Magnetic branes in third order Lovelock-Born-Infeld gravity

M.H. Dehghani,1,2,* N. Bostani,1 and S. H. Hendi3,+

1Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran

3Physics Department, College of Sciences, Yasouj University, Yasouj 75914, Iran(Received 16 June 2008; published 11 September 2008)

Considering both the nonlinear invariant terms constructed by the electromagnetic field and the

Riemann tensor in gravity action, we obtain a new class of ðnþ 1Þ-dimensional magnetic brane solutions

in third order Lovelock-Born-Infeld gravity. This class of solutions yields a spacetime with a longitudinal

nonlinear magnetic field generated by a static source. These solutions have no curvature singularity and no

horizons but have a conic geometry with a deficit angle �. We find that, as the Born-Infeld parameter

decreases, which is a measure of the increase of the nonlinearity of the electromagnetic field, the deficit

angle increases. We generalize this class of solutions to the case of spinning magnetic solutions and find

that, when one or more rotation parameters are nonzero, the brane has a net electric charge which is

proportional to the magnitude of the rotation parameters. Finally, we use the counterterm method in third

order Lovelock gravity and compute the conserved quantities of these spacetimes. We found that the

conserved quantities do not depend on the Born-Infeld parameter, which is evident from the fact that the

effects of the nonlinearity of the electromagnetic fields on the boundary at infinity are wiped away. We

also find that the properties of our solution, such as deficit angle, are independent of Lovelock coefficients.

DOI: 10.1103/PhysRevD.78.064031 PACS numbers: 04.50.�h, 04.20.Jb, 04.40.Nr

I. INTRODUCTION

Actions of Lovelock gravity and Born-Infeld electro-dynamics have been the subject of wide interest in recentyears. This is due to the fact that both of them emerge in thelow-energy limit of string theory [1]. On the gravity sidestring theories in their low-energy limit give rise to effec-tive models of gravity in higher dimensions which involvehigher curvature terms, while on the electrodynamics sidethe effective action for the open string ending on D-branescan be written in a Born-Infeld form. While Lovelockgravity [2] was proposed to have field equations with atmost second order derivatives of the metric [3], the non-linear electrodynamics was proposed, by Born and Infeld,with the aim of obtaining a finite value for the self-energyof a pointlike charge [4]. Lovelock gravity reduces toEinstein gravity in four dimensions and also in the weakfield limit, while the Lagrangian of the Born-Infeld (BI)electrodynamics reduces to the Maxwell Lagrangian in theweak field limit. There have been considerable works onboth of these theories. In Lovelock gravity, there have beensome attempts for understanding the role of the highercurvature terms from various points of view. For example,exact static spherically symmetric black hole solutions ofthe second order Lovelock gravity have been found inRef. [5] and of the Gauss-Bonnet-Maxwell model inRef. [6]. An exact static solution in third order Lovelockgravity was introduced in Ref. [7] and of the chargedrotating black brane solution in Ref. [8], and the magnetic

solutions with longitudinal and angular magnetic fieldwere considered in Ref. [9]. All of these works were inthe presence of a linear electromagnetic field. The firstattempt to relate the nonlinear electrodynamics and gravityhas been done by Hoffmann [10]. He obtained a solution ofthe Einstein equations for a pointlike Born-Infeld charge,which is devoid of the divergence of the metric at the originthat characterizes the Reissner-Nordstrom solution.However, a conical singularity remained there, as it waslater objected by Einstein and Rosen. The sphericallysymmetric solutions in Einstein-Born-Infeld gravity withor without a cosmological constant have been consideredby many authors [11,12].In this paper, we are dealing with the issue of the space-

times generated by static and spinning brane sources whichare horizonless and have nontrivial external solutions.These kinds of solutions have been investigated by manyauthors in four dimensions. Static uncharged cylindricallysymmetric solutions of Einstein gravity in four dimensionswere considered in Ref. [13]. Similar static solutions in thecontext of cosmic string theory were found in Ref. [14]. Allof these solutions [13,14] are horizonless and have a coni-cal geometry; they are everywhere flat except at the loca-tion of the line source. The extension to include theelectromagnetic field has also been done [15,16]. In threedimensions, the rotating magnetic solutions of Einsteingravity have been studied in Ref. [17], while those ofBrans-Dicke gravity have been considered in Ref. [18].The generalization of these solutions in Einstein gravity inthe presence of a dilaton and Born-Infeld electromagneticfield has been done in Ref. [19]. Our aim in this paper is toconstruct ðnþ 1Þ-dimensional horizonless solutions of

*mhd@shirazu.ac.ir+hendi@mail.yu.ac.ir

PHYSICAL REVIEW D 78, 064031 (2008)

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third order Lovelock gravity in the presence of a nonlinearelectromagnetic field and investigate the effects of thenonlinearity of the electromagnetic field on the propertiesof the spacetime such as the deficit angle of the spacetime.The first reason for studying higher-dimensional solutionsof gravity, which invite higher order Lovelock terms, is dueto the scenario of string theory and brane cosmology. Theidea of brane cosmology, which is also consistent withstring theory, suggests that matter and gauge interaction(described by an open string) may be localized on a braneembedded into a higher-dimensional spacetime, and allgravitational objects are higher-dimensional [20,21]. Thesecond reason for studying higher-dimensional solutions isthe fact that four-dimensional solutions have a number ofremarkable properties. It is natural to ask whether theseproperties are general features of the solutions or whetherthey crucially depend on the world being four-dimensional.Finally, we want to check whether the counterterm methodintroduced in Refs. [8,22] can be applied to the case ofsolutions in the presence of a nonlinear electromagneticfield, too.

The outline of our paper is as follows. We give a briefreview of the field equations of Lovelock gravity in thepresence of a Born-Infeld electromagnetic field in Sec. II.In Sec. III, we present static horizonless solutions whichproduce a longitudinal magnetic field and investigate theeffects of the nonlinearity of the electromagnetic field andLovelock terms on the deficit angle of the spacetime.Section IV will be devoted to the generalization of thesesolutions to the case of rotating solutions and use of thecounterterm method to compute the conserved quantitiesof them. We finish our paper with some concludingremarks.

II. FIELD EQUATIONS

A natural generalization of general relativity in higher-dimensional spacetimes with the assumption of Einstein,that the left-hand side of the field equations is the mostgeneral symmetric conserved tensor containing no morethan second derivatives of the metric, is Lovelock theory.Lovelock found the most general symmetric conservedtensor satisfying this property. The resultant tensor is non-linear in the Riemann tensor and differs from the Einsteintensor only if the spacetime has more than 4 dimensions.The Lovelock tensor in ðnþ 1Þ dimensions may be writtenas [2]

X½n=2�i¼1

�0i

�HðiÞ

�� � 1

2g��LðiÞ

�þ�g�� ¼ 8�T��; (1)

where � ¼ �nðn� 1Þ=2l2 is the cosmological constantfor asymptotically anti-de Sitter (AdS) solutions, ½n=2�denotes the integer part of n=2, �0

1 ¼ 1, �0i’s for i � 2

are Lovelock coefficients, T�� is the energy-momentum

tensor, and H ðiÞ�� and LðiÞ are

H ðiÞ�� ¼ i

2i��1�2...�2i��2...�2i

R�1�2��2R�3�4

�3�4 . . .R�2i�1�2i

�2i�1�2i ;

(2)

L ðiÞ ¼ 1

2i��1�2...�2i�1�2...�2i

R�1�2

�1�2 . . .R�2i�1�2i

�2i�1�2i : (3)

In this paper, we consider up to third order Lovelockterms. The first term is just the Einstein tensor, and thesecond and third terms are Gauss-Bonnet and third orderLovelock tensors, respectively, which are written in termsof Riemann tensor explicitly in Ref. [23]. In the presenceof a Born-Infeld electromagnetic field, the energy-momentum tensor of Eq. (1) is

T�� ¼ 1

4�

�1

4g��LðFÞ þ

F��F��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ F2

2�2

q �; (4)

where F�� is the nonlinear electromagnetic field tensor

which satisfies the BI equation

@�

� ffiffiffiffiffiffiffi�gp

F��ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ F2

2�2

q �¼ 0; (5)

and F2 ¼ F��F��. The parameter � is called the Born-

Infeld parameter which has the dimension of length�2, and,as it goes to 1, the BI equation reduces to the standardMaxwell equation.

III. STATIC MAGNETIC BRANES

Here we want to obtain the ðnþ 1Þ-dimensional solu-tions of Eqs. (1)–(5) which produce longitudinal magneticfields in the Euclidean submanifold spans by xi coordinates(i ¼ 1; . . . ; n� 2). We will work with the following ansatzfor the metric [16]:

ds2 ¼ ��2

l2dt2 þ d�2

fð�Þ þ l2fð�Þd2 þ �2

l2dX2; (6)

where dX2 ¼ Pn�2i¼1 ðdxiÞ2 is the Euclidean metric on the

ðn� 2Þ-dimensional submanifold. The angular coordinate is dimensionless as usual and ranges in ½0; 2��, whilexi’s range in ð�1;1Þ. The motivation for this metricgauge [gtt / ��2 and ðg��Þ�1 / g] instead of the usual

Schwarzschild gauge [ðg��Þ�1 / gtt and g / �2] comes

from the fact that we are looking for a horizonless solution.The electromagnetic field equation (5) can be integrated

immediately to give

F� ¼ 2qln�1

�n�1ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p ; (7)

where q is the charge parameter of the solution and

¼ 4q2l2n�4

�2�2ðn�1Þ : (8)

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Equation (7) shows that � should be greater than �0 ¼ð2qln�2�Þ1=ðn�1Þ in order to have a real nonlinear electro-magnetic field and consequently a real spacetime. To findthe function fð�Þ, one may use any components of Eq. (1).The simplest equation is the �� component of these equa-tions which can be written as

ð�3�f2 � 2�2�

3fþ �5Þf0 þ n� 6

3�3f

3

� ðn� 4Þ�2�2f2 þ ðn� 2Þ�4f� n

l2�6

¼ �2�6

2ðn� 1Þ ð1�ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p Þ; (9)

where the prime denotes the derivative with respect to �and

�2 ¼ ðn� 2Þðn� 3Þ�02; �3 ¼ 72

n� 24

� ��03: (10)

The only real solution of Eq. (9) is

fð�Þ ¼ �2�2

�3

f1� ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ j2ð�Þ

qþ jð�ÞÞ1=3

þ �1=3ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ j2ð�Þ

qþ jð�ÞÞ�1=3g; (11)

where

� ¼��3

�22

� 1

�3; (12)

jð�Þ ¼ �1þ 3�3

2�22

� 3�23

2l2�32

kð�Þ; (13)

kð�Þ ¼ 1þ 2ml3

�n þ �2l2hðÞ2nðn� 1Þ : (14)

The constants m and q in Eq. (14) are the mass and chargeparameters, respectively, of the metric which are related tothe mass and charge density of the solution, and hðÞ isgiven as

hðÞ ¼ 1� ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p� ðn� 1Þ

ðn� 2Þ 2F1

��1

2;n� 2

2n� 2

�;

�3n� 4

2n� 2

�;

�;

(15)

where 2F1ð½a; b�; ½c�; zÞ is a hypergeometric function.Using the fact that 2F1ð½a; b�; ½c�; zÞ has a convergent seriesexpansion for jzj< 1, we can find the behavior of themetric for large � and �. The function jð�Þ approachesthe constant

� ¼ �1þ 3�3

2�22

� 3�23

2l2�32

;

as � goes to infinity, and the effective cosmological con-

stant for the spacetime is

�eff ¼ � nðn� 1Þ�2

2�3

f1þ ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ �2

qþ �Þ1=3

� �1=3ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ �2

qþ �Þ�1=3g: (16)

In the rest of the paper, we investigate only the case of � �0, for which fð�Þ is real. The function fð�Þ is negative forlarge values of �, if�eff > 0. Since g�� and g are related

by fð�Þ ¼ g�1�� ¼ l�2g, when g�� becomes negative

(which occurs for large �), so does g. This leads to an

apparent change of signature of the metric from ðn� 1Þþto ðn� 2Þþ as � goes to infinity, which is not allowed.Thus, �eff should be negative which occurs provided theLovelock coefficients are assumed to be positive.In order to study the general structure of the solution

given in Eq. (11), we first look for curvature singularities. Itis easy to show that the Kretschmann scalar R����R

����

diverges at � ¼ 0, and therefore one might think that thereis a curvature singularity located at � ¼ 0. However, as wewill see below, the spacetime will never achieve � ¼ 0.The function fð�Þ is negative for � < rþ and positive for� > rþ, where rþ is the largest real root of fð�Þ ¼ 0whichreduces to

kðrþÞ ¼ 1þ 2ml3

rnþþ �2l2hðþÞ

2nðn� 1Þ ¼ 0: (17)

Again g�� cannot be negative (which occurs for � < rþ),because of the change of signature of the metric from ðn�1Þþ to ðn� 2Þ þ . Thus, one cannot extend the spacetimeto � < rþ. To get rid of this incorrect extension, weintroduce the new radial coordinate r as

r2 ¼ �2 � r2þ ) d�2 ¼ r2

r2 þ r2þdr2: (18)

With this new coordinate, the metric (6) is

ds2 ¼ � r2 þ r2þl2

dt2 þ r2

ðr2 þ r2þÞfðrÞdr2 þ l2fðrÞd2

þ r2 þ r2þl2

dX2; (19)

where the coordinates r and assume the values 0 � r <1 and 0 � < 2�, respectively. The function fðrÞ is nowgiven as

fðrÞ ¼ �2ðr2 þ r2þÞ�3

f1� ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ j2ðrÞ

qþ jðrÞÞ1=3

þ �1=3ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�þ j2ðrÞ

qþ jðrÞÞ�1=3g; (20)

where

¼ 4q2l2n�4

�2ðr2 þ r2þÞðn�1Þ ; (21)

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jðrÞ ¼ �1þ 3�3

2�22

� 3�23

2l2�32

kðrÞ; (22)

kðrÞ ¼ 1þ ml3

ðr2 þ r2þÞn=2þ �2l2hðÞ

4�nðn� 1Þ : (23)

The electromagnetic field equation in the new coordinate is

Fr ¼ 2qln�1

ðr2 þ r2þÞðn�1Þ=2 ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p : (24)

The function fðrÞ given in Eq. (20) is positive in thewhole spacetime and is zero at r ¼ 0. One can easily showthat the Kretschmann scalar does not diverge in the range0 � r <1. However, the spacetime has a conic geometryand has a conical singularity at r ¼ 0, since

limr!0

1

r

ffiffiffiffiffiffiffiffiffig

grr

s� 1: (25)

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 ¼ 0. The canonicalsingularity can be removed if one identifies the coordinate with the period

Period ¼ 2�

�limr!0

1

r

ffiffiffiffiffiffiffiffiffig

grr

s ��1 ¼ 2�ð1� 4 Þ; (26)

where is given by

¼ 1

4

�1� 2

lrþf000

�: (27)

In Eq. (27), f000 is the value of the second derivative of fðrÞat r ¼ 0, which can be calculated as

f000 ¼ 2r2þk000 ; (28)

where k000 ¼ k00ðr ¼ 0Þ. By the above analysis, one con-

cludes that near the origin r ¼ 0 the metric (19) may bewritten as

r2þl2

ð�dt2 þ dX2Þ þ 1

r4þk000½dr2 þ ðlr3þk000Þ2r2d2�: (29)

This metric describes a spacetime that is locally flat but hasa conical singularity at r ¼ 0with a deficit angle � ¼ 8� ,which is proportional to the brane tension at r ¼ 0 [24]. Ofcourse, one may ask for the completeness of the spacetimewith r � 0 (or � � rþ). It is easy to see that the spacetimedescribed by Eq. (19) is both null and timelike geodesicallycomplete as in the case of four-dimensional solutions[16,25]. In fact, one can show that every null or timelikegeodesic starting from an arbitrary point can either extendto infinite values of the affine parameter along the geodesicor end on a singularity at r ¼ 0 [9].

Now we investigate the effects of different parameters ofthe solution on the deficit angle of the spacetime. First, we

investigate the effect of the nonlinearity of the magneticfield on the deficit angle. In order to do this, we plot �versus the parameter �. This is shown in Fig. 1, whichshows that as � decreases (the nonlinearity of the electro-magnetic field becomes more evident) the deficit angleincreases. Since for fixed values of mass, charge, and thecosmological constant the parameter � has a minimumvalue which can be evaluated by the fact that ðr ¼ 0Þ<1, the deficit angle starts from a maximum value �max for�min and goes to its minimum value as � goes to infinity.Second, we consider the effect of Lovelock terms on the

deficit angle in the following subsection.

A. Static magnetic branes in lower order Lovelockgravity

As two special cases, we introduce the static magneticsolutions of Einstein and Gauss-Bonnet gravity in thepresence of a Born-Infeld electromagnetic field. The firstspecial case is when both of the Lovelock coefficients arezero. Solving Eq. (9) in the case of �2 ¼ �3 ¼ 0 andperforming the transformation (18), one obtains

fEinðrÞ ¼ r2 þ r2þl2

kðrÞ; (30)

where kðrÞ is given by Eq. (23) and rþ is again the largestreal root of kðr ¼ 0Þ ¼ 0. Of course, one may also obtainthe above solution by calculating the limit of fðrÞ of thirdorder Lovelock gravity given in Eq. (20) as �2 and�3 go tozero.As the second special case, we introduce the magnetic

solution in Gauss-Bonnet gravity which may be obtainedby solving Eq. (9) in the case of�3 ¼ 0 and performing thetransformation (18). In this case, one obtains

fGBðrÞ ¼ r2 þ r2þ2�2

�1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 4�2

l2kðrÞ

s �; (31)

where again kðrÞ and rþ are the same as that of Einstein or

3.23

3.24

3.25

3.26

3.27

3 4 5 6 7 8 9 10

FIG. 1. � versus � for n ¼ 6, q ¼ 0:2, rþ ¼ :67, l ¼ 1, �2 ¼0:2, and �3 ¼ 0:1.

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Lovelock gravities. Now we investigate the effects ofLovelock terms on the properties of the spacetime suchas the deficit angle. According to Eq. (29), the deficit angledepends on k000 ¼ k00ðr ¼ 0Þ and rþ, which are the same for

Einstein, Gauss-Bonnet, and third order Lovelock gravitiesand are independent of Lovelock coefficients. Thus, onemay expect that Lovelock terms of any order have no effecton the properties of our spacetime, which is a warpedproduct of an ðn� 2Þ-dimensional zero-curvature space-time and a curved two-dimensional space. In this relation,we will give some comments in the last section.

IV. SPINNING MAGNETIC BRANES

Nowwewant to endow our spacetime solution (6) with aglobal rotation. It should be mentioned that the (t ¼ const,r ¼ const) boundary of our spacetime is curvature-free,and therefore this solution is not a counterpart of the Kerr-type solution in Lovelock gravity. We first consider thesolutions with one rotation parameter. In order to addangular momentum to the spacetime, we perform the fol-lowing rotation boost in the t- plane:

t � �t� a; � �� a

l2t; (32)

where a is the rotation parameter and � ¼ 1þ a2=l2.Substituting Eq. (32) into Eq. (6), we obtain

ds2 ¼ � r2 þ r2þl2

ð�dt� adÞ2 þ r2dr2

ðr2 þ r2þÞfðrÞ

þ l2fðrÞ�a

l2dt��d

�2 þ r2 þ r2þ

l2dX2; (33)

where fðrÞ is the same as fðrÞ given in Eqs. (20)–(23). Thenonvanishing electromagnetic field components become

Frt ¼ � a

�l2Fr ¼ 2ln�3aq

rn�1ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p : (34)

The transformation (32) generates a new metric, because itis not a permitted global coordinate transformation. Thistransformation can be done locally but not globally [26].Therefore, the metrics (6) and (33) can be locally mappedinto each other but not globally, and so they are distinct.Again, this spacetime has no horizon and curvature singu-larity. However, it has a conical singularity at r ¼ 0.

Second, we study the rotating solutions with more rota-tion parameters. The rotation group in nþ 1 dimensions isSOðnÞ, and therefore the number of independent rotationparameters is ½n=2�, where ½x� is the integer part of x. Wenow generalize the above solution given in Eq. (6) withk � ½n=2� rotation parameters. This generalized solutioncan be written as

ds2 ¼� r2 þ r2þl2

��dt�Xk

i¼1

aidi

�2 þ fðrÞ

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

pdt

� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p Xki¼1

aidi

�2 þ r2

ðr2 þ r2þÞfðrÞdr2

þ r2 þ r2þl2ð�2 � 1Þ

Xki<j

ðaidj � ajdiÞ2 þ ðr2 þ r2þÞl2

dX2;

(35)

where� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þP

ki a

2i =l

2q

, dX2 is the Euclidean metric on

the ðn� k� 1Þ-dimensional submanifold with volumeVn�k�1, and fðrÞ is the same as fðrÞ given in Eq. (11).The nonvanishing components of the electromagnetic fieldtensor are

Frt ¼ �ð�2 � 1Þ�ai

Fri ¼ 2qln�2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p

rn�1ffiffiffiffiffiffiffiffiffiffiffiffiffi1�

p : (36)

In the remaining part of this section, we compute theconserved quantities of the solution. In general, the con-served quantities are divergent when evaluated on thesolutions. A systematic method of dealing with this diver-gence for asymptotically AdS solutions of Einstein gravityis through the use of the counterterms method inspired bythe anti-de Sitter conformal field theory correspondence[27]. For asymptotically AdS solutions of Lovelock gravity

with flat boundary Rabcdð�Þ ¼ 0, the finite energy-momentum tensor is [8,22]

Tab ¼ 1

8�

�ðKab � K�abÞ þ 2�0

2ð3Jab � J�abÞ

þ 3�03ð5Pab � P�abÞ þ n� 1

L�ab

�; (37)

where L is a constant which depends on l, �02, and �0

3 that

reduces to l as the Lovelock coefficients vanish. For thespecial case that �3 ¼ �2

2, L becomes

L ¼ 15l2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2ð1� �Þp

5l2 þ 9�2 � l2�2 � 4l2�;

where � ¼ ð1� 3�2=l2Þ�1=3. In Eq. (37), Kab is the ex-

trinsic curvature of the boundary, K is its trace, �ab is theinduced metric of the boundary, and J and P are traces ofJab and Pab given as

Jab ¼ 13ð2KKacK

cb þ KcdK

cdKab � 2KacKcdKdb

� K2KabÞ; (38)

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Pab ¼ 15f½K4 � 6K2KcdKcd þ 8KKcdK

deK

ec

� 6KcdKdeKefK

fc þ 3ðKcdKcdÞ2�Kab

� ð4K3 � 12KKedKed þ 8KdeK

efK

fdÞKacKcb

� 24KKacKcdKdeK

eb þ ð12K2 � 12KefK

ef� KacK

cdKdb þ 24KacKcdKdeK

efKbfg: (39)

One may note that, when �i’s go to zero, the finite stress-energy tensor (37) reduces to that of asymptotically AdSsolutions of Einstein gravity with a flat boundary.

To compute the conserved charges of the spacetime, wechoose a spacelike surface B in @M with metric �ij and

write the boundary metric in Arnowitt-Deser-Misner form:

�abdxadxa ¼ �N2dt2 þ �ijðd’i þ VidtÞðd’j þ VjdtÞ;

(40)

where the coordinates ’i are the angular variables parame-trizing the hypersurface of constant r around the origin andN 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. (37) can be written as

Q ð�Þ ¼ZBdn�1’

ffiffiffiffi�

pTabn

a�b; (41)

where � is the determinant of the metric �ij and na is the

timelike unit normal vector to the boundary B. In thecontext of counterterm method, the limit in which theboundary B becomes infinite (B1) is taken, and the coun-terterm prescription ensures that the action and conservedcharges are finite. For our case of horizonless rotatingspacetimes, the first Killing vector is � ¼ @=@t, and there-fore its associated conserved charge of the brane is themass per unit volume Vn�k�1 calculated as

M ¼ZBdn�1x

ffiffiffiffi�

pTabn

a�b ¼ ð2�Þk4

½nð�2 � 1Þ þ 1�m:

(42)

The second class of conserved quantities is the angularmomentum per unit volume Vn�k�1 associated with therotational Killing vectors �i ¼ @=@i, which may be cal-culated as

Ji ¼ZBdn�1x

ffiffiffiffi�

pTabn

a�bi ¼ ð2�Þk4

nm�ai: (43)

Next, we calculate the electric charge of the solutions.To determine the electric field, we should consider theprojections of the electromagnetic field tensor on specialhypersurfaces. The normal to such hypersurfaces for thespacetimes with a longitudinal magnetic field is

u0 ¼ 1

N; ur ¼ 0; ui ¼ �Ni

N;

and the electric field is E� ¼ g��F��u�. Then the electric

charge per unit volume Vn�k�2 can be found by calculatingthe flux of the electromagnetic field at infinity, yielding

Q ¼ ð2�Þk2

qlffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p: (44)

Note that the electric charge is proportional to the rotationparameter and is zero for the case of static solutions. Thisquantity and the conserved quantities given by Eqs. (42)and (43) show that the metrics (6) and (35) cannot bemapped into each other globally.

V. CLOSING REMARKS

In this paper, we investigated the effects of the nonline-arity of the electromagnetic fields on the properties of thespacetime by finding a new class of magnetic solutions inthird order Lovelock gravity in the presence of a nonlinearBorn-Infeld electromagnetic field. This class of solutionsyields an ðnþ 1Þ-dimensional spacetime with a longitudi-nal nonlinear magnetic field [the only nonzero componentof the vector potential is AðrÞ] generated by a static

magnetic brane. We found that these solutions have nocurvature singularity and no horizons but have conic sin-gularity at r ¼ 0with a deficit angle �which is sensitive tothe nonlinearity of the electromagnetic field. We foundthat, as the effects of the nonlinearity of the electromag-netic fields become larger, the deficit angle increases. Inthese static spacetimes, the electric field vanishes, andtherefore the brane has no net electric charge. Next, weendow the spacetime with rotation. For the spinning brane,when the rotation parameters are nonzero, the brane has anet electric charge density which is proportional to the

magnitude of the rotation parameters given byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2 � 1

p.

We also applied the counterterm method in order to calcu-late the conserved quantities of the spacetime and foundthat these conserved quantities do not depend on the Born-Infeld parameter �. This can be understand easily, since atthe boundary at infinity the effects of the nonlinearity ofthe electromagnetic fields vanish.On the effects of Lovelock terms on the properties of our

spacetime, we found that the Lovelock terms have noeffects on the deficit angle of the spacetime. This alsohappens for the mass, angular momenta, and charge givenby Eqs. (41)–(43). This feature of our solution is a genericproperty of any kind of solutions of any order of Lovelockgravity for spacetimes which are warped products of anðn� 2Þ-dimensional zero-curvature space (spacetime) anda two-dimensional curved spacetime (space). For example,all of the thermodynamic quantities of black holes ofLovelock gravity with a flat horizon, such as the entropy,the temperature, the free energy, the mass, etc., are inde-pendent of the Lovelock coefficients [28]. Of course, onemay note that the thermodynamic quantities of black holesof Lovelock gravity with a nonflat horizon depend on theLovelock coefficients [28], and therefore one may predictthat the deficit angle of a spacetime with conic geometry

M.H. DEHGHANI, N. BOSTANI, AND S.H. HENDI PHYSICAL REVIEW D 78, 064031 (2008)

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which is a warped product of an ðn� 2Þ-dimensionalnonzero-curvature spacetime and a two-dimensionalcurved space should depend on the Lovelock coefficients.This has been shown for Gauss-Bonnet gravity in thepresence of codimension two branes [29]. Finally, wegive some comments on the number of independent solu-tions in various order of Lovelock gravity. For static solu-tions of Lovelock gravity, the rr components of the fieldequation can be integrated in order to get an algebraicequation for fðrÞ. This algebraic equation is linear for theEinstein equation, quadratic for Gauss-Bonnet gravity,cubic for third order Lovelock gravity, quartic for fourth

order Lovelock gravity, and so on. This suggests that fðrÞhas only one branch in Einstein, two branches in Gauss-Bonnet, three branches in third order Lovelock gravity, andso on. But the point which should be regarded carefully isthe number of real branches of fðrÞ. Although the numberof real branches is two in Gauss-Bonnet gravity, it is one inthird order Lovelock gravity.

ACKNOWLEDGMENTS

This work has been supported by Research Institute forAstrophysics and Astronomy of Maragha.

[1] E. S. Fradkin and A.A. Tseytlin, Phys. Lett. 163B, 123(1985); E. Bergeshoeff, E. Sezgin, C.N. Pope, and P. K.Townsend, ibid. 188, 70 (1987); R. R. Metsaev, M.A.Rahmanov, and A.A. Tseytlin, ibid. 193, 207 (1987).

[2] D. Lovelock, J. Math. Phys. (N.Y.) 12, 498 (1971).[3] B. Zwiebach, Phys. Lett. 156B, 315 (1985); B. Zumino,

Phys. Rep. 137, 109 (1986).[4] M. Born and L. Infeld, Proc. R. Soc. A 144, 425 (1934).[5] D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656

(1985); J. T. Wheeler, Nucl. Phys. B268, 737 (1986).[6] D. L. Wiltshire, Phys. Lett. 169B, 36 (1986).[7] M.H. Dehghani and M. Shamirzaie, Phys. Rev. D 72,

124015 (2005).[8] M.H. Dehghani, N. Alinejadi, and S. H. Hendi, Phys. Rev.

D 77, 104025 (2008); S. H. Hendi and M.H. Dehghani,Phys. Lett. B 666, 116 (2008).

[9] M.H. Dehghani and N. Bostani, Phys. Rev. D 75, 084013(2007).

[10] B. Hoffmann, Phys. Rev. 47, 877 (1935).[11] H. D. de Oliveira, Classical Quantum Gravity 11, 1469

(1994); G.W. Gibbons and D.A. Rasheed, Nucl. Phys.B454, 185 (1995).

[12] R. G. Cai, D.W. Pang, and A. Wang, Phys. Rev. D 70,124034 (2004); T. K. Dey, Phys. Lett. B 595, 484 (2004).

[13] T. Levi-Civita, Rend. Reale Accad. Lincei Cl. Sci. Fis.Mat. Nat. 28, 3 (1919); L. Marder, Proc. R. Soc. A 244,524 (1958).

[14] A. Vilenkin, Phys. Rev. D 23, 852 (1981); A. Banerjee, N.Banerjee, and A.A. Sen, ibid. 53, 5508 (1996).

[15] B. C. Mukherji, Bull. Calcutta Math. Soc. 30, 95 (1938);E. Witten, Nucl. Phys. B249, 557 (1985).

[16] O. J. C. Dias and J. P. S. Lemos, Classical QuantumGravity 19, 2265 (2002).

[17] O. J. C. Dias and J. P. S. Lemos, J. High Energy Phys. 01(2002) 006.

[18] O. J. C. Dias and J. P. S. Lemos, Phys. Rev. D 66, 024034(2002).

[19] M.H. Dehghani, A. Sheykhi, and S.H. Hendi, Phys. Lett.B 659, 476 (2008).

[20] L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370(1999); 83, 4690 (1999).

[21] G. Dvali, G. Gabadadze, and M. Porrati, Phys. Lett. B 485,208 (2000); G. Dvali and G. Gabadadze, Phys. Rev. D 63,065007 (2001).

[22] M.H. Dehghani, N. Bostani, and A. Sheykhi, Phys. Rev. D73, 104013 (2006).

[23] F. Muller-Hoissen, Phys. Lett. 163B, 106 (1985).[24] S. Randjbar-Daemi and V. Rubakov, J. High Energy Phys.

10 (2004) 054; H.M. Lee and A. Papazoglou, Nucl. Phys.B705, 152 (2005).

[25] J. H. Horne and G. T. Horowitz, Nucl. Phys. B368, 444(1992).

[26] J. Stachel, Phys. Rev. D 26, 1281 (1982).[27] J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998).[28] T. Torii and H. Maeda, Phys. Rev. D 71, 124002 (2005);

72, 064007 (2005); R. G. Cai, ibid. 65, 084014 (2002);M.H. Dehghani and R. B. Mann, ibid. 73, 104003 (2006);M.H. Dehghani, G.H. Bordbar, and M. Shamirzaie, ibid.74, 064023 (2006); M.H. Dehghani and S. H. Hendi, Int.J. Mod. Phys. D 16, 1829 (2007).

[29] P. Wang and X.H. Meng, Phys. Rev. D 71, 024023 (2005).

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