# Thermodynamics of charged rotating dilaton black branes coupled to
logarithmic nonlinear electrodynamics

A. Sheykhi<EMAIL_ADDRESS>M. H. Dehghani<EMAIL_ADDRESS>Physics
Department and Biruni Observatory, College of Sciences, Shiraz University,
Shiraz 71454, Iran  Research Institute for Astronomy and Astrophysics of
Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran  M. Kord Zangeneh
<EMAIL_ADDRESS>Physics Department and Biruni Observatory, College of
Sciences, Shiraz University, Shiraz 71454, Iran

###### Abstract

We construct a new class of charged rotating black brane solutions in the
presence of logarithmic nonlinear electrodynamics with complete set of the
rotation parameters in arbitrary dimensions. The topology of the horizon of
these rotating black branes are flat, while, due to the presence of the
dilaton field the asymptotic behaviour of them are neither flat nor (anti)-de
Sitter [(A)dS]. We investigate the physical properties of the solutions. The
mass and angular momentum of the spacetime are obtained by using the
counterterm method inspired by AdS/CFT correspondence. We derive temperature,
electric potential and entropy associated with the horizon and check the
validity of the first law of thermodynamics on the black brane horizon. We
study thermal stability of the solutions in both canonical and grand canonical
ensemble and disclose the effects of the rotation parameter, nonlinearity of
electrodynamics and dilaton field on the thermal stability conditions. We find
the solutions are thermally stable for $\alpha<1$, while for $\alpha>1$ the
solutions may encounter an unstable phase where $\alpha$ is dilaton-
electromagnetic coupling constant.

###### pacs:

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

##  I Introduction

Thermodynamics of black holes plays a central role in the attractive modern
method relating classical gravity and high energy physics namely gauge/gravity
duality. The issue was first taken under consideration by Bekenstein Bek  and
Hawking Haw  and encountered increasing interests rapidly. Among different
frameworks, the thermodynamics of black solutions has been studied within,
dilaton gravity possesses a significant predominancy. This preference has at
least two reasons. From one side, the dilaton gravity which is one of the
modified gravities, is able to justify the accelerating expansion of the
Universe confirmed from observations ExpUni  while Einstein gravity (General
Relativity) requires exotic matter violating energy conditions to justify this
phase of universe. From another side, the dilaton gravity appears in the low
energy limit of string theory Green  and therefore can provide a good
laboratory for testing this theory in the low energy limit through
gauge/gravity duality. Since string theory proposes higher than four
dimensions Green , it is natural to consider higher-dimensional solutions
within the gravity theories come from string theory.

Exact asymptotically flat solutions of Einstein-Maxwell-dilaton gravity have
been constructed in the absence of dilaton potential in Gibbons ; Gar ; Mig ;
Pol . However, breaking of spacetime supersymmetry in ten dimensions may cause
one or more Liouville-type potentials in the action of dilaton gravity. This
type of dilaton potential change the asymptotic behavior of solutions MW ; Cai
; yaz2 ; CHM ; Clem . In general, these solutions are neither asymptotically
flat nor (anti) de Sitter [(A)dS]. Thermodynamics of topological dilaton black
holes in Einstein-Maxwell gravity has been explored in Shey3 . Asymptotically
non-flat and non-(A)dS linearly charged rotating black branes was taken under
investigation in rotBB . Slowly rotating charged black holes have been studied
from thermodynamics point of view as well Slrot .

A natural interesting extension of such solutions is to change the
electrodynamics Lagrangian from linear Maxwell to nonlinear ones. Some efforts
have been done to construct exact solution in dilaton gravity with nonlinear
electrodynamics. For example, thermodynamics of static black hole solutions in
the presence of nonlinear power-law Maxwell (PLM) topPLM , Born-Infeld (BI)
topBI  and exponential HendiJHEP ; BHExp  electrodynamics have been
investigated. As well, thermodynamic properties of rotating black brane
solutions have been studied in the presence of PLM rotPLM , BI rotBI  and
exponential rotExp  nonlinear electrodynamics. Any one of the above mentioned
nonlinear electrodynamics has its own importance and motivations. For
instance, the PLM electrodynamics extends the conformal invariance property of
Linear Maxwell Lagrangian in four dimensions to higher dimensional spacetimes,
while BI BI  electrodynamics, which comes from open string theory BI1 ; BI2 ,
solves the problem of infinite self-energy of charged point-particle that
appears in linear Maxwell case. The latter problem is also overcome by
logarithmic nonlinear electrodynamics proposed for the first time in Soleng .
Although this type of nonlinear electrodynamics has no direct relation with
superstring theory, it can be motivated from different sides. First, it can be
regarded as a toy model showing that certain nonlinear field theories can
produce particle-like solutions that can realize the limiting curvature
hypothesis in cosmological theories Soleng . Second, the behavior of
logarithmic electrodynamics and BI electrodynamics Lagrangians are the same
for large values of nonlinear parameter $\beta$. Third, from gauge/gravity
duality point of view, the ratio of holographic viscosity to entropy density
is not affected by nonlinear terms rise from a nonlinear electrodynamics in
contrast with gravitational corrections vis . Fourth, the values of important
parameters of a holographic superconductor system such as critical temperature
and order parameter are firmly sensitive to choice of electrodynamics ord ;
ord2 . Other studies on the nonlinear electrodynamics have been carried out in
Frad ; Cal ; Hendi ; HendiSlowly ; SheyHaj ; SheyKaz ; Shey1 ; DHSR .

The above pointed out motivations are convincingly enough to satisfy one to
seek for the effects of logarithmic electrodynamics on the solutions. Till
now, exact rotating solutions of logarithmic electrodynamics in the context of
dilaton gravity has not been constructed. In this paper, we would like to
construct the rotating dilaton black branes in the presence of logarithmic
nonlinear electrodynamics and investigate their thermodynamics as well as
their thermal stability.

The outline of this paper is as follows. In the next section, we present the
basic field equations. In section III, we construct the rotating dilaton black
branes with a complete set of rotation parameters in all higher dimensions and
investigate their properties. In section IV, we study thermodynamics of the
spacetime, by calculating the conserved and thermodynamic quantities. In
section V, we perform a stability analysis and show that the dilaton creates
an unstable phase for the solutions. The last section is devoted to
conclusions and discussions.

##  II Basic Field Equations

We consider an $n$-dimensional action in which gravity is coupled to a dilaton
and a nonlinear electrodynamic field

$\displaystyle I$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\int_{\mathcal{M}}d^{n}x\sqrt{-g}\left(\mathcal{\
R}\text{ }-\frac{4}{n-2}(\nabla\Phi)^{2}-V(\Phi)+L(F,\Phi)\right)$ (1)
$\displaystyle-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n-1}x\sqrt{-h}\Theta(h),$

where the Lagrangian of the logarithmic nonlinear electrodynamics coupled to
the dilaton field (LNd) is chosen in the following form

$L(F,\Phi)=-8\beta^{2}e^{4\alpha\Phi/(n-2)}\ln\Bigg{(}1+\frac{e^{-8\alpha\Phi/(n-2))}F^{2}}{8\beta^{2}}\Bigg{)}.$
(2)

In action (1), $\mathcal{R}$ is the Ricci scalar curvature, $\Phi$ is the
dilaton field, and $V(\Phi)$ is a potential for $\Phi$. The dilaton parameter
$\alpha$ determines the strength of coupling of the scalar and LNd fields,
$F^{2}=F^{\mu\nu}F_{\mu\nu}$, where
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic tensor field,$\ A_{\mu}$ is the vector potential, and $\beta$
is the nonlinear parameter with dimension of mass. The last term in Eq. (1) is
the Gibbons-Hawking boundary term which is chosen such that the variational
principle is well-defined. The manifold $\mathcal{M}$ has metric $g_{\mu\nu}$
and covariant derivative $\nabla_{\mu}$. $\Theta$ is the trace of the
extrinsic curvature $\Theta^{ab}$ of any boundary(ies) $\partial\mathcal{M}$
of the manifold $\mathcal{M}$, with induced metric(s) $h_{ab}$. In this paper,
we consider the action (1) with a Liouville type potential,

$V(\Phi)=2\Lambda e^{4\alpha\Phi/(n-2)},$ (3)

where $\Lambda$ is a constant which may be referred to as the cosmological
constant, since in the absence of the dilaton field ($\Phi=0$) the action (1)
reduces to the action of Einstein gravity in the presence of nonlinear
electrodynamics with cosmological constant. For later convenience, we redefine
it as $\Lambda=-(n-1)(n-2)/2l^{2}$, where $l$ is a constant with dimension of
length. The series expansion of (2) for large $\beta$, leads to

$L_{\mathrm{LNd}}(F,\Phi)=-e^{-4\alpha\Phi/(n-2)}F^{2}+\frac{e^{-12\alpha\Phi/(n-2)}F^{4}}{16\beta^{2}}-\frac{e^{-20\alpha\Phi/(n-2)}F^{6}}{192\beta^{4}}+\mathcal{O}\Bigg{(}\frac{1}{\beta^{6}}\Bigg{)}.$

For latter convenience we rewrite

$L_{\mathrm{LNd}}(F,\Phi)=-8\beta^{2}e^{4\alpha\Phi/(n-2)}\mathcal{L}(Y),$

where we have defined

$\mathcal{L}(Y)=\ln(1+Y)\mathrm{,}$
$Y=\frac{e^{-8\alpha\Phi/(n-2)}F^{2}}{8\beta^{2}}.$

By varying the action (1) with respect to the gravitational field
$g_{\mu\nu}$, the dilaton field $\Phi$ and the gauge field $A_{\mu}$. We find

$\displaystyle\mathcal{R}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{4}{n-2}\left(\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{1}{4}g_{\mu\nu}V(\Phi)\right)+2e^{-4\alpha\Phi/(n-2)}\partial_{Y}{\mathcal{L}}(Y)F_{\mu\eta}F_{\nu}^{\text{
}\eta}$ (4)
$\displaystyle-\frac{8\beta^{2}}{n-2}e^{4\alpha\Phi/(n-2)}\left[2Y\partial_{Y}{\mathcal{L}}(Y)-{\mathcal{L}}(Y)\right]g_{\mu\nu},$
$\nabla^{2}\Phi=\frac{n-2}{8}\frac{\partial
V}{\partial\Phi}-4\alpha\beta^{2}e^{4\alpha\Phi/(n-2)}\left[2{\
Y}\partial_{Y}{\mathcal{L}}(Y)-\mathcal{L}(Y)\right],$ (5)
$\nabla_{\mu}\left(e^{-4\alpha\Phi/(n-2)}\partial_{Y}{\mathcal{L}}(Y)F^{\mu\nu}\right)=0.$
(6)

In the limiting case where $\beta\rightarrow\infty$, we have
$\mathcal{L}(Y)=Y$. In this case the system of field equations (4)-(6) restore
the well-known equations of EMd gravity CHM ; Shey3 ; Cai ; yaz2 ; Clem , as
expected.

###  II.1 FINITE ACTION IN CANONICAL AND GRAND-CANONICAL ENSEMBLES

In general, the total action $I$ given in Eq. (1) is divergent when evaluated
on a solution. One way of dealing with the divergences of the action is adding
some counterterms to the action (1). The counterterms should contain a part
which removes the divergence of the gravity part of the action and a part for
dealing with the divergence of the matter action. Since the horizon of our
solution is flat, the counterterm which removes the divergence of the gravity
part should be proportional to $\sqrt{-h}$. The counterterm for the matter
part of the action in the presence of the dilaton is given by

$I_{\mathrm{ct}}=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}\
d^{n-1}x\sqrt{-h}\left(\frac{n-2}{l_{\mathrm{eff}}}\right)+I_{\mathrm{deriv}},$
(7)

where $l_{\mathrm{eff}}$ is given by (13) and $I_{\mathrm{deriv}}$ is a
collection of terms involving derivatives of the boundary fields that could
involve the curvature tensor constructed from the boundary metric. Since in
our case the boundary is flat so $I_{\mathrm{deriv}}$ is zero on the boundary.
The variation of the total action $\left(I_{\mathrm{tot}}\
=I+I_{\mathrm{ct}}\right)$ about the solutions of the equations of motion is

$\delta I_{\mathrm{tot}}=\int d^{n-1}xS_{ab}\delta h^{ab}-\frac{1}{16\pi}\int
d^{n-1}x\sqrt{-h}e^{-4\alpha\Phi/(n-2)}\partial_{Y}{\mathcal{L}}(Y)n^{a}F_{ab}\delta
A^{b},$ (8)

where

$S_{ab}=\frac{\sqrt{-h}}{16\pi}\left\\{\Theta_{ab}-\Theta
h_{ab}+\frac{n-2}{l_{\mathrm{eff}}}h_{ab}\right\\}.$ (9)

Equation (8) shows that the variation of the total action with respect to
$A^{a}$ will only give the equation of motion of the nonlinear massless field
$A^{a}$ provided the variation is at fixed nonlinear massless gauge potential
on the boundary. Thus, the total action, $I_{\mathrm{tot}}\
=I+I_{\mathrm{ct}}$, given in Eq. (8) is appropriate for the grand-canonical
ensemble, where $\delta A^{a}=0$ on the boundary. But in the canonical
ensemble, where the electric charge
$\left[-e^{-4\alpha\Phi/(n-2)}\partial_{Y}{\mathcal{L}}(Y)n^{a}F_{ab}\right]$
is fixed on the boundary, the appropriate action is

$I_{\mathrm{tot}}=I+I_{\mathrm{ct}}+\frac{1}{16\pi}\int_{\partial\mathcal{M}`}d^{n-1}x\
\sqrt[\
]{-h}e^{-4\alpha\Phi/(n-2)}\partial_{Y}{\mathcal{L}}(Y)n^{a}F_{ab}\delta
A^{b}.$ (10)

The last term in Eq. (10) is the generalization of the boundary term
introduced by Hawking for linear electromagnetic field S. W. Hawking grand
and the results of DSV ; mhd2  for the nonlinear Lifshitz black holes to the
exponential nonlinear gauge field coupled to the dilaton field. Thus, both in
canonical and grand-canonical ensemble, the variation of total action about
the solutions of the field equations is

$\delta I_{\mathrm{tot}}=\int d^{n-1}xS_{ab}\delta h^{ab}.$ (11)

That is, the nonlinear gauge field is absent in the variation of the total
action both in canonical and grand-canonical ensembles.

In order to obtain the conserved charges of the spacetime, we use the
counterterm method BY ; Mal  inspired by (A)dS/CFT correspondence. For
asymptotically AdS solutions this method works very well Mal . However, in our
paper we have the scalar dilaton field with a Liouville potential. It was
argued that the presence of Liouville-type dilaton potential, which is
regarded as the generalization of the cosmological constant, changes the
asymptotic behavior of the solutions to be neither asymptotically flat nor
(A)dS. It has been shown that no dilaton dS or AdS black hole solution exists
with the presence of only one Liouville-type dilaton potential MW . But, as in
the case of asymptotically AdS spacetimes, according to the domain-wall/QFT
(quantum field theory) correspondence Sken , there may be a suitable
counterterm for the stress-energy tensor which removes the divergences. In
this paper, we deal with the spacetimes with zero curvature boundary
[$R_{abcd}(h)=0$], and therefore the counterterm for the stress-energy tensor
should be proportional to $h^{ab}$. We find the finite stress-energy tensor in
$n$-dimensional Einstein-dilaton gravity with Liouville-type in the form DHSR

$T^{ab}=\frac{1}{8\pi}\left[\Theta^{ab}-\Theta
h^{ab}+\frac{n-2}{l_{\mathrm{eff}}}h^{ab}\right],$ (12)

where $l_{\mathrm{eff}}$ is given by

$l_{\mathrm{eff}}^{2}=\frac{(n-2)(\alpha^{2}-n+1)}{2\Lambda}e^{-4\alpha\Phi/(n-2)}.$
(13)

In the particular case $\alpha=0$, the effective $l_{\mathrm{eff}}^{2}$ of Eq.
(13) reduces to $l^{2}=-(n-1)(n-2)/2\Lambda$ of the AdS spacetimes. The first
two terms in Eq. (12) is the variation of the action (1) with respect to
$h_{ab}$, and the last term is the counterterm which removes the divergences.
One may note that the counterterm has the same form as in the case of
asymptotically AdS solutions with zero curvature boundary, where $l$ is
replaced by $l_{\mathrm{eff}}$. If we choose the Killing vector field
$\mathcal{\xi}$ on spacelike surface $\mathcal{B}$ in $\partial\mathcal{M}$
with metric $\sigma_{ij}$, then the quasilocal conserved quantities may be
obtained from the following relation DHSR

$Q(\mathcal{\xi)}=\int_{\mathcal{B}}d^{n-2}x\sqrt{\sigma}T_{ab}n^{a}\mathcal{\
\xi}^{b},$ (14)

where $\sigma$ is the determinant of the boundary metric $\sigma_{ij}$ and
$n^{a}$ is the unit normal vector on the boundary $\mathcal{B}$. In our case,
the boundary $\mathcal{B}$ has two Killing vector fields timelike
($\partial/\partial t$) and rotational ($\partial/\partial\varphi$). The
corresponding conserved charges are the quasilocal mass and angular momentum
may be obtained as

$\displaystyle M$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{n-2}x\sqrt{\sigma}T_{ab}n^{a}\xi^{b},$ (15)
$\displaystyle J$ $\displaystyle=$
$\displaystyle\int_{\mathcal{B}}d^{n-2}x\sqrt{\sigma}T_{ab}n^{a}\varsigma^{b}.$
(16)

##  III Rotating dilaton black branes in higher dimensions

In this section, we would like to construct the rotating black brane solutions
of the field equations (4)-(6) with $k$ rotation parameters. The number of
independent rotation parameters for an $n$-dimensional localized object is
equal to the number of Casimir operators, which is $[(n-1)/2]\equiv k$, where
$[x]$ is the integer part of $x$ SDRP . The metric of $n$-dimensional rotating
solution with cylindrical or toroidal horizons and $k$ rotation parameters can
be written as Lemos ; awad

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-f(r)\left(\Xi
dt-{{\sum_{i=1}^{k}}}a_{i}d\phi_{i}\right)^{2}+\frac{r^{2}}{l^{4}}R^{2}(r){{\sum_{i=1}^{k}}}\left(a_{i}dt-\Xi
l^{2}d\phi_{i}\right)^{2}$
$\displaystyle-\frac{r^{2}}{l^{2}}R^{2}(r){\sum_{i<j}^{k}}(a_{i}d\phi_{j}-a_{j}d\phi_{i})^{2}+\frac{dr^{2}}{f(r)}+\frac{r^{2}}{l^{2}}R^{2}(r)dX^{2},$
$\displaystyle\Xi^{2}$ $\displaystyle=$ $\displaystyle
1+\sum_{i=1}^{k}\frac{a_{i}^{2}}{l^{2}},$ (17)

where $a_{i}$’s are $k$ rotation parameters. There are two unknown functions
$f(r)$ and $R(r)$ in the above metric which should be determined by solving
the field equations. The range of the angular coordinates are
$0\leq\phi_{i}\leq 2\pi$ and $dX^{2}$ is the Euclidean metric on the
$(n-k-2)$-dimensional submanifold with volume $\Sigma_{n-k-2}$.

First of all, we integrate the electromagnetic field equation (6). The result
is

$\displaystyle F_{{tr}}$ $\displaystyle=$ $\displaystyle\frac{2q\Xi
e^{4\alpha\Phi/(n-2)}}{(rR(r))^{n-2}}\Bigg{(}1+\sqrt{1+\frac{q^{2}}{\beta^{2}(rR(r))^{2n-4}}}\Bigg{)}^{-1},$
(18) $\displaystyle F_{\phi_{i}r}$ $\displaystyle=$
$\displaystyle-\frac{a_{i}}{\Xi}F_{tr},$ (19)

where $q$, is an integration constant related to the electric charge of the
brane. When $\beta\rightarrow\infty$, $F_{tr}$ reduces to the electric field
of $n$-dimensional black brane of Einstein-Maxwell-dilaton gravity SDRP

$F_{tr}=\frac{q\Xi
e^{4\alpha\Phi/(n-2)}}{(rR(r))^{n-2}}+O\left(\frac{1}{\beta^{2}}\right).$ (20)

In order to solve the system of equations (4) and (5) for three unknown
functions $f(r)$, $R(r)$ and $\Phi(r)$, we make the ansatz SDRP

$R(r)=e^{2\alpha\Phi/(n-2)}.$ (21)

In order to justify this choice for the metric function $R(r)$, let us note
that $R(r)$ is indeed added to the metric (17) in order to increase the
degrees of freedom for obtaining solutions in the presence of the dilaton
field. Choosing $R(r)$ in the form of Eq. (21), is an ansatz. However, it is
chosen such that in the absence of the dilaton field $\Phi=0$, we have
$R(r)=1$, as expected. With this ansatz, we are able to solve the field
equation, analytically.

Substituting (21), the electromagnetic fields (18)- (19) and the metric (17)
into the field equations (4) and (5), one can obtain the following solutions

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{2(\alpha^{2}+1)^{2}(\Lambda-4\beta^{2})b^{\gamma}}{(n-2)(\alpha^{2}-n+1)}r^{2-\gamma}-\frac{m}{r^{n-3-(n-2)\gamma/2}}$
(22)
$\displaystyle-\frac{8\beta^{2}(\alpha^{2}+1)b^{\gamma}}{(n-2)r^{n-3-(n-2)\gamma/2}}\int
r^{n(1-\frac{\gamma}{2})-2}\Bigg{\\{}\sqrt{1+\eta}-\ln\bigg{(}\frac{\eta}{2}\bigg{)}+\ln\Big{(}-1+\sqrt{1+\eta}\,\Big{)}\Bigg{\\}}dr,$
$\Phi(r)=\frac{(n-2)\alpha}{2(\alpha^{2}+1)}\ln\left(c+\frac{b}{r}\right),$
(23)

where $c$ and $b$ are constant of integration. We find that these solutions
will fully satisfy the system of equations (4) and (5) provided we choose
$c=0$. Note that $b$ has the dimension of [Length] to make the argument of
logarithmic function dimensionless. In the above solutions
$\gamma=2\alpha^{2}/(1+\alpha^{2})$, and

$\eta=\frac{q^{2}b^{(2-n)\gamma}}{\beta^{2}r^{(n-2)(2-\gamma)}}.$ (24)

In the above expression, $m$ appears as an integration constant and is related
to the mass of the black hole. The integration of Eq. (22) can be performed
using the MATHEMATICA software. The resulting solution can be written

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{2(\Lambda-4\beta^{2})(\alpha^{2}+1)^{2}b^{\gamma}}{(n-2)(\alpha^{2}-n+1)}r^{2-\gamma}-\frac{m}{r^{n-3-(n-2)\gamma/2}}$
(25)
$\displaystyle+\frac{8\beta^{2}(\alpha^{2}+1)^{2}}{(\alpha^{2}-n+1)^{2}}b^{\gamma}r^{2-\gamma}\Bigg{\\{}1-{}_{2}F_{1}\Bigg{(}\Bigg{[}\frac{-1}{2}\,,\frac{\alpha^{2}-n+1}{2n-4}\Bigg{]}\,,\Bigg{[}\frac{\alpha^{2}+n-3}{2n-4}\Bigg{]}\,,-\eta\Bigg{)}\Bigg{\\}}$
$\displaystyle+\frac{8\beta^{2}(\alpha^{2}+1)^{2}}{(n-2)(\alpha^{2}-n+1)}\,b^{\gamma}r^{2-\gamma}\Bigg{\\{}\sqrt{1+\eta}-\ln\left({\frac{\eta}{2}}\right)+\ln\Big{(}-1+\sqrt{1+\eta}\,\Big{)}\Bigg{\\}},$

where ${}_{2}F_{1}([a,b],[c],z)$ is the hypergeometric function Lambert . It
is worth mentioning that the solutions are ill-defined for
$\alpha=\sqrt{n-1}$. We expect that for large $\beta$, the function $f(r)$
reduces to the $n$-dimensional charged rotating dilaton black brane solutions
given in Ref. SDRP . Indeed, if we expand Eq. (25) for large $\beta$, we
arrive at

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{2\Lambda(\alpha^{2}+1)^{2}}{(n-2)(\alpha^{2}-n+1)}b^{\gamma}r^{2-\gamma}-\frac{m}{r^{n-3-(n-2)\gamma/2}}$
(26)
$\displaystyle+\frac{2q^{2}(\alpha^{2}+1)^{2}b^{-(n-3)\gamma}}{(n-2)(\alpha^{2}+n-3)r^{(n-3)(2-\gamma)}}-\frac{q^{4}(\alpha^{2}+1)^{2}b^{-(2n-5)\gamma}}{4\beta^{2}(n-2)(\alpha^{2}+3n-7)r^{(2n-5)(2-\gamma)}}+\mathcal{O}\Bigg{(}\frac{1}{\beta^{4}}\Bigg{)}.$

Setting $\alpha=\gamma=0$ in (26), we reach

$f(r)=\frac{r^{2}}{l^{2}}-\frac{m}{r^{n-3}}+\frac{2q^{2}}{(n-2)(n-3)r^{2n-6}}-\frac{1}{4\beta^{2}(n-2)(3n-7)}\frac{q^{4}}{r^{4n-10}}+\mathcal{O}\Bigg{(}\frac{1}{\beta^{4}}\Bigg{)}.$

The last term in the right hand side of the above expression is the leading
nonlinear correction to the AdS black brane with dilaton field. In the absence
of a nontrivial dilaton ($\alpha=\gamma=0$), the above solutions reduce to the
asymptotically AdS charged rotating black brane solutions of Einstein gravity
in the presence of EN electrodynamics Hendi . Finally, in the limit
$\beta^{2}\rightarrow\infty$ and $\alpha=0=\gamma$, the solution given by Eq.
(26) has the form of the asymptotically AdS black brane solutions awad ; Deh4
. Figs. (1) and (2) depict the behavior of $f(r)$ given by Eq. (25) for
different $\alpha$’s and $\beta$’s respectively.

Figure 1: The behavior of $f(r)$ versus $r$ with $l=b=1$, $q=0.5$, $\Xi=1.25$,
$n=5$, $\beta=2$ and $m=1.5$.

Figure 2: The behavior of $f(r)$ versus $r$ with $l=b=1$, $q=0.8$, $\Xi=1.25$,
$n=5$, $\alpha=0.2$ and $m=0.5$.

###  III.1 Asymptotic behavior of the spacetime

Next, we study the geometry of this spacetime. For this purpose, we first seek
for the curvature singularities in the presence of dilaton and nonlinear
electrodynamic fields. It is a matter of calculation to show that the Ricci
scalar and the Kretschmann invariant behave as

$\displaystyle\lim_{r\longrightarrow 0^{+}}R=\infty,$ (27)
$\displaystyle\lim_{r\longrightarrow
0^{+}}R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\infty,$ (28)

which indicate that there is an essential singularity at $r=0$. In order to
study the asymptotic behavior of the solutions, we expand the metric function
$f(r)$ for $r\rightarrow\infty$ limit. We find

$\lim_{r\longrightarrow\infty}f(r)=\frac{2\Lambda(\alpha^{2}+1)^{2}}{(n-2)(\alpha^{2}-n+1)}b^{\gamma}r^{2-\gamma}.$
(29)

Let us note that in the absence of the dilaton field ($\alpha=0=\gamma$), the
metric function becomes

$\lim_{r\longrightarrow\infty}f(r)=-\frac{2\Lambda r^{2}}{(n-1)(n-2)},$ (30)

which describes an asymptotically AdS ($\Lambda<0$) or dS ($\Lambda>0$)
spacetimes. However, as one can see from Eq. (29), in the presence of the
dilaton field, the asymptotic behavior is neither flat nor (A)dS. For example,
taking $\alpha=\sqrt{2}$, $n=6$ and $b=1$, we have

$\lim_{r\longrightarrow\infty}f(r)=-\frac{3\Lambda}{2}r^{2/3}.$ (31)

Clearly, the metric function (31) is neither flat nor (A)dS. This is
consistent with the argument given in MW , which states that no dilaton dS or
AdS black hole solution exists with the presence of only one or two Liouville-
type dilaton potential. It is important to note that this asymptotic behavior
is not due to the nonlinear nature of the electrodynamic field, since as
$r\rightarrow\infty$ the effects of the nonlinearity disappear. Besides, from
the dilaton field (23) we see that as $r\rightarrow\infty$, the dilaton field
does not vanishes, while in case of asymptotic flat or (A)dS we expect to have
$\lim_{r\longrightarrow\infty}\Phi(r)=0.$ Indeed, by solving the field
equation (5) we find

$\Phi(r)=\frac{(n-2)\alpha}{2(\alpha^{2}+1)}\ln\left(c+\frac{b}{r}\right),$
(32)

however, the system of equation (4)-(6) will be fully satisfied provided we
choose $c=0$. From the above arguments we conclude that the asymptotic
behavior of the obtained solutions is neither flat nor (A)dS.

##  IV Thermodynamics of black branes

It is easy to show that the metric given by (17) and (25) has both Killing and
event horizons SDRP . The Killing horizon is a null surface whose null
generators are tangent to a Killing field. It is easy to see that the Killing
vector

$\chi=\partial_{t}+{{{\sum_{i=1}^{k}}}}\Omega_{i}\partial_{\phi_{i}},$ (33)

is the null generator of the event horizon, where $\Omega_{i}$ is the $i$th
component of angular velocity of the outer horizon which may be obtained by
analytic continuation of the metric. The Hawking temperature and the angular
velocities of the outer event horizon can be obtained as

$\displaystyle T_{+}$ $\displaystyle=$ $\displaystyle\frac{f^{\text{
}^{\prime}}(r_{+})}{4\pi\Xi}=-\frac{\alpha^{2}+1}{4\pi\Xi}r_{+}^{1-\gamma}\Bigg{\\{}\frac{2(\Lambda-4\beta^{2})b^{\gamma}}{(n-2)}$
(34)
$\displaystyle+\frac{8\beta^{2}b^{\gamma}}{n-2}\Bigg{[}\sqrt{1+\eta_{+}}-\ln\bigg{(}\frac{\eta_{+}}{2}\bigg{)}+\ln\Big{(}-1+\sqrt{1+\eta_{+}}\Big{)}\Bigg{]}\Bigg{\\}},$
$\displaystyle\Omega_{i}$ $\displaystyle=$ $\displaystyle\frac{a_{i}}{\Xi
l^{2}},$ (35)

where $\eta_{+}=\eta(r=r_{+})$ and we have used $f(r_{+})=0$ for deleting $m$.
For large $\beta$, we can expand $T_{+}$ and arrive at the temperature of the
higher dimensional black branes in EMd gravity SDRP

$T_{+}=-\frac{\Lambda(\alpha^{2}+1)b^{\gamma}}{2\pi\Xi(n-2)}r_{+}^{1-\gamma}-\frac{q^{2}(\alpha^{2}+1)b^{-\gamma(n-3)}}{2\pi\Xi(n-2)}r_{+}^{5-2n-3\gamma+n\gamma}+\mathcal{O}\Bigg{(}\frac{1}{\beta^{2}}\Bigg{)}.$
(36)

The mass and angular momentum of the black branes ($\alpha<\sqrt{n-1}$) can be
calculated through the use of Eqs. (15) and (16). Denoting the volume of the
hypersurface boundary at constant $t$ and $r$ by
$V_{n-2}=(2\pi)^{k}\Sigma_{n-k-2}$, the mass and angular momentum per unit
volume $V_{n-2}$ of the black branes can be obtained as

${M}=\frac{b^{(n-2)\gamma/2}}{16\pi
l^{n-3}}\left\\{\frac{(n-1-\alpha^{2})\Xi^{2}+\alpha^{2}-1}{1+\alpha^{2}}\right\\}m,$
(37) $J_{i}=\frac{b^{(n-2)\gamma/2}}{16\pi
l^{n-3}}\left(\frac{n-1-\alpha^{2}}{1+\alpha^{2}}\right)\Xi ma_{i}.$ (38)

Note that, in order to avoid repeating the factor $V_{n-2}$, we calculate, in
this paper, the mass $M$ and extensive quantities such as angular momentum
$J_{i}$, entropy $S$ and charge $Q$ appearing in first law of thermodynamics
per unit volume. For the static case where $a_{i}=0$ ($\Xi=1$), the angular
momentum per unit volume vanishes, and therefore $a_{i}$’s are the rotational
parameters of the black branes.

Black hole entropy typically satisfies the so called area law of the entropy
Beck . This near universal law applies to almost all kinds of black holes and
black branes in Einstein gravity hunt . It is easy to show that the entropy
per unit volume $V_{n-2}$ of the black brane can be written as

${S}=\frac{\Xi b^{(n-2)\gamma/2}r_{+}^{(n-2)(1-\gamma/2)}}{4l^{n-3}},$ (39)

The electric charge per unit volume $V_{n-1}$ can be found by calculating the
flux of the electric field at infinity, yielding

${Q}=-\frac{1}{4\pi
V_{n-1}}\int_{\Sigma}\nabla_{\mu}\left(\partial_{Y}{\mathcal{L}}(Y)F^{\mu\nu}\right)dS_{\nu}=-\frac{1}{8\pi
V_{n-1}}\oint_{\partial\Sigma}\partial_{Y}{\mathcal{L}}(Y)F^{\mu\nu}dS_{\mu\nu}=\frac{\Xi
q}{4\pi l^{n-3}},$ (40)

where the volume is replaced by an arbitrary spacelike hypersurface $\Sigma$
(partial Cauchy surface) with boundary $\partial\Sigma$. In addition, the
volume element on $\Sigma$ is a non-spacelike covector ($1$-form) $dS_{\nu}$
and $dS_{\mu\nu}$ is the area element of $\partial\Sigma$. We should note that
for linear Maxwell case ($\beta\longrightarrow\infty$), one obtains
$\partial_{Y}{\mathcal{L}}(Y)=-1$.

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

$U=A_{\mu}\chi^{\mu}\left|{}_{r\rightarrow\infty}-A_{\mu}\chi^{\mu}\right|_{r=r_{+}},$
(41)

where $\chi$ is the null generator of the horizon given by Eq. (33). One can
easily show that the vector potential $A_{\mu}$ corresponding to the
electromagnetic tensor (18) and (19) can be written as

$\displaystyle A_{\mu}$ $\displaystyle=$
$\displaystyle\left(\Xi\delta_{\mu}^{t}-a_{i}\delta_{\mu}^{i}\right)\times\frac{q(\alpha^{2}+1)b^{(4-n)\gamma/2}}{\alpha^{2}+n-3}r^{3-n-(4-n)\gamma/2}{}$
$\displaystyle\times{}_{3}F_{2}\Bigg{(}\Bigg{[}\frac{1}{2}\,,1\,,\frac{3-n-\alpha^{2}}{4-2n}\Bigg{]}\,,\Bigg{[}2\,,\frac{7-3n-\alpha^{2}}{4-2n}\Bigg{]}\,,-\eta\Bigg{)},$

where ${}_{3}F_{2}$ is the hypergeometric function and we have set the
constant of integration equal to zero. Therefore, the electric potential may
be obtained as

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{q(\alpha^{2}+1)b^{(4-n)\gamma/2}}{\Xi(\alpha^{2}+n-3)}r_{+}^{3-n-(4-n)\gamma/2}{}$
$\displaystyle\times{}_{3}F_{2}\Bigg{(}\Bigg{[}\frac{1}{2}\,,1\,,\frac{3-n-\alpha^{2}}{4-2n}\Bigg{]}\,,\Bigg{[}2\,,\frac{7-3n-\alpha^{2}}{4-2n}\Bigg{]}\,,-\eta_{+}\Bigg{)}.$

Now, we are in a position to verify the first law of thermodynamics. In order
to do this, we obtain the mass $M$ as a function of extensive quantities $S$,
$\mathbf{J}$ and $Q$. Using the expression for the mass, the angular momenta,
the entropy, and the charge given in Eqs. (37), (38), (39), (40) and the fact
that $f(r_{+})=0$, one can obtain a Smarr-type formula as

$M(S,\mathbf{J},Q)=\frac{\left[(n-1-\alpha^{2})Z+\alpha^{2}-1\right]\mathbf{J}}{l(n-1-\alpha^{2})\sqrt{Z(Z-1)}},$
(44)

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

$\displaystyle\mathbf{J}+\frac{\beta^{2}l^{4-n}(\alpha^{2}+1)}{2\pi(n-2)(n-1-\alpha^{2})}b^{\alpha^{2}}\sqrt{Z(Z-1)}\left(\frac{4Sl^{n-3}}{\sqrt{Z}}\right)^{\frac{n-1-\alpha^{2}}{n-2}}\bigg{\\{}(n-1-\alpha^{2})\bigg{[}\ln\left(-1+\sqrt{1+\zeta}\right)-\ln\left(\frac{\zeta}{2}\right)+\sqrt{1+\zeta}\bigg{]}$
$\displaystyle+(n-2)\
{}_{2}F_{1}\left(\left[-\frac{1}{2}\,,\frac{\alpha^{2}-n+1}{2n-4}\right]\,,\left[\frac{\alpha^{2}+n-3}{2n-4}\right]\,,-\zeta\right)+\frac{(n-1)(n-2)}{8l^{2}\beta^{2}}(\alpha^{2}-n+1)+\alpha^{2}-2n+3\bigg{\\}}=0.$
(45)

where $\zeta=\pi^{2}Q^{2}/\left(S^{2}\beta^{2}\right)$. We can regard the
parameters $S$, $\mathbf{J}$, and $Q$ as a complete set of extensive
parameters for the mass $M(S,\mathbf{J},Q)$ and define the intensive
parameters conjugate to $S$, $\mathbf{J}$ and $Q$. This parameters are,
respectively, the temperature, the angular velocities, and the electric
potential, which are defined as

$T=\left(\frac{\partial M}{\partial S}\right)_{J,Q},\ \
\Omega_{i}=\left(\frac{\partial M}{\partial J_{i}}\right)_{S,Q},\ \
U=\left(\frac{\partial M}{\partial Q}\right)_{S,\mathbf{J}}.$ (46)

Numerical calculations show that the intensive quantities calculated by Eq.
(46) coincide with Eqs. (34), (35) and (IV). Thus, these thermodynamics
quantities satisfy the first law of thermodynamics

$dM=TdS+{{{\sum_{i=1}^{k}}}}\Omega_{i}d{J}_{i}+Ud{Q}.$ (47)

##  V Thermal Stability of the black branes in canonical and grand-canonical
ensembles

Figure 3: The behavior of $(\partial^{2}M/\partial S^{2})_{Q,\mathbf{J}}$
versus $\alpha$ with $l=b=1$, $q=0.8$, $\Xi=1.25$, $n=5$ and $\beta=2$.

Figure 4: The behavior of $\mathbf{H}_{SQ\mathbf{J}}^{M}$ versus $\alpha$ with
$l=b=1$, $q=0.8$, $\Xi=1.25$, $n=5$ and $\beta=2$. Note that the curve
corresponding to $r_{+}=1$ rescaled by a factor $10^{-1}$.

Figure 5: The behavior of $T$ versus $\alpha$ with $l=b=1$, $q=0.8$,
$\Xi=1.25$, $n=5$ and $\beta=2$.

Figure 6: The behavior of $(\partial^{2}M/\partial S^{2})_{Q,\mathbf{J}}$
versus $\alpha$ with $l=b=1$, $r_{+}=1.5$, $\Xi=1.25$, $n=5$ and $\beta=2$.

Figure 7: The behavior of $\mathbf{H}_{SQ\mathbf{J}}^{M}$ versus $\alpha$ with
$l=b=1$, $r_{+}=1.5$, $\Xi=1.25$, $n=5$ and $\beta=2$.

Figure 8: The behavior of $T$ versus $\alpha$ with $l=b=1$, $r_{+}=1.5$,
$\Xi=1.25$, $n=5$ and $\beta=2$.

Figure 9: The behaviors of $10^{-1}T$ and $(\partial^{2}M/\partial
S^{2})_{Q,\mathbf{J}}$ versus $r_{+}$ with $l=1$, $b=2$, $\alpha=1.5$,
$\Xi=1.25$, $n=5$, $\beta=5$ and $q=1.1$.

Figure 10: The behaviors of $T$ and $(\partial^{2}M/\partial
S^{2})_{Q,\mathbf{J}}$ versus $q$ with $l=b=1$, $\alpha=1.5$, $\Xi=1.25$,
$n=5$, $\beta=5$ and $r_{+}=1.5$.

Figure 11: The behaviors of $T$ and $10^{-2}(\partial^{2}M/\partial
S^{2})_{Q,\mathbf{J}}$ versus $\beta$ with $l=1$, $b=0.5$, $\alpha=3$,
$\Xi=1.25$, $n=6$, $q=10$ and $r_{+}=1$.

In this section, we intend to investigate thermal stability of our nonlinearly
charged rotating black brane solutions in both canonical and grand-canonical
ensembles. We know that the entropy of a thermally stable system is at local
maximum. The aim of thermal stability analysis is to find the situations under
which the system is stable thermally i.e. its entropy is a local maximum.
Therefore, the stability of charged rotating black brane is studied in terms
of entropy $S(M,Q,\mathbf{J})$. However, the thermal stability can also be
discussed in terms of internal energy. When the entropy is at local maximum,
the internal energy is at local minimum. Hence, we can equivalently analyse
thermal stability in terms of Legendre transformation of entropy namely
internal energy $M(S,Q,\mathbf{J})$. This analysis is commonly done by
studying the determinant of the Hessian matrix of $M(S,Q,\mathbf{J})$ with
respect to its extensive variables $X_{i}$,
$\mathbf{H}_{X_{i}X_{j}}^{M}=\left[\partial^{2}M/\partial X_{i}\partial
X_{j}\right]$ Cal1 ; Gub . The positivity of $\mathbf{H}_{X_{i}X_{j}}^{M}$
shows that the system is thermally stable. The number of thermodynamic
variables depends on the ensemble in which the system is studied. For
instance, in canonical ensemble where the charge and angular momenta are
fixed, the entropy is the only variable and consequently
$\mathbf{H}_{X_{i}X_{j}}^{M}$ reduces to $(\partial^{2}M/\partial
S^{2})_{Q,\mathbf{J}}$. Thus, in this ensemble, the positivity of
$(\partial^{2}M/\partial S^{2})_{Q,\mathbf{J}}$ is sufficient to ensure the
thermal stability of course in the ranges the temperature $T$ is positive as
well. In grand-canonical ensemble $Q$ and $\mathbf{J}$ are no longer fixed.

Since the presence of charge does not change stable solutions to unstable ones
Deh4 , we first study thermal stability for uncharged case i.e. $q\rightarrow
0$. In this case

$\left(\frac{\partial^{2}M}{\partial
S^{2}}\right)_{\mathbf{J}}=\frac{(n-1)\left({\alpha}^{2}+1\right)\left[({\Xi}^{2}-1)(n-2{\alpha}^{2})+{\Xi}^{2}\left(1-{\alpha}^{2}\right)\right]}{{\pi\Xi}^{2}{l}^{5-n}{b}^{{\left(n-4\right)\gamma/2}}\left[({\alpha}^{2}+n-3){\Xi}^{2}+1-{\alpha}^{2}\right]}{r}_{+}^{{(3-n-{\alpha}^{2})/(\alpha}^{2}+1)},$
(48)

and

$\mathbf{H}_{S\mathbf{J}}^{M}=\frac{16\left(1-\,{\alpha}^{2}\right){l}^{2(n-4)}{r}_{+}^{2{(2-\,n)/(\alpha}^{2}+1)}}{{b}^{{\left(n-2\right)\gamma}}{\Xi}^{4}\left[\left({\Xi}^{2}-1\right){\alpha}^{2}+1+\left(n-3\right){\Xi}^{2}\right]}.$
(49)

Since $\Xi^{2}\geq 1$, both Eq. (48) and $\mathbf{H}_{S\mathbf{J}}^{M}$ are
positive for $\alpha\leq 1$, therefore the uncharged rotating solutions are
stable in both canonical and grand-canonical ensembles provided $\alpha\leq
1$. For this case, the temperature is also always positive as one can see from
(36). As pointed out before, the charge cannot change thermal stability and
therefore we always have thermally stable rotating black brane solutions for
$\alpha\leq 1$. This fact is illustrated in Figs. (3) and (4) for different
values of $r_{+}$. The positivity of temperature for them is shown in Fig.
(5). For different $q$’s, Figs. (6) and (7) show that charge does not affect
the thermal stability and therefore charged solutions are still stable for
$\alpha\leq 1$. The positivity of $T$ for mentioned parameters in Figs. (6)
and (7) is shown in Fig. (8).

Now, we discuss the stability for nonlinearly charged rotating black brane
solutions for $\alpha>1$. One can see from (49) that $\alpha=1$ is the root of
$\mathbf{H}_{S\mathbf{J}}^{M}$. Numerical investigations show that $\alpha=1$
is the root of determinant of Hessian matrix in charged case too. Also, for
$\alpha>1$, $\mathbf{H}_{SQ\mathbf{J}}^{M}$ is always negative as
$\mathbf{H}_{S\mathbf{J}}^{M}$ obviously is (see (49)). Therefore, we have
unstable solutions for $\alpha>1$ in grand-canonical ensemble. Figs. (4) and
(7) illustrate this fact. However, in canonical ensemble we have both stable
and unstable solutions for $\alpha>1$. Figures (3) and (6) show that there is
an $\alpha_{\max}(>1)$ that we have stable solutions for values lower than it
(note that $\alpha<\sqrt{n-1}$; see sentences above (37)). There is also a
$r_{+\max}$ that for $r_{+}>r_{+\max}$ solutions are unstable (see Fig. (9)).
The behavior of $(\partial^{2}M/\partial S^{2})_{Q,\mathbf{J}}$ in terms of
$q$ and $\beta$ are depicted in Figs. (10) and (11) respectively. These
figures show that there are $q_{\min}$ and $\beta_{\min}$ that for values
greater than them black branes are thermally stable.

##  VI Conclusions and discussions

In this paper, we studied the higher dimensional action in the context of
dilaton gravity and in the presence of the logarithmic nonlinear
electrodynamics. By varying the action, we found the field equations of this
theory. Then, we constructed a new class of charged, rotating black brane
solutions, with $k=[(n-1)/2]$ rotation parameters, in an arbitrary dimension.
We found that the presence of the dilaton field changes the asymptotic
behavior of the obtained solutions to be neither flat nor (A)dS. We presented
the suitable counterterm which remove the divergences of the action in the
presence of the dilaton field. In the absence of a non-trivial dilaton
($\alpha=\gamma=0$), these solutions reduce to the asymptotically AdS charged
rotating black brane solutions of Einstein theory in the presence of
logarithmic nonlinear electrodynamics Hendi . When
$\beta\longrightarrow\infty$, these solutions reduce to the charged rotating
dilaton black brane solutions given in Ref. SDRP . We also calculated the
conserved and thermodynamic quantities of the spacetime such as mass, angular
momentum, temperature, entropy and electric potential and checked that the
first law of thermodynamics holds on the black brane horizon.

Then, we explored thermal stability of the nonlinearly charged rotating black
brane solutions in both canonical and grand-canonical ensembles. We found that
in both ensembles the solutions are thermally stable for $\alpha\leq 1$, while
for $\alpha>1$ the solutions are always thermally unstable in the grand-
canonical ensemble where $\alpha$ is the dilaton-electromagnetic coupling
constant. In the canonical ensemble, however, we can have both stable and
unstable solutions for $\alpha>1$. We found that, in this ensemble, there is
an $\alpha_{\max}>1$ for which the solutions are thermally stable provided
$\alpha<\alpha_{\max}$. The pointed out results implies that the thermal
stability is ensemble-dependent and $\alpha$ influences the stability under
thermal perturbations. These results are expectable since different ensembles
allow different sets of quantities to be variable and a thermally stable
system is one which is stable under varying variable quantities. On the other
hand, values of conserved and thermodynamic quantities depend on values of
parameters such as $\alpha$ and therefore the fact that dilaton-
electromagnetic coupling has direct effect on thermal stability of the system
seems natural.

It is notable to mention that in this paper, we only constructed the charged
rotating dilaton black branes of nonlinear electrodynamics with flat horizon.
One can try to construct the rotating dilaton black holes of this theory with
curved horizon. One can laso investigate the thermodynamic geometry of these
solutions. The latter issue is now under investigation and the results will be
presented elsewhere.

Conflict of Interests

The author declares no conflict of interests for the present paper.

###### Acknowledgements.

We thank the Research Council of Shiraz University. This work has been
supported financially by Research Institute for Astronomy & Astrophysics of
Maragha (RIAAM), Iran.

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