# Topological Black Holes in Lovelock-Born-Infeld Gravity

M. H. Dehghani1,2111email address<EMAIL_ADDRESS>N. Alinejadi1 and S. H.
Hendi3222email address<EMAIL_ADDRESS>1Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
2Research Institute for Astrophysics and Astronomy of Maragha (RIAAM),
Maragha, Iran
3 Physics Department, College of Sciences, Yasouj University, Yasouj 75914,
Iran

###### Abstract

In this paper, we present topological black holes of third order Lovelock
gravity in the presence of cosmological constant and nonlinear electromagnetic
Born-Infeld field. Depending on the metric parameters, these solutions may be
interpreted as black hole solutions with inner and outer event horizons, an
extreme black hole or naked singularity. We investigate the thermodynamics of
asymptotically flat solutions and show that the thermodynamic and conserved
quantities of these black holes satisfy the first law of thermodynamic. We
also endow the Ricci flat solutions with a global rotation and calculate the
finite action and conserved quantities of these class of solutions by using
the counterterm method. We compute the entropy through the use of the Gibbs-
Duhem relation and find that the entropy obeys the area law. We obtain a
Smarr-type formula for the mass as a function of the entropy, the angular
momenta, and the charge, and compute temperature, angular velocities, and
electric potential and show that these thermodynamic quantities coincide with
their values which are computed through the use of geometry. Finally, we
perform a stability analysis for this class of solutions in both the canonical
and the grand-canonical ensemble and show that the presence of a nonlinear
electromagnetic field and higher curvature terms has no effect on the
stability of the black branes, and they are stable in the whole phase space.

###### pacs:

04.40.Nr, 04.20.Jb, 04.70.Bw, 04.70.Dy

##  I Introduction

Over the last few years, several extra-dimensional models have been introduced
in an attempt to deal with the hierarchy problem. These models can lead to
rather unique and spectacular signatures at Terascale colliders such as the
LHC and ILC. In higher dimensions, it is known that the Einstein-Hilbert (EH)
Lagrangian, $R$, can only be regarded as the first order term in an effective
action, so one may on general grounds expect that as one probes energies
approaching the fundamental scale, significant deviations from EH expectations
are likely to appear. This motivates one to consider the more general class of
gravitational action:

$I_{G}=\frac{1}{16\pi}\int
d^{n+1}x\sqrt{-g}\mathcal{F}(R,R_{\mu\nu}R^{\mu\nu},R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}).$

The presence of higher curvature terms can also be seen in the renormalization
of quantum field theory in curved spacetime Birrel , or in the construction of
low energy effective action of string theory Green . Among the higher
curvature gravity theories, the so-called Lovelock gravity is quite special,
whose Lagrangian consist of the dimensionally extended Euler densities. This
Lagrangian is obtained by Lovelock as he tried to calculate the most general
tensor that satisfies properties of Einstein’s tensor in higher dimensions
Lovelock . Since the Lovelock tensor contains derivatives of metrics of order
not higher than two, the quantization of linearized Lovelock theory is free of
ghosts Boulware . Thus, it is natural to study the effects of higher curvature
terms on the properties and thermodynamics of black holes.

Accepting the nonlinear terms of the invariants constructed by Riemann tensor
on the gravity side of the action, it seems natural to add the nonlinear terms
in the matter action too. Thus, in the presence of an electromagnetic field,
it is worthwhile to apply the action of Born-Infeld BI  instead of the Maxwell
action. In this paper, we generalize static and rotating black hole solutions
of third order Lovelock gravity in the presence of Maxwell field DSh ; DM1  to
the case of these solutions in the presence of nonlinear electromagnetic
fields. Indeed, it is interesting to explore new black hole solutions in
higher curvature gravity and investigate which properties of black holes are
peculiar to Einstein gravity, and which are robust features of all generally
covariant theories of gravity. The first aim to relate the nonlinear
electrodynamics and gravity has been done by Hoffmann Hoffmann . 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-Nordström 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 EBI ; Dey , while the rotating
solutions of this theory is investigated in Rastegar . Also, these kinds of
solutions in the presence of a dilaton field have been introduced in DHSR .
The static black hole solutions of Gauss-Bonnet-Born-Infeld gravity have been
constructed in Ref. Wil2 , and the rotating solution of this theory has been
considered in DH .

The out line of our paper is as follows. We present the topological black
holes of third order Lovelock gravity in the presence of Born-Infeld field in
Sec. II. In Sec. III, we calculate the thermodynamic quantities of
asymptotically flat solutions and investigate the first law of thermodynamics.
In Sec. IV we introduce the rotating solutions with flat horizon and compute
the thermodynamic and conserved quantities of them. We also perform a
stability analysis of the solutions both in canonical and grand canonical
ensemble. We finish our paper with some concluding remarks.

##  II  Topological Black Holes

The action of third order Lovelock gravity in the presence of nonlinear Born-
Infeld electromagnetic field is

$I_{G}=\frac{1}{16\pi}\int
d^{n+1}x\sqrt{-g}\left(-2\Lambda+\mathcal{L}_{1}+\alpha_{2}\mathcal{L}_{2}+\alpha_{3}\mathcal{L}_{3}+L(F)\right),$
(1)

where $\Lambda$ is the cosmological constant, $\alpha_{2}$ and $\alpha_{3}$
are the second and third order Lovelock coefficients, $\mathcal{L}_{1}=R$ is
just the Einstein-Hilbert Lagrangian,
$\mathcal{L}_{2}=R_{\mu\nu\gamma\delta}R^{\mu\nu\gamma\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$
is the Gauss-Bonnet Lagrangian,

$\displaystyle\mathcal{L}_{3}$ $\displaystyle=$ $\displaystyle
2R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\rho\tau}R_{\phantom{\rho\tau}{\mu\nu}}^{\rho\tau}+8R_{\phantom{\mu\nu}{\sigma\rho}}^{\mu\nu}R_{\phantom{\sigma\kappa}{\nu\tau}}^{\sigma\kappa}R_{\phantom{\rho\tau}{\mu\kappa}}^{\rho\tau}+24R^{\mu\nu\sigma\kappa}R_{\sigma\kappa\nu\rho}R_{\phantom{\rho}{\mu}}^{\rho}$
(2)
$\displaystyle+3RR^{\mu\nu\sigma\kappa}R_{\sigma\kappa\mu\nu}+24R^{\mu\nu\sigma\kappa}R_{\sigma\mu}R_{\kappa\nu}+16R^{\mu\nu}R_{\nu\sigma}R_{\phantom{\sigma}{\mu}}^{\sigma}-12RR^{\mu\nu}R_{\mu\nu}+R^{3}$

is the third order Lovelock Lagrangian, and $L(F)$ is the Born-Infeld
Lagrangian given as

$L(F)=4\beta^{2}\left(1-\sqrt{1+\frac{F^{2}}{2\beta^{2}}}\right).$ (3)

In the limit $\beta\rightarrow\infty$, $L(F)$ reduces to the standard Maxwell
form $L(F)=-F^{2}$, where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Varying the action
(1) with respect to the metric tensor $g_{\mu\nu}$ and electromagnetic vector
field $A_{\mu}$ the equations of gravitation and electromagnetic fields are
obtained as:

$G_{\mu\nu}^{(1)}+\Lambda
g_{\mu\nu}+\alpha_{2}G_{\mu\nu}^{(2)}+\alpha_{3}G_{\mu\nu}^{(3)}=\frac{1}{2}g_{\mu\nu}L(F)+\frac{2F_{\mu\lambda}F_{{\nu}}^{\lambda}}{\sqrt{1+\frac{F^{2}}{2\beta^{2}}}},$
(4)
$\partial_{\mu}\left(\frac{\sqrt{-g}F^{\mu\nu}}{\sqrt{1+\frac{F^{2}}{2\beta^{2}}}}\right)=0,$
(5)

where $G_{\mu\nu}^{(1)}$ is the Einstein tensor, and $G_{\mu\nu}^{(2)}$ and
$G_{\mu\nu}^{(3)}$ are the second and third order Lovelock tensors given as
Muller :

$G_{\mu\nu}^{(2)}=2(R_{\mu\sigma\kappa\tau}R_{\nu}^{\phantom{\nu}\sigma\kappa\tau}-2R_{\mu\rho\nu\sigma}R^{\rho\sigma}-2R_{\mu\sigma}R_{\phantom{\sigma}\nu}^{\sigma}+RR_{\mu\nu})-\frac{1}{2}\mathcal{L}_{2}g_{\mu\nu},$
(6) $\displaystyle G_{\mu\nu}^{(3)}$ $\displaystyle=$
$\displaystyle-3(4R^{\tau\rho\sigma\kappa}R_{\sigma\kappa\lambda\rho}R_{\phantom{\lambda}{\nu\tau\mu}}^{\lambda}-8R_{\phantom{\tau\rho}{\lambda\sigma}}^{\tau\rho}R_{\phantom{\sigma\kappa}{\tau\mu}}^{\sigma\kappa}R_{\phantom{\lambda}{\nu\rho\kappa}}^{\lambda}+2R_{\nu}^{\phantom{\nu}{\tau\sigma\kappa}}R_{\sigma\kappa\lambda\rho}R_{\phantom{\lambda\rho}{\tau\mu}}^{\lambda\rho}$
(7)
$\displaystyle-R^{\tau\rho\sigma\kappa}R_{\sigma\kappa\tau\rho}R_{\nu\mu}+8R_{\phantom{\tau}{\nu\sigma\rho}}^{\tau}R_{\phantom{\sigma\kappa}{\tau\mu}}^{\sigma\kappa}R_{\phantom{\rho}\kappa}^{\rho}+8R_{\phantom{\sigma}{\nu\tau\kappa}}^{\sigma}R_{\phantom{\tau\rho}{\sigma\mu}}^{\tau\rho}R_{\phantom{\kappa}{\rho}}^{\kappa}$
$\displaystyle+4R_{\nu}^{\phantom{\nu}{\tau\sigma\kappa}}R_{\sigma\kappa\mu\rho}R_{\phantom{\rho}{\tau}}^{\rho}-4R_{\nu}^{\phantom{\nu}{\tau\sigma\kappa}}R_{\sigma\kappa\tau\rho}R_{\phantom{\rho}{\mu}}^{\rho}+4R^{\tau\rho\sigma\kappa}R_{\sigma\kappa\tau\mu}R_{\nu\rho}+2RR_{\nu}^{\phantom{\nu}{\kappa\tau\rho}}R_{\tau\rho\kappa\mu}$
$\displaystyle+8R_{\phantom{\tau}{\nu\mu\rho}}^{\tau}R_{\phantom{\rho}{\sigma}}^{\rho}R_{\phantom{\sigma}{\tau}}^{\sigma}-8R_{\phantom{\sigma}{\nu\tau\rho}}^{\sigma}R_{\phantom{\tau}{\sigma}}^{\tau}R_{\mu}^{\rho}-8R_{\phantom{\tau}{\sigma\mu}}^{\tau\rho}R_{\phantom{\sigma}{\tau}}^{\sigma}R_{\nu\rho}-4RR_{\phantom{\tau}{\nu\mu\rho}}^{\tau}R_{\phantom{\rho}\tau}^{\rho}$
$\displaystyle+4R^{\tau\rho}R_{\rho\tau}R_{\nu\mu}-8R_{\phantom{\tau}{\nu}}^{\tau}R_{\tau\rho}R_{\phantom{\rho}{\mu}}^{\rho}+4RR_{\nu\rho}R_{\phantom{\rho}{\mu}}^{\rho}-R^{2}R_{\nu\mu})-\frac{1}{2}\mathcal{L}_{3}g_{\mu\nu}.$

Here we want to obtain the $(n+1)$-dimensional static solutions of Eqs. (4)
and (5). We assume that the metric has the following form:

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (8)

where

$d\Omega^{2}=\left\\{\begin{array}[]{cc}d\theta_{1}^{2}+\sum\limits_{i=2}^{n-1}\prod\limits_{j=1}^{i-1}\sin^{2}\theta_{j}d\theta_{i}^{2}&k=1\\\
d\theta_{1}^{2}+\sinh^{2}\theta_{1}d\theta_{2}^{2}+\sinh^{2}\theta_{1}\sum\limits_{i=3}^{n-1}\prod\limits_{j=2}^{i-1}\sin^{2}\theta_{j}d\theta_{i}^{2}&k=-1\\\
\sum\limits_{i=1}^{n-1}d\phi_{i}^{2}&k=0\end{array}\right\\}$

represents the line element of an $(n-1)$-dimensional hypersurface with
constant curvature $(n-1)(n-2)k$ and volume $V_{n-1}$.

Using Eq. (5), one can show that the vector potential can be written as

$A_{\mu}=-\sqrt{\frac{(n-1)}{2n-4}}\frac{q}{r^{n-2}}\digamma(\eta)\delta_{\mu}^{0},$
(9)

where $q$ is an integration constant which is related to the charge parameter
and

$\eta=\frac{{(n-1)(n-2)q^{2}}}{2\beta^{2}r^{2n-2}}.$

In Eq. (9) and throughout the paper, we use the following abbreviation for the
hypergeometric function

${{}_{2}F_{1}}\left({\left[\frac{1}{2},\frac{n-2}{2n-2}\right],\left[\frac{3n-4}{2n-2}\right],-z}\right)=\digamma(z).$
(10)

The hypergeometric function $\digamma(\eta){\rightarrow 1}$ as
$\eta\rightarrow 0$ ($\beta\rightarrow\infty$) and therefore $A_{\mu}$ of Eq.
(9) reduces to the gauge potential of Maxwell field. One may show that the
metric function

$f(r)=k+\frac{r^{2}}{\alpha}\left(1-g(r)^{1/3}\right),$ (11)
$g(r)=1+\frac{3\alpha
m}{r^{n}}-\frac{12\alpha\beta^{2}}{n(n-1)}\left[1-\sqrt{1+\eta}-\frac{\Lambda}{2\beta^{2}}+\frac{(n-1)\eta}{(n-2)}\digamma(\eta)\right]$
(12)

satisfies the field equations (4) in the special case

$\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{(n-2)(n-3)},$ $\displaystyle\alpha_{3}$
$\displaystyle=$
$\displaystyle\frac{\alpha^{2}}{72(_{\phantom{-}{4}}^{n-2})},$

where $m$ is the mass parameter. Solutions of Gauss-Bonnet gravity are not
real in the whole range $0\leq r<\infty$ and one needs a transformation to
make them real DH ; DehBord . But, here the metric function $f(r)$ is real in
the whole range $0\leq r<\infty$.

In order to consider the asymptotic behavior of the solution, we put $m=q=0$
where the metric function reduces to

$f(r)=k+\frac{r^{2}}{\alpha}\left[1-\left(1+\frac{6\Lambda\alpha}{n(n-1)}\right)^{1/3}\right].$
(13)

Equation (13) shows that the asymptotic behavior of the solution is AdS or dS
provided $\Lambda<0$ or $\Lambda>0$. The case of asymptotic flat solutions
($\Lambda=0$) is permitted only for $k=1$.

As in the case of black holes of Gauss-Bonnet-Born-Infeld gravity Wil2 ; DH ,
the above metric given by Eqs. (8), (11) and (12) has an essential timelike
singularity at $r=0$. Seeking possible black hole solutions, we turn to
looking for the existence of horizons. The event horizon(s), if there exists
any, is (are) located at the root(s) of $g^{rr}=f(r)=0$. Denoting the largest
real root of $f(r)$ by $r_{+}$, we consider first the case that $f(r)$ has
only one real root. In this case $f(r)$ is minimum at $r_{+}$ and therefore
$f^{\prime}(r_{+})=0$. That is,

$(n-1)k\left[3(n-2)r_{+}^{4}+3(n-4)k\alpha
r_{+}^{2}+(n-6)k^{2}\alpha^{2}\right]+12r_{+}^{6}\beta^{2}\left(1-\sqrt{1+\eta_{+}}\right)-6{\Lambda}r_{+}^{6}=0.$
(14)

One can find the extremal value of mass, $m_{\mathrm{ext}}$, in terms of
parameters of metric function by finding $r_{+}$ from Eq. (14) and inserting
it into equation $f(r_{+})=0$. Then, the metric of Eqs. (8), (11) and (12)
presents a black hole solution with inner and outer event horizons provided
$m>m_{\mathrm{ext}}$, an extreme black hole for $m=m_{\mathrm{ext}}$
[temperature is zero since it is proportional to $f^{\prime}(r_{+})$] and a
naked singularity otherwise. It is a matter of calculation to show that
$m_{\mathrm{ext}}$ for $k=0$ becomes

$m_{\mathrm{ext}}=\frac{2(n-1)q_{\mathrm{ext}}^{2}}{n}\left(\frac{\Lambda(\Lambda-4\beta^{2})}{2(n-1)(n-2)\beta^{2}q_{\mathrm{ext}}^{2}}\right)^{(n-2)/(2n-2)}\digamma(\frac{\Lambda(\Lambda-4\beta^{2})}{4\beta^{4}}).$
(15)

The Hawking temperature of the black holes can be easily obtained by requiring
the absence of conical singularity at the horizon in the Euclidean sector of
the black hole solutions. One obtains

${T}_{+}=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{(n-1)k\left[3(n-2)r_{+}^{4}+3(n-4)k\alpha
r_{+}^{2}+(n-6)k^{2}\alpha^{2}\right]+12r_{+}^{6}\beta^{2}\left(1-\sqrt{1+\eta_{+}}\right)-6{\Lambda}r_{+}^{6}}{12\pi(n-1)r_{+}(r_{+}^{2}+k\alpha)^{2}}.$
(16)

It is worthwhile to note that $T_{+}$ is zero for $m=m_{\mathrm{ext}}$.

##  III Thermodynamics of Asymptotically Flat Black Holes for $k=1$

In this section, we consider the thermodynamics of spherically symmetric black
holes which are asymptotically flat. This is due to the fact that only the
entropy of asymptotically black holes of Lovelock gravity is well known Myer .
Usually entropy of black holes satisfies the so-called area law of entropy
which states that the black hole entropy equals one-quarter of the horizon
area Bek . One of the surprising and impressive features of this area law of
entropy is its universality. It applies to all kinds of black holes and black
strings of Einstein gravity Haw . However, in higher derivative gravity the
area law of entropy is not satisfied in general fails . It is known that the
entropy of asymptotically flat black holes of Lovelock gravity is Myer

$S=\frac{1}{4}\sum_{k=1}^{[(n-2)/2]}k\alpha_{k}\int
d^{n-1}x\sqrt{\tilde{g}}\tilde{\mathcal{L}}_{k-1},$ (17)

where the integration is done on the $(n-1)$-dimensional spacelike
hypersurface of the Killing horizon, $\tilde{g}_{\mu\nu}$ is the induced
metric on it, $\tilde{g}$ is the determinant of $\tilde{g}_{\mu\nu}$, and
$\tilde{\mathcal{L}}_{k}$ is the $k$th order Lovelock Lagrangian of
$\tilde{g}_{\mu\nu}$. Thus, the entropy for asymptotically flat black holes in
third order Lovelock gravity is

$S=\frac{1}{4}\int
d^{n-1}x\sqrt{\tilde{g}}\left(1+2\alpha_{2}\tilde{R}+3\alpha_{3}(\tilde{R}_{\mu\nu\sigma\kappa}\tilde{R}^{\mu\nu\sigma\kappa}-4\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu}+\tilde{R}^{2})\right),$
(18)

where $\tilde{R}_{\mu\nu\rho\sigma}$ and $\tilde{R}_{\mu\nu}$ are Riemann and
Ricci tensors and $\tilde{R}$ is the Ricci scalar for the induced metric
$\tilde{g}_{ab}$ on the $(n-1)$-dimensional horizon. It is a matter of
calculation to show that the entropy of black holes is

$S=\frac{V_{n-1}}{4}\left(r_{+}^{4}+\frac{2(n-1)}{n-3}\alpha
r_{+}^{2}+\frac{n-1}{n-5}\alpha^{2}\right)r_{+}^{n-5}.$ (19)

The charge of the black hole can be found by calculating the flux of the
electric field at infinity, yielding

$Q=\frac{V_{n-1}}{4\pi}\sqrt{\frac{(n-1)(n-2)}{2}}q.$ (20)

The electric potential $\Phi$, measured at infinity with respect to the
horizon, is defined by

$\Phi=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(21)

where $\chi=\partial/\partial t$ is the null generator of the horizon. One
finds

$\Phi=\sqrt{\frac{(n-1)}{2(n-2)}}\frac{q}{r_{+}^{n-2}}\digamma(\eta_{+}).$
(22)

The ADM (Arnowitt-Deser-Misner) mass of black hole can be obtained by using
the behavior of the metric at large $r$. It is easy to show that the mass of
the black hole is

$M=\frac{V_{n-1}}{16\pi}\left(n-1\right)m.$ (23)

We now investigate the first law of thermodynamics. Using the expression for
the entropy, the charge, and the mass given in Eqs. (19), (20) and (23), and
the fact that $f(r_{+})=0$, one obtains

$\displaystyle M(S,Q)$ $\displaystyle=$
$\displaystyle\frac{\left(n-1\right)}{16\pi}\left\\{\frac{2r_{+}^{n}}{n(n-1)}\left(2\beta^{2}\left[1-\sqrt{1+\Im}+\frac{(n-1)\Im}{(n-2)}\digamma(\Im)\right]-\Lambda\right)\right.$
(24) $\displaystyle\left.-r_{+}^{n-2}+\alpha
r_{+}^{n-4}-\frac{\alpha^{2}r_{+}^{n-6}}{3}\right\\},$

where

$\Im=\frac{16\pi^{2}Q^{2}}{\beta^{2}r_{+}^{2n-2}}.$

In Eq. (24) $r_{+}$ is the real root of Eq. (19) which is a function of $S$.
One may then regard the parameters $S$ and $Q$ as a complete set of extensive
parameters for the mass $M(S,Q)$ and define the intensive parameters conjugate
to them. These quantities are the temperature and the electric potential

$T=\left(\frac{\partial M}{\partial S}\right)_{Q},\ \ \ \
\Phi=\left(\frac{\partial M}{\partial Q}\right)_{S}.$ (25)

Computing $\partial M/\partial r_{+}$ and $\partial S/\partial r_{+}$ and
using the chain rule, it is easy to show that the intensive quantities
calculated by Eq. (25) coincide with Eqs. (16) and (22), respectively. Thus,
the thermodynamic quantities calculated in Eqs. (16) and (22) satisfy the
first law of thermodynamics,

$dM=TdS+\Phi dQ.$ (26)

##  IV Thermodynamics of Asymptotically AdS Rotating Black Branes with Flat
Horizon

Now, we want to endow our spacetime solution (8) for $k=0$ with a global
rotation. These kinds of rotating solutions in Einstein gravity have been
introduced in Lem . In order to add angular momentum to the spacetime, we
perform the following rotation boost in the $t-\phi_{i}$ planes

$t\mapsto\Xi t-a_{i}\phi_{i},\hskip
14.22636pt\phi_{i}\mapsto\Xi\phi_{i}-\frac{a_{i}}{l^{2}}t$ (27)

for $i=1...[n/2]$, where $[x]$ is the integer part of $x$. The maximum number
of rotation parameters is due to the fact that the rotation group in $n+1$
dimensions is $SO(n)$ and therefore the number of independent rotation
parameters is $[n/2]$. Thus the metric of an asymptotically AdS rotating
solution with $p\leq[n/2]$ rotation parameters for flat horizon can be written
as

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-f(r)\left(\Xi
dt-{{\sum_{i=1}^{p}}}a_{i}d\phi_{i}\right)^{2}+\frac{r^{2}}{l^{4}}{{\sum_{i=1}^{p}}}\left(a_{i}dt-\Xi
l^{2}d\phi_{i}\right)^{2}$ (28) $\displaystyle\ \text{
}+\frac{dr^{2}}{f(r)}-\frac{r^{2}}{l^{2}}{\sum_{i<j}^{p}}(a_{i}d\phi_{j}-a_{j}d\phi_{i})^{2}+r^{2}{{\sum_{i=p+1}^{n-1}}}d\phi_{i},$

where $\Xi=\sqrt{1+\sum_{i}^{k}a_{i}^{2}/l^{2}}$. Using Eq. (5), one can show
that the vector potential can be written as

$A_{\mu}=-\sqrt{\frac{(n-1)}{2n-4}}\frac{q}{r^{n-2}}\digamma(\eta)\left(\Xi\delta_{\mu}^{0}-\delta_{\mu}^{i}a_{i}\right)\text{(no
sum on }i\text{)}.$ (29)

One can obtain the temperature and angular momentum of the event horizon by
analytic continuation of the metric. One obtains

${T}_{+}{=}\frac{f^{\prime}(r_{+})}{4\pi\Xi}={{\frac{r_{+}}{2(n-1)\pi\Xi}}}\left(2\beta^{2}(1-\sqrt{1+\eta_{+}})-{\Lambda}\right),$
(30) $\Omega_{i}=\frac{a_{i}}{\Xi l^{2}},$ (31)

where $\eta_{+}=\eta(r=r_{+})$. Next, we calculate the electric charge and
potential of the solutions. The electric charge per unit volume $V_{n-1}$ can
be found by calculating the flux of the electric field at infinity, yielding

$Q=\frac{1}{4\pi}\sqrt{\frac{(n-1)(n-2)}{2}}\,\,\Xi q.$ (32)

Using Eq. (21) and the fact that
$\chi=\partial_{t}+{\sum_{i}^{k}}\Omega_{i}\partial_{\phi_{i}}$ is the null
generator of the horizon, the electric potential $\Phi$ is obtained as

$\Phi=\sqrt{\frac{(n-1)}{2(n-2)}}\frac{q}{\Xi r_{+}^{n-2}}\digamma(\eta_{+}).$
(33)

###  IV.1 Conserved quantities of the solutions

Here, we calculate the action and conserved quantities of the black brane
solutions. In general the action and conserved quantities of the spacetime are
divergent when evaluated on the solutions. A systematic method of dealing with
this divergence for asymptotically AdS solutions of Einstein gravity is
through the use of the counterterms method inspired by the anti-de Sitter
conformal field theory (AdS/CFT) correspondence Mal . For asymptotically AdS
solutions of Lovelock gravity with flat boundary,
$\widehat{R}_{abcd}(\gamma)=0$, the finite action is DBS ; DM1

$I=I_{G}+\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\left\\{L_{1b}+\alpha_{2}L_{2b}+\alpha_{3}L_{3b}\right\\}+\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\left(\frac{n-1}{L}\right),$
(34)

where $L$ is

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{15l^{2}\sqrt{\alpha(1-\lambda)}}{5l^{2}+9\alpha-l^{2}\lambda^{2}-4l^{2}\lambda},$
(35) $\displaystyle\lambda$ $\displaystyle=$
$\displaystyle(1-\frac{3\alpha}{l^{2}})^{1/3}.$ (36)

One may note that $L$ reduces to $l$ as $\alpha$ goes to zero. The first
integral in Eq. (34) is a boundary term which is chosen such that the
variational principle is well defined. In this integral $L_{1b}=K$,
$L_{2b}=2(J-2\widehat{G}_{ab}^{(1)}K^{ab})$ and

$L_{3b}=3(P-2\widehat{G}_{ab}^{(2)}K^{ab}-12\widehat{R}_{ab}J^{ab}+2\widehat{R}J-4K\widehat{R}_{abcd}K^{ac}K^{bd}-8\widehat{R}_{abcd}K^{ac}K_{e}^{b}K^{ed}),$

where $\gamma_{\mu\nu}$ and $K$ are induced metric and trace of extrinsic
curvature of boundary, $\widehat{G}_{ab}^{(1)}$ and $\widehat{G}_{ab}^{(2)}$
are the $n$-dimensional Einstein and second order Lovelock tensors (Eq. (6))
of the metric $\gamma_{ab}$ and $J$ and $P$ are the trace of

$J_{ab}=\frac{1}{3}(2KK_{ac}K_{b}^{c}+K_{cd}K^{cd}K_{ab}-2K_{ac}K^{cd}K_{db}-K^{2}K_{ab}),$
(37)

and

$\displaystyle P_{ab}$ $\displaystyle=$
$\displaystyle\frac{1}{5}\\{[K^{4}-6K^{2}K^{cd}K_{cd}+8KK_{cd}K_{e}^{d}K^{ec}-6K_{cd}K^{de}K_{ef}K^{fc}+3(K_{cd}K^{cd})^{2}]K_{ab}$
(38)
$\displaystyle-(4K^{3}-12KK_{ed}K^{ed}+8K_{de}K_{f}^{e}K^{fd})K_{ac}K_{b}^{c}-24KK_{ac}K^{cd}K_{de}K_{b}^{e}$
$\displaystyle+(12K^{2}-12K_{ef}K^{ef})K_{ac}K^{cd}K_{db}+24K_{ac}K^{cd}K_{de}K^{ef}K_{bf}\\}.$

Using Eqs. (1) and (34), the finite action per unit volume $V_{n-1}$ can be
calculated as

$I=-\frac{1}{T_{+}}\left\\{\frac{r_{+}^{n}}{16\pi
l^{2}}-\frac{r_{+}^{n}\beta^{2}(\sqrt{1+\eta_{+}}-1)}{4n(n-1)\pi}+\frac{(n-1)q^{2}}{8n\pi
r_{+}^{(n-2)}}\digamma(\eta_{+})\right\\}.$ (39)

Using the Brown-York method Brown , the finite energy-momentum tensor is

$T^{ab}=\frac{1}{8\pi}\\{(K^{ab}-K\gamma^{ab})+2\alpha_{2}(3J^{ab}-J\gamma^{ab})+3\alpha_{3}(5P^{ab}-P\gamma^{ab})+\frac{n-1}{L}\gamma^{ab}\
\\},$ (40)

and the conserved quantities associated with the Killing vectors
$\partial/\partial t$ and $\partial/\partial\phi^{i}$ are

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{1}{16\pi}m\left(n\Xi^{2}-1\right),$ (41) $\displaystyle
J_{i}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi}n\Xi ma_{i},$ (42)

which are the mass and angular momentum of the solution.

Now using Gibbs-Duhem relation

$S=\frac{1}{T}(M-Q\Phi-{{\sum_{i=1}^{k}}}\Omega_{i}J_{i})-I,$ (43)

and Eqs. (33), (39) and (41)-(42) one obtains

$S=\frac{\Xi}{4}r_{+}^{n-1}$ (44)

for the entropy per unit volume $V_{n-1}$. This shows that the entropy obeys
the area law for our case where the horizon is flat.

###  IV.2 Stability of the solutions

Calculating all the thermodynamic and conserved quantities of the black brane
solutions, we now check the first law of thermodynamics for our solutions with
flat horizon. We obtain the mass as a function of the extensive quantities
$S$, $\mathbf{J}$, and $Q$. Using the expression for charge mass, angular
momenta and entropy given in Eqs. (32), (41), (42), (44) and the fact that
$f(r_{+})=0$, one can obtain a Smarr-type formula as

$M(S,\mathbf{J},Q)=\frac{(nZ-1)J}{nl\sqrt{Z(Z-1)}},$ (45)

where $J=\left|\mathbf{J}\right|=\sqrt{\sum_{i}^{k}J_{i}^{2}}$ and $Z=\Xi^{2}$
is the positive real root of the following equation:

$\frac{Z^{1/2(n-1)}}{\sqrt{Z-1}}=\frac{\left[4l^{2}\beta^{2}\left(1-\sqrt{1+\eta_{+}}\right)+n(n-1)\right]S^{n/(n-1)}}{(n-1)\pi
lJ}+\frac{4\pi
lQ^{2}\digamma(\frac{\pi^{2}Q^{2}}{\beta^{2}S^{2}})}{(n-2)J(4S)^{(n-2)/(n-1)}}.$
(46)

One may then regard the parameters $S$, $J_{i}$’s, and $Q$ as a complete set
of extensive parameters for the mass $M(S,\mathbf{J},Q)$ and define the
intensive parameters conjugate to them. These quantities are the temperature,
the angular velocities, and the electric potential

$T=\left(\frac{\partial M}{\partial S}\right)_{J,Q},\ \
\Omega_{i}=\left(\frac{\partial M}{\partial J_{i}}\right)_{S,Q},\ \
\Phi=\left(\frac{\partial M}{\partial Q}\right)_{S,J}.$ (47)

Straightforward calculations show that the intensive quantities calculated by
Eq. (47) coincide with Eqs. (30), (31) and (33). Thus, these quantities
satisfy the first law of thermodynamics:

$dM=TdS+{{{\sum_{i=1}^{k}}}}\Omega_{i}dJ_{i}+\Phi dQ.$

Finally, we investigate the local stability of charged rotating black brane
solutions of third order Lovelock gravity in the presence of nonlinear
electrodynamic Born-Infeld field in the canonical and grand canonical
ensembles. In the canonical ensemble, the positivity of the heat capacity
$C_{\mathbf{J},Q}=T_{+}/(\partial^{2}M/\partial S^{2})_{\mathbf{J},Q}$ and
therefore the positivity of $(\partial^{2}M/\partial S^{2})_{\mathbf{J},Q}$ is
sufficient to ensure the local stability. Using the fact that

${{}_{2}F_{1}}\left(\left[{\frac{3}{2},\frac{3{n-4}}{{2n-2}}}\right],\left[{\frac{5{n-6}}{{2n-2}}}\right],-z\right)=\frac{(3n-4)}{(n-1)z}\left\\{\digamma(\frac{\pi^{2}Q^{2}}{\beta^{2}S^{2}})-\frac{1}{\sqrt{1+z}}\right\\},$

it is easy to show that

$\displaystyle\frac{\partial^{2}M}{\partial
S^{2}}=\frac{2[(n-1)(n-2)^{2}q^{2}-\Lambda
r_{+}^{(2n-2)}\sqrt{1+\eta_{+}}+2\beta^{2}r_{+}^{(2n-2)}\left(\sqrt{1+\eta_{+}}-1\right)]}{(n-1)^{2}\pi\Xi^{2}r_{+}^{(3n-4)}\sqrt{1+\eta_{+}}}-$
$\displaystyle\frac{8(\Xi^{2}-1)\Lambda\left(-r_{+}^{2n-2}\sqrt{1+\eta_{+}}(\frac{n(n-1)}{2}+2\beta^{2}l^{2})+(n-1)(n-2)\beta
l^{2}q^{2}+2\beta^{3}l^{2}r_{+}^{(2n-2)}\right)^{2}}{\pi
ml^{2}\beta^{2}\Xi^{2}(n-1)^{4}(4\Xi^{2}+1)(1+\eta_{+})r^{(4n-6)}}.$ (48)

Both of the two terms of Eq. (48) are positive, and therefore the condition
for thermal equilibrium in the canonical ensemble is satisfied.

In the grand canonical ensemble, the positivity of the determinant of the
Hessian matrix of $M(S,Q,\mathbf{J})$ with respect to its extensive variables
$X_{i}$, $\mathbf{H}_{X_{i}X_{j}}^{M}=\left(\partial^{2}M/\partial
X_{i}\partial X_{j}\right)$, is sufficient to ensure the local stability. It
is a matter of calculation to show that the determinant of
$\mathbf{H}_{S,Q,\mathbf{J}}^{M}$ is:

$\left|\mathbf{H}_{SJQ}^{M}\right|=\frac{64\pi\left(2(n-2)\beta^{2}\eta_{+}-\Lambda\sqrt{1+\eta_{+}}+2\beta^{2}\left(\sqrt{1+\eta_{+}}-1\right)\right)}{(n-2)(n-1)^{3}ml^{2}\Xi^{6}r_{+}^{2(n-2)}\left[(n-2)\Xi^{2}+1\right]\sqrt{1+\eta_{+}}}\digamma({\eta}_{+})+\frac{8(n-1)\Xi}{r_{+}}T_{+}.$
(49)

Equation (49) shows that the determinant of the Hessian matrix is positive,
and therefore the solution is stable in the grand canonical ensemble too. The
stability analysis given here shows that the higher curvature and nonlinear
Maxwell terms in the action have no effect on the stability of black holes
with flat horizon, and these kinds of black holes are thermodynamically stable
as in the case of toroidal black holes of Einstein-Maxwell gravity Lem2 . This
phase behavior is also commensurate with the fact that there is no Hawking-
Page transition for a black object whose horizon is diffeomorphic to
$\mathbb{R}^{p}$ and therefore the system is always in the high temperature
phase Wit2 .

##  V  CLOSING REMARKS

In this paper we considered both the nonlinear scalar terms constructing from
the curvature tensor and electromagnetic field tensor in gravitational action,
which are on similar footing with regard to the string corrections on gravity
and electrodynamic sides. We presented static topological black hole solutions
of third order Lovelock gravity in the presence of Born-Infeld gravity, which
are asymptotically AdS for negative cosmological constant, dS for positive
$\Lambda$. For the case of solutions with positive curvature horizon ($k=1$),
one can also have asymptotically flat solutions provided $\Lambda=0$. The
topological solutions obtained in this paper may be interpreted as black holes
with two inner and outer event horizons for $m>m_{\mathrm{ext}}$ , extreme
black holes for $m=m_{\mathrm{ext}}$ or naked singularity otherwise. We found
that these solutions reduce to the solutions of Einstein-Born-Infeld gravity
as the Lovelock coefficients vanish, and reduce to the solutions of third
order Lovelock gravity in the presence of Maxwell field as $\beta$ goes to
infinity DSh . We consider thermodynamics of asymptotically flat solutions and
found that the first law of thermodynamics is satisfied by the conserved and
thermodynamic quantities of the black hole. We also consider the rotating
solution with flat horizon and computed the action and conserved quantities of
it through the use of counterterm method. We found that the entropy obeys the
area law for black branes with flat horizon. We obtained a Smarr-type formula
for the mass of the black brane as a function of the entropy, the charge, and
the angular momenta, and found that the conserved and thermodynamics
quantities satisfy the first law of thermodynamics. We also studied the phase
behavior of the $(n+1)$-dimensional rotating black branes in third order
Lovelock gravity and showed that there is no Hawking-Page phase transition in
spite of the angular momenta of the branes and the presence of a nonlinear
electromagnetic field. Indeed, we calculated the heat capacity and the
determinant of the Hessian matrix of the mass with respect to $S$,
$\mathbf{J}$, and $Q$ of the black branes and found that they are positive for
all the phase space, which means that the brane is locally stable for all the
allowed values of the metric parameters.

###### Acknowledgements.

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

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