Ž .Physics Letters B 468 1999 201–207

Spinning charged solutionsin 2q1 dimensional Einstein–Maxwell–dilaton gravity

Sharmanthie Fernando 1

Physics Department, UniÕersity of Cincinnati, Cincinnati, OH 45221, USA

Received 28 September 1999; accepted 25 October 1999Editor: L. Alvarez-Gaume

Abstract

We report a new class of rotating charged solutions in 2q1 dimensions. These solutions are obtained for Einstein–Maxwell gravity coupled to a dilaton field in which the electromagnetic fields are selfdual. The mass and the angularmomentum of the solutions computed at spatial infinity are finite. The class of solutions considered here have nakedsingularities and are asymptotically flat. q 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords: 2q1 dimensions; Dilaton; Selfdual

1. Introduction

Interest in 2q1 dimensional gravity has beenheighten by the discovery of a black hole solution by

w x Ž .Banados et al. 1 . This black hole, named BTZ has˜anti-de Sitter structure locally and globally differ toanti-de Sitter by identifications done with a discretesubgroup of the isometry group of anti-de Sitter

Ž . w xspace, SO 2,2 2 . It enjoys many black hole proper-ties of its counterparts in higher dimensions whichmakes BTZ a suitable model to understand blackhole physics in a technically simpler setting.

Extension of the BTZ black hole with charge havebeen met with mixed success. The first investigationinto static charged black holes was done by Banados˜

1 E-mail: fernando@physung.phy.uc.edu

w xet al. 1 . Due to the logarithmic nature of theelectromagnetic potential, these solutions give rise to

w xunphysical properties 3 . The horizonless static solu-tion with magnetic charge were studied by Hirsh-

w xmann et al. 3 , and the persistence of these unphysi-w xcal properties was highlighted by Chan 4 . Kamata

w xet al. 5 presented a rotating charged black hole withŽ .self anti-self duality imposed on the electromag-

netic fields. The resulting solutions were asymptoticto an extreme BTZ black hole solution but had

w xdiverging mass and angular momentum 4 . Clement´w x w x6 , Fernando and Mansouri 7 , introduced aChern–Simons term as a regulator to screen theelectromagnetic potential and obtained horizonlesscharged solutions.

In this work, we couple a dilaton to Einstein–Maxwell gravity to obtain rotating charged solutionswith finite mass and finite angular momentum. It is

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 01245-9

well known that the introduction of a dilaton fielddrastically changes the space-time structure in 3q1dimensions. Furthermore, studying dilaton gravity isimportant since it arises in low energy string theory.Therefore it is worthwhile to see how the presence ofthe dilaton would modify the BTZ black hole solu-tions and how it would help in curing the diver-gences occurred in the previous charged solutions.

Einstein–Maxwell–dilaton action in 2q1 dimen-sions is written as follows:

g3 bf m< <(Is d x g Ry2 Le y , f , fŽ .H m2

y4 af 2ye F 1Ž .

Here L is the cosmological constant and we con-sider the case for L-0 which corresponds to anti-deSitter spaces. b ,g ,a are coupling constant. In thisexpression 1r16p G is taken to be 1. The aboveaction is the low energy string action when gs8,bs4, as1. There are several work related to

w xcharged solutions to the above action. Chan et al. 8have studied a one parameter family of static chargeddilaton black holes which were non-asymptoticallyanti-de sitter and solutions with cosmological hori-

w xzons when 4asb. Park et al. 9 obtained axiallysymmetric static solutions by using a dimensionalreduction method. A more general magneticallycharged solution to dilaton gravity was found by

w xKoikawa et al. 10 . By applying T-duality to thew xstatic electric charged black holes of Chan et al. 14 ,

w xChen 11 obtained rotating charged black hole solu-tions to Einstein–Maxwell–dilaton gravity. How-ever, these solutions are not the most general rotatingcharged solutions to Einstein–Maxwell–dilaton grav-ity.

The aim of the present paper is to search forrotating charged solutions to Einstein–Maxwell–di-laton gravity with self duality imposed on the elec-tromagnetic fields. The plan of the paper is follows.In Section 2 we will compute the most generalsolutions with self duality imposed. In Section 3 wewill impose restrictions on coupling constants so thatthe mass and angular momentum are finite. In Sec-tion 4 we will discuss the properties of theses solu-tions and finally give concluding remarks.

2. General solutions

By extremizing the Lagrangian of the action inŽ .Eq. 1 with respect to the metric g , the electro-mn

magnetic potential A and the dilaton field f, onem

obtains the corresponding field equations for gravita-tional, electromagnetic and dilaton respectively asfollows:

R sy2 Lg e bfmn mn

qey4 af 2 F F r yg F F lsŽ .mr n mn ls

gq , f , f 2Ž . Ž .Ž .m n2

, ey4 afF mn s0 3Ž . Ž .m

gm y4af 2 bf, , fq2ae F ybe Ls0 4Ž .m2

Let us assume that the three dimensional space-timeis stationary and circularly symmetric, having two

E Ecommuting Killing vectors, and . Such a space-Ef E t

time could be parameterized with a line element asfollows:

22 2 2 y2 2 2 uds syN dt qL dr qK duqN dt 5Ž . Ž .The functions N, L, K and Nu depends only on radialcoordinate r. We use the tetrad formalism and Car-tan structure equations to look for solutions. Theobvious non coordinate basis for the above metricwould be

e0 sNdt ; e1 sK duqNudt ; e2 sLy1drŽ .6Ž .

The indices a,bs0,1,2 are for the orthonormal ba-sis and m,ns0,1,2 for the coordinate basis withx 0 s t; x1 su ; x 2 sr. The non-vanishing compo-nents of the electromagnetic field tensor in the coor-dinate basis are given by F sE, F sB, and in thetr ru

ˆorthonormal basis they are given by F sE, F s02 21

B. They are related by

ˆ ˆ u ˆENyBKN BKŽ .Es ; Bs 7Ž .

L L

Ž .From electromagnetic field Eq. 3 ,

C e4af C e4af1 2tr ruF s ; F s 8Ž .< < < <( (g g

In order to seek solutions to the gravitational fieldequations, we make the ansatz that, the electric andthe magnetic fields are self dual in the orthonormalbasis. Therefore,

ˆ ˆEsBsu r 9Ž . Ž .Ž . Ž .With the use of Eqs. 7 – 9 ,

w xexp 4af N C2uu r syC ; N s y 10Ž . Ž .1 K K C1

ˆConsidering the behavior of usE for flat space-times, we can represent C with electric charge Q .1 e

A value for C will be assigned later.2

Having used the ansatz of self duality, the gravita-tional field equations in the orthonormal basis takesthe following form for a circularly symmetric spacetime:

X 2XX X X X X uN N K LL N KLN2R sL q q y200 ž / ž /N NK N 2 N

sy2 Le bf q2ey4 af u2 11Ž .X 2XX X X X X uK N K LL K KLN

2R syL q y y211 ž / ž /K NK K 2 N

s2 Le bf q2ey4 af u2 12Ž .X X3 uL K LN

y4 af 2R sy sy2e u 13Ž .01 2 ž /2 NK

X 2X X uN K KLN2G sL q22 ž / ž /NK 2 N

g 2bf ˆsyLe q , f 14Ž .Ž .24

Ž . Ž .The gravitational field Eqs. 11 – 14 , the dilatonŽ . Ž .field Eq. 4 and the conditions 10 leads to the final

equations to be solved as follows:

XXX bfLX sy4Le 15Ž . Ž .L

X2 2 4afL HK s2Q e 16Ž . Ž .e

X 2X L g 2X bfs f L yLe 17Ž . Ž .ž /2 X 4

XX bfg fLX s2 Lbe 18Ž . Ž .L

where

L N X K X

XsNK ; Hs y 19Ž .ž /2 N K

Ž .Here, the prime is given by drdr. Eq. 15 isobtained by observing that R q2 Le bf sR y00 11

bf Ž .2 Le . Eq. 16 is obtained by the fact that R q00Ž . Ž .R sy2 R . Eq. 17 is just Eq. 14 rewritten. Eq.11 01

Ž .18 is the dilaton field equations with the selfduality imposed. The solution to the above set ofequations are

bfexp ž /y2gf 2

Xsa exp ; Ls ;X1 ž /b b f0

y2gf 2b a ybf0 32K sa exp y exp2 ž /ž / 2g bb 2yž /b 2

4Q2 b2e 0

yb 2g

4ay 4aybqž / ž /2 b

=exp 4ayb f 20Ž . Ž .Ž .

Ž 2 2 .2 4g ygbHere b s and the above solutions are valid02Ž < < .4 L b

b g22 Ž . Ž .only when b )0, 4ay /0, 4aybq /0,0 2 b

and b/0. Here a ,a ,a are integrating constants.1 2 3

From the above equations it is clear that the allthe functions N, L, K , Nu depend on the dilaton fieldf. In order to compare these solutions with thepreviously obtained dilaton solutions and also to seethe correspondence with the BTZ black hole we pickL to be the following:

v1 2 22< <L r yrŽ .0Ls 21Ž .

r

Here, v and r 2 are constants which would be0

determined by imposing some restrictions on thesolutions to give finite mass and finite angular mo-

Ž .mentum. Note that Eq. 21 is equivalent to thedilaton field f and X being

Ž .2 vy1

b2 2fs ln b r yr ;Ž .1 0ž /Ž .4g 1yv

b 21 2 22< <Xs L r yr 22Ž .Ž .0

where

y2rbybb s1 1r2< <4b yvq1 LŽ .0

gy2 1r2b < <and a is normalized so that a b s L . By1 1 1

Ž .substituting f into Eq. 20 ,

Ž .4g yvq1

b 2 Ž .yvq12 2 2 2 2K sa r yr qa r yrŽ . Ž .2 0 3 0

Ž . Ž .2 4ayb vy1

b2 2 2qQ r yr 23Ž .Ž .e 0

where

y2b bŽyb r2.a0 1 3y2g r ba sa b ; a s ;2 2 1 3 2g byž /b 2

y4Q2 b2 b4aybe 0 12Q se b 2g

4ay 4aybqž / ž /2 b

We restrict v/1. Otherwise the dilaton would be aconstant and the solutions would be trivial.

3. Quasilocal mass and angular momentum

In this section, we will impose restrictions on thevalues of a ,b ,g ,v so that the above solutions havefinite mass and finite angular momentum. Also wewill interpret the integrating constants a ,a ,a as1 2 3

appropriate physical constants. Here we will adoptw xthe prescription of Brown et al. 12,13 in computing

mass and angular momentum.

3.1. Quasilocal angular momentum

The quasilocal angular momentum according tow xthe prescription given by Brown et al. 12,13 is

K 3NuX

LJ r s 24Ž . Ž .

N

Ž .For the metric in Eq. 20 it is equivalent to

J r sy2 HK 2 25Ž . Ž .Ž . Ž .From Eqs. 19 and 23 ,

4a1 12 22 2< < < <K Hsy L a y2 L Q y13 e ž /b

=

Ž . Ž .8 ayb vy1

b2 2vy1 r yr 26Ž . Ž .Ž .0

Ž .Hence for large r, J r becomes finite only if

8avy1 y1 F0 27Ž . Ž .ž /b

Ž .Since v/1 and 8ayb /0, the present class ofsolutions will obey strictly smaller condition for Eq.Ž .27 . Hence,

12< <lim J r sJs2 L a 28Ž . Ž .3

r™`

3.2. Quasilocal mass

w xFrom the definition of Brown et al. 12,13 , thequasilocal mass is given by

uM r s2 N r L r yL r yJ r N rŽ . Ž . Ž . Ž . Ž . Ž .0

29Ž .

Ž .Here, L r is chosen to be the reference when there0

is zero mass. In comparison with the BTZ black holewe choose

r 2 v12< <L s L s L 30Ž . Ž .r s00 0

r

Ž .Consider the first term in Eq. 29 ,12< <2 L X

Ž .2 2 vy32 N L yL s vr rŽ . Ž0 0K

qlower order terms of r 31Ž ..

Ž . nFor finite mass, XrK should behave as r withŽ . Ž .nFy 2vy3 . If n-y 2vy3 , then this solu-

tion will correspond to the ‘‘massless’’ BTZ solutionfor appropriate limits. Hence to avoid such extreme

Ž .cases, we would pick n to be y2vq3 . Therefore,since at large r

Ž .8g yvq1

b 2X™r 32Ž .K should behave as

Ž .8g yvq1Ž ..Ž q 2 vy3K™r 33Ž .b 2

Ž .for large r for ns y2vq3 . Now, we will con-sider the Nu term at large r:

X C2uN s y 34Ž .2 CK 1

u Ž . 2To impose the boundary condition N ` s0, XrKhas to converge for r™`. With the constraint in

Ž .Eq. 33 it means

8gyvq1 y q4 q2F0 35Ž . Ž .2ž /b

To approximate the behavior of K 2 to be r 2 at largeand small r, we assume all terms in K 2 has positive

2 2Žpowers of r . Considering the fact that the a r y32 .Žyvq1. 2r term in K is proportional to the angular0

momentum, to take appropriate limits, we take22 2 4g Žyvq1.r bŽ .a r yr term to be the dominant2 0

power of K 2. Without loss of generality we assumea s1. Hence,2

4g yvq1Ž .) yvq1 )0;Ž .2b

2 4ayb vy1Ž . Ž .)0 36Ž .

b

Ž . Ž .and from Eq. 33 and assumptions in Eq. 36 , vŽ 2 .4g y3bs . With all these preliminaries,

2Ž < < .4 L b

1u 2< <lim N r s L yC rC 37Ž . Ž .Ž .2 1

r™`

1u 2Ž . < <To impose N ` s 0, we let C s L C . Hence,2 1

< < 2lim M r sMs2 L vr 38Ž . Ž .0r™`

M2leading to r s .< <0 2 L v

4. Exact solutions with finite mass and finiteangular momentum

With the above preliminaries, the final form ofthe solutions with finite mass and finite angularmomentum is

v1 2 22< <L r yrŽ .0Ls ;

r

1 Ž .y2 vq32 22< <L r yr NŽ . 10 u 2< <Ns ; N s y L ;K K

Ž . Ž .y2 vq3 yvq12 2 2 2 2K s r yr qa r yrŽ . Ž .0 3 0

Ž .Ž .2 4ayb vy1

b2 2 2qQ r yr 39Ž .Ž .e 0

Ž .2 vy1

b2 2fs ln b r yr 40Ž .Ž .1 0ž /With

4gy3b 2 MŽ .2vs ; r s ;02 < <2 L v4gy2bŽ .

Ja s 13

2< <2 L

The electromagnetic potential is

m2 2r yrŽ . 10m 2< <AsA dx s L dtqdu . 41Ž .Ž .m 2m

Ž .b 8 a y bwhere ms . From the constraint imposed2Ž .2 b y2g

Ž .in Eq. 27 , it is obvious that m-0. Hence, thepotential is finite at large r. Therefore, the presenceof the dilaton ‘‘screens’’ the electromagnetic poten-tial. To clarify this further, if we look at dilaton fieldin flat space for self-dual case as considered above,

ˆ ˆ w xthe values for E and B would be Q exp 4a b rr.e 04a b0 Ž .Hence the potential A sQ r r 4a b is finitet e 0

for 4a b -0. Therefore the presence of the dilaton0

modifies the Coulomb force in 2q1 dimensions.w xWe may recall that in Refs. 6,7 , a topological mass

term m e abgA F was introduced to Einstein–p a bg

Maxwell gravity to cure the divergences of the

quasilocal mass M which gives a similar effect tothe Coulomb force. If we compare the electric fieldin three separate cases,

Q Q r 4a b0e e

E s ; E s ;Coulomb dilatonr r

Q eym p re

E s 42Ž .topologicalr

the topological mass term has a better regulatingeffect in comparison with the dilaton field.

4.1. Causal structure

The curvature scalars in 2q1 dimensions are R,R Rab and det R rdet g . For the above solu-ab a b a b

tions, with g ,b ,a/0 and for M)0, all of themdiverge at rsr and finite everywhere else. Hence0

the curvature singularity at rsr is a naked0

singularity without horizons. However, K 2 whichis the g term in the metric has to be positiveuu

to avoid closed time like curves since u is aperiodic coordinate. Even with the constraints im-posed on the parameters a , b , g , there is a possibil-ity that K 2 would become negative. Hence one hasto include the possibility of closed time like curvesfor these solutions. The scalar curvature R is

Ž .v y12 2g 2Ž .Ž . Ž .6Lq r yr . Since 1yv )0, R™0

24b 0

0 for large r. Therefore the solution becomes flatasymptotically.

4.2. The relation with the BTZ black hole

Note that above discussion is for when g , b anda is non vanishing. To see the correspondence of theabove solutions with the BTZ black hole, let us takethe limit v™1 and Q ™0. Then,e

2 2 2 2< <M™2 L r ; f™0; K ™ r yr qaŽ .0 0 3

43Ž .12 2< <If r sa then Ms L J with the following met-0 3

ric:

M1 22< <L r y 1ž /2< <2 LLsN™ ;

r

JuK™r ; N ™ 44Ž .22 r

Hence the solution obtained in this work approachesan extreme BTZ black hole as a special case. Thepresence of the dilaton and charge have left the BTZspace-time horizonless and asymptotically flat.

5. Conclusions

We have obtained a family of rotating chargedsolutions to Einstein–Maxwell–dilaton gravity in 2q1 dimensions. Here we have imposed self dualityon the electric and magnetic field to facilitate solvethe field equations exactly. With certain constraintson the coupling constants g , b and a , we obtainedsolutions with finite mass and finite angular momen-tum. For non-zero values of a b and g , the class ofsolutions considered in this paper are horizonless,have naked singularities and are asymptotically flat.These solutions approaches the extreme BTZ blackhole solutions as a special case for a ,b ,g™0. Thepresence of the dilaton ‘‘screens’’ the electromag-netic potential and modifies the structure of thespace-time considerably. However, since the metric

Ž .depends on the dilaton as it is clear from Eq. 20 ,one may use other polynomial functions for thedilaton field in these solutions to understand how thespace-time structure changes accordingly. It is also aquestion how a massive dilaton would effect thespace-time structure of the above solutions. In ex-tending this work it may be possible to include a

Ž . b1f b 2 fpotential of the form V f s2 L e q2 L e1 2

to the action considered in this paper. These kind ofpotentials are investigated in dimensions nG4 andhave shown the possibility of having three horizons

w xby Chan et al. 14 . It would be interesting to seewhether one can construct rotating charged dilatonblack holes with the above potential in 2q1 dimen-sions.

w xWe may recall that Chen 11 obtained rotatingcharged black hole solutions for dilaton gravity withg s 4 and 4asb by a T-duality transformation on

w xthe static charged black holes of Chan et al. 8 . Thesolutions obtained in this paper are for more generalvalues of a , b , g and they are not T-dual to the

w xsolutions of Chen 11 . Furthermore instead ofMaxwell fields, one can include Yang-Mills fields to

study the space-time structure for dilaton gravity in2q1 dimensions which we hope to report else-where.

Acknowledgements

I wish to thank F. Mansouri and M. Muhkerjeefor helpful comments. This work was supported inpart by the Department of Energy under contractnumber DOE-FG02-84ER40153.

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