Magnetic strings in Einstein–Born–Infeld-dilaton gravity

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    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.

    1. Introduction

    The BornInfeld [1] 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 [2].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[3], and has distinct properties from Bardeen black holes [4].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: (M.H. Dehghani), (A. Sheykhi), (S.H. Hendi).

    Infeld (EBI) gravity in arbitrary dimensions with positive, zeroor negative constant curvature horizons have been constructed[57]. 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 [1015],exact rotating black holes solutions with curved horizons havebeen obtained only for some limited values of the coupling con-stant[1618]. For general dilaton coupling, the properties ofrotating charged dilaton black holes only with infinitesimallysmall charge [19] or small angular momentum have been inves-tigated [2022]. When the horizons are flat, rotating solutionsof EMd gravity with Liouville-type potential in four [23] and(n + 1)-dimensions have been constructed [24]. The studieson the black hole solutions of EBId gravity in three and fourdimensions have been carried out in [25] and [2628], respec-tively. Thermodynamics of (n+1)-dimensional EBId solutionswith flat [29] and curved horizons have also been explored [30].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

    Available online



    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)

    ornInfeld-dilaton gravity

    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 [31]. 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 [32]. 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 [33]. 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 [36] and [37], 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 [38]. The properties of these superconductingcosmic strings have been investigated in [39]. Superconduct-ing cosmic strings have also been studied in BransDicke the-ory [40], and in dilaton gravity [41]. Exact magnetic rotatingsolutions in three dimensions have been considered in [42]while, two classes of magnetic rotating solutions in four andhigher dimensional EMd gravity with Liouville-type potentialhave been explored in [43] and [44], 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 [45].

    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


    (R 4

    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 limit ,L(F,) reduces to the standard Maxwell field coupled to adilaton field

    (4)L(F,) = e4/(n1)F 2,and L(F,) 0 as 0. It is convenient to set

    (5)L(F,) = 42e4/(n1)L(Y ),


    (6)L(Y ) = 1 1 + Y ,

  • s L478 M.H. Dehghani et al. / Physic

    (7)Y = e8/(n1)F 2


    The equations of motion can be obtained by varying the ac-tion (1) with respect to the gravitational field g , the dilatonfield and the gauge field A which yields the following fieldequations

    R = 4n 1

    ( + 14gV ()


    4e4/(n1)YL(Y )FF(8)+ 4


    n 1e4/(n1)[2YYL(Y ) L(Y )]g,

    2 = n 18


    (9)+ 22e4/(n1)[2 YYL(Y ) L(Y )],(10)

    (e4/(n1)YL(Y )F

    )= 0.In particular, in the case of the linear electrodynamics withL(Y ) = 12Y , the system of Eqs. (8)(10) reduce to the well-known equations of EMd gravity [14].

    The conserved mass and angular momentum of the solutionsof the above field equations can be calculated through the useof the substraction method of Brown and York [48]. Such a pro-cedure causes the resulting physical quantities to depend on thechoice of reference background. For asymptotically (A)dS solu-tions, the way that one deals with these divergences is throughthe use of counterterm method inspired by (A)dS/CFT corre-spondence [49]. However, in the presence of a non-trivial dila-ton field, the spacetime may not behave as either dS (> 0) orAdS ( < 0). In fact, it has been shown that with the exceptionof a pure cosmological constant potential where = 0, no AdSor dS static spherically symmetric solution exist for Liouville-type potential [13]. But, as in the case of asymptotically AdSspacetimes, according to the domain-wall/QFT (quantum fieldtheory) correspondence [50], there may be a suitable countert-erm for the stress energy tensor which removes the divergences.In this paper, we deal with the spacetimes with zero curva-ture boundary [Rabcd(h) = 0], and therefore the countertermfor the stress energy tensor should be proportional to hab . Thus,the finite stress-energy tensor in (n + 1)-dimensional Einstein-dilaton gravity with Liouville-type potential may be written as

    (11)T ab = 18

    [ab hab + n 1



    where leff is given by

    (12)l2eff =(n 1)(2 n)


    In the particular case = 0, the effective l2eff of Eq. (12) reducesto l2 = n(n 1)/2 of the AdS spacetimes. The first twoterms in Eq. (11) is the variation of the action (1) with respectto hab , and the last term is the counterterm which removes the

    divergences. One may note that the counterterm has the sameform as in the case of asymptotically AdS solutions with zerocurvature boundary, where l is replaced by leff. To compute theetters B 659 (2008) 476482

    conserved charges of the spacetime, one should choose a space-like surface B in M with metric ij , and write the boundarymetric in ADM (ArnowittDeserMisner) form:hab dx

    a dxa = N2 dt2(13)+ ij

    (di + V i dt)(dj + V j dt),

    where the coordinates i are the angular variables parameteriz-ing the hypersurface of constant r around the origin, and N andV i are the lapse and shift functions, respectively. When there isa Killing vector field on the boundary, then the quasilocal con-served quantities associated with the stress tensors of Eq. (11)can be written as

    (14)Q() =B



    where is the determinant of the metric ij , and na are theKilling vector field and the unit normal vector on the bound-ary B. For boundaries with timelike ( = /t) and rotationalKilling vector field (i = /i ), one obtains the quasilocalmass and components of the total angular momentum as

    (15)M =B



    (16)Ji =B


    Tabnabi .

    Note that these quantities depend on the location of the bound-ary B in the spacetime, although each is independent of theparticular choice of foliation B within the surface M.

    3. Magnetic rotating solutions

    In this section we are going to obtain rotating horizonlesssolutions of the field equations (8)(10). First, we constructthe rotating 4-dimensional spacetimes generated by a magneticsource which produces a longitudinal magnetic field. Second,we generalize these 4-dimensional solutions to the case of(n + 1)-dimensional solutions.

    3.1. Longitudinal magnetic field solutions

    Here we want to obtain the 4-dimensional solution ofEqs. (8)(10) which produces a longitudinal magnetic fieldsalong the z direction. We assume the following form for themetric

    ds2 = 2

    l2R2()( dt a d)2 + f ()


    ldt l d


    (17)+ d2

    f ()+


    l2R2() dz2,

    where a is the rotation parameter and = 1 + a2/l2. The

    functions f () and R() should be determined and l has thedimension of length which is related to the cosmological con-stant for the case of Liouville-type potential with constant .

  • cs LM.H. Dehghani et al. / Physi

    The angular coordinate is dimensionless as usual and rangesin [0,2], while and z have dimension of length.

    The electromagnetic field equation (10) can be integratedimmediately to give

    (18)F = qle2


    1 q22(R)4

    , Ft = al2


    where q is the charge parameter of the string. To solve the sys-tem of Eqs. (8) and (9) for three unknown functions f (), R()and (), we make the ansatz

    (19)R() = e.Using (19), the electromagnetic fields (18) and the metric (17),one can show that Eqs. (8) and (9) have solutions of the form

    (20)f () = ( 22)(2 + 1)2b22 3

    2(1 ) + m12

    + 22(2 + 1)b2 21(21)

    2(12 )

    (1 ) d,

    (22)() = 1 + 2 ln



    where = 2/(1 + 2) and

    (23) q2

    2b4 4(1 ).

    Eq. (18) shows that s...