Magnetic Branes in Third Order Lovelock-Born-Infeld Gravity

Abstract

Considering both the nonlinear invariant terms constructed by the electromagnetic field and the Riemann tensor in gravity action, we obtain a new class of $(n+1)$-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 $\delta$. 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.

Full text

Page 1

arXiv:0806.1429v2 [gr-qc] 28 Aug 2008
Magnetic Branes in Third Order Lovelock-Born-Infeld Gravity
M. H. Dehghani1,2∗N. Bostani1and S. H. Hendi3†
1Physics Department and Biruni Observatory,
College of Sciences, Shiraz University, Shiraz 71454, Iran
2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM), Maragha, Iran
3Physics Department, College of Sciences,
Yasouj University, Yasouj 75914, Iran
Abstract
Considering both the nonlinear invariant terms constructed by the electromagnetic field and the
Riemann tensor in gravity action, we obtain a new class of (n + 1)-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 δ. 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.
∗email address: mhd@shirazu.ac.ir
†hendi@mail.yu.ac.ir
1

Page 2

I. INTRODUCTION
Actions of Lovelock gravity and Born-Infeld electrodynamics have been the subject of
wide interest in recent years. This is due to the fact that both of them emerge in the low-
energy limit of string theory [1]. On the gravity side, string theories in their low-energy limit
give rise to effective models of gravity in higher dimensions which involve higher curvature
terms, while on the electrodynamics side the effective action for the open string ending on
D-branes can be written in a Born-Infeld form. While Lovelock gravity [2] was proposed to
have field equations with at most second order derivatives of the metric [3], the nonlinear
electrodynamics was proposed, by Born and Infeld, with the aim of obtaining a finite value
for the self-energy of a pointlike charge [4]. Lovelock gravity reduces to Einstein gravity in
four dimensions and also in the weak field limit, while the Lagrangian of the Born-Infeld
(BI) electrodynamics reduces to the Maxwell Lagrangian in the weak field limit. There have
been considerable works on both of these theories. In Lovelock gravity, there have been
some attempts for understanding the role of the higher curvature terms from various points
of view. For example, exact static spherically symmetric black hole solutions of the second
order Lovelock gravity have been found in Ref. [5], and of the Gauss-Bonnet-Maxwell model
in Ref. [6]. An exact static solution in third order Lovelock gravity was introduced in Ref. [7]
and of the charged rotating black brane solution in Ref. [8], and the magnetic solutions with
longitudinal and angular magnetic field were considered in Ref. [9]. All of these works were
in the presence of a linear electromagnetic field. The first attempt to relate the nonlinear
electrodynamics and gravity has been done by Hoffmann [10]. He obtained a solution of the
Einstein equations for a pointlike Born-Infeld charge, which is devoid of the divergence of the
metric at the origin that characterizes the Reissner-Nordstr¨ om solution. However, a conical
singularity remained there, as it was later objected by Einstein and Rosen. The spherically
symmetric solutions in Einstein-Born-Infeld gravity with or without a cosmological constant
have been considered by many authors [11, 12].
In this paper we are dealing with the issue of the spacetimes generated by static and spin-
ning brane sources which are horizonless and have non-trivial external solutions. These kinds
of solutions have been investigated by many authors in four dimensions. Static uncharged
cylindrically symmetric solutions of Einstein gravity in four dimensions were considered in
Ref. [13]. Similar static solutions in the context of cosmic string theory were found in Ref.
2

Page 3

[14]. All of these solutions [13, 14] are horizonless and have a conical geometry; they are
everywhere flat except at the location of the line source. The extension to include the elec-
tromagnetic field has also been done [15, 16]. In three dimensions, the rotating magnetic
solutions of Einstein gravity have been studied in Ref. [17], while those of Brans-Dicke
gravity have been considered in Ref. [18]. The generalization of these solutions in Einstein
gravity in the presence of a dilaton and Born-Infeld electromagnetic field has been done
in Ref. [19]. Our aim in this paper is to construct (n + 1)-dimensional horizonless solu-
tions of third order Lovelock gravity in the presence of a nonlinear electromagnetic field
and investigate the effects of the nonlinearity of the electromagnetic field on the properties
of the spacetime such as the deficit angle of the spacetime. The first reason for studying
higher-dimensional solutions of gravity, which invite higher order Lovelock terms, is due to
the scenario of string theory and brane cosmology. The idea of brane cosmology, which is
also consistent with string theory, suggests that matter and gauge interaction (described by
an open string) may be localized on a brane embedded into a higher-dimensional spacetime,
and all gravitational objects are higher-dimensional [20, 21]. The second reason for studying
higher-dimensional solutions is the fact that four-dimensional solutions have a number of
remarkable properties. It is natural to ask whether these properties are general features of
the solutions or whether they crucially depend on the world being four-dimensional. Finally,
we want to check whether the counterterm method introduced in Refs. [8, 22] can be applied
to the case of solutions in the presence of a nonlinear electromagnetic field, too.
The outline of our paper is as follows. We give a brief review of the field equations of
Lovelock gravity in the presence of a Born-Infeld electromagnetic field in Sec. II. In Sec.
III we present static horizonless solutions which produce a longitudinal magnetic field and
investigate the effects of the nonlinearity of the electromagnetic field and Lovelock terms on
the deficit angle of the spacetime. Section IV will be devoted to the generalization of these
solutions to the case of rotating solutions and use of the counterterm method to compute
the conserved quantities of them. We finish our paper with some concluding remarks.
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 most general
3

Page 4

symmetric conserved tensor containing no more than second derivatives of the metric, is
Lovelock theory. Lovelock found the most general symmetric conserved tensor satisfying
this property. The resultant tensor is nonlinear in the Riemann tensor and differs from the
Einstein tensor only if the spacetime has more than 4 dimensions. The Lovelock tensor in
(n + 1) dimensions may be written as [2]
[n/2]
?
i=1
α′
i[H(i)
µν−1
2gµνL(i)] + Λgµν= 8πTµν, (1)
where Λ = −n(n − 1)/2l2is the cosmological constant for asymptotically anti-de Sitter
(AdS) solutions, [n/2] denotes the integer part of n/2, α′
coefficients, Tµνis the energy-momentum tensor, and H(i)
µν=i
2iδµ1µ2···µ2i
L(i)=1
1= 1, α′
i’s for i ≥ 2 are Lovelock
µν and L(i)are
H(i)
µ ν2···ν2iR
ν2
µ1µ2ν R
ν3ν4
µ3µ4
···R
ν2i−1ν2i
µ2i−1µ2i
,(2)
2iδµ1µ2···µ2i
ν1 ν2···ν2iR
ν1ν2
µ1µ2
···R
ν2i−1ν2i
µ2i−1µ2i
.(3)
In this paper, we consider up to third order Lovelock terms. The first term is just the
Einstein tensor, and the second and third terms are Gauss-Bonnet and third order Lovelock
tensors, respectively, which are written in terms of Riemann tensor explicitly in Ref. [23].
In the presence of a Born-Infeld electromagnetic field, the energy-momentum tensor of Eq.
(1) is
Tµν=
1
4π

1
4gµνL(F) +
FµλFνλ
?
1 +F2
2β2

,(4)
where Fµνis the nonlinear electromagnetic field tensor which satisfies the BI equation

1 +F2
∂µ

√−gFµν
?
2β2

= 0,(5)
and F2= FµνFµν. The parameter β is called the Born-Infeld parameter which has the
dimension of length−2, and, as it goes to ∞, the BI equation reduces to the standard
Maxwell equation.
III.STATIC MAGNETIC BRANES
Here we want to obtain the (n + 1)-dimensional solutions of Eqs. (1-5) which produce
longitudinal magnetic fields in the Euclidean submanifold spans by xicoordinates (i =
4

Page 5

1,...,n − 2). We will work with the following ansatz for the metric [16]:
ds2= −ρ2
where dX2=?n−2
The angular coordinate φ is dimensionless as usual and ranges in [0,2π], while xi’s range in
(−∞,∞). The motivation for this metric gauge [gtt∝ −ρ2and (gρρ)−1∝ gφφ] instead of
the usual Schwarzschild gauge [(gρρ)−1∝ gttand gφφ∝ ρ2] comes from the fact that we are
l2dt2+dρ2
f(ρ)+ l2f(ρ)dφ2+ρ2
l2dX2,(6)
i=1(dxi)2is the Euclidean metric on the (n − 2)-dimensional submanifold.
looking for a horizonless solution.
The electromagnetic field equation (5) can be integrated immediately to give
Fφρ=
2qln−1
ρn−1√1 − η,
(7)
where q is the charge parameter of the solution and
η =4q2l2n−4
β2ρ2(n−1).(8)
Equation (7) shows that ρ should be greater than ρ0= (2qln−2β)1/(n−1)in order to have a
real nonlinear electromagnetic field and consequently a real spacetime. To find the function
f(ρ), one may use any components of Eq. (1). The simplest equation is the ρρ component
of these equations which can be written as
?α3ρf2− 2α2ρ3f + ρ5?f′+n − 6
+(n − 2)ρ4f −n
3
α3f3− (n − 4)α2ρ2f2,
?1 − η),
l2ρ6=
β2ρ6
2(n − 1)(1 −
(9)
where the prime denotes the derivative with respect to ρ and
α2= (n − 2)(n − 3)α′
2,α3= 72(n−2
4)α′
3. (10)
The only real solution of Eq. (9) is
f(ρ) =α2ρ2
α3
?
1 −
??
γ + j2(ρ) + j(ρ)
?1/3
+ γ1/3??
γ + j2(ρ) + j(ρ)
?−1/3?
,(11)
where
γ = (α3
α2
2
− 1)3,(12)
j(ρ) = −1 +3α3
k(ρ) = 1 +2ml3
2α2
2
−
3α2
2l2α3
β2l2h(η)
2n(n − 1).
3
2
k(ρ),(13)
ρn
+ (14)
5