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\begin{document}
\title{Topological Black Holes in Brans-Dicke-Maxwell Theory }
\author{A. Sheykhi$^{a,b}$\footnote{sheykhi@mail.uk.ac.ir} and H. Alavirad$^{a}$}
\address{$^a$Department of Physics, Shahid Bahonar University, PO Box 76175, Kerman, Iran\\
$^b$Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran}
\begin{abstract}
We derive a new analytic solution of $(n+1)$-dimensional $(n\geq
4)$ Brans-Dikce-Maxwell theory in the presence of a potential for
the scalar field, by applying a conformal transformation to the
dilaton gravity theory. These solutions describe topological
charged black holes with unusual asymptotics. We obtain the
conserved and thermodynamic quantities through the use of the
Euclidean action method. We also study the thermodynamics of the
solutions and verify that the conserved and thermodynamic
quantities of the solutions satisfy the first law of black hole
thermodynamics.
\end{abstract}
\maketitle
\section{Introduction\label{I}}
Among the alternative theories of general relativity, perhaps the
most well-known theory is the scalar-tensor theory which was
pioneered several decades ago by Brans-Dicke \cite{BD}, who sought
to incorporate Mach's principle into gravity. Compared to
Einstein's general relativity, Brans-Dicke (BD) theory describes
the gravitation in terms of the metric as well as scalar fields
and accommodates both Mach's principle and Dirac's large number
hypothesis as new ingredients. Although BD theory has passed all
the possible observational tests \cite{Will}, however, the
singularity problem remains yet in this theory. In recent years
this theory got a new impetus as it arises naturally as the low
energy limit of many theories of quantum gravity such as the
supersymmetric string theory or the Kaluza-Klein theory. Besides,
recent observations show that at the present epoch, our Universe
expands with acceleration instead of deceleration along the scheme
of standard Friedmann models and since general relativity could
not describe such Universe correctly, cosmologists have attended
to alternative theories of gravity such as BD theory. Due to
highly nonlinear character of BD theory, a desirable pre-requisite
for studying strong field situation is to have knowledge of exact
explicit solutions of the field equations. And as black holes are
very important both in classical and quantum gravity, many authors
have investigated various aspects of them in BD theory \cite{Sen}.
It turned out that the dynamic scalar field in the BD theory plays
an important role in the process of collapse and critical
phenomenon. The first four-dimensional black hole solutions of BD
theory was obtained by Brans in four classes \cite{Brans}. It has
been shown that among these four classes of the static spherically
symmetric solutions of the vacuum BD theory of gravity only two
are really independent, and only one of them is permitted for all
values of $\omega$. It has been proved that in four dimensions,
the stationary and vacuum BD solution is just the Kerr solution
with constant scalar field everywhere \cite{Hawking}. It has been
shown that the charged black hole solution in four-dimensional
Brans-Dicke-Maxwell (BDM) theory is just the Reissner-Nordstrom
solution with a constant scalar field, however, in higher
dimensions, one obtains the black hole solutions with a nontrivial
scalar field \cite{Cai1}. This is because the stress energy tensor
of Maxwell field is not traceless in the higher dimensions and the
action of Maxwell field is not invariant under conformal
transformations. Accordingly, the Maxwell field can be regarded as
the source of the scalar field in the BD theory \cite{Cai1}. Other
studies on black hole solutions in BD theory have been carried out
in \cite{Kim,Deh1,Gao}. In this Letter, we would like to study
topological black hole solutions in $(n+1)$-dimensional BDM theory
for an arbitrary value of $\omega$ and investigate their
properties.
In the next section, we review the basic equations and the
conformal transformation between the action of the dilaton gravity
theory and the BD theory. In section \ref{III}, we construct
charged topological black hole solutions in BDM theory and
investigate their properties. In section \ref{IV}, we study the
thermodynamical properties of the solutions and calculate the
conserved quantities through the use of the Euclidean action
method. The last section is devoted to summary and conclusion.
\section{ Basic equations and Conformal Transformations}\label{II}
The action of the $(n+1)$-dimensional Brans-Dicke-Maxwell theory
with one scalar field $\Phi$ and a self-interacting potential
$V(\Phi)$ can be written as
\begin{equation}
I_{G}=-\frac{1}{16\pi}\int_{\mathcal{M}}
d^{n+1}x\sqrt{-g}\left(\Phi {R}-\frac{\omega}{\Phi}(\nabla\Phi)^2
-V(\Phi)-F_{\mu \nu}F^{\mu \nu}\right),\label{act1}
\end{equation}
where ${R}$ is the scalar curvature, $V(\Phi )$ is a potential for
the scalar field $\Phi $, $F_{\mu \nu }=\partial _{\mu }A_{\nu
}-\partial _{\nu }A_{\mu }$ is the electromagnetic field tensor,
and $A_{\mu }$ is the electromagnetic potential. The factor
$\omega$ is the coupling constant. The equations of motion can be
obtained by varying the action (\ref{act1}) with respect to the
gravitational field $g_{\mu \nu }$, the scalar field $\Phi $ and
the gauge field $A_{\mu }$ which yields the following field
equations
\begin{eqnarray}
&&G_{\mu
\nu}=\frac{\omega}{\Phi^2}\left(\nabla_{\mu}\Phi\nabla_{\nu}\Phi-\frac{1}{2}g_{\mu
\nu}(\nabla\Phi)^2\right)
-\frac{V(\Phi)}{2\Phi}g_{\mu \nu}+\frac{1}{\Phi}\left(\nabla_{\mu}\nabla_{\nu}\Phi-g_{\mu \nu}\nabla^2\Phi\right)\nonumber \\
&&+\frac{2}{\Phi}\left(F_{\mu \lambda}F_{ \nu}^{\
\lambda}-\frac{1}{4}F_{\rho \sigma}F^{\rho
\sigma}g_{\mu \nu}\right), \label{Eq1}\\
&&\nabla^2\Phi=-\frac{n-3}{2(n-1)\omega+2n}F^2+\frac{1}{2(n-1)\omega+2n}\left((n-1)\Phi\frac{dV(\Phi)}{d\Phi}
-(n+1)V(\Phi)\right),\label{Eq2} \\
&&\nabla_{\mu}F^{\mu \nu}=0, \label{Eq3}
\end{eqnarray}
where $G_{\mu \nu}$ and $\nabla$ are, respectively, the Einstein
tensor and covariant differentiation in the spacetime metric
$g_{\mu \nu}$. It is apparent that the right hand side of Eq.
(\ref{Eq1}) includes the second derivatives of the scalar field,
so it is hard to solve the field equations (\ref{Eq1})-(\ref{Eq3})
directly. We can remove this difficulty by a conformal
transformation. Indeed, the BDM theory (\ref{act1}) can be
transformed into the Einstein-Maxwell theory with a minimally
coupled scalar field via the conformal transformation
\begin{eqnarray}
&&\bar{g}_{\mu \nu}=\Phi^{\frac{2}{n-1}}g_{\mu \nu}, \nonumber \\
&&\bar{\Phi}=\frac{n-3}{4\alpha}\ln \Phi, \label{conf}
\end{eqnarray}
where
\begin{equation}
\alpha=\frac{n-3}{\sqrt{4(n-1)\omega+4n}}. \label{6}
\end{equation}
Using this conformal transformation, the action (\ref{act1})
transforms to
\begin{equation}
\bar{I}_{G}=-\frac{1}{16\pi}\int_{\mathcal{M}}
d^{n+1}x\sqrt{-\bar{g}}\left({\bar{R}}-\frac{4}{n-1}(\bar{\nabla}\
\bar{\Phi})^2-\bar{V}(\bar{\Phi})-e^{-\frac{4\alpha\bar{\Phi}}{n-1}}\bar{F}_{\mu
\nu}\bar{F}^{\mu \nu}\right), \label{act2}
\end{equation}
where ${\bar{R}}$ and $\bar{\nabla}$ are, respectively, the Ricci
scalar and covariant differentiation in the spacetime metric
$\bar{g}_{\mu \nu}$, and $\bar{V}(\bar{\Phi})$ is
\begin{equation}
\bar{V}(\bar{\Phi})=\Phi^{-\frac{n+1}{n-1}}V(\Phi).\label{8}
\end{equation}
This action is just the action of the $(n+1)$-dimensional
Einstein-Maxwell-dilaton gravity, where $\bar{\Phi}$ is the
dilaton field and $\bar{V}(\bar{\Phi})$ is a potential for
$\bar{\Phi}$. $\alpha $ is an arbitrary constant governing the
strength of the coupling between the dilaton and the Maxwell
field. Varying the action (\ref{act2}), we can obtain equations of
motion
\begin{eqnarray}
&&\bar{{R}}_{\mu
\nu}=\frac{4}{n-1}\left(\bar{\nabla}_{\mu}\bar{\Phi}\bar{\nabla}
_{\nu}\bar{\Phi}+\frac{1}{4}\bar{V}(\bar{\Phi})\bar{g}_{\mu
\nu}\right)+ 2e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\left(\bar{F}_{\mu
\lambda}\bar{F}_{\nu}^{ \ \lambda} -\frac{1}{2(n-1)}\bar{F}_{\rho
\sigma}\bar{F}^{\rho \sigma}\bar{g}_{\mu \nu}\right), \label{Eqd1}\\
&&
\bar{\nabla}^2\bar{\Phi}=\frac{n-1}{8}\frac{\partial\bar{V}}{\partial\bar{\Phi}}
-\frac{\alpha}{2}e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\bar{F}_{\rho
\sigma}\bar{F}^{\rho \sigma},\label{Eqd2}\\
&&
\bar{\nabla}_{\mu}\left(e^{\frac{-4\alpha\bar{\Phi}}{n-1}}\bar{F}^{\mu
\nu}\right)=0. \label{Eqd3}
\end{eqnarray}
Comparing Eqs. (\ref{Eq1})-(\ref{Eq3}) with Eqs.
(\ref{Eqd1})-(\ref{Eqd3}), we find that if $\left(\bar{g}_{\mu
\nu},\bar{F}_{\mu \nu},\bar{\Phi}\right)$ is the solution of Eqs.
(\ref{Eq1})-(\ref{Eq3}) with potential $\bar{V}(\bar{\Phi})$, then
\begin{equation}
\left[{g}_{\mu \nu},{F}_{\mu
\nu},{\Phi}\right]=\left[\exp\left({\frac{-8\alpha
\bar{\Phi}}{(n-1)(n-3)}}\right)\bar{g}_{\mu \nu},\bar{F}_{\mu
\nu},\exp\left({\frac{4\alpha \bar{\Phi}}{n-3}}\right)\right],
\end{equation}\label{conf2}
is the solution of Eqs. (\ref{Eqd1})-(\ref{Eqd3}) with potential
$V(\Phi)$.
\section{Topological black holes in BDM theory\label{III}}
The solutions of the field equations (\ref{Eqd1})-(\ref{Eqd3}) for
various metrics have been constructed by many authors (see e.g.
\cite{MW,CHM,Cai2,Clem,DF,SDR,sheykhi1}). Here we would like to
obtain the topological black hole solutions of the field equations
(\ref{Eq1})-(\ref{Eq3}) in BDM theory, by applying the conformal
transformations (\ref{conf2}) on the corresponding solutions in
the dilaton gravity theory. The $(n+1)$-dimensional topological
black hole solution of the field equations
(\ref{Eqd1})-(\ref{Eqd3}) has been obtained by one of us in
\cite{sheykhi1} for two Liouville-type dilaton potentials
\begin{equation}\label{v2}
\bar{V}(\bar{\Phi}) = 2\Lambda_{0} e^{2\zeta_{0}\bar{\Phi}} + 2
\Lambda e^{2\zeta \bar{\Phi}},
\end{equation}
where $\Lambda_{0}$, $\Lambda$, $ \zeta_{0}$ and $ \zeta$ are
constants. In \cite{sheykhi1} the spacetime metric was written in
the form
\begin{equation}
d\bar{s}^2=-f(r)dt^2+\frac{dr^2}{f(r)}+r^2{R^2(r)}h_{ij}dx^{i}dx^j,
\label{met1}
\end{equation}
where $f(r)$ and $R(r)$ are functions of $r$ which should be
determined, and $h_{ij}$ is a function of coordinate $x_i$ which
spanned an $(n-1)-$dimensional hypersurface with constant scalar
curvature $(n-1)(n-1)k$. Here $k$ is a constant which
characterized the hypersurface. Without loss of generality, one
can take $k=0,1,-1$, such that the black hole horizon or
cosmological horizon in (\ref{met1}) can be a zero (flat),
positive (elliptic) or negative (hyperbolic) constant curvature
hypersurface. The Maxwell equations can be integrated immediately
to give
\begin{equation}
\bar{F}_{tr}=\frac{qe^{\frac{4\alpha\bar{\Phi}}{n-1}}}{(rR)^{n-1}},
\label{13}
\end{equation}
where $q$, an integration constant, is related to the electric
charge of black hole. Defining the electric charge via $ Q =
\frac{1}{4\pi} \int \exp\left[{-4\alpha\bar{\Phi}/(n-1)}\right]
\text{ }^{*} \bar{F} d{\Omega}, $ we get
\begin{equation}
{Q}=\frac{q\Omega _{n-1}}{4\pi}, \label{Charge}
\end{equation}
where $\Omega_{n-1}$ represents the volume of constant curvature
hypersurface described by $h_{ij}dx^idx^j$. Notice that $Q$ is
invariant under the conformal transformation (\ref{conf2}). Using
the ansatz
\begin{equation}
R(r)=e^{\frac{2\alpha\bar{\Phi}}{n-1}}, \label{ansa}
\end{equation}
one can show that the system of equations
(\ref{Eqd1})-(\ref{Eqd2}) have solutions of the form
\cite{sheykhi1}
\begin{eqnarray}
&&f(r)=-\frac{k(n-2)(\alpha^2+1)^2b^{-2\gamma}r^{2\gamma}}{(\alpha^2-1)(\alpha^2+n-2)}-
\frac{m}{r^{(n-1)(1-\gamma)-1}}+
\frac{2q^{2}(\alpha^2+1)^{2}b^{-2(n-2)\gamma}}{(n-1)(\alpha^2+n-2)}r^{2(n-2)(\gamma-1)} \nonumber \\
&+&
\frac{2\Lambda(\alpha^2+1)^{2}b^{2\gamma}}{(n-1)(\alpha^2-n)}r^{2(1-\gamma)},
\label{f1}\\
&& R(r)=\left(\frac{b}{r}\right)^{\gamma},\label{R1}\\
&&
\bar{\Phi}=\frac{(n-1)\alpha}{2(1+\alpha^2)}\ln\left(\frac{b}{r}\right),
\label{phibar1}
\end{eqnarray}
where $b$ is an arbitrary constant and $\gamma =\alpha
^{2}/(\alpha ^{2}+1)$. In the above expression, $m$ appears as an
integration constant and is related to the mass of the black hole.
In order to fully satisfy the system of equations, we must have
\cite{sheykhi1}
\begin{equation}\label{lam}
\zeta_{0} =\frac{2}{\alpha(n-1)}, \hspace{.8cm}
\zeta=\frac{2\alpha}{n-1}, \hspace{.8cm} \Lambda_{0} =
\frac{k(n-1)(n-2)\alpha^2 }{2b^2(\alpha^2-1)}.
\end{equation}
Notice that here $\Lambda$ is a free parameter which plays the
role of the cosmological constant. Using the conformal
transformation (\ref{conf2}), the $(n+1)$-dimensional topological
black hole solutions of BDM theory can be obtained as
\begin{equation}
ds^2=-U(r)dt^2+\frac{dr^2}{V(r)}+r^2{H^2(r)}h_{ij}dx^{i}dx^j,
\label{met2}
\end{equation}
where $U(r)$, $V(r)$, $H(r)$ and $\Phi(r)$ are
\begin{eqnarray}
&&U(r)=-\frac{k(n-2)(\alpha^2+1)^2b^{-2\gamma(\frac{n-1}{n-3})}r^{2\gamma(\frac{n-1}{n-3})}}
{(\alpha^2-1)(\alpha^2+n-2)}-
\frac{mb^{(\frac{-4\gamma}{n-3})}}{r^{n-2}}r^{\gamma\left(n-1+\frac{4}{n-3}\right)}\nonumber \\
&&+\frac{2q^{2}(\alpha^2+1)^{2}b^{-2\gamma\left(n-2+\frac{2}{n-3}\right)}}{(n-1)\left(\alpha^2+n-2\right)r^{2[(n-2)(1-\gamma)-
\frac{2\gamma}{n-3}]}}+
\frac{2\Lambda(\alpha^2+1)^{2}b^{2\gamma(\frac{n-5}{n-3})}}{(n-1)(\alpha^2-n)}r^{2(1-\frac{\gamma(n-5)}{n-3})},
\label{U1}\\
&&V(r)=-\frac{k(n-2)(\alpha^2+1)^2b^{-2\gamma(\frac{n-5}{n-3})}r^{2\gamma(\frac{n-5}{n-3})}}
{(\alpha^2-1)(\alpha^2+n-2)}-\frac{mb^{(\frac{4\gamma}{n-3})}}{r^{n-2}}r^{\gamma(n-1-\frac{4}{n-3})}
\nonumber \\
&&+\frac{2q^{2}(\alpha^2+1)^{2}b^{-2\gamma\left(n-2-\frac{2}{n-3}\right)}}{(n-1)(\alpha^2+n-2)r^{2[(n-2)(1-\gamma)+
\frac{2\gamma}{n-3}]}}+
\frac{2\Lambda(\alpha^2+1)^{2}b^{2\gamma(\frac{n-1}{n-3})}}{(n-1)(\alpha^2-n)}r^{2(1-\gamma(\frac{n-1}{n-3}))}
,
\label{V1}\\
&& H(r)=\left(\frac{b}{r}\right)^{\frac{(n-5)\gamma}{n-3}},
\label{H1}\\
&&\Phi(r)=\left(\frac{b}{r}\right)^{\frac{2(n-1)\gamma}{n-3}}.
\label{Phi1}
\end{eqnarray}
Applying the conformal transformation, the electromagnetic field
and the scalar potential become
\begin{eqnarray}
F_{tr}&=&\frac{qb^{(3-n)\gamma}}{r^{(n-3)(1-\gamma)+2}},
\label{Ftr2} \\
V(\Phi)&=&2\Lambda_{0}\Phi^{\frac{n(\alpha^2+1)+\alpha^2-3}{\alpha^2(n-1)}}+2\Lambda\Phi^2.
\label{V2}
\end{eqnarray}
It is worth noting that in the case $k\neq0$, these solutions are
ill-defined for the string case where $\alpha=1$ (this is
corresponding to $\omega=(n-9)/4$). It is also notable to mention
that the electric field $F_{tr}$ and the scalar field $\Phi(r)$ go
to zero as $r\rightarrow\infty$. When $\omega\rightarrow\infty$
($\alpha=0=\gamma$), these solutions reduce to
\begin{eqnarray}
U(r)=V(r)=k-\frac{m}{r^{n-2}}+\frac{2q^2}{(n-1)(n-2)r^{2(n-2)}}-\frac{2\Lambda}{n(n-1)}r^2,
\end{eqnarray}
which describes an $(n+1)$-dimensional asymptotically (anti)-de
Sitter topological black holes with a positive, zero or negative
constant curvature hypersurface (see e.g. \cite{Bril1,Cai3}). It
easy to show that the Kretschmann scalar $R_{\mu \nu \lambda
\kappa}R^{\mu \nu \lambda \kappa}$ diverge at $r=0$ and therefore
there is an essential singularity at $r=0$. As one can see from
Eqs. (\ref{U1})-(\ref{V1}), the solutions are also ill-defined for
$\alpha=\sqrt{n}$ with $\Lambda\neq0$ (corresponding to
$\omega=-3(n+3)/4n$). The cases with $\alpha<\sqrt{n}$ and
$\alpha>\sqrt{n}$ should be considered separately. In the first
case where $ \alpha <\sqrt{n}$, there exist a cosmological
horizon for $\Lambda >0$, while there is no cosmological horizons
if $\Lambda <0$. Indeed, in the latter case ($\alpha <\sqrt{n}$
and $\Lambda <0$) the spacetimes exhibit a variety of possible
casual structures depending on the values of the metric parameters
$\alpha $, $m$, $q$ and $k$. Therefore, our solutions can
represent topological black hole, with inner and outer event
horizons, an extreme topological black hole, or a naked
singularity provided the parameters of the solutions are chosen
suitably. In the second case where $\alpha
>\sqrt{n}$, the spacetime has a cosmological horizon for $\Lambda <0$ despite the value of
curvature constant $k$, while for $\Lambda>0$ we have cosmological
horizon in the case $k=1$ and naked singularity for $k=0,-1$.
\section{Thermodynamics of topological BD black holes \label{IV}}
We now turn to the investigation of the thermodynamics of
charged topological BD black hole solutions we have just found. With the Euclidean action
method we obtain the conserved and thermodynamic quantities of
black holes \cite{Cai4}.
In the context of BD gravity, where we have the additional
gravitational scalar degree of freedom, the entropy of the black
hole does not follow the area law. This is due to the fact that
the black hole entropy comes from the boundary term in the
Euclidean action formalism. The boundary term in the BD theory
leads to
\begin{equation}
I_b=-\frac{1}{8\pi}\int_{\partial\mathcal{M}}d^{n}x\sqrt{-\gamma}\Phi
\left(K-K_0\right), \label{GH}
\end{equation}
where $\gamma$ and $K$ are, respectively, the determinant of the
induced metric on the boundary $\partial\mathcal{M}$ in the
spacetime metric $g_{\mu \nu}$, and the trace of the extrinsic
curvature of boundary. $K_0$ is the extrinsic curvature of vacuum
background (here it is the $(n+1)$-dimensional Minkowski
spacetime). The black hole entropy in the BD theory is found to be
\begin{equation}
S=-\frac{1}{8\pi}\int_{r_{+}}d^{n}x\sqrt{-\gamma}\Phi
\left(K-K_0\right)=\frac{1}{4}\Phi(r_{+}) A, \label{ent}
\end{equation}
where $A$ is the area of the the outer horizon $r_+$ in BD theory.
It is apparent from (\ref{ent}) that the area formula is no longer
valid in the BD theory \cite{kang}. The explicit form of the
entropy can be found as
\begin{equation}
{S}=\frac{b^{(n-1)\gamma}r_{+}^{(n-1)(1-\gamma )}}{4}\Omega
_{n-1},\label{Entropy}
\end{equation}
which shows that the entropy remains unchanged under conformal
transformations \cite{sheykhi1}. The Hawking temperature of the
topological black hole on the outer horizon $r_+$ can be
calculated using the relation
\begin{equation}
T_{+}=\frac{\kappa}{2\pi}= \frac{f^{\text{ }^{\prime
}}(r_{+})}{4\pi},
\end{equation}
where $\kappa$ is the surface gravity. Then, one can easily show
that
\begin{eqnarray}\label{Tem}
T_{+}&=&-\frac{(\alpha ^2+1)}{2\pi (n-1)}\left(
\frac{k(n-2)(n-1)b^{-2\gamma}}{2(\alpha^2-1)}r_{+}^{2\gamma-1}
+\Lambda b^{2\gamma}r_{+}^{1-2\gamma}+q^{2}b^{-2(n-2)\gamma
}r_{+}^{(2n-3)(\gamma -1)-\gamma}\right)\nonumber\\
&=&-\frac{k(n-2)(\alpha ^2+1)b^{-2\gamma}}{2\pi(\alpha
^2+n-2)}r_{+}^{2\gamma-1}+\frac{(n-\alpha ^{2})m}{4\pi(\alpha
^{2}+1)}{r_{+}}^{(n-1)(\gamma -1)} \nonumber\\
&&-\frac{q^{2}(\alpha ^{2}+1)b^{-2(n-2)\gamma }}{\pi(\alpha
^{2}+n-2)}{r_{+}}^{(2n-3)(\gamma -1)-\gamma}.
\end{eqnarray}
If we compare Eq. (\ref{Tem}) with the temperature obtained in the
dilaton gravity theory \cite{sheykhi1}, we find that the
temperature is invariant under the conformal transformation
(\ref{conf2}). This is due to the fact that the conformal
parameter is regular at the horizon. Equation (\ref{Tem}) shows
that when $k=0$, the temperature is negative for the two cases of
(\emph{i}) $\alpha
>\sqrt{n}$ despite the sign of $\Lambda $, and (\emph{ii})
positive $\Lambda $ despite the value of $\alpha $. As we argued
above in these two cases we encounter with cosmological horizons,
and therefore the cosmological horizons have negative temperature
in this case.
The ADM (Arnowitt-Deser-Misner) mass of the topological black hole
can be calculated through the use of the Euclidean action method
\cite{Cai4}. We obtain
\begin{equation}
{M}=\frac{b^{(n-1)\gamma}(n-1) \Omega _{n-1}}{16\pi(\alpha^2+1)}m,
\label{Mass}
\end{equation}
which is invariant under the conformal transformation
\cite{sheykhi1}.
The electric potential $U$, measured at infinity with respect to
the horizon, is defined by
\begin{equation}
U=A_{\mu }\chi ^{\mu }\left| _{r\rightarrow \infty }-A_{\mu }\chi
^{\mu }\right| _{r=r_{+}}, \label{Pot}
\end{equation}
where $\chi=\partial_{t}$ is the null generator of the horizon.
One can easily show that the gauge potential $A_{t }$
corresponding to the electromagnetic field (\ref{Ftr2}) can be
written as
\begin{eqnarray}\label{vectorpot}
A_{t}&=&\frac{qb^{(3-n)\gamma }}{\Upsilon r^{\Upsilon }},
\end{eqnarray}
where $\Upsilon =(n-3)(1-\gamma )+1$. Therefore, the electric
potential may be obtained as
\begin{equation}
U=\frac{qb^{(3-n)\gamma }}{ \Upsilon{r_{+}}^{\Upsilon }}.
\label{Pot}
\end{equation}
Finally, we consider the first law of thermodynamics for the
topological black hole. Numerical calculations show that the the
conserved and thermodynamic quantities calculated above satisfy
the first law of black hole thermodynamics
\begin{equation}
dM = TdS+Ud{Q}.
\end{equation}
\section{Summary and Conclusions \label{V}}
To conclude, in $(n+1)$-dimensions, when the $(n-1)$-sphere of
black hole event horizn is replaced by an $(n-1)$-dimensional
hypersurface with positive, zero or negative constant curvature,
the black hole is called as a topological black hole. The
construction and analysis of these exotic black holes in anti-de
Sitter (AdS) space is a subject of much recent interest. This
interest is motivated by the correspondence between the
gravitating fields in an AdS spacetime and conformal field theory
on the boundary of the AdS spacetime. In this Letter, we further
generalized these exotic solutions by constructing a class of
$(n+1)$-dimensional $(n\geq4)$ topological black holes in BDM
theory in the presence of a potential for the scalar field. In
contrast to the topological black holes in the Einstein-Maxwell
theory, which are asymptotically AdS, the topological black holes
we found here, are neither asymptotically flat nor (A)dS. Indeed,
the scalar potential plays a crucial role in the existence of
these black holes, as the negative cosmological constant does in
the Einstein-Maxwell theory. When $k=\pm 1 $, these solutions do
not exist for the string case where $\alpha=1$ (corresponding to
$\omega=(n-9)/4$). Besides they are ill-defined for
$\alpha=\sqrt{n}$ with $\Lambda\neq0$ (corresponding to
$\omega=-3(n+3)/4n$). We obtained the conserved and thermodynamic
quantities through the use of the Euclidean action method, and
verified that the conserved and thermodynamic quantities of the
solutions satisfy the first law of black hole thermodynamics. We
found that the entropy does not satisfy the area law. We also
found that the conserved and thermodynamic quantities are
invariant under the conformal transformation.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments{This work has been supported financially by
Research Institute for Astronomy and Astrophysics of Maragha,
Iran.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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