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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=20

First, 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.+h

I. INTRODUCTION

It seems likely, at least at sufficiently high energy scales,that gravity is not governed by the Einstein’s 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 Mach’s 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 cosmic

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05=71(6)=064010(8)$23.00 064010

string, 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

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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 � 1�dimensions is [15]

IG � �1

16�

ZM

dn�1x��g

p�R�

4

n � 1�r��2

� 2�e2� � e�4��=�n�1�F �F ��

�1

8�

Z@M

dnx��

p��; (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 �:

@ ��g

pe�4��=�n�1�F � � 0; (2)

R � �2

n�1�2@ �@��e2�g ��

�2e�4��=�n�1�

�F �F�

� �1

2�n�1�F��F��g �

�;

(3)

r2� �n � 1

2�e2� �

�2

e�4��=�n�1�F��F��: (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 pure

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cosmological 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 �1

8�

��ab ��ab �

n � 1

leff�ab

�; (5)

where leff is given by

l2eff �4n�n � 1� � �n � 1�32

8�e�2�: (6)

As goes to zero, the effective l2eff of Eq. (6) reduces tol2 � n�n � 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 � �ij�d’i � Vidt��d’j � 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 �&� �ZB

dn�1’��

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 �ZB

dn�1’��

pTabna&b; (8)

J �ZB

dn�1’��

pTabna&b; (9)

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

P �ZB

dn�1’��

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

l2R2��dt2 �

d�2

f�� l2f��d’2

��2

l2R2��dz2; (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 <z < 1) has the dimension of length, while the angularcoordinate ’ is dimensionless as usual and ranges in 0 �’ < 2�. The motivation for this metric gauge [gtt / ��2

and �g��1 / g’’] instead of the usual Schwarzschild

gauge [�g��1 / gtt and g’’ / �2] comes from the fact

that I am looking for a magnetic solution instead of anelectric one.

The Maxwell equation (2) for the metric (11) is@ ��

2R2�� exp��2��F � � 0, which shows that, ifone chooses

R�� exp��; (12)

then the vector potential is

A � �ql�

0’ ; (13)

where q is the charge parameter. The field equations (3)and (4) for the metric (11) can be written as

f0�1� ��0� � �f��00 � 2��02 � 4�0 � ��1

� q2��3e�2�� e2� � 0; (14)

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�2

f00 � f0�1� ��0� � 2f��00 � ��1� �2��02

� 2��0 �q2

�3 e�2�� e2� � 0; (15)

�2

f00 � f0�1� ��0� � q2��3e�2�� e2� � 0;

(16)

f�00 � f0�0 � 2�f�02 � 2��1f�0

� �q2��4e�2�� e2� � 0; (17)

where ‘‘prime’’ denotes differentiation with respect to �.Subtracting Eq. (15) from Eq. (16) gives

��00 � 2��0 � ��1� �2��02 � 0;

which shows that �� can be written as

��

1� �2 ln�b�� c

�; (18)

where b and c are two arbitrary constants. Using theexpression (18) for �� in Eqs. (14)–(17), one finds thatthese equations are inconsistent for c � 0. Thus, c shouldvanish.

The only case where I find exact solutions for arbitraryvalues of � (including � � 0) with R�� and �� ofEqs. (12) and (18) is when � �. It is easy, then, to obtainthe function f�� as

f�� 2��V0�1� �2�2

�2 � 3�2�1�2��

m��

�1� �2�q2

V0�2

�;

(19)

where � � �2=�1� �2� and V0 � b2�. In the absence of anontrivial dilaton (� � 0 � �), the solution reduces to theasymptotically AdS horizonless magnetic string for � ��3=l2 [6]. As one can see from Eq. (19), there is nosolution for � �

��3

pwith a Liouville potential (� � 0).

In order to study the general structure of this solution,one may first look for curvature singularities. It is easy toshow that the Kretschmann scalar R ��2R ��2 diverges at� � 0 and therefore one might think that there is a curva-ture singularity located at � � 0. However, as will be seenbelow, the spacetime will never achieve � � 0. Second,one looks for the existence of horizons, and therefore onesearches for possible black hole solutions. The horizons, ifany exist, are given by the zeros of the function f�� g��. Let us denote the smallest positive root of f�� 0 byr�. The function f�� is negative for � < r�, and thereforeone may think that the hypersurface of constant time and� � r� is the horizon. However, this analysis is not cor-rect. Indeed, one may note that g�� and g33 are related byf�� g�1

�� l�2g33, and therefore when g�� becomesnegative (which occurs for � < r�) so does g33. Thisleads to an apparent change of signature of the metric

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from �2 to �2, and therefore indicates that an incorrectextension is used. To get rid of this incorrect extension, onemay introduce the new radial coordinate r as

r2 � �2 � r2� ) d�2 �r2

r2 � r2�dr2: (20)

With this new coordinate, the metric (11) is

ds2 � �r2 � r2�

l2e2��dt2 � l2f�r�d32

�r2

�r2 � r2��f�r�dr2 �

r2 � r2�l2

e2��dz2; (21)

where f�r� is now given as

f�r� � �r2 � r2��V0�1� �2�2

��2 � 3��r2 � r2��

�1�2��

�m

�r2 � r2��1=2

��1� �2�q2

V0�r2 � r2��

�: (22)

The gauge potential in the new coordinate is

A �ql

�r2 � r2��1=2

0’ : (23)

Now it is a matter of calculation to show that theKretschmann scalar does not diverge in the range 0 � r <1. However, the spacetime has a conic geometry and has aconical singularity at r � 0, since

limr!0

1

r

��g’’

grr

s�

1

2mlr2��1�

� �2�1� �2�

��2 � 3��lV0r

1�2�� 1:

That is, as the radius r tends to zero, the limit of the ratio‘‘circumference/radius’’ is not 2� and therefore the space-time has a conical singularity at r � 0. In order to inves-tigate the casual structure of the spacetime, I consider it fordifferent ranges of � separately.

For � >��3

p, as r goes to infinity the dominant term in

Eq. (22) is the second term, and therefore the function f�r�is positive in the whole spacetime, despite the sign of thecosmological constant �, and is zero at r � 0. Thus, thesolution given by Eqs. (21) and (22) exhibits a spacetimewith conic singularity at r � 0.

For � <��3

p, the dominant term for large values of r is

the first term, and therefore the function f�r� given inEq. (22) is positive in the whole spacetime only for nega-tive values of �. In this case the solution presents a space-time with conic singularity at r � 0. The solution is notacceptable for � <

��3

pwith positive values of �, since the

function f�r� is negative for large values of r.Of course, one may ask for the completeness of the

spacetime with r � 0 [26]. It is easy to see that the space-time described by Eq. (21) is both null and timelike geo-desically complete for r � 0. To do this, one may showthat every null or timelike geodesic starting from an arbi-

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trary point either can be extended to infinite values of theaffine parameter along the geodesic or will end on asingularity at r � 0. Using the geodesic equation, oneobtains

_t �l2

V0�r2 � r2��1�� E; _z �

l2

V0�r2 � r2��1�� P;

_3 �1

l2f�r�L;

(24)

r2 _r2 � �r2 � r2��f�r��

l2�E2 � P2�

V0�r2 � r2��1�� 2

�

�r2 � r2�

l2L2; (25)

where the overdot denotes the derivative with respect to anaffine parameter, and 2 is zero for null geodesics and �1for timelike geodesics. E, L, and P are the conservedquantities associated with the coordinates t, 3, and z,respectively. Notice that f�r� is always positive for r > 0and zero for r � 0.

First, I consider the null geodesics (2 � 0). (i) If E > P,the spiraling particles (L > 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 l2�E2 � P2� >V0r

2�1�� . In this case the turning points for the nonspiral-

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

r2tp ��V�1

0 l2�E2 � P2�1=�1�� r2�

q; (26)

while the spiraling (L � 0) timelike particles are boundbetween ra

tp and rbtp given by 0 < ra

tp � rbtp < r2tp. Thus, I

confirmed 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 t � 3 plane:

t � �t � a’; ’ � �’ �a

l2t; (27)

where a is a rotation parameter and � ��1� a2=l2

p.

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

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

ds2 � �r2 � r2�

l2e2��dt � ad’�2 �

r2dr2

�r2 � r2��f�r�

� l2f�r��a

l2dt ��d’

�2�

r2 � r2�l2

e2��dz2;

(28)

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

A � �q

�r2 � r2��1=2

�al0t

��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 �V0��3� �2��2 � 2

8l�1� �2�m;

J ��3� �2�V0

8l�1� �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 � r2�

l2e2��dt2 �

r2dr2

�r2 � r2��f�r�� r2

� r2��e2��d’2 � f�r�dz2; (31)

where f�r� is given in Eq. (22). The angular coordinate ’ranges in 0 � ’ < 2�. The gauge potential is now givenby

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A �q

�r2 � r2��1=2

0z : (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=l�z, z ��x � �v=l�t] in the t � z plane and obtain

ds2��r2�r2�

l2e2��

��dt�

vldz�2�f�r�

�vldt��dz

�2

�r2dr2

�r2�r2��f�r��r2�r2��e

2��d’2; (33)

where v is a boost parameter and � ��1� v2=l2

p. The

gauge potential is given by

A � �q

�r2 � r2��1=2

�vl0t

��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 �V0��3� �2��2 � 2

8l�1� �2�m;

P ��3� �2�V0

8l�1� �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 �1

N; ur � 0; ui � �

Vi

N;

and the electric field is E � g � exp��2��F��u�.Then the electric charge per unit length Q can be foundby calculating the flux of the electric field at infinity,yielding

Q ��2 � 1�q

2l: (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

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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

l2e2�dt2 �

d�2

f�� l2f��d32 �

�2

l2e2�d�2;

(36)

where d�2 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

f0�1� ��0� � f��00 � �n � 1��2�0 � ��02�

� �n � 2��1 �2��n � 1

e2� � 0; (37)

�n � 1

f00 � f0�1� ��0� �4��

�n � 1�2e2� � 2�n � 2��f

�

��00 � 2�0 � �

�2 �

4

�n � 1�2

��02

�� 0;

(38)

�n � 1

f00 � f0�1� ��0� �4��

�n � 1�2e2� � 0; (39)

f�00 �f0�0 ��n�1�

�

�f�02��1f�0 �

1

2�e2�

��0: (40)

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

��00 � 2�0 � ��2 �

4

�n � 1�2

��02 � 0;

which shows that �� can be written as

�� n � 1�2

4� �n � 1�22 ln�c�� 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�� 8�V0

�n � 1�32 � 4n�n � 1��2� � m�1��n�1��;

(42)

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where

V0 � ��2c2�1��; � � 4f�n � 1�22 � 4g�1: (43)

One may note that there is no solution for �n � 1�22 �4n � 0. For the two cases of �n � 1�22 � 4n > 0 withpositive �, and �n � 1�22 � 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 � 1�22 � 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 � 1�22 � 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 as

ds2 � �r2 � r2�

l2e2�dt2 � l2f�r�d32

�r2

�r2 � r2��f�r�dr2 �

r2 � r2�l2

e2�dz2; (44)

f�r� �8�V0

�n � 1�32 � 4n�n � 1�

� �r2 � r2�� m�r2 � r2��

�1=2��1��n�1��; (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 SO�n� 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

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PHYSICAL REVIEW D 71, 064010 (2005)

ds2 � �r2 � r2�

l2e2�

��dt �

Xki�1

aid3i�2� f�r�

�

� ��2 � 1

pdt �

��2 � 1

pXki�1

aid3i�2

�r2dr2

�r2 � r2��f�r��

r2 � r2�l2��2 � 1�

� e2�Xki<j

�aid3j � ajd3i�2 �

r2 � r2�l2

e2�dX2;

�2 � 1�Xki�1

a2i

l2;

(46)

where ai’s are k rotation parameters, f�r� is given inEq. (45), and dX2 is now the Euclidean metric on the �n �k � 1�-dimensional submanifold with volume Vn�k�1.

The conserved mass and angular momentum per unitvolume Vn�k�1 of the solution calculated on the boundaryB at infinity can be calculated through the use of Eqs. (8)and (9),

M ��2��k�nV�n�1�=2

0

16�ln�k�1

��n�

�n�1�22

4

��2��n�1�

�m;

(47)

J i � �2��k�nV�n�1�=2

0

16�ln�k�1

�n �

�n � 1�2�2

4

��mai: (48)

Note that for Einstein gravity the above computed con-served quantities reduce to those given in [7].

V. CLOSING REMARKS

Until now, no explicit rotating charged dilaton solutionshave been found except for some dilaton coupling such as� �

��3

p[18] or � � 1 when the string three-form Habc is

included [28]. For general dilaton coupling, the propertiesof charged dilaton black holes have been investigated onlyfor rotating solutions with infinitesimally small angularmomentum [19] or small charge [20]. It has also beenshown numerically that for the case of rotating solutions� �

��3

pis a critical value while for larger values of � a

new effect can appear [21]. In this paper, I obtained twoclasses of exact horizonless rotating solutions with

MAGNETIC STRINGS IN DILATON GRAVITY

064010

Liouville-type potentials in four dimensions, provided � � �

��3

p. These solutions are neither asymptotically

flat nor (A)dS. The first class of solutions yields a four-dimensional spacetime with a longitudinal magnetic field[the only nonzero component of the vector potential isA’�r�] generated by a static magnetic string. I also foundthe rotating spacetime with a longitudinal magnetic fieldby a rotational boost transformation. These solutions haveno curvature singularity and no horizons, but have conicsingularity at r � 0. In these spacetimes, when the rotationparameter is zero (static case), the electric field vanishes,and therefore the string has no net electric charge. For thespinning string, when the rotation parameter is nonzero,the string has a net electric charge density which is pro-portional to the rotation parameter a. The second class ofsolutions yields a spacetime with angular magnetic field.These solutions have no curvature singularity, no horizon,and no conic singularity. Again, I found that the string inthese spacetimes has no net electric charge when the boostparameter vanishes. I also showed that, for the case of thetraveling string with nonzero boost parameter, the netelectric charge density of the string is proportional to themagnitude of its velocity (v). These solutions reduce to themagnetic rotating string of [6] as � ! 0. I also computedthe conserved quantities of the four-dimensional magneticstring through the use of the counterterm method.

Next, I generalized the rotating uncharged solutions toarbitrary n � 1 dimensions with all rotation parameters.These spacetimes present naked singularities for ��n �1�22 � 4n < 0;� < 0 and ��n � 1�22 � 4n > 0;� >0. The only case where the spacetime exhibits a horizon-less dilaton string with conic singularity is when �n �1�22 � 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.

ACKNOWLEDGMENTS

This work has been supported by Research Institute forAstronomy and Astrophysics of Maragha, Iran.

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