Magnetic strings in dilaton gravity

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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 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.III. FOUR-DIMENSIONAL HORIZONLESSDILATON SOLUTIONS

    In this section, I want to obtain the four-dimensionalhorizonless solutions of Eqs. (2)(4). First, a spacetimegenerated by a magnetic source which produces a longitu-dinal magnetic field along the z axis is constructed, and,second, I obtain a spacetime generated by a magneticsource that produces angular magnetic fields along the coordinate.

    A. The longitudinal magnetic field solutions

    I assume that the metric has the following form:

    ds2 2

    l2R2dt2 d

    2

    f l2fd2

    2

    l2R2dz2; (11)

    where the constant l has dimension of length which isrelated to the cosmological constant in the absence ofa dilaton field ( 0). Note that the coordinate z ( 1 0) coming from infinity have aturning point at rtp > 0, while the nonspiraling particles(L 0) have a turning point at rtp 0. (ii) If E P andL 0, whatever the value of r, _r and _3 vanish and there-fore the null particles move on the z axis. (iii) For E Pand L 0, and also for E< P and any values of L, there isno possible null geodesic.

    Second, I analyze the timelike geodesics (2 1).Timelike geodesics are possible only if l2E2 P2>V0r

    21 . In this case the turning points for the nonspiral-

    ing particles (L 0) are r1tp 0 and r2tp given as

    r2tp V10 l2E2 P21=1 r2

    q; (26)

    while the spiraling (L 0) timelike particles are boundbetween ratp and rbtp given by 0< ratp rbtp < r2tp. Thus, Iconfirmed that the spacetime described by Eq. (21) is bothnull and timelike geodesically complete.

    B. The rotating longitudinal magnetic field solutions

    Now, I want to endow the spacetime solution (21) with aglobal rotation. In order to add angular momentum to thespacetime, one may perform the following rotation boost inthe t3 plane:

    t t a; al2t; (27)

    where a is a rotation parameter and 1 a2=l2

    p.

    Substituting Eq. (27) into Eq. (21), one obtains-4

  • MAGNETIC STRINGS IN DILATON GRAVITY PHYSICAL REVIEW D 71, 064010 (2005)ds2 r2 r2l2

    e2dt ad2 r2dr2

    r2 r2fr

    l2fra

    l2dtd

    2 r

    2 r2l2

    e2dz2;

    (28)

    where fr is the same as fr given in Eq. (22). The gaugepotential is now given by

    A q

    r2 r21=2al0t l0

    : (29)

    The transformation (27) generates a new metric, because itis not a permitted global coordinate transformation [27].This transformation can be done locally but not globally.Therefore, the metrics (21) and (28) can be locally mappedinto each other but not globally, and so they are distinct.Note that this spacetime has no horizon and curvaturesingularity. However, it has a conical singularity at r 0. It is notable to mention that this solution reduces to thesolution of Einstein-Maxwell equation introduced in [6] as goes to zero.

    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. (8) and (9),

    M V03 22 2

    8l1 2 m;

    J 3 2V0

    8l1 2 ma:(30)

    For a 0 ( 1), the angular momentum per unit lengthvanishes, and therefore a is the rotational parameter of thespacetime. Of course, one may note that these conservedcharges reduce to the conserved charges of the rotatingblack string obtained in Ref. [8] as ! 0.

    C. The angular magnetic field solutions

    In Sec. III A, I found a spacetime generated by a mag-netic source which produces a longitudinal magnetic fieldalong the z axis. Now, I want to obtain a spacetime gen-erated by a magnetic source that produces angular mag-netic fields along the coordinate. Following the steps ofSec. III A but now with the roles of and z interchanged,one can directly write the metric and vector potentialsatisfying the field equations (14)(17) as

    ds2 r2 r2l2

    e2dt2 r2dr2

    r2 r2fr r2

    r2e2d2 frdz2; (31)

    where fr is given in Eq. (22). The angular coordinate ranges in 0 < 2. The gauge potential is now givenby064010A q

    r2 r21=20z: (32)

    The Kretschmann scalar does not diverge for any r andtherefore there is no curvature singularity. The spacetime(31) is also free of conic singularity.

    To add linear momentum to the spacetime, one mayperform the boost transformation [t t v=lz, z x v=lt] in the t z plane and obtain

    ds2r2r2l2

    e2dtv

    ldz2fr

    vldtdz

    2

    r2dr2

    r2r2frr2r2e2d2; (33)

    where v is a boost parameter and 1 v2=l2

    p. The

    gauge potential is given by

    A q

    r2 r21=2vl0t 0z

    : (34)

    Contrary to transformation (27), this boost transformationis permitted globally since z is not an angular coordinate.Thus the boosted solution (33) is not a new solution.However, it generates an electric field.

    The conserved quantities of the spacetime (33) are themass and linear momentum along the z axis given as

    M V03 22 2

    8l1 2 m;

    P 3 2V0

    8l1 2 mv:

    Now, I calculate the electric charge of the solutions (28)and (33) obtained in this section. To determine the electricfield, one should consider the projections of the electro-magnetic field tensor on special hypersurfaces. The normalto such hypersurfaces for the spacetimes with a longitudi-nal magnetic field is

    u0 1N; ur 0; ui V

    i

    N;

    and the electric field is E g exp2Fu.Then the electric charge per unit length Q can be foundby calculating the flux of the electric field at infinity,yielding

    Q 2 1q2l

    : (35)

    Note that the electric charge of the string vanishes, whenthe string has no angular or linear momenta.IV. THE ROTATING SOLUTIONS IN VARIOUSDIMENSIONS

    In this section, I look for the uncharged rotating solu-tions of field equations (2)(4) in n 1 dimensions. I first-5

  • M. H. DEHGHANI PHYSICAL REVIEW D 71, 064010 (2005)obtain the uncharged static solution and then generalize itto the case of rotating solution with all the rotationparameters.

    A. Static solutions

    I assume the metric has the following form:

    ds2 2

    l2e2dt2 d

    2

    f l2fd32

    2

    l2e2d2;

    (36)

    where d2 is the metric of the n 1-dimensional hyper-surface which has zero curvature. The field equa-tions (2)(2)(4) in the absence of electromagnetic fieldbecome

    f01 0 f00 n 120 02

    n 21 2n 1 e

    2 0; (37)

    n 1f

    00 f01 0 4n 12 e2 2n 2f

    00 20

    2 4n 12

    02

    0;

    (38)

    n 1 f

    00 f01 0 4n 12 e2 0; (39)

    f00 f00 n1

    f021f0 1

    2

    e2

    0: (40)

    Subtracting Eq. (38) from Eq. (39) gives

    00 20

    2 4n 12

    02 0;

    which shows that can be written as

    n 12

    4 n 122 lnc d

    ; (41)

    where c and d are two arbitrary constants. Substituting of Eq. (41) into the field equations (37)(40), onefinds that they are consistent only for d 0. Putting d 0,then f can be written as

    f 8V0n 132 4nn 12 m1n1;

    (42)064010whereV0 2c21; 4fn 122 4g1: (43)One may note that there is no solution for n 122 4n 0. For the two cases of n 122 4n > 0 withpositive , and n 122 4n < 0 with negative , thefunction f is positive in the whole spacetime. In thesetwo cases, since the Kretschmann scalar diverges at 0and f is positive for > 0, the spacetimes presentnaked singularities. For n 122 4n < 0 with positive, the function f is negative for > rc, where rc is theroot of f 0. In this case, the signature of the metric(36) will be changed as becomes larger than rc, andtherefore this case is not acceptable.

    The only case that there exist horizonless solutions withconic singularity is when n 122 4n > 0 with nega-tive . In this case, as r goes to infinity the dominant termis the second term of Eq. (42), and therefore f> 0 for > r, where r is the root of f 0. Again, oneshould perform the transformation (20). Thus, the horizon-less solution can be written asds2 r2 r2l2

    e2dt2 l2frd32

    r2

    r2 r2frdr2 r

    2 r2l2

    e2dz2; (44)fr 8V0n 132 4nn 1 r2 r2 mr2 r21=21n1; (45)where and V0 are given in Eq. (43). Again this spacetimehas no horizon and no curvature singularity. However, ithas a conical singularity at r 0. One should note that thissolution reduces to the n 1-dimensional unchargedsolution of Einstein gravity given in [7] for 0 ( 1 V0).

    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 (44) with allrotation parameters is-6

  • PHYSICAL REVIEW D 71, 064010 (2005)ds2 r2 r2l2

    e2dt

    Xki1

    aid3i2 fr

    2 1p

    dt 2 1

    p Xki1

    aid3i2

    r2dr2

    r2 r2fr r

    2 r2l22 1

    e2Xki 0;>0. The only case where the spacetime exhibits a horizon-less dilaton string with conic singularity is when n122 4n > 0 and < 0.

    Note that the n 1-dimensional rotating solutionsobtained here are uncharged. Thus, it would be interestingif one can construct rotating solutions in n 1 dimen-sions in the presence of dilaton and electromagnetic fields.One may also attempt to generalize these kinds of solutionsobtained here to the case of two-term Liouville potential[15]. The case of a charged rotating black string in fourdimensions and n 1-dimensional rotating black braneswill be present elsewhere.

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