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PHYSICAL REVIEW D 71, 064010 (2005)Magnetic strings in dilaton gravity

M. H. Dehghani1,2,*1Physics Department and Biruni Observatory, Shiraz University, Shiraz 71454, Iran

2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran(Received 5 December 2004; published 11 March 2005)*Electronic

1550-7998=20First, I present two new classes of magnetic rotating solutions in four-dimensional Einstein-Maxwell-dilaton gravity with Liouville-type potential. The first class of solutions yields a four-dimensionalspacetime with a longitudinal magnetic field generated by a static or spinning magnetic string. I findthat these solutions have no curvature singularity and no horizons, but have a conic geometry. In thesespacetimes, when the rotation parameter does not vanish, there exists an electric field, and therefore thespinning string has a net electric charge which is proportional to the rotation parameter. The second classof solutions yields a spacetime with an angular magnetic field. These solutions have no curvaturesingularity, no horizon, and no conical singularity. The net electric charge of the strings in thesespacetimes is proportional to their velocities. Second, I obtain the (n 1)-dimensional rotating solutionsin Einstein-dilaton gravity with Liouville-type potential. I argue that these solutions can presenthorizonless spacetimes with conic singularity, if one chooses the parameters of the solutions suitable. Ialso use the counterterm method and compute the conserved quantities of these spacetimes.

DOI: 10.1103/PhysRevD.71.064010 PACS numbers: 04.20.Jb, 04.40.Nr, 04.50.+hI. INTRODUCTION

It seems likely, at least at sufficiently high energy scales,that gravity is not governed by the Einsteins action, and ismodified by the superstring terms which are scalar tensorin nature. In the low energy limit of the string theory, onerecovers Einstein gravity along with a scalar dilaton fieldwhich is nonminimally coupled to the gravity [1]. Scalar-tensor theories are not new, and it was pioneered by Bransand Dicke [2], who sought to incorporate Machs principleinto gravity.

In this paper, I want to obtain new exact horizonlesssolutions of four-dimensional Einstein-Maxwell-dilatongravity and higher dimensional Einstein-dilaton gravity.There are many papers which are dealing directly withthe issue of spacetimes generated by string source thatare horizonless and have nontrivial external solutions.Static uncharged cylindrically symmetric solutions ofEinstein gravity in four dimensions with vanishing cosmo-logical constant have been considered in [3]. Similar staticsolutions in the context of cosmic string theory have beenfound in [4]. All of these solutions [3,4] are horizonless andhave a conical geometry; they are everywhere flat except atthe location of the line source. An extension to include theelectromagnetic field has also been done [5].Asymptotically anti-de Sitter (AdS) spacetimes generatedby static and spinning magnetic sources in three- and four-dimensional Einstein-Maxwell gravity with negative cos-mological constant have been investigated in [6]. Thegeneralization of these asymptotically AdS magnetic rotat-ing solutions of the Einstein-Maxwell equation to higherdimensions [7] and higher derivative gravity [8] has alsobeen done. In the context of the electromagnetic cosmicaddress: mhd@shirazu.ac.ir

05=71(6)=064010(8)$23.00 064010string, it was shown that there are cosmic strings, known assuperconducting cosmic string, that behave as supercon-ductors and have interesting interactions with astrophysicalmagnetic fields [9]. The properties of these superconduct-ing cosmic strings have been investigated in [10].Superconducting cosmic strings have also been studied inBrans-Dicke theory [11], and in dilaton gravity [12].

On the other side, some efforts have been done to con-struct exact solutions of Einstein-Maxwell-dilaton gravity.Exact charged dilaton black hole solutions in the absenceof dilaton potential have been constructed by many authors[13,14]. In the presence of Liouville potential, staticcharged black hole solutions have also been discoveredwith a positive constant curvature event horizon [15], andzero or negative constant curvature horizons [16]. Theseexact solutions are all static. Until now, charged rotatingdilaton solutions for an arbitrary coupling constant havenot been constructed in four or higher dimensions. Indeed,exact magnetic rotating solutions have been considered inthree dimensions [17], while exact rotating black holesolutions in four dimensions have been obtained only forsome limited values of the coupling constant [18]. Forgeneral dilaton coupling, the properties of rotating chargeddilaton black holes only with infinitesimally small angularmomentum [19] or small charge [20] have been investi-gated, while for arbitrary values of angular momentum andcharge only a numerical investigation has been done [21].My aim here is to construct exact rotating charged dilatonsolutions for an arbitrary value of the coupling constant.

The outline of the paper is as follows. Section II isdevoted to a brief review of the field equations and generalformalism of calculating the conserved quantities. In Sec.III, I obtain the four-dimensional rotating dilaton stringswhich produce longitudinal and angular magnetic fields,and investigate their properties. In Sec. IV, I present the-1 2005 The American Physical Society

M. H. DEHGHANI PHYSICAL REVIEW D 71, 064010 (2005)n 1-dimensional rotating dilaton solutions with conicsingularity and compute their conserved charges. I finishthe paper with some concluding remarks.II. FIELD EQUATIONS AND CONSERVEDQUANTITIES

The action of dilaton Einstein-Maxwell gravity with onescalar field with Liouville-type potential in n 1dimensions is [15]

IG 1

16

ZM

dn1xgp R 4

n 1 r2

2e2 e4=n1FF

18

Z@M

dnxp ; (1)

where R is the Ricci scalar, is a constant determining thestrength of coupling of the scalar and electromagneticfields, F @A @A is the electromagnetic tensorfield, and A is the vector potential. The last term in Eq. (1)is the Gibbons-Hawking boundary term. The manifold Mhas metric g and covariant derivative r. is the traceof the extrinsic curvature of any boundary(ies) @M ofthe manifold M, with induced metric(s) ij. One may referto as the cosmological constant, since in the absence ofthe dilaton field ( 0) the action (1) reduces to theaction of Einstein-Maxwell gravity with cosmologicalconstant.

The field equations in n 1 dimensions are obtainedby varying the action (1) with respect to the dynamicalvariables A, g, and :

@gp e4=n1F 0; (2)

R 2

n12@@e2g

2e4=n1FF

1

2n1FFg

;

(3)

r2 n 12

e2 2e4=n1FF: (4)

The conserved mass and angular momentum of the solu-tions of the above field equations can be calculated throughthe use of the subtraction method of Brown and York [22].Such a procedure causes the resulting physical quantities todepend on the choice of reference background. For asymp-totically (A)dS solutions, the way that one deals with thesedivergences is through the use of the counterterm methodinspired by (A)dS/CFT correspondence [23]. However, inthe presence of a nontrivial dilaton field, the spacetimemay not behave as either dS (> 0) or AdS (< 0). Infact, it has been shown that, with the exception of a pure064010cosmological constant potential, where 0, no AdS ordS static spherically symmetric solution exists forLiouville-type potential [24]. But, as in the case of asymp-totically AdS spacetimes, according to the domain-wall/QFT (quantum field theory) correspondence [25], theremay be a suitable counterterm for the stress-energy tensorwhich removes the divergences. In this paper, I deal withthe spacetimes with zero curvature boundary [Rabcd 0], and therefore the counterterm for the stress-energytensor should be proportional to ab. Thus, the finitestress-energy tensor in n 1 dimensions may be writtenas

Tab 18

ab ab n 1

leffab

; (5)

where leff is given by

l2eff 4nn 1 n 132

8e2: (6)

As goes to zero, the effective l2eff of Eq. (6) reduces tol2 nn 1=2 of the (A)dS spacetimes. The first twoterms in Eq. (5) are 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 Arnowitt-Deser-Misner form:

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 & on the boundary, thenthe quasilocal conserved quantities associated with thestress tensors of Eq. (5) can be written as

Q & ZBdn1

pTabn

a&b; (7)

where is the determinant of the metric ij, and & and na

are the Killing vector field and the unit normal vector onthe boundary B. For boundaries with timelike (& @=@t),rotational (& @=@), and translational Killing vectorfields (( @=@x), one obtains the quasilocal mass, angu-lar, and linear momenta

M ZBdn1

pTabna&b; (8)

J ZBdn1

pTabna&b; (9)-2

MAGNETIC STRINGS IN DILATON GRAVITY PHYSICAL REVIEW D 71, 064010 (2005)P ZBdn1

pTabn

a(b; (10)

provided the surface B contains the orbits of &. Thesequantities are, respectively, the conserved mass, angular,and linear momenta of the system enclosed by the bound-ary B. Note that