~r2�~r21

: (55)

We see that the dilaton field and the Kaluza-Klein asymp-totic structure modify the value of the gyromagnetic ratioof the five-dimensional Kerr-Newman black hole in theslow rotation limit [7] (see also [55]) through the couplingparameter ~�, which measures the strength of the dilaton-electromagnetic coupling, and the squashing parameter ~r1,which is proportional to the size of compact extradimension.

In Fig. 1, we show the behavior of the gyromagneticratio g versus ~� in the range of parameters 0< ~r� < ~rþ <~r1. From Fig. 1, we find out that the gyromagnetic ratiodecreases with increasing ~� for any size of the compactextra dimension. In particular, when r1 ! 1, the gyro-magnetic ratio (55) reduces to

g ¼ 6½ð2þ ~�2Þ~r2þ þ ð2� ~�2Þ~r2��2ð2þ ~�2Þ~r2þ þ ð4� ~�2Þ~r2�

; (56)

which is the gyromagnetic ratio of the five-dimensionalslowly rotating charged dilaton black hole with two equalangular momenta [29]. Moreover, in the absence of anontrivial dilaton, ~� ! 0, the gyromagnetic ratio (56)reduces to

g ¼ 3; (57)

which is the gyromagnetic ratio of the asymptotically flatslowly rotating charged black hole with two equal angularmomenta [7].

VI. SUMMARYAND DISCUSSION

To sum up, we have obtained a class of slowly rotatingcharged Kaluza-Klein black hole solutions of Einstein-Maxwell-dilaton theory with arbitrary dilaton couplingconstant in five dimensions. Our investigations are re-stricted to black holes with two equal angular momenta.At infinity, the metric asymptotically approaches a twistedS1 bundle over the four-dimensional Minkowski space-time. Our strategy for obtaining this solution is the pertur-bative method proposed by Horne and Horowitz [22] andsolving the equations of motion up to the linear order of theangular momentum parameter. We have started from thenonrotating charged Kaluza-Klein dilaton black hole solu-tions in five dimensions [38]. Then, we have considered theeffect of adding a small amount of rotation parameter a tothe solution. We have discarded any terms involving a2 orhigher powers in a. Inspection of the known rotating blackhole solutions shows that the only terms in the metricwhich change to OðaÞ are gt� and gtc . Similarly, the

dilaton field does not change to OðaÞ. In the absence ofthe dilaton field (� ¼ 0), our solution reduces to the slowlyrotating charged Kaluza-Klein black hole solution. Wehave calculated the angular momentum J and the gyro-magnetic ratio g which appear up to the linear order of theangular momentum parameter a. Interestingly enough, wefound that the dilaton field and the Kaluza-Klein asymp-totic structure modify the value of the gyromagnetic ratio gthrough the coupling parameter �, which measures thestrength of the dilaton-electromagnetic coupling, and thesquashing parameter r1, which is proportional to the sizeof a compact extra dimension. We have also seen that thegyromagnetic ratio crucially depends on the dilaton cou-pling constant and decreases with increasing � for any sizeof the compact extra dimension.

ACKNOWLEDGMENTS

M.A. would like to thank Jutta Kunz for fruitful dis-cussions and a revision of the paper. K.M. would like tothank Tsuyoshi Houri, Hideki Ishihara, Masashi Kimura,Takeshi Oota, and Yukinori Yasui for fruitful comments.

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