nonlinear electromagnetic field is presented. These solutions have no curvature singularity and no horizon, but have a conic geometry. In thesespacetimes, when the rotation parameter does not vanish, there exists an electric field, and therefore the spinning string has a net electric chargewhich is proportional to the rotation parameter. Although the asymptotic behavior of these solutions are neither flat nor (A)dS, we calculatethe conserved quantities of these solutions by using the counterterm method. We also generalize these four-dimensional solutions to the case of(n + 1)-dimensional rotating solutions with k [n/2] rotation parameters, and calculate the conserved quantities and electric charge of them. 2007 Elsevier B.V. All rights reserved.
The BornInfeld  type of generalizations of Abelian andnon-Abelian gauge theories have received a lot of interest inrecent years. This is due to the fact that such generalizationsappear naturally in the context of the superstring theory .The nonlinearity of the electromagnetic field brings remarkableproperties to avoid the black hole singularity problem whichmay contradict the strong version of the Penrose cosmic cen-sorship conjecture in some cases. Actually a new nonlinearelectromagnetism was proposed, which produces a nonsingularexact black hole solution satisfying the weak energy condition, and has distinct properties from Bardeen black holes .The BornInfeld action including a dilaton and an axion field,appears in the couplings of an open superstring and an Abeliangauge field. This action, describing a BornInfeld-dilaton-axionsystem coupled to Einstein gravity, can be considered as anonlinear extension of the Abelian field of EinsteinMaxwell-dilaton-axion gravity. Exact static solutions of EinsteinBorn
* Corresponding author.E-mail addresses: firstname.lastname@example.org (M.H. Dehghani),
email@example.com (A. Sheykhi), firstname.lastname@example.org (S.H. Hendi).
Infeld (EBI) gravity in arbitrary dimensions with positive, zeroor negative constant curvature horizons have been constructed. Rotating solutions of Einstein (GaussBonnet)BornInfeld in various dimensions with flat horizons have also beenobtained [8,9]. When a dilaton field is coupled to gravity, ithas profound consequences for the black hole/string solutions.Many attempts have been done to construct exact solutionsof EinsteinMaxwell-dilaton (EMd) and EinsteinBornInfeld-dilaton (EBId) gravity. While exact static dilaton black holesolutions of EMd gravity have been constructed in ,exact rotating black holes solutions with curved horizons havebeen obtained only for some limited values of the coupling con-stant. For general dilaton coupling, the properties ofrotating charged dilaton black holes only with infinitesimallysmall charge  or small angular momentum have been inves-tigated . When the horizons are flat, rotating solutionsof EMd gravity with Liouville-type potential in four  and(n + 1)-dimensions have been constructed . The studieson the black hole solutions of EBId gravity in three and fourdimensions have been carried out in  and , respec-tively. Thermodynamics of (n+1)-dimensional EBId solutionswith flat  and curved horizons have also been explored .The appearance of dilaton changes the asymptotic behavior ofPhysics Letters B 659
Magnetic strings in Einstein
M.H. Dehghani a,b, A. Sa Physics Department and Biruni Observ
b Research Institute for Astrophysics and Asc Department of Physics, Shahid Bahonar U
d Department of Physic, College of ScieReceived 1 October 2007; received in revised fo
A class of spinning magnetic string in 4-dimensional Einstein-dila0370-2693/$ see front matter 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.11.0158) 476482www.elsevier.com/locate/physletb
eykhi c,, S.H. Hendi d
y, Shiraz University, Shiraz 71454, Iranomy of Maragha (RIAAM), Maragha, Iranersity, P.O. Box 76175-132, Kerman, Iran
, Yasouj University, Yasouj 75914, Iran5 November 2007; accepted 9 November 2007November 2007
gravity with Liouville type potential which produces a longitudinal
cs LM.H. Dehghani et al. / Physi
the solutions to be neither asymptotically flat nor (anti)-de Sitter[(A)dS]. There are two motivations for exploring nonasymptot-ically flat nor (A)dS solutions of Einstein gravity. First, thesesolutions can shed some light on the possible extensions ofAdS/CFT correspondence. Indeed, it has been speculated thatthe linear dilaton spacetimes, which arise as near-horizon lim-its of dilatonic black holes, might exhibit holography . Thesecond motivation comes from the fact that such solutions maybe used to extend the range of validity of methods and toolsoriginally developed for, and tested in the case of, asymptoti-cally flat or asymptotically AdS black holes.
On the other hand, there are many papers which are dealingdirectly with the issue of spacetimes in the context of cosmicstring theory . All of these solutions are horizonless andhave a conical geometry; they are everywhere flat except at thelocation of the line source. An extension to include the elec-tromagnetic field has also been done . Asymptotically AdSspacetimes generated by static and spinning magnetic sourcesin three and four-dimensional EinsteinMaxwell gravity withnegative cosmological constant have been investigated in [34,35]. The generalization of these rotating solutions to higherdimensions and higher derivative gravity have also been donein  and , respectively. In the context of electromag-netic cosmic string, it was shown that there are cosmic strings,known as superconducting cosmic string, that behave as super-conductors and have interesting interactions with astrophysicalmagnetic fields . The properties of these superconductingcosmic strings have been investigated in . Superconduct-ing cosmic strings have also been studied in BransDicke the-ory , and in dilaton gravity . Exact magnetic rotatingsolutions in three dimensions have been considered in while, two classes of magnetic rotating solutions in four andhigher dimensional EMd gravity with Liouville-type potentialhave been explored in  and , respectively. These solu-tions are not black holes, and represent spacetimes with conicsingularities. In the absence of a dilaton field, magnetic rotat-ing solutions of (n+ 1)-dimensional EBI theory have also beenconstructed .
Our aim in this Letter is to construct (n + 1)-dimensionalhorizonless solutions of EBId gravity. The motivation for study-ing these kinds of solutions is that they may be interpreted ascosmic strings. Cosmic strings are topological defects that arisefrom the possible phase transitions in the early universe, andmay play an important role in the formation of primordial struc-tures. Besides there are two main reasons for studying higherdimensional solutions of EBId gravity. The first originates fromstring theory, which is a promising approach to quantum grav-ity. String theory predicts that spacetime has more than four-dimensions. For a while it was thought that the extra spatialdimensions would be of the order of the Planck scale, mak-ing a geometric description unreliable, but it has recently beenrealized that there is a way to make the extra dimensions rel-atively large and still be unobservable. This is if we live ona three-dimensional surface (brane) in a higher dimensional
spacetime (bulk) [46,47]. In such a scenario, all gravitationalobjects are higher dimensional. The second reason for studyinghigher dimensional solutions has nothing to do with string the-etters B 659 (2008) 476482 477
ory. Four-dimensional solutions have a number of remarkableproperties. It is natural to ask whether these properties are gen-eral features of the solutions or whether they crucially dependon the world being four-dimensional.
The outline of our Letter is as follows: In Section 2, wepresent the basic field equations and general formalism of cal-culating the conserved quantities. In Section 3, we obtain themagnetic rotating solutions of Einstein equation in the presenceof dilaton and nonlinear electromagnetic fields, and exploretheir properties. The last section is devoted to summary andconclusions.
2. Field equations and conserved quantities
We consider the (n+1)-dimensional action in which gravityis coupled to dilaton and BornInfeld fields with an action
IG = 116M
n 1 ()2
(1) V () + L(F,))
where R is the Ricci scalar curvature, is the dilaton field,V () is a potential for and F 2 = FF (F = A A is the electromagnetic field tensor and A is the electro-magnetic potential). The last term in Eq. (1) is the GibbonsHawking boundary term which is chosen such that the varia-tional principle is well-defined. The manifold M has metricg and covariant derivative . is the trace of the extrin-sic curvature ab of any boundary(ies) M of the manifoldM, with induced metric(s) hab . In this Letter, we consider theaction (1) with a Liouville type potential,
(2)V () = 2e4/(n1),where is a constant which may be referred to as the cos-mological constant, since in the absence of the dilaton field( = 0) the action (1) reduces to the action of EinsteinBornInfeld gravity with cosmological constant [6,7]. The BornInfeld, L(F,), part of the action is given by
(3)L(F,) = 42e4/(n1)(
1 + e8/(n1)F 2
Here, is a constant determining the strength of coupling ofthe scalar and electromagnetic field and is called the BornInfeld parameter with dimension of mass. In the li