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PHYSICAL REVIEW D 71, 044008 (2005)Charged rotating dilaton black strings

M. H. Dehghani1,2,* and N. Farhangkhah11Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran

2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran(Received 13 December 2004; published 8 February 2005)*Electronic

1550-7998=20In this paper we, first, present a class of charged rotating solutions in four-dimensional Einstein-Maxwell-dilaton gravity with zero and Liouville-type potentials. We find that these solutions can present ablack hole/string with two regular horizons, an extreme black hole or a naked singularity provided theparameters of the solutions are chosen suitable. We also compute the conserved and thermodynamicquantities, and show that they satisfy the first law of thermodynamics. Second, we obtain the (n 1)-dimensional rotating solutions in Einstein-dilaton gravity with Liouville-type potential. We find that thesesolutions can present black branes, naked singularities or spacetimes with cosmological horizon if onechooses the parameters of the solutions correctly. Again, we find that the thermodynamic quantities ofthese solutions satisfy the first law of thermodynamics.

DOI: 10.1103/PhysRevD.71.044008 PACS numbers: 04.20.Jb, 04.40.Nr, 04.50.+h, 04.70.BwI. INTRODUCTION

The Einstein equation without a cosmological constanthas asymptotically flat black hole solutions with eventhorizon being a positive constant curvature n-sphere.However, for the Einstein equation with positive or nega-tive cosmological constant [1,2] or Einstein-Gauss-Bonnetequation with or without cosmological constant [3], onecan have also asymptotically anti de Sitter (AdS) or deSitter (dS) black hole solutions with horizons being zero ornegative constant curvature hypersurfaces. These blackhole solutions, whose horizons are not n-sphere, are oftenreferred to as the topological black holes in the literature.Various properties of these topological black holes havebeen investigated in recent years. For example, the ther-modynamics of rotating charged solutions of the Einstein-Maxwell equation with a negative cosmological constantwith zero curvature horizons in various dimensions havebeen studied in Ref. [4], while the thermodynamics ofthese kind of solutions in Gauss-Bonnet gravity havebeen investigated in [5].

All of the above mentioned black holes are the solutionsof field equations in the presence of a long-range gravita-tional tensor field g and a long-range electromagneticvector field A. It is natural then to suppose the existenceof a long-range scalar field too. This leads us to the scalar-tensor theories of gravity, where, there exist one or severallong-range scalar fields. Scalar-tensor theories are not new,and it was pioneered by Brans and Dicke [6], who sought toincorporate Machs principle into gravity. Also in the con-text of string theory, the action of gravity is given by theEinstein action along with a scalar dilaton field which isnon minimally coupled to the gravity [7]. The action of(n 1)-dimensional dilaton Einstein-Maxwell gravitywith one scalar field and potential V can be writtenas [8]address: mhd@shirazu.ac.ir

05=71(4)=044008(7)$23.00 044008IG1

16

ZM

dn1xgp R 4

n1r2V

e4n1FF 18

Z@M

dnxp ; (1)

where R is the Ricci scalar, is a constant determining thestrength of coupling of the scalar and electromagnetic field,F @A @A is the electromagnetic tensor fieldand A is the vector potential. The last term in Eq. (1) is theGibbons-Hawking boundary term. The manifold M hasmetric g and covariant derivative r. is the trace ofthe extrinsic curvature of any boundary(ies) @M ofthe manifold M, with induced metric(s) ij. Exact chargeddilaton black hole solutions of the action (1) in the absenceof a dilaton potential [V 0] have been constructed bymany authors [9,10]. The dilaton changes the casual struc-ture of the spacetime and leads to curvature singularities atfinite radii. In the presence of Liouville-type potential[V 2 exp2], static charged black hole solu-tions have also been discovered with a positive constantcurvature event horizons and zero or negative constantcurvature horizons [8,11]. Recently, the properties of theseblack hole solutions which are not asymptotically AdS ordS, have been studied [12].

These exact solutions are all static. Recently, one of ushas constructed two classes of magnetic rotating solutionsin four-dimensional Einstein-Maxwell-dilaton gravity withLiouville-type potential [13]. These solutions are not blackholes, and present spacetimes with conic singularities. Tillnow, charged rotating dilaton black hole solutions for anarbitrary coupling constant have not been constructed.Indeed, exact rotating black hole solutions have been ob-tained only for some limited values of the coupling con-stant [14]. For general dilaton coupling, the properties ofcharged dilaton black holes only with infinitesimally smallangular momentum [15] or small charge [16] have beeninvestigated. Our aim here is to construct exact rotating-1 2005 The American Physical Society

M. H. DEHGHANI AND N. FARHANGKHAH PHYSICAL REVIEW D 71, 044008 (2005)charged dilaton black holes for an arbitrary value of cou-pling constant and investigate their properties.

The outline of our paper is as follows. In Sec. II, weobtain the four-dimensional charged rotating dilaton blackholes/strings, which are not asymptotically flat, AdS or dS,and show that the thermodynamic quantities of these blackstrings satisfy the first law of thermodynamics. Section IIIis devoted to a brief review of the general formalism ofcalculating the conserved quantities, and investigation ofthe first law of thermodynamics for the charged rotatingblack string. In Sec. IV, we construct the (n 1)-dimensional rotating dilaton black branes, and investigatetheir properties. We finish our paper with some concludingremarks.II. ROTATING CHARGED DILATON BLACKSTRINGS

The field equation of (n 1)-dimensional Einstein-Maxwell-dilaton gravity in the presence of one scalar field with the potential V can be written as:

@gp e4n1F 0; (2)

R 4

n 1rr

1

4Vg

2e4n1FF

1

2n 1FFg

;

(3)

r2 n 18

@V@

2e4n1FF: (4)

In this section we want to obtain the four-dimensionalcharged rotating black hole solutions of the field Eqs. (2)(4) with cylindrical or toroidal horizons. The metric ofsuch a spacetime with cylindrical symmetry can be writtenas

ds2 frdt ad2 r2R2ra

l2dtd

2

dr2

fr r2

l2R2rdz2;

2 1 a2

l2; (5)

where the constants a and l have dimensions of length andas we will see later, a is the rotation parameter and l isrelated to the cosmological constant for the case ofLiouville-type potential with constant . In metric (5),the ranges of the time and radial coordinates are 1 1. Thisspacetime presents a naked singularity with a regular cos-mological horizon at

rc 1 2q2

CV0;

provided C> 0, and no cosmological horizon for C< 0.

B. Solutions with Liouville-type potentials

Now we consider the solutions of Eqs. (8)-(11) with aLiouville-type potential V 2 exp2. One mayrefer to as the cosmological constant, since in theabsence of the dilaton field the action (1) reduces tothe action of Einstein-Maxwell gravity with cosmologicalconstant. The only case that we find exact solutions for anarbitrary values of with Rr and r given in Eqs. (6)and (12) is when . It is easy then to obtain thefunction fr as

fr rV01 22

2 3 r21 m

r 1

2q2V0r2

:

(13)

In the absence of a nontrivial dilaton ( 0 ), thesolution reduces to the asymptotically AdS and dS chargedrotating black string for 3=l2 and 3=l2,respectively [2,4]. As one can see from Eq. (13), thereis no solution for 3p with a Liouville potential ( 0). In order to investigate the casual structure of the space-time, we consider it for different ranges of separately.

For >3

p, as r goes to infinity the dominant term is

the second term, and therefore the spacetime has a cosmo-logical horizon for positive values of the mass parameter,despite the sign of the cosmological constant .

For 0, while there is no cosmological horizonsif < 0 . Indeed, in the latter case (

3

pdespite the sign of , and (ii) positive

despite the value of . As we argued above in these twocases we encounter with cosmological horizons, and there-fore the cosmological horizons have negative temperature.Numerical calculations shows that the temperature of theevent horizon goes to zero as the black hole approaches theextreme case. Thus, one can see from Eq. (14) that thereexist extreme black holes only for negative and 0) orAdS (< 0). In fact, it has been shown that with theexception of a pure cosmological constant potential, where 0, no AdS or dS static spherically symmetric solutionexist for Liouville-type potential [19]. But, as in the case ofasymptotically AdS spacetimes, according to the domain-wall/QFT (quantum field theory) correspondence [20],there may be a suitable counterterm for the stress-energytensor which removes the divergences. In this paper, wedeal with the spacetimes with zero curvature boundary[Rabcd 0], and therefore the counterterm for thestress-energy tensor should be proportional to ab. Thus,the finite stress-energy tensor in n 1-dimensionalEinstein-dilaton gravity with Liouville-type potentialmay be written as

Tab 18

ab ab n 1

leffab

; (18)

where leff is given by

l2eff n 132 4nn 1

8e2: (19)044008As goes to zero, the effective leff of Eq. (19) reduces tol nn 1=2 of the (A)dS spacetimes. The first twoterms in Eq. (18) is the variation of the action (1) withrespect to ab, and the last term is the counterterm whichremoves the divergences. One may note that the counter-term has the same form as in the case of asymptoticallyAdS solutions with zero curvature boundary, where l isreplaced by leff . To compute the conserved charges of thespacetime, one should choose a spacelike surface B in @Mwith metric ij, and write the boundary metric in ADMform:

abdxadxa N2dt2 ijdi Vidtdj Vjdt;where the coordinates i are the angular variables parame-trizing the hypersurface of constant r around the origin,and N and Vi are the lapse and shift functions, respectively.When there is a Killing vector field 1 on the boundary, thenthe quasilocal conserved quantities associated with thestress tensors of Eq. (18) can be written as

Q 1 ZBdn1

pTabn

a1b; (20)

where is the determinant of the metric ij, 1 and na arethe Killing vector field and the unit normal vector on theboundary B . For boundaries with timelike (1 @=@t) androtational (& @=@) Killing vector fields, one obtains thequasilocal mass and angular momentum

M ZBdn1

pTabna1b; (21)

J ZBdn1

pTabn

a&b; (22)

provided the surface B contains the orbits of &. Thesequantities are, respectively, the conserved mass, angularand linear momenta of the system enclosed by the bound-ary B. Note that they will both be dependent on thelocation of the boundary B in the spacetime, althougheach is independent of the particular choice of foliationB within the surface @M.

The mass and angular momentum per unit length of thestring when the boundary B goes to infinity can be calcu-lated through the use of Eqs. (21) and (22),

M V08l

3 22 2 11 2

m;

J 3 2V0

8l1 2 ma:(23)

For a 0 ( 1), the angular momentum per unit lengthvanishes, and therefore a is the rotational parameter of thespacetime. Note that Eq. (23) is valid only for 0,then the spacetime has a cosmological horizon with radiusgiven in Eq. (34) for positive values of . The Hawkingtemperature of the event or cosmological horizon is:

Th V02

r21h ;-5

M. H. DEHGHANI AND N. FARHANGKHAH PHYSICAL REVIEW D 71, 044008 (2005)which is positive for event horizon (< 0) and negativefor cosmological horizon (> 0).

B. Rotating solutions with all the rotation parameters

The rotation group in n 1-dimensions is SOn andtherefore the number of independent rotation parametersfor a localized object is equal to the number of Casimiroperators, which is n=2 k, where n=2 is the integerpart of n=2. The generalization of the metric (27) with allrotation parameters is

ds2fr

dt

Xki1

aid

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