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\preprint{NTUTP-03/A07}

\author{Ishwaree P. Neupane}

%% \email{ishwaree.neupane@cern.ch}

\title{\large \bf Thermodynamic and Gravitational Instability on Hyperbolic
Spaces}

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\affiliation{ Department of Physics,
National Taiwan University, Taipei 106, Taiwan, R.O.C.\\
{\scriptsize \bf ishwaree@phys.ntu.edu.tw, ~
Ishwaree.Neupane@cern.ch }\\
~~~~~}

% \altaffiliation[Also at ]{Theory Division, CERN, Switzerland.}
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\begin{abstract}

We study the properties of anti-de Sitter black holes for various
horizon topologies ($k=0,\,\pm 1$) and for various dimensions. We
explore the thermodynamic and classical (in)stability of higher
dimensional black holes with emphasis on the less well understood
$k=-1$ solution. In particular, we show that there exists a unique
$k=-1$ extremal black hole solution which has the lowest energy
for all spacetimes in its asymptotic class. What looks encouraging
is that the Gauss-Bonnet type curvature corrections to the
Einstein action not only admit exact solutions but they might be
crucial for stability of hyperbolic $(k=-1)$ black holes. This in
turn implies that the hyperbolic spacetimes can be stable
thermodynamically and classically if the background is defined by
an extremal solution and the extremal entropy is non-negative. For
the ground state metric taken from the Einstein-Gauss-Bonnet
theory, the gravitational potential can be positive and bounded
from below for a small coupling $\alpha~(<<l^2)$, with $l$ being
the effective curvature radius of anti-de Sitter space.

\end{abstract}

\pacs{04.70.-s, 04.50.+h, 11.10.Kk, 11.25.-w}

\maketitle

\section{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In parallel to the development of Anti de Sitter (AdS) holography
(more generally, the gravity -- gauge theory
correspondence~\cite{Maldacena97a,Witten98}), the AdS black holes
in higher dimensional spacetimes are known to play an important
role in the dual field theory defined on the AdS (conformal)
boundary; viz., a black hole in AdS space is described as a
thermal state of the dual conformal field theory
(CFT)~\cite{Witten98a}. In the presence of a negative cosmological
constant the Einstein field equations admit AdS black holes as
exact vacuum solutions whose event horizons are Einstein manifolds
with positive, zero or negative scalar curvature. Many efforts
have been concentrated to explore AdS black holes with
non-spherical event horizons, analyzing black hole solutions in
four-dimensions~\cite{Vanzo} (see~\cite{Lemos95a,Mann96a} for
relevant discussions) and in higher
dimensions~\cite{Birmingham98a}. The analysis
in~\cite{Birmingham98a} (see also~\cite{Emparan99b} in the spirit
of holographic counter-term prescription proposed in
~\cite{Vijay99a,Emparan99a}) is well motivated from the AdS/CFT
correspondence. The study of hyperbolic (also called
"topological") black holes~\cite{Mann96a,Vanzo}) has provided a
new angle of interest to the study of non-supersymmetric version
of the AdS/CFT correspondence. No much has been done to understand
the thermodynamic and classical instability of these solutions in
higher dimensions when there are Gauss-Bonnet (GB) type curvature
corrections to the black hole thermodynamics. So we study this
problem in detail in this paper.

One of the important issues that one has to address on a
hyperbolic manifold is the positivity of energy. In the low energy
limit of the superstring theory (namely, supergravity), the
underlying supersymmetry and constant positive (scalar) curvature
ensure that solutions which approach AdS globally or flat locally
respect the known positive energy theorems. In general, the
positive energy theorems only show stability of the spacetimes
which saturate the bounds, i.e., they become asymptotically either
locally flat or AdS. However, the known theorems do not extend to
include the Horowitz-Myers (HM) soliton~\cite{Horowitz98a}, whose
AdS asymptotic is a toroidal space with zero scalar curvature. The
HM soliton is a non-supersymmetric background, but it is
conjectured that an AdS soliton is the ground state for planar
($k=0$) black holes. This means the absence of sypersymmetry does
not necessarily imply instability of the background.

On the other hand, there arises the question of background choice
for black hole spacetimes with hyperbolic event horizons. In a
gravitational theory, it is necessary to make the Euclideanized
action finite. One can do so by assigning classically stable
lowest energy configurations to the AdS black holes with
$k=0,\,\pm 1$ event horizons. For a spherical black hole (i.e.,
$k=+1$), the background is simply the global AdS space, which is a
solution at finite temperature. Of course, a spherical event
horizon topology is the most convenient choice to explain the dual
dynamics, such as, a thermal phase transition in large N gauge
theories~\cite{Witten98a}. Also, the all specific examples of CFTs
that arise in AdS/CFT are well behaved when $k=+1$. Nonetheless,
there are some good reasons to explore physics behind the $k=-1$
topology, e.g., the ground state for $k=-1$ is different from the
solution that is locally isometric to AdS, and the hyperbolic AdS
black holes exhibit some new and interesting features that other
two cousins ($k=0,+1$) of AdS simply do not share, such as
enhancements in the value of the entropy that are not accompanied
by increment in the energy~\cite{Emparan99b}. It is also
known~\cite{Emparan98a,Klemm99a} that the AdS space with $k=-1$ is
relevant to study CFTs with less than maximal (or no)
supersymmetry.

Maximally symmetric hyperbolic spaces~\cite{Kehagias00a} naturally
arise as the near-horizon region of certain
$p$-branes~\cite{Emparan98a} and black hole
geometries~\cite{Horowitz91a,Gibbons93a} in supergravity theories.
Therefore, the background choice in the $k=-1$ case and the
classical stability of hyperbolic spaces both are interesting
issues. It is argued in~\cite{Galloway02a} that a conformal
boundary with cross sections $k=0,1$ does not serve as the
conformal boundary of $k=-1$ non-singular asymptotically locally
AdS spacetimes. This means, for hyperbolic black holes, a global
AdS space as the choice of background might be inconsistent. This
looks reasonable because the conformal boundary of a pure AdS
space is isometric to $I\!\!R^{1}\times {\cal M}^{n-1}$, where the
base manifold ${\cal M}^{n-1}$ is a round-sphere for a spherical
$(k=1)$ horizon, which, of course, has constant positive scalar
curvature, but, for hyperbolic black holes, ${\cal M}^{n-1}$ is
either a hyperbolic manifold $H^{n-1}$ or its quotient space
$H^{n-1}/\Gamma$. Therefore, one has to assign a background to
hyperbolic ($k=-1$) black holes different from a pure AdS
metric~\cite{Vanzo,Birmingham98a,Klemm99a}.

AdS spacetime whose conformal boundary (in the sense of Penrose)
is a hypersurface of negative scalar curvature could be unstable
under metric perturbations~\cite{Gibbons02a}, if the background is
a zero mass topological black hole. One may expect instability in
some cases, even if the background is the negative mass extremal
state, but not in others, depending upon the higher order
corrections. In Einstein gravity, one must restrict the spacetime
region $r>r_{extr}$, with $r_{extr}$ being the extremal horizon,
for stability. We will be led to entertain the somewhat radical
proposition that {\it the hyperbolic black hole solutions can be
stable, thermodynamically and classically, if the ground state
metric receives higher order curvature corrections and the
background is defined by a negative mass extremal state}. We show
that there exists a unique $k=-1$ extremal black hole solution
which has lowest energy for all spacetimes in its asymptotic
class.

The layout of the paper is as follows. In section II we give black
hole solutions in AdS spacetimes, compute extremal parameters and
define different reference backgrounds in Einstein gravity
modified with a Gauss-Bonnet term. In section III we compute
Euclideanized actions applicable to the curvature $k=0,\,\pm 1$ of
the black hole horizons, part of these results had been obtained
in~\cite{IPN02b}. In section IV we relate the free energy and
specific heat curves and discuss the thermal phase transitions. In
section V we turn to stability analysis of the background metrics
(vacuum solutions) in Einstein gravity under metric perturbations,
by setting up a Sturm-Liouville problem. We extend this analysis
in section VI for the background metrics in the
Einstein-Gauss-Bonnet theory. Finally, section VII contains the
discussion, outlook and future problems.

\section{Black holes in AdS Space}

We shall be interested in the gravitational action of a general
form that might include the contribution from higher curvature
terms. Our starting point is therefore the Lagrangian of gravity
including a Gauss-Bonnet term \bea\label{action1}
I&=&\frac{1}{16\pi G_{n+1}}\int
d^{n+1}x\,\sqrt{-g}\,\left(R-2\Lambda\right)+ {\mbox{surface
terms}} \nn \\
&+& \alpha^\prime \int d^{n+1}x\,\sqrt{-g}\,
\left(R_{abcd}R^{abcd}-4 R_{cd}R^{cd}+R^2\right)\,.\nn \\
&{}&~~~~~ \eea The Gauss-Bonnet type curvature corrections to the
Einstein action in a pure gravity theory were originally proposed
by Lanczos~\cite{Lanczos} and Lovelock~\cite{Lovelock}, longtime
before the realization that a GB term can indeed arise in some
string theory as leading $\alpha^\p$-corrections~\cite{Zumino}.
The black hole solutions with a GB term were first given by
Boulware and Deser~\cite{Deser85a}. Those solutions were nicely
generalized by Myers and Simon~\cite{Myers88a} regularizing the
classical action (see also~\cite{Wiltshire} for charged
Gauss-Bonnet black holes). These calculations, however, were
limited to a vanishing cosmological constant and for a spherical
event horizon. Recently, there has been considerable interest in
generalizing those results for $ \Lambda < 0$ and $k \neq 1$, and
some progress have been made
in~\cite{Cai01a,Nojiri01c,IPN02a,IPN02b}, see
also~\cite{Cai98a,Zanelli00a} within the context of dimensionally
extended (higher derivative) Lovelock gravity.

\subsection{Black holes for $\alpha^\p=0$}

The Einstein field equations modified by a Gauss-Bonnet term take
the following form \be \label{newEE} R_{ab}-\frac{2\Lambda}{n-1}\,
g_{ab}=-\,32 \pi G_{n+1}\,\alpha^\p
\left(H_{ab}-\frac{1}{n-1}\,g_{ab} H\right)\,.\ee Here $H_{ab}=R
R_{ab}-2R_{acbd}R^{bd}+R_{acde}R_b\,^{cde}-2R_{ac} R_b^c$ and $H$
is its trace. For generality, throughout this paper we work in
$AdS_{n+1}$ spacetimes with arbitrary number of spatial dimensions
$n$. For $\alpha^\p=0$, we have the well known AdS black hole
solutions \be ds^2=-\,f(r)\,dt^2+\frac{dr^2}{f(r)}+{r^2}\,
d\Sigma_{k,n-1}^2\,,\ee with
$$ f(r)=k-\frac{\mu}{r^{n-2}}
+\frac{r^2}{\ell^2}\,$$ where $\mu$ is an integration constant and
$\ell$ is a length scale that fixes the cosmological term
$\Lambda=-\,n(n-1)/2\ell^2$. The metric of a $(n-1)$-dimensional
manifold ${\cal M}^{n-1}$, whose Ricci scalar equals to
$(n-1)(n-2)k$, is denoted as $d\Sigma_{k,n-1}^2$, the latter is
the unit metric on $S^{n-1}$, $I\!\!R^{n-1}$, or $H^{n-1}$,
respectively, for $k=1,\,0$ or $-1$. Defining a discrete subgroup
$\Gamma\subset S0(n-1,1)$ that acts on a $(n-1)$-dimensional
hyperbolic space, one can obtain a compact hyperbolic manifold
$H^{n-1}/\Gamma$. The spacetime metric for $\mu=0$ reduces to
$AdS$ in global coordinate when $k=+1$, and to a massless
topological black hole when $k=-1$.

One can fix the periodicity of the Euclidean time $\tau=it\to
\beta$, using $\left(f(r)\right)^{\prime} (r_+)=4\pi/\beta$, to be
\be {\beta}=\frac{4\pi\, r_+\, \ell^2}
{n\,r_+^2+k\,(n-2)\,\ell^2}\,.\ee where $r_+$ is the largest real
positive root of $f(r)$. When $k=+1$, the Euclidean period $\beta$
is always non-negative, but this is not so when $k=-1$. The notion
that an inverse of Euclidean period is the Hawking temperature of
a black hole is unique only up to the sign, that is, $T=1/\beta$
or $T=-\,1/\beta$. This is because the temperature cannot be
negative but the sign for the Euclidean period is a convention,
which depends on whether we measure it clockwise or
anti-clockwise, so $T=|\beta^{-1}|$. Similar situation would arise
in de Sitter spacetimes.

If $\mu=0$ is taken as the background for $k=-1$, then the
coordinate patch breaks down at $r=l$. This is partly resolved by
defining a negative mass extremal background (which has zero
temperature)~\cite{Vanzo,Birmingham98a}, the latter is defined by
the degenerate horizon at $r=r_{extr}$, with a mass parameter
$\mu=\mu_{extr}$, namely, \be
r^2_{extr}=\left(\frac{n-2}{n}\right) \ell^2\,,\quad
\mu_{extr}=-2\,\ell^{n-2}\sqrt{\frac{(n-2)^{n-2}}{n^n}}\,. \ee
These are the minimum values of $\mu$ and $r_+$ that are
compatible with cosmic censorship. Obviously, $\mu_{extr}<0$ for
$n>2$. When $n=2$, the extremal state is a pure $AdS_3$ soliton
with a singularity at $r_{extr}=0$, analogous to the BTZ
solution~\cite{BMTZ93a}. For a massless background, there appear
states, namely, those for $\mu_{extr}\leq \mu< 0$, with total
negative energy, and the gravitational potential is not well
behaved for small $r$. This means that a reference spacetime with
$\alpha^\p=0$ might be incomplete for $k=-1$. We may overcome to
this difficulty by activating higher curvature corrections, such
as a GB term.

\subsection{Gauss-Bonnet black holes}

We include a Gauss-Bonnet term into the effective action not only
because the black hole solutions are more general (than in
Einstein gravity), but also for a gravitational stability of AdS
black hole spacetimes with hyperbolic horizons, whose ground state
may be defined by a negative mass extremal state, the curvature
corrections to the Einstein action might be important.

For $\alpha^\p>0$, the metric solution is modified to \be
\label{adsBHn} f(r)= k+\frac{r^2}{2\alpha}\pm \,
\frac{r^2}{2\alpha}\sqrt{1+\frac{8\alpha\Lambda}{n(n-1)}
+\frac{4\alpha\,\mu}{r^n}} \,, \ee where the coupling
$\alpha=16\pi G_{n+1}\, (n-2)(n-3)\alpha^\prime$. We can fix the
cosmological constant $\Lambda=-\,n(n-1)/2 l^2$, where $l$ is the
effective curvature radius of AdS space. The fixing of $\Lambda$
should be consistent with the solution of the equations of motion.
For example, in a constant negative (or positive) curvature
background, $R_{ab}=-\,n c\,g_{ab}$, the field
equations~(\ref{newEE}) amount to fix the cosmological term
$\Lambda=-\,n(n-1)\,c\left(1-c\alpha\right)/2\equiv \mp\,n(n-1)/2
l^2$, where $c=1/\ell^2>0$ for $\Lambda< 0$, $c=0$ for
$\Lambda=0$, and $c=-\,1/\ell^2$ for $\Lambda>0$. In this case,
the length scale $\ell$ may be re-scaled, namely, $\ell^2\to l^2=
\ell^2/\left(1-\alpha/\ell^2\right)$.

For generality, henceforth, we use a common scale $l^2$, unless
otherwise stated, but a convention that $l^2\to \ell^2$, in the
limit $\alpha\to 0$, is to be understood. One also notes that the
coupling $\alpha$ is not the bare coupling $\alpha^\p$, although
they are proportional. For example, when $n=5$, $\alpha=96\pi
G_{6}\,\alpha^\p$, and the dimension of $\alpha$ is
$(\mbox{length})^2$.

\subsection{Extremal Gauss-Bonnet black holes}

For $\alpha>0$, the periodicity of the Euclidean time is \be
\label{HawkingT} {\beta}=\frac{4\pi\, r_+\, l^2\,
\left(r_+^2+2\alpha\,k\right)} {n\,r_+^4+k\,(n-2)\,r_+^2\,l^2
+(n-4)\alpha\,k^2\,l^2}\,. \ee The mass parameter $\mu$ may be
expressed in terms of the horizon radius $r_+$, namely, \be
\label{BHMass} \mu (r=r_+) = r_+^{n-2}\left(k+\frac{r_+^2}{l^2}
+\frac{\alpha k^2}{r_+^2}\right)\equiv \frac{16\pi
G_{n+1}\,M}{(n-1)V_{n-1}}\,, \ee where $M$ is the
Arnowitt-Deser-Misner (ADM) mass of a black hole, and $V_{n-1}$ is
the volume of $d\Sigma^2_{k,n-1}$.

With $\alpha>0$, the extremal black hole interpretation exists
only for $n\geq 4$, because the Gauss-Bonnet term is a topological
for $n=3$, and there is no extremal solution for $n=2$. When
$n=4$, there exists the degenerate extremal horizon at
$r=r_{extr}$ satisfying \be r_{extr}^2=\frac{l^2}{2}\,,\quad
\mu_{extr}=-\frac{l^2}{4}+\alpha\, \label{4dextr}\ee given that
$\alpha \neq  l^2/4$. In five dimensions, there is no black hole
interpretation for $\alpha>l^2/4$. One should also note that the
Hawking temperature at $\alpha=l^2/4$, when $n=4$, \be
T=\frac{2n\,r_+^2+(n-4)k l^2}{8\pi r_+l^2}= \frac{r_+}{\pi\,l^2}
\ee is non-zero and positive. In $n+1>5$ spacetime dimensions,
when $k=-1$, the extremal parameters, such that $\beta\to \infty$,
are~\cite{IPN02b} \be
r_{extr,\,crit}^2=\left(\frac{n-2}{2n}\right)l^2 \left(1\pm
\sqrt{1-\frac{4n(n-4)}{(n-2)^2}
\,\frac{\alpha}{l^2}}\right)\,,\label{rextr} \ee
%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea
\mu_{extr,\,crit}&=&\left(\frac{2\,r_{extr,crit}^{n-2}}{n-4}\right)
\nn \\
&{}& \times\Bigg[\frac{2}{n}\mp
\sqrt{\left(\frac{n-2}{n}\right)^2-\frac{4(n-4)}{n}\,
\frac{\alpha}{l^2}}\Bigg]\,.\label{mextr} \eea Again, for
$\alpha=l^2/4$, the negative root in~(\ref{mextr}) gives
$\mu_{extr}=0$ independent of $n$. More precisely, \bea
&& r_{extr}^2=\frac{l^2}{2}\,, \quad \mu_{extr}=0\,, \nn \\
&& ~~~~~~~~ T=\frac{n}{4\pi r_+
l^2}\,\left(r_+^2-\frac{n-4}{n}\,r_{extr}^2\right)\,,\eea
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bea r_{crit}^2&=&\frac{(n-4)}{2n}\,l^2\,, \nn \\
\mu_{crit}&=&l^{n-2}\,\sqrt{\frac{(n-4)^{n-4}}{n^n}\,2^{(8-n)}}\,,\nn
\\
 T&=&\frac{n}{4\pi r_+ l^2}\big(r_+^2-r_c^2\big)\,. \eea
It will be somewhat of misnomer to call $\mu_{extr}=0$ state as an
extremal state because extremal black holes are defined to have
zero temperature, which require $\alpha<l^2/4$, but the
$\mu_{extr}=0$ state defined by the coupling $\alpha=l^2/4$ has
non-zero temperature~\cite{Cai01a,IPN02a}. We therefore call the
$\mu_{extr}=0$ state as a massless state, but this should not be
mixed up with a massless topological black hole, the latter simply
means $\mu=0$ and $k=-1$.

For $\alpha=l^2/4$, only the black hole of critical size
$r_+=r_{crit}$ has zero temperature but this is not a massless
state, rather $\mu_{crit}>0$. Notice that the $k=-1$ solution with
$\mu_{extr}=0$ saturates the bound $r_+^2 \geq
2\alpha=l^2/2=r_{extr}^2$. The proper extremal black holes are
those which satisfy $\alpha < l^2/4$ and have zero Hawking
temperature. We will show that the AdS extremal solution gives a
local minimum of the AdS energy functional.

\subsection{Choice of backgrounds}

For $\alpha^\p=0$, the harmonic function $f(r)$ is defined by \be
\label{bhmetric2} f(r)=k+\frac{r^2}{l^2}-\frac{\mu}{r^{n-2}}
\,.\ee It is established that a pure AdS space ($k=1$, $\mu=0$),
whose (Euclidean) boundary is a conformal sphere, is the reference
background for AdS Schwarzschild black holes. For the case of
toroidal $k=0$ horizon, a candidate ground state was proposed by
Horowitz and Myers~\cite{Horowitz98a}. The latter background, for
example, in five dimensions, has the form  \bea
ds^2&=&\frac{r^2}{l^2}\,\left(1-\frac{r_0^4}{r^4}\right)\,d\phi^2
+\frac{l^2}{r^2}\left(1-\frac{r_0^4}{r^4}\right)^{-1}\,dr^2 \nn \\
&{}&+\,r^2
\left(-\,dt^2+\sum_{i=1}^{2}\left(d\theta^i\right)^2\right)\,,\eea
where $r_0$ is some constant related to AdS soliton mass. We are
interested here in AdS black holes whose event horizon is a
hyperbolic ($k=-1$) space. When $\alpha=0$, $n+1=5$, the extremal
mass parameter is $\mu_{extr}=-\,l^2/4$, so the extremal metric
has the form \be
ds^2=-\,f(r)\,dt^2+\frac{dr^2}{f(r)}+r^2\,dH_{3,k=-1}^2\,\ee with
\be
 f(r)=\left(\frac{r}{l}-\frac{l}{2r}\right)^2 =
\frac{r^2}{l^2}\left(1-\frac{l^2}{2r^2}\right)^2 \,.\ee This is a
candidate ground state for hyperbolic black holes, in restricting
the spacetime region $r\geq r_e=l/\sqrt{2}$ for the background.
The gravitational potential is not well behaved in the region
$r<l/\sqrt{2}$ (see, section V), but the curvature corrections
(i.e., $\alpha^\p>0$) to the Einstein action, and hence to the
extremal background, should improve the situation.

We therefore consider the metric solution with $\alpha>0$: \be
\label{newk=-1} f(r)= k+\frac{r^2}{2\alpha}\mp
\frac{r^2}{2\alpha}\sqrt{1-\frac{4\alpha}{l^2}
+\frac{4\alpha\,\mu}{r^n}} \,. \ee For generality, we kept the
both signs in~(\ref{newk=-1}) but a more physical result is given
by the negative root, which reduces to that of Einstein gravity in
the limit $\alpha\to 0$. For the spherical black hole (i.e.,
$k=1$), $\mu=0$ itself defines a reference background (AdS vacuum
solution). In this case, it is obvious that the constraint
$\alpha\leq l^2/4$ must hold. Interestingly, a gravitational
action defined with $\alpha=l^2/4$, in some cases, is equivalent
to the Einstein gravity. This branch corresponds to a class of
solutions originally proposed in~\cite{Chamseddine89a} and studied
in~\cite{Cai98a,Zanelli00a}. For $k=0$, a massless ground state is
still legitimate. In this case, for the particular coupling
$\alpha=l^2/4$, the graviton propagators in $AdS_{n+1}$ spacetime,
when $n\geq 4$, do not receive any corrections from the massive
(Kaluza-Klein) modes (see, e.g.~\cite{IPN02e} in the braneworld
context but this result holds in general). The $k=0$, $\mu=0$
background is stable under metric perturbations.

Turn now to background choice for $k=-1$. It is known that a
solution with $\mu=0$ cannot be the ground state (background) for
hyperbolic ($k=-1$) black holes. This was first observed
in~\cite{Vanzo} with $\alpha=0$, but this is true also for
$\alpha>0$. For the coupling $(n-2)^2/[4n(n-4)]>\alpha/l^2\geq
1/4$, when $n>4$, one might achieve $\mu\geq 0$ such that the
terms inside the square root in~(\ref{newk=-1}) is non-negative.
However, for this coupling the background itself will be unstable
classically and thermodynamically. Thus, one requires $\alpha <
l^2/4 $ at the background, and the extremal mass $\mu_{extr}$ can
only take a negative value as in Einstein
gravity~\cite{Birmingham98a,Emparan99a}. With $\alpha=l^2/4$, for
example, when $n+1=5$, the Ricci-scalar and Kretschmann scalar
$K~(=R_{abcd}R^{abcd})$ read \bea
R&=&-\,\frac{40}{l^2}+\frac{12\sqrt{\mu\,l^2}}{r^2 l^2}\,,\nn \\
K&=&\frac{160}{l^4}+\frac{48\mu}{r^4\,l^2}\mp
\frac{96\sqrt{\mu\,l^2}}{r^2\,l^4}\,.\eea Therefore, the metric
spacetime is only asymptotically AdS when $\mu\neq 0$, and, of
course, $\mu<0$ is not allowed in this case. Next, consider, for
example, a coupling $\alpha=l^2/12$. For a large $r$, the
curvature scalars read \bea R&=&-\frac{40}{l^2}\,\Big(3\mp
\sqrt{6}\Big)-\frac{10(15\pm 4\sqrt{6})\,\mu}{r^4}+{\cal
O}\left(\frac{1}{r^6}\right)\nn \\
K&=&\frac{480(5\mp 2\sqrt{6})}{l^4}+{\cal
O}\left(\frac{1}{r^8}\right)\,,\eea in satisfying
$\mu>-\,2r^4/l^2$ and $r^2>l^2/2\sqrt{3}$. We should note that $K$
diverges when $\mu=-\,2r^4/l^2$. Our proposition is that one has
to take $\mu_{extr}<0$, which certainly requires $\alpha<l^2/4$,
as the background value for $k=-1$. A candidate ground state is
then the extremal metric obtained by replacing $\mu$
in~(\ref{newk=-1}) with $\mu_{extr}$. This is a physically
motivated choice of background for topological black
holes~\cite{Vanzo,Birmingham98a,IPN02b}. An extremal background is
not (maximally) supersymmetric because the geometry
$H^{n-1}/\Gamma$ often breaks the supersymmetry but it may be
stable under gravitational (tensor) perturbations when $\alpha<<
l^2$.


\section{Background subtraction and thermodynamic
quantities}

For a solution of the equations of motion, the classical
action~(\ref{action1}) becomes \be I=\frac{1}{16\pi G_{n+1}}\int
d^{n+1}x\,\sqrt{-g}\left(-\frac{2}{n-3}\, R
+\frac{8\Lambda}{n-3}\right)\,. \ee This action diverges for the
classical solution when integrated from zero to infinity. A choice
of the background is the key to this problem. As is known, there
should be a regularized setting -- a cut off in the radial
integration and subtraction of a suitably chosen background in
order to cancel divergences, so that the action (energy) is
finite. In the $k=+1$ case, one subtracts energy of the reference
geometry which is simply AdS space produced by a setting $\mu=0$.
For the $k=0$ case, a zero mass ground state is still legitimate,
and is an acceptable background~\cite{Vanzo,Birmingham98a}.
However, it has been known that this prescription would not
directly apply to the $k=-\,1$ case. The background subtraction
technique for $k=-\,1$ may follow similar to the Horowitz-Myers
soliton. For HM soliton, one subtracts a background for the
non-zero negative mass soliton, so that the AdS soliton itself is
a local minimum of the energy. While, for the $k=-1$ case,
subtraction of an extremal state (again non-zero mass background)
would be physical, because the extremal ground state will have a
local minimum of the energy at zero
temperature~\cite{Vanzo,Birmingham98a,Emparan99b,IPN02b}.

One notes that for the global AdS (or AdS extremal metric) ($X_1$)
any value of the periodicity $\beta^\p$ is possible, while the
black hole spacetime ($X_2$) has a fixed periodicity $\beta$. One
can therefore adjust $\beta^\p$ such that the geometry of the
hypersurface at $r=R\to \infty$ is the same for AdS space and
AdS-Schwarzschild when $k=1$~\cite{Witten98a}, and for the
extremal state and hyperbolic AdS black hole spacetime when
$k=-1$~\cite{Birmingham98a}. The surface term has no role in the
large $r$ limit because the black hole correction to the AdS
metric ($k=+1$) or extremal state ($k=-1$) vanishes too rapidly at
$r=R\to \infty$~\cite{Hawking83a,Witten98a}. Thus one can fix
$\beta^\p$ by demanding \be \beta'\sqrt{k-\frac{\mu
_{extr}}{r^2}+\frac{r^2}{l^2}}\simeq \beta \sqrt{k-\frac{\mu
}{r^2}+\frac{r^2}{l^2}}\,. \ee In doing this, one finds the
(Euclidean) action difference $\widehat{I}=I(X_2)-I(X_1)$ in the
form \bea \widehat{I}&=& -\frac{V_{n-1}\,r_+^{n-4}\,\beta}{16\pi
G_{n+1}\,(n-3)} \Bigg[2(n-1)\,k r_+^2-(n-1) \mu \,r_+^{4-n}\nn \\
&{}&~~~~~~~ -\,\frac{8\pi
r_+^3}{\beta}+\frac{4(n-1)\,r_+^4}{l^2}\Bigg]-\beta\,M_{extr}\delta_{k,-1}
\nn \\
&=&-\frac{V_{n-1}\,r_+^{n-4}\,\beta}{16\pi G_{n+1}\,(n-3)}\nn \\
&{}& ~~~~~~~~~\times
\Bigg[(n-1)\left(k\,r_+^2-\alpha\,k^2\right)+\frac{3(n-1)\,r_+^4}{l^2}\Bigg]
\nn \\
&{}&~~~~~~~~~~ +\,\frac{V_{n-1}\,r_+^{n-1}}{2(n-3)G_{n+1}}
-\beta\,M_{extr}\delta_{k,-1}\,. \eea The free energy of the black
hole is then given by $F=\widehat{I}/\beta$. Hence \bea F &=& -
\frac{(n-1) V_{n-1}\,r_+^{n-4}}{16\pi G_{n+1}\,(n-3)}
\left[\left(k\,r_+^2-\alpha\,k^2\right)
+\frac{3\,r_+^4}{l^2}\right]\nn \\
&{}&+\frac{V_{n-1}r_+^{n-1}}{2(n-3)G_{n+1}}\,\frac{1}{\beta}
-M_{extr}\delta_{k,-1} \nn \\
%% intermediate step %%%%
%%&=& \frac{V_{n-1}\,r_+^{n-2}}{16\pi G_{n+1}}
%%\frac{1}{(r_+^2+2\alpha k)}\bigg[
%%r_+^2\left(k-\frac{r_+^2}{l^2}\right)-\frac{6(n-1)\alpha k
%%r_+^2}{(n-3)l^2} +\frac{(n-7)\alpha k^2}{n-3}+\frac{2(n-1)\alpha^2
%%k}{(n-3)r_+^2} \bigg]
%%-M_{extr} \nn \\
&=&\frac{V_{n-1}\,r_+^{n-2}}{16\pi G_{n+1}}
\Bigg[\left(k-\frac{r_+^2}{l^2}\right) +\frac{n-1}{n-3}\,
\frac{\alpha k^2}{r_+^2}\nn \\
&{}&-\frac{2n}{n-3}\, \frac{\alpha k\left(2r_+^2+k
l^2\right)}{l^2\left(r_+^2+2\alpha
k\right)}\Bigg]-M_{extr}\delta_{k,-1} \,.\eea The condition
$\alpha=l^2/4$ is an exceptional one. In this case, $M_{extr}=0$,
and the last term inside the square bracket becomes a constant. On
the other hand, the solutions with $k=0$ are qualitatively similar
to the $\alpha=0$ case. For $k=0$, the free energy scales as $
V_{n-1}/\beta^n$ for all temperatures. This may be traced back to
the observation that the planar black hole in $AdS_5$ is exactly
dual to a thermal state of ${\cal N}=4$ supersymmetric Yang-Mills
theory in four-dimensional Minkowski space~\cite{Witten98a}.
While, for $k=\pm 1$, this scaling is valid only in the limit of
high temperatures.

A local minimum of the Hawking temperature is given by solving
$\partial \beta/\partial r_+=0$, where \bea\label{Tminima}
\frac{1}{\beta^2}\,\frac{\partial \beta}{\partial r_+}&=&
\frac{1}{4\pi} \Bigg[\frac{(n-2) k}{r_+^2} - \frac{n}{l^2}-
\frac{n\alpha k}{r_+^2 l^2}\, \frac{(2r_+^2+k l^2)}{(r_+^2+2\alpha
k)}\nn
\\&{}& ~~~~~-\,\frac{2n\alpha k^2}{(r_+^2+2\alpha k)^2}
\left(1-\frac{4\alpha}{l^2}\right) \Bigg]\,. \eea For $\alpha=0$,
it is obvious that there is no minimum of temperature for $k=\{0,
-1\}$ family, thus such black holes exist for all temperatures but
the situation could be different for $\alpha> 0$ as we exhibit
below.

The resulting black hole entropy is~\cite{IPN02a,IPN02b} \bea
\label{GBentropy} {\cal S}&=&\beta^2\,\frac{\partial F}{\partial
\beta}=\left(\frac{1}{\beta^2}\,\frac{\partial \beta}{\partial
r_+}\right)^{-1}\,
\frac{\partial F}{\partial r_+}\nn \\
&=&\frac{V_{n-1} r_+^{n-3}}{4G_{n+1}}
\left(r_+^2+\frac{2(n-1)\alpha\,k}{(n-3)}\right)\,. \eea The
entropy flow is given by \bea d {\cal S}&=&\frac{(n-1)V_{n-1}
r_+^{n-4}}{4G_{n+1}}
\Big(r_+^2+2\alpha\,k\Big)\nn \\
&=&\beta\,\frac{(n-1)V_{n-1}}{16\pi G_{n+1}} \times\nn \\
&{}&\left[\frac{n\, r_+^{n-1}}{l^2} +(n-2)k r_+^{n-3}+(n-4)\alpha
k^2 r_+^{n-5}\right]\,. \eea In the second step above we have made
use of~(\ref{HawkingT}). Using the thermodynamic relation $d{\cal
S} =\beta\,dE$, we arrive at \be\label{totalE1} E=M+E_0\,. \ee One
reads $M$ from Eq.~(\ref{BHMass}), and $E_0$ is an integration
constant. We are able to do something better than fixing $E_0$
using the criterion $E\vert_{\beta\to \infty}=0$. The energy $E$,
obtained directly using the Euclideanized action, takes a
remarkably simple form \be \label{totalE2}
E=\frac{\partial\widehat{I}}{\partial \beta}=M
-M_{extr}\,\delta_{k,-1}\,.\ee Comparing (\ref{totalE1}) and
(\ref{totalE2}), we can easily identify that
$E_0=-\,M_{extr}\,\delta_{k,-1}\geq 0$. For $k=0,\,1$, the
thermodynamic energy of AdS black holes is given by the black hole
mass $M$, since AdS solution with $\mu=0$ is the one with lowest
action and energy. This is generally not the case for $k=-1$
geometry, rather, $E=-M_{extr}>0$ when $\mu=0$, $E=0$ when
$\mu=\mu_{extr}$, and $E>0$ otherwise. The total energy is a
positive, concave function of the black hole's temperature for all
values $k=0,\,\pm1$.

The black hole solutions obtained for flat or spherical horizons,
with $\alpha^\p>0$, share many properties of the simpler
Schwarzschild-type solution of the Einstein gravity. These are the
cases studied essentially in more detail in the literature.
However, not much has been done for solutions involving the
hyperbolic geometries, so we are primarily interested here in the
$k=-1$ case.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Extremal state as the ground state}

A hyperbolic $AdS_5$ black hole has some special feature, viz.,
the total energy \bea E&=&\frac{3V_3\,r_+^2}{16\pi
G}\,\left(-1+\frac{r_+^2}{l^2}+\frac{\alpha}{r_+^2}\right)
+\frac{3l^2\,V_3}{64\pi G}\left(1-\frac{4\alpha}{l^2}\right)\nn \\
&=&\frac{3\,V_3}{16\pi G}\left(-\, r_+^2+\frac{r_+^4}{l^2}
+\frac{l^2}{4}\right)\, \eea is independent of the coupling
$\alpha$. This simple observation might have some new and
interesting consequences in the field theory dual, if the latter
exists with a GB term.

We can perhaps make use of the relation
$$ \frac{\alpha^\p}{l^2}=\left(2 g_{YM}^2 N\right)^{-1/2}\,.$$
(In the units $16\pi G_5=1$, $\alpha^\p=\alpha/2$). Then a small
$\alpha^\p$ corresponds to the strong coupling limit, (i.e.,
$g_{YM}^2 N$ is large.) And also note that the entropy is  $$
{\cal S}=|\beta|\left(E-F\right)\,. $$ As shown above, for $n=4$
and $k=-1$, the free energy $F$ is dependent of the coupling
$\alpha$, but the energy $E$ (or energy density $E/V_3$) is not.
That is, free energy and entropy both scale with the coupling
$\alpha$. For the planar or spherical black holes,
$\mbox{lim}_{\beta\to \infty}\beta F= 0$ holds, but in the
hyperbolic $AdS_5$ case one observes that $\mbox{lim}_{\beta\to
\infty}\beta F= \mbox{finite}$ for a small coupling $\alpha^\p$.
This result may be traced back to the discussions
in~\cite{Emparan99b}, where similar results that support the
scaling behavior of the free energy (and entropy) on hyperbolic
space -- when going from strong to weak coupling -- are presented
(see~\cite{Horowitz97a} for original ideas on black-hole/string
correspondence).

\begin{figure}[ht]
\begin{center}
\epsfig{figure=5DMextr.eps, height=4.0cm, width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=7DMextr.eps, height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The ADM mass $M$ (curved lines) and extremal mass
$M_{extr}$ (horizontal lines) in the function of horizon ($r_+$).
The values are fixed at $l=1$, $16\pi G=1$, and (a) (upper plot)
$n=4$, $V_3=2\pi^2$, $k=-1$ and $\alpha=0,~1/12,~1/4$ (up to
down); (b) (lower plot) $n=6$, $V_3=\pi^3$, $k=-1$ and
$\alpha=0,17/100,\,1/4$.} \label{figure1}
\end{figure}

In spacetime dimensions $n+1>5$, however, the energy $E$ also
depends upon the coupling $\alpha$, independently, we choose
$\mu_{extr}=0$ or $\mu_{extr}\neq 0$ or $\mu=0$ as a reference
background for hyperbolic $(k=-1)$ black holes. What is more
interesting to note is that in all spacetime dimensions the energy
$E$ has a unique local minimum at the extremal horizon position
$r_+=r_{extr}$. These are clearly seen in the plots of
Figs.~(\ref{figure1}) and (\ref{figure2}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\epsfig{figure=5DEnergy.eps, height=4.0cm, width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=7DEnergy.eps, height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The energy $E$ in the function of horizon ($r_+$). The
values are fixed at $l=1$, $16\pi G=1$, and (a) $n=4$ (upper
plot), $V_3=2\pi^2$, $k=+1$ (curves that grow with $r_+$) at
$\alpha=1/4,~1/12$ (up to down), and $k=-1$ (curve that has
minimum at $r_+=r_{extr}$); (b) $n=6$ (lower plot), $V_3=\pi^3$,
$k=-1$ and $\alpha=0,\,17/100,\,1/4$ (top to bottom curves --
along the $E$-axis).} \label{figure2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

As we noted above, specially, for the coupling $\alpha=l^2/4$,
$M_{extr}=0$. Take, for simplicity, $n=6$, then one finds \be
E=\frac{5V_{5}\,r_+^{4}}{16\pi G_{7}} \left( -1+\frac{r_+^2}{l^2}
+\frac{l^2}{4r_+^2}\right)\geq 0\,.\ee There are two minima -- one
at $r_+=0$ and another at $r_+=r_{extr}=l/\sqrt{2}$ -- only the
latter is a stable minimum.

\subsection{Thermodynamic instability for $k=-1$}

To justify that $M_{extr}=0$ is not the ground state for the
$k=-1$ case, we may adopt an Euclidean path integral formulation
along with a consideration that the (extremal) entropy and
specific heat should be non-negative at the background. The black
hole entropy is always positive for $k=\{0,~1\}$ horizons.
However, for $k=-1$, the inequality $r_+^2\geq 2\alpha$, which
must hold in order to have a black hole interpretation, is not
enough to guarantee that the (extremal) entropy and specific heat
are always positive, rather one must satisfy $r_+^2\geq
2(n-1)\alpha/(n-3)$. That is, there must be an upper bound in the
strength of the coupling $\alpha$ for a thermodynamically stable
background. For example, when $n+1=7$, the positivity of extremal
entropy requires that $r_+^2>10\,\alpha/3$ and hence
$\alpha/l^2<17/100$, which further implies that $4\alpha<l^2$.
When one approaches a massless state at $4\alpha=l^2$, so
$M_{extr}=0$, the specific heat could be negative, which mimics a
thermodynamic instability.

In the canonical ensemble, which should be the case here as we are
considering uncharged black holes, the second derivative of the
Euclidean action $\widehat{I}~(=\beta E-{\cal S})$ along the path
parameterized by $x$ is \be
\left(\frac{\partial^2\hat{I}}{\partial^2\,
x}\right)_T=\frac{1}{\beta^2}\,\left(\frac{d\beta}{d x
}\right)^{2}\, \frac{d E}{d T}\,, \ee where $x=x(T)$ is the
parameter that labels the path in the Euclidean path integral
formulation. The black hole cannot be a local minimum of the
action when the specific heat $dE/dT$ is negative~\cite{Reall01a}.

Though the arguments for thermodynamic stability of black hole
spacetimes are not directly available to the classical
stability~\cite{Reall01a}, they might be related. A study of
gravitational perturbations shows that stability of the $k=-1$
spacetimes against metric perturbations with $\alpha>0$ indeed
requires $\alpha<<l^2$. As we noted above, when $n+1=7$, the
thermodynamic stability of the solution requires $\alpha\leq
17l^2/100$. We will need a slightly stronger bound, namely,
$\alpha\lesssim 2l^2/15$, at the classical level, such that the
gravitational potential is positive and bounded from below for the
$k=\pm 1$ geometry. So for the thermodynamic stability of
hyperbolic black hole spacetimes, not only the specific heat but
also the extremal entropy must be non-negative at the background.


\section{Specific heat and free energy curves and thermal
phase transition}

In Einstein gravity, small spherical black holes can have negative
specific heat but large size blacks holes have positive specific
heat~\cite{Gregory93a}. There exists a discontinuity for the
specific heat at $r_+=l/\sqrt{2}$, so small and large black holes
are somehow disjoint. However, this is not essentially the case in
the Einstein-Gauss-Bonnet theory, and specially for
Schwarzschild-$AdS_5$ black hole, also the small black holes can
have positive specific heat~\cite{IPN02a}, which are not only
thermodynamically stable but also globally preferred. Therefore, a
Gauss-Bonnet type corrections to Einstein gravity look promising
even from the view of thermodynamic stability of small spherical
black holes. In this section, however, we will try to learn more
about the hyperbolic black holes, which have some peculiar
features unavailable to other two classes of $AdS$ black holes --
planar ($k=0$) and spherical ($k=+1$).

For $k=+1$, as first observed by Hawking and
Page~\cite{Hawking83a} in Einstein gravity, the AdS black hole of
horizon $r_+<l$ has positive and $r_+>l$ has negative free
energies, and there occurs a first order phase transition between
a thermal AdS space and an AdS black hole at the scale $r_+\sim
l$. As we see shortly, for the spherical black holes, the
Hawking-Page type phase transition is available only for a small
Gauss-Bonnet coupling.

The $AdS_5$ and $AdS_7$ black holes are of some particular
interest because the string or M-theory provides dual for
$AdS_{n+1}$, $n=4,~6$ as the CFTs describing the world-volume
theory for $N$ parallel $D3$- and $M5$-branes. Though it is not
explored yet whether a dual description should exist with a
Gauss-Bonnet term, it is certainly interesting to explore a
possibility of thermal phase transition in this theory.
In~\cite{Gubser98a,Klemm99a}, the string or M-theory corrections,
of order ${\alpha^\p}^3\,{\cal R}$, where ${\cal R}$ is a scalar
constructed out of contraction of four Weyl tensors, to the black
hole thermodynamic are presented. With perturbative calculations,
it is shown in~\cite{Klemm99a} that such corrections do not give
rise to a thermal phase transition for flat and hyperbolic
horizons, although to a quotient of hyperbolic space
$H^{n-1}/\Gamma$, and also to a flat space $I\!\!R^{n-1}$
compactified down to a torus, there may arise new phase
transitions. The implications of finite size and finite coupling
effects on the appearance of strong/weak coupling phase
transitions are studied in the literature,
e.g.,~\cite{Mli99a,MLi99b,Kogan99a}.


\subsection{Specific heat for hyperbolic black holes}

Figs.~(\ref{figure3}) and (\ref{figure4}) show the plots of
specific heat in function of horizon position, when $k=-1$. One
again has to make a distinction between the two cases:
$\alpha=l^2/4$ and $\alpha < l^2/4 $. For a hyperbolic $AdS_5$
black hole, the small horizon regime $r<r_{extr}$ has a single
branch for $\alpha=l^2/4$ and two branches for $\alpha < l^2/4 $.
The first branch (cusp) on the left, which will almost coincide
with the $r_+$-axis when $\alpha<<l^2$, has negative specific
heat, so is unstable.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=5Dheat1.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=5Dheat2.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=5Dheat3.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The Euclidean period $\beta$ (curves that asymptote to
$C$ or/and $r_+$-axes) and specific heat (curves with one or more
cusps) {\it vs} horizon position $r_+$. The values are fixed at
$l=1$, $n=4$, $V_3/16\pi G=2\pi^2$, $k=-1$, and $\alpha=1/4$,
$\alpha=1/8$ and $\alpha=1/12$ (top to bottom plot).}
\label{figure3}
\end{figure}

When the negative mass extremal state is taken as a reference
background, the energy condition $E\geq 0$ always holds. When
$\alpha=l^2/4$, the Euclidean time has a fixed and positive period
$\beta=\pi\,l^2/r_+$, thus the Hawking temperature and specific
heat are defined, respectively, by $T=1/\beta$ and $C=dE/dT$.
However, for the coupling, $\alpha < l^2/4 $, the sign of $\beta$
is important.

For $n=4$ and $k=-1$, an interesting situation arises when
$\alpha=l^2/12$. For this particular coupling, the extremal
entropy is zero, hence the Euclidean period is \be \beta=\frac{\pi
l^2\left(6r_+^2-l^2\right)}{r_+\left(2r_+^2-l^2\right)} \,.\ee
This is negative in the region $0.408<r_+<0.707$, where one still
has a black hole interpretation because $r_+^2\geq 2\alpha$ holds,
but the specific heat must be defined by \be \label{newdefC}
C=-\,\beta^2\,\frac{dE}{d(-\beta)}=\frac{dE}{dT}\,.\ee This
reveals that the second cusp in the bottom plot of
Fig.~(\ref{figure3}) actually has positive specific heat. One also
notes that the free energy depends upon a regularization
(background choice) but the Hawking temperature does not. And, the
extremal mass $M_{extr}$ is $r_+$ or $T$ independent, thus one
gets the same answer for specific heat from $C=dE/dT$ and
$C=dM/dT$.

One has to make sure that a negative (extremal) entropy is not due
to a wrong sign of $\beta$, while deriving the entropy formula.
For $n=4$, one finds \be \label{entropy5} {\cal
S}=\beta^2\,\frac{dF}{d\beta}=\frac{V_{3,k=-1}\,r_+^{3}}{4G_{5}}
\left(1-\frac{6\alpha}{r_+^2}\right)\,. \ee The entropy at the
extremal state takes the form~\cite{IPN02b} \be \label{5dentropy}
{\cal S}_{extr}=\frac{V_3}{G_5}\,\frac{l^3}{2^{7/2}}
\left(1-\frac{12\alpha}{l^2}\right)\,. \ee This is non-negative
only if $\alpha\leq l^2/12$. As noted before, for the coupling
$\alpha=l^2/4$, the Euclidean period $\beta=\pi l^2/r_+$ is always
positive, so the formula ${\cal S}=\beta^2\,(dF/d\beta)$ is
effective. It is then obvious that the extremal entropy can be
negative for $\alpha=l^2/4$; this branch therefore belongs to an
unstable region where the specific heat is also negative (see the
upper most plot in Fig.~(\ref{figure3})). This particular feature
is not totally new; even for $k=+1$, when $\alpha=0$, small black
holes have negative specific heat, so are unstable. There is
however a difference here. For $k=+1$, a thermodynamic instability
arises only due to a finite size effect, viz., a black hole of
size $r_+\leq l/\sqrt{2}$ is unstable, $r_+>l$ is stable and
$l>r_+>l/\sqrt{2}$ is only locally preferred but globally
unstable. In the $k=-1$ case, however, a thermodynamic instability
could arise either due to a large coupling effect or a small size
effect.

For $k=-1$, any negative entropy state though may arise,
typically, when the higher derivative couplings are in the order
of the AdS length scale $l^2$, can be ruled out as being
unphysical (see, the discussion in~\cite{Nojiri02a} in the context
of dual field theories), because neither such states are stable
classically nor the supergravity approximation allows one to take
$\alpha$ in the same order as $l^2$. Indeed, the coupling $\alpha$
should be at least one order magnitude smaller than the length
scale $l^2$.

For $n=4$, $\alpha=l^2/12$, the entropy of the extremal black hole
is zero. It is then relevant to ask what would happen in the limit
$l^2/4>\alpha>l^2/12$. As a specific example, for the coupling
$\alpha=l^2/8$, the period is \be \beta=\frac{\pi
l^2\left(4r_+^2-l^2\right)}{2r_+\left(2r_+^2-l^2\right)}\,.\ee
This is negative only in the range $l^2/4<r_+^2<l^2/2$, i.e.,
$0.5<r_+<0.707$. Nevertheless, for the coupling $\alpha=l^2/8$,
the entropy~(\ref{entropy5}) can be negative in the range
$l^2/2<r_+^2<3l^2/4$ where $\beta$ is positive and the formula
${\cal S}=\beta^2\,(dF/d\beta)$ is effective. Thus, in five
dimensions, black hole solutions with $\alpha>0$ are
thermodynamically stable only if $\alpha\leq l^2/12$, that is,
${\cal S}_{extr}\geq 0$.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=7Dheat1.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=7Dheat2.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=7Dheat3.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The Euclidean period (curves that asymptote to $C$ or/and
$r_+$-axes) and specific heat (curves with two or more cusps) {\it
vs} horizon position $r_+$. The values are fixed at $l=1$, $n=6$,
$V_5/16\pi G=\pi^3$, $k=-1$, and $\alpha=0.25$, $\alpha=0.23$ and
$\alpha=17/100$ (top to bottom plots).} \label{figure4}
\end{figure}

As the bottom plot in Fig.~(\ref{figure3}) or Fig.~(\ref{figure4})
shows, for small $\alpha$, the black hole with small horizon
(second cusp), which has negative specific heat and positive free
energy, is separated from a large non-extremal black hole
$(r_+>r_{extr})$, by a critical size black hole (third cusp),
which itself belongs to an unstable region. So only the black
holes of extremal size or bigger than this are thermodynamically
stable.

For $n+1=7$, the extremal entropy vanishes for $\alpha=17
l^2/100$. In this case, the Euclidean period is \be \beta
=\frac{4\pi r_+ l^2(50 r_+^2-17l^2)}{(10r_+^2-l^2)(30r_+^2-17
l^2)}\,.\ee Note that $\beta$ is negative in the range
$0<r_+<0.3162$ and $0.5831<r_+<0.7528$, where the
formula~(\ref{newdefC}) does apply. The free energy is also
negative when $0.5831<r_+<0.7528$. Hence, the first and third
cusps in the bottom plot of Fig.~(\ref{figure4}) should be mirror
reflected. However, as we noted above, one has no black hole
interpretation for the first cusp. For $\alpha=l^2/4$, there are
only two cusps, because in this case two unstable branches (the
second and third cusps that appear for $\alpha<l^2/4$) merge to a
single cusp, which has negative specific heat. A clear message
behind these observations is that the thermodynamic stability in
$n=6$ requires $\alpha/l^2\leq 17/100$. In other words, the
specific heat and extremal entropy should be non-negative at the
background. Similar arguments would apply to all hyperbolic black
holes in $n+1 \geq 5$ spacetime dimensions.


\subsection{Free energy for hyperbolic black holes}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\epsfig{figure=hyper1.eps, height=4.0cm, width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper2.eps, height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper3.eps,height=4.0cm,width=7.0cm}
\end{center}
\caption{The Euclidean period (curves that asymptote to $F$ or/and
$r_+$-axis) and free energy (curves that are bounded from up or
that take finite values at $r_+=0$) {\it vs} horizon position
$r_+$. The values are fixed at $k=-1$, $n=4$, $l=1$, $16\pi G=1$,
$V_3=2\pi^2$, and, $\alpha=1/6,\,0.1305,\,1/8$ (top to bottom
plots).} \label{figure5}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\begin{center}
\epsfig{figure=hyper4.eps, height=4.0cm, width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper5.eps, height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper6.eps, height=4.0cm,width=7.0cm}
\end{center}
\caption{The Euclidean period (curves that asymptote to $F$ or/and
$r_+$-axis) and free energy (curves that have finite values at
$r_+=0$ or bounded from up) {\it vs} horizon upper two plots. The
values are fixed at $k=-1$, $n=6$, $l=1$, $16\pi G=1$,
$V_5=\pi^3$; and $\alpha=0.25,~0.2162,\,0.2072$ (top to bottom
plots)} \label{figure6}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We now want to use the obtained thermodynamic quantities to
determine the thermal phase structure. Figs.~(\ref{figure5}) and
(\ref{figure6}) show the plots of Euclidean period $\beta$ and
free energy $F$ as functions of horizon size $r_+$, when $k=-1$.
The first plot in Fig.~(\ref{figure6}) corresponds to the
$\mu_{extr}=0$ state with $\alpha=l^2/4$. In lower two plots, the
extremal state is shown by an asymptotic on the right, and the
critical state, which is absent in $AdS_5$ (c.f.,
Fig.~(\ref{figure5})), is shown by another asymptotic on the left.
The middle vertical line, where the free energy diverges and
$\beta=0$, corresponds to the horizon at $r_+^2=2\alpha$.

To interpret these plots, we must keep in mind that, except for
the coupling $\alpha=l^2/4$ for which $M_{extr}=0$, the
subtraction has been done with respect to a zero-temperature,
zero-energy density state. Therefore, a net sign difference
between the free energies of the extremal state and the
corresponding black hole is important in order to explain any
possible phase transition.

It is perhaps relevant to note that in the region $0<r_+^2\leq
2(n-1)\alpha/(n-3)$, where the behavior of extremal entropy is
somewhat exotic (unusual), the second law of thermodynamics does
not necessarily hold. Besides, for a hyperbolic $AdS_5$ black
hole, the free energy scales with the coupling $\alpha$, but the
energy (or energy density) does not. The energy is vanishing at
the extremal horizon $r_+=l/\sqrt{2}$, where $\beta\to \infty$,
but the free energy can be positive, zero or negative depending
upon the coupling constant. Unlike a naive expectation, the
entropy is non-zero at the extremal state, in particular, for a
small coupling $\alpha$. This is consistent with the earlier
observations made in~\cite{Emparan99a,Emparan99b} for
$\alpha^\p=0$; a common thread in these results is that
$\mbox{lim}_{\beta\to \infty} \beta F=finite$. This further helped
the authors of~\cite{Emparan99a,Emparan99b} to trace back their
result of a non-zero extremal entropy to the discussion of
"precursor states" first noticed in~\cite{Susskind99a}, when
studying a short distance behavior of $AdS{\slash} CFT$ duality.
(Further comments on it is beyond the scope of this paper.)

One can smoothly approach the negative mass extremal limit from a
non-zero temperature, and the regime $r>r_{extr}$ has a unique
black hole interpretation for all temperature $T>0$. As the lower
two plots of Figs.~(\ref{figure5}) and (\ref{figure6}) show the
Euclidean period is zero at $r_+=\sqrt{2\alpha}=r_1$, where the
free energy diverges, and $\beta$ is negative in the range $r_{1}<
r_+<r_{extr}$, thus the Hawking temperature is defined by
$T={|\beta|}^{-1}$. In the range $r_{1}< r_+<r_{extr}$, the
Hawking temperature decreases until it approaches the extremal
state with zero temperature. A singularity at
$r_+=\sqrt{2\alpha}=r_1$, where free energy diverges and
$\beta=0$, is hidden inside the extremal horizon, so is harmless.

The extremal entropy is vanishing when $\alpha/l^2=1/12$ for
$n=4$, and $\alpha/l^2=17/100$ for $n=6$. At these particular
couplings, as one approaches the extremal state ($T=0$) from the
right the free energy curve approaches a maximum value which is
less than zero. Thus, even at the zero temperature, the
thermodynamic ensemble is dominated by an extremal black hole.
While the physics for high enough temperature for all cases is
dominated by non-extremal black holes.

In Einstein gravity, as first discussed in~\cite{Birmingham98a},
there is no Hawking-Page type phase transition for flat and
hyperbolic horizons, hence the spatial Wilson loop cannot exhibit
confinement in a dual description. Though this is purely a gauge
theory intuition, with some perturbative calculations, this result
is shown to hold with string or M-theory curvature corrections to
the black hole thermodynamics~\cite{Klemm99a}. For a Gauss-Bonnet
theory, which admits the exact solution, we might need further
clarifications.

When $\alpha=l^2/4$ and $n=4$, no distinction can be made between
the small (critical or extremal) and large (non-extremal) black
holes. When $n>4$, there appears a critical black hole at zero
temperature. The point where the free energy curve crosses the
$F=0$ axis falls in the high temperature regime, and a thermal
phase transition is not seen as a function of temperature.

As argued in~\cite{Klemm99a,Surya01a}, for hyperbolic and flat
horizons, any phase transition, if exists, should be of different
nature than for spherical horizons. It could be either due to a
large coupling $\alpha^\p$ effect or due to a non-trivial horizon
topology, such as $H^3/\Gamma$ or $I\!\!R^{n-1}/\Gamma$.
Specifically, a new phase transition might occur as the coupling
$\alpha^\p$ is increased~\cite{Emparan99b,Klemm99a}. One also
observes a similar behavior here, for example, when $n=4$, at some
finite coupling $0.25\,l^2>\alpha>0.1305\,l^2$, in the range
$r_2>r_+>r_{extr}$, there exists a region where the free energy is
positive and the Hawking-Page temperature is finite.

For an extremal background having a zero or positive entropy
${\cal S}_{extr}\geq 0$ (i.e., for small $\alpha$) the free energy
$F(\beta)$ can only be negative for positive $\beta$. The free
energy takes a maximum value at the extremal state only if ${\cal
S}_{extr}=0$ (for example, when $n=4$ and $\alpha=l^2/12$). In
other words, when $k=-1$ and $n=4$, there is no thermal AdS phase
at all for the coupling $\alpha<0.1305\,l^2$, hence no any phase
transition occurs in this case. Unlike in the conventional
Hawking-Page thermal phase transition, where a first order phase
transition takes place between a thermal AdS phase ($r_+<l$) and a
Schwarzschild AdS black hole ($r_+>l$), for the $k=-1$ topology,
there does not occur any phase transition as a function of
temperature.

\subsection{Free energy for spherical black holes}

Next we examine the behavior of free energy and the thermodynamic
stability of spherical black holes. In fact, there would exist
relatively more stable black hole solutions for spherical
horizons. In this case, the reference background itself is a
global AdS space, so $E=M$, and hence the free energy is zero when
\bea \alpha_{\pm}&=&\pm \frac{r_+^2}{2}
\sqrt{\frac{9\,r_+^4}{l^4}+\frac{(15-n)\,r_+^2}{(n-1)\,l^2}
-\frac{(n+1)(7n-25)}{4(n-1)^2}}\nn \\
&{}&
+\,r_+^2\left(\frac{3\,r_+^2}{2\,l^2}-\frac{(n-7)}{4(n-1)}\right)
\,. \eea The corresponding zero free energy curves, each for
$\alpha_-$ and $\alpha_+$, coincide when \bea \label{stableF}
r_+^2&=&\left(\frac{n+8\sqrt{n(n-3)}-15}{18(n-1)}\right)l^2 \,,
\nn \\
\alpha&=&r_+^2\left[\frac{4\sqrt{n(n-3)}-(n-3)}{6(n-1)}\right]\
.\eea Thus, for example, when $n+1=5$, one has
$r_+^2=18\alpha/7>2\alpha$. Only for coupling $\alpha_-$ one would
expect a stable AdS solution with $F=0$, in satisfying $\alpha <<
l^2$.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=hyper7.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper8.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper9.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{Inverse temperature $\beta$ (curves that asymptote to $F$
and $r_+$ axes) and free energy (curves that cross $r_+$-axis)
{\it vs} horizon position $r_+$ for spherical black holes. The
values are fixed at $k=+1$, $n=4$, $l=1$, and, from top to bottom,
$\alpha=0.0278$, $0.01$, and $\alpha=0.003$. A new branch of
stable black hole appears in the $AdS_5$ case.} \label{figure7}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[ht]
\begin{center}
\epsfig{figure=hyper10.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper11.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=hyper12.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{Inverse temperature $\beta$ (curves that asymptote to
$r_+$-axis) and free energy (curves that cross the $r_+$-axis)
{\it vs} horizon position $r_+$ for spherical black holes. The
values are fixed at $k=+1$, $n=6$, $l=1$, and, from top to bottom,
$\alpha=0.1$, $0.05$, and $\alpha=0.01$.} \label{figure8}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For spherical black holes, we have computed the free energy with
respect to AdS space, so positive free energy means a thermal AdS
phase is thermodynamically preferred, while negative free energy
means a black hole phase is preferred. And, in general, a finite
low temperature phase corresponds to a thermal AdS phase. A
relatively more stable region is that where the free energy is
negative and the Hawking-Page temperature is large.

The spherical black holes indeed present an interesting thermal
feature, namely, a Hawking-Page thermal phase transition at finite
temperature~\cite{Hawking83a}. Looking to the extreme left of each
plots in Fig~(\ref{figure7}) and Fig.~(\ref{figure8}), the free
energy is positive for $AdS_5$ spherical black holes and is
vanishing for $AdS_7$ spherical black holes.

In the $AdS_5$ case, as the coupling strength of $\alpha$ is
decreased the free energy further lowers towards zero at low
temperature. For an extremely small coupling $\alpha<<l^2$, in the
small $r_+$ region, the free energy nearly approaches zero but it
never touches the $F=0$ axis. For example, when
$\alpha/l^2=3\times 10^{-7}$, the free energy is $F\sim 4\times
10^{-6}$. That is, for $\alpha<<l^2$, the free energy curve
crosses the $F=0$ only once, namely when $r_+\lesssim l$, which
corresponds to a Hawking-Page phase transition.

For the coupling $\alpha/l^2\gtrsim 0.0278$, the free energy is
monotonously decreasing function of $r_+$ and the second branch
(peak) on the right disappears. Making use of the relation $
{\alpha^\p}/l^2=1/\sqrt{2 g_{YM}^2 N}$ (in the units $16\pi
G_{n+1}=1$, $\alpha=(n-2)(n-3)\alpha^\p$), one observes that the
Hawking-Page transition might disappear at sufficiently small 't
Hooft coupling $g_{YM}^2 N$, but there might appear a new phase
transition in small $r_+$ region, as in~\cite{Mli99a,Klemm99a}.

For $AdS_5$ spherical black holes, the plots of inverse
temperature and free energy would remind us of the behavior of a
Van der Waals gas in the Clapeyron $(V,P)$ phase, where one
observes the point of inflection in $\beta(r)$ to signal a
critical point. Therefore, these graphics can be compared to those
in Refs.~\cite{Chamblin99a,Klemm99a}. In~\cite{Chamblin99a}, the
plane of the Clapeyron phase was observed for charged AdS black
holes, and in~\cite{Klemm99a}, analogous behavior was noted for
AdS black holes that receive stringy corrections of the form
${\alpha^\p}^3 R^4$. These results are available also to AdS black
hole thermodynamic that receives a $R_{abcd}R^{abcd}$ type
correction~\cite{IPN02c} (see~\cite{Odintsov01b} for a related
discussion on the confining (deconfining) phase transitions).
In~\cite{Klemm99a}, the free energy curves, however, diverge for
small $r_+$ and the solutions are only perturbative. With a
Gauss-Bonnet type correction we have exact solutions and the free
energy curves are smooth functions of the horizon position $r_+$,
and hence of the Hawking-Page temperature.

In $AdS_7$ spacetime, a thermal AdS phase always dominates over a
black hole, so there are no stable small black holes. Nonetheless,
the low temperature corresponds to a thermal AdS phase with zero
or positive free energy, and the high temperature corresponds to
an AdS black hole with negative free energy. Unlike in the $AdS_5$
case, for $AdS_7$, the qualitative features of the Hawking-Page
transitions are not changed by the coupling strength $(\alpha)$,
except that the free energy becomes more and more positive as the
coupling $\alpha$ is decreased.

Indeed, for the $\alpha>0$ solution, the $AdS_7$ black hole
behaves differently from the $AdS_5$ black hole. But, in all
spacetime dimensions, a maximum of free energy with $F>0$
corresponds to the low temperature phase and the $F<0$ phase
corresponds to the high temperature phase, separated by the
Hawking-Page transition at $F=0$. At low temperature and small
coupling $\alpha$ (i.e., $g_{YM}^2 N$ is large), a pure AdS space
dominates the thermodynamic ensemble, and at high temperature,
where free energy is negative, the Schwarzschild-AdS black hole
dominates the thermodynamic ensemble.

Translating plots for the free energy to the field theory, one
would expect that, for the $k=1$ event horizon topology, the
four-dimensional dual field theory can have a vanishing or
positive free energy at low temperature. In the spirit of
Witten-Hawking-Page type phase transition~\cite{Witten98a}, it is
plausible that the low temperature thermal AdS phase with $F=0$
corresponds to a confining phase in the dual large $N$ Yang-Mills
theory, and the high temperature black hole phase with $F<0$
corresponds to a de-confining phase of the gauge theory.

\section{Stability analysis: metric perturbations}

Stability of the spacetime matrices under gravitational
perturbations can normally be checked from an analysis that
whether the bound states with negative eigen state exist. If the
corresponding Schrodinger equations allows a state with $E<0$
or/and there are growing (quasi-) normal modes at the event
horizon, then the spacetime metrics are usually unstable. This
will be the main focus in what follows henceforth.

\subsection{Linearized field equations}

The Einstein field equations supplemented with a Gauss-Bonnet term
read \be \label{newEE2} R_{ab}=\frac{2\Lambda}{n-1}\,
g_{ab}-32\alpha^\p \pi G_{n+1} \left(H_{ab}-\frac{1}{n-1}\,g_{ab}
H\right)\,.\ee In maximally symmetric spaces, for a background
metric $\bar{g}_{ab}$, the Ricci curvature is $\bar{R}_{ab}=-\,n c
\,\bar{g}_{ab}$, with $c$ being a constant of dimension
$(\mbox{length})^{-2}$. One considers a small perturbation to the
metric $\bar{g}_{ab}\to g_{ab}= \bar{g}_{ab}+h_{ab}$,
$g^{ab}=\bar{g}^{ab}-h^{ab}$, such that $\vert h^a\,_b\vert <<1$.
Then, for $\alpha^\p=0$, the linearized vacuum Einstein equations
read \be \delta R_{ab}=-\,n c\, h_{ab}=-\,\frac{n}{l^2}\,h_{ab}\,.
\ee

Next, for $\alpha^\p>0$, as evaluated in the Appendix, the
perturbations $\delta g_{ab}~(=h_{ab})$ must satisfy \bea
&{}&\left(1-2 C\alpha\right) \delta R_{ab}= -\,n\,C\,
\left(1-\frac{(3n^2-3n+2) C
\alpha}{n(n-3)}\right)h_{ab}\nn \\
&{}& ~~~~ -\,\frac{2C\alpha}{n-3}\,\bar{g}_{ab}\left(\bna^c\bna^d
h_{cd}-\bna^2 h+C(n+1)h\right)\,. \label{A3} \eea For the
transverse trace-free (de Donder) perturbations, $h_c^c=\bna^a
h_{ab}=0$, we obtain \bea 2\delta
R_{ab}&=&\left(\Delta_L h\right)_{ab}\nn \\
&=&-\,\frac{2n C}{1-2C\alpha}\left[1-\frac{C \alpha
(3n^2-3n+2)}{n(n-3)}\right]h_{ab} \,.\nn \\
&{}& ~~~~ \eea For the $k=-1$ horizon, it is not {\it a priori}
sufficient to prove stability for the tensor modes only. In these
cases, the scalar and vector modes also can be unstable. However,
it is generally the case for a spherical horizon topology. In
particular, when the vacuum solutions of the Einstein field
equations are taken into account, the vector and scalar
perturbations, in the background of higher dimensional black
holes, are insensitive to probe stabilities on the base manifold
${\cal S}^{n-1}$ (see, for
example,~\cite{Gubser00a,Shiromizu,Gibbons02a} for relevant
discussion). As argued in~\cite{Gibbons02a}, for the
$k=\{0,\,-1\}$ geometry, instability could arise also due to
scalar fluctuations. We will not study such modes here, which
might require knowledge of couplings between the curvature and
conformal scalars on the boundary other than the scalar modes of
the metric fluctuations. A linear perturbation analysis for
stability of dilatonic black holes with a Gauss-Bonnet term has
been presented in~\cite{Torii98}.

We wish to study stability of the background metrics under the
conditions \be h_{0a}=h_{1a}=0\,. \ee Alternatively, one may
impose the conditions such that $|h_a\,^b|<<1$ and \bea h_{cd}&=&
{\cal O}\left(r^{-2}\right)\,, \quad h_{c
r}=\left(r^{-4}\right)\,, \quad h_{rr}=\left(r^{-6}\right)\,,\nn
\\
&{}& ~~~~~ \mbox{with}\,,~ c,\,d\neq r\,.\nn \eea With these
constraints, along with a requirement that the base manifold
${\cal M}^{n-1}$ is itself an Einstein space $\widetilde{R}_{i
j}(h)=(n-2)k\,\tilde{g}_{i j}$, the transverse tracefree property
of $h_{ab}$ is characterized by $h_{i j}$, where $i\,\,j,\cdots$
run from $2$ to $n~(\equiv D-1)$, \bea 2\delta R_{i
j}&=&\left(\Delta_L h\right)_{i j}\nn \\
&=&-\,\frac{2n\,C}{1-2C\alpha}\,\left(1-\frac{C \alpha
(3n^2-3n+2)}{n(n-3)}\right)h_{i j}\nn \\
&\equiv& -\,2\,n\,A\,h_{i j}\,, \label{mainlinear} \eea with \bea
&{}&\left(\Delta_L h\right)_{i
j}\,=\,\frac{1}{r^2}\,\tilde{\Delta}_L h_{i j} -f
\left(\frac{d^2}{d r^2}
+\frac{4}{r^2}\right)h_{i j} \nn \\
&{}& ~~~~~-\,\frac{r f^\p+(n-5)f}{r}\,\frac{d}{d r}\,h_{i
j}+\frac{1}{f}\, \frac{d^2}{d t^2}\,h_{i j} \,, \eea where
$\tilde{\Delta}_L h_{i j}$ is the Lichnerowicz operator acting on
${\cal M}^{n-1}$. One may look for unstable tensor modes of the
form \be h_{i j}(x) = r^2\,\phi(r)\, e^{\omega t}\,\wth_{i j}
(\tilde{x})\,, \ee where $(\tilde{x})$ are coordinates on ${\cal
M}^{n-1}$. It is convenient to assume that \be
\left(\widetilde{\Delta}_L \wth\right)_{i j}=\lambda\, \wth_{i
j}\,,\ee where $\lambda$ is the eigenvalue of the Lichnerowicz
operator. Eq.~(\ref{mainlinear}) then takes the form of
Sturm-Liouville type problem~\cite{Gibbons02a} \bea
&{}&-\,f\frac{d}{dr}\left(f\,
r^{n-1}\,\frac{d\phi(r)}{dr}\right)-\,\frac{2f}{r^2}\,\Big((n-2)f
-\frac{\lambda}{2}\nn \\
&{}&+\,r\,f^\p-nA\,r^2\Big)\phi(r)=\omega^2\,\phi(r)
\,r^{n-1}\,.\eea In terms of Regge-Wheeler type coordinates:
$dr\equiv f\,dr_*$ and $\phi(r)\equiv \Phi\,r^{-(n-1)/2}$, this
takes the form \be \frac{d^2\Phi}{dr_*^2}- V(r(r_*))\Phi=\omega^2
\Phi \ee where \bea \label{mainpoten}
V(r)&=&\frac{\lambda\,f}{r^2}+\frac{\left((n-2)(n-10)-1\right)
f^2}{4r^2} + \frac{(n-5)ff^\p}{2r}\nn \\
&{}& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ + 2nA\,f \,.\eea A
requirement of finite energy is equivalent to the normalization
condition that $\int \Phi^2\,dr_*=\int f(r)^{-1}\,\Phi^2\,dr$ is
finite~\cite{Gibbons02a}. In general, stability of the AdS vacuum,
and stability the above potential, in particular, depends on the
eigenvalues of the Lichnerowicz operator $\Delta_L$.

\subsection{Stability of massless state in general relativity}

We specialize the above case for $\alpha^\p=0$. In fact, for the
$k=+1$ geometry, the metric background with a zero mass is always
stable in AdS space. This is reminiscent of the fact that the
ground state of the Schwarzschild AdS black hole is itself given
by a global AdS space with $\mu=0$. The issue of instability
arises only for $n+1>4$, because in four spacetime dimensions the
Schwarzschild black holes are stable~\cite{Vishves70a}, and also
the symmetric transverse trace-free tensor modes are not available
in stability analysis, when $n=3$.

Consider first the case with a vanishing cosmological constant
$\Lambda=0$, so $A=0$. In this case, the spacetime is flat if the
base is a sphere and if the base is not a sphere then the full
spacetime is a cone which is singular at the origin. So it is
relevant to consider the $k=1$ geometry. In this case, one has
$f(r)\to 1$ asymptotically and $r=r_*$. The asymptotic potential
is \be \label{inftypoten}
V_\infty(r)=\frac{4\lambda+(n-2)(n-10)-1}{4\,r^2}
=\frac{(2\nu-1)(2\nu+1)}{4\,r^2}\,. \ee  Thus the
potential~(\ref{inftypoten}) is non-negative only if \bea
\label{lambdarel.} \lambda &\geq & \frac{1+(n-2)(10-n)}{4}\,,\nn
\\ or, \quad \nu &=&\frac{1}{2}\,\sqrt{(n-2)(n-10)+4\lambda}\geq
\frac{1}{2} \eea where $V_\infty (r)=0$ if equality holds. In
general, stability of the background metric requires that there is
no any bound negative energy states or no growing modes with
$\omega>0$ at the horizon, if latter exists. A requirement that
$\lambda>0$ indeed constraints the spacetime dimensions $D=(n+1)$
by $n\leq 10$, because $n>10$ may allow a negative $\lambda$,
which should not be the case for the $k=+1$ topology. The
dimensions $D=n+1\leq 11$ itself is interesting from the view
point of low energy supergravity.

The asymptotic solution for $\Phi(r)$ that decays as $r$ goes to
$+\infty$ is \bea \Phi_\infty &=& C_1\,\sqrt{r}\,
J_\nu\left(\sqrt{-\omega^2}\,r\right)+
C_2\,\sqrt{r}\,~  Y_\nu \left(\sqrt{-\omega^2}\,r\right)\nn \\
&=& Re \,\left(\sqrt{r}\,K_\nu(\omega r)\right) \,. \eea For small
$r$, and a real positive $\nu$, $\Phi_\infty(r)$ behaves as $\sim
r^{-\nu+1/2}$. The solution is divergent but normalizable for
$1>\nu >1/2$, and divergent and non-normalizable for $\nu\geq 1$.
In this case, there are no localized negative energy modes due to
an imaginary $\nu$. For $n\leq 10$, an imaginary $\nu$ does not
exist for a positive potential. For $n>10$, the eigenvalue
$\lambda$ may take a negative value, so $\nu>1$, but this solution
is always non-normalizable.

Above we have analyzed only the $\Lambda=0$ case. It has been
known that a massless solution with $k=+1$, $\mu\geq 0$ is usually
stable under metric perturbations, and this is true also for the
$\Lambda<0$ solution. In this case, the
potential~(\ref{inftypoten}) is modified to \bea
\label{fullpotenk=1}
V(r)&=&\left(\frac{4\lambda+(n-2)(n-10)-1}{4\,r^2}
+\frac{(n-1)(n+1)}{4\,l^2}\right)\nn \\
&{}& ~~~~~~~~~~~~~~~~~~~~~~\times
\left(1+\frac{r^2}{l^2}\right)\nn \\
V(x) &=&\frac{1}{l^2} \Bigg(\frac{4(\lambda-n+1)+(n-3)(n-5)}{4\,
x} \nn \\
&{}& ~~~~~~~~~~~~~~ +\,
\frac{(n-1)(n+1)}{4}\Bigg)\left(1+x\right)\,, \eea where $x\equiv
r^2/l^2$. The potential~(\ref{fullpotenk=1}) is always positive
and bounded from below in satisfying~(\ref{lambdarel.}). The
Schr\"odinger equation under the above potential may be expressed
as a hypergeometric equation \bea \label{Schro} &{}&
x(x+1)^2\,\Phi^{\p\p}(x)+
\frac{(x+1)(3x+1)}{2}\,\Phi^\p(x)\nn \\
&{}&~~ -\, \frac{x+1}{4}\times
\Bigg(\frac{4\lambda+(n-2)(n-10)-1}{4x}\nn \\
&{}& ~~~~~+\,
\frac{(n-1)(n+1)}{4}+\frac{\omega^2\,l^2}{(x+1)}\Bigg)\Phi(x)=0\,,
\eea with no singularities in the range $0<x<\infty$. To
simplify~(\ref{Schro}) one can make the following substitutions
\bea 2\nu &=&\sqrt{4\lambda+(n-2)(n-10)}\,,
\quad \sigma=-\,\frac{i\omega\,l}{2}\\
a&=&\sigma+\frac{2\nu-(n-2)}{4}\,, \quad
b=\sigma+\frac{2\nu+(n+2)}{4}\,, \nn \\
&{}&~~~~~~~~~~~~~~~~~ c=\nu+1\,. \eea Under the decomposition
$\Phi(x)=x^{(1+2\nu)/4}\, (1+x)^\sigma\,\varphi(x)$, two
independent solutions of~(\ref{Schro}) are \bea &{}&\varphi(x)=
{}_2F_1(a,\, b;\, c;\, -\,x)\, \quad \mbox{and}\nn \\
&{}&~~ x^{1-c}\,~{}_2F_1(a-c+1,\, b-c+1;\, 2-c;\, -\,x)\,, \eea
where ${}_2F_1(a, b; c;x)$ is the hypergeometric function, and the
parameter $c\neq 0, -1, -2, -3, \cdots$ for the first solution,
and $c\neq 2, 3, 4,\cdots$ for the second solution. A general
solution of~(\ref{Schro}) is thus given by a linear combination of
\bea &{}& \Phi_\pm (x)= x^{(1\pm
2\nu)/4}\,\left(1+x\right)^{-\,i\omega l/2}\times \nn \\
&{}& {}_2F_1\Bigg(\frac{\pm\, 2\nu-(n-2)}{4}-\frac{i\omega
l}{2},\,
\frac{\pm\, 2\nu+(n+2)}{4}-\frac{i\omega l}{2};\nn \\
&{}& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\,\pm\,\nu+1;\,-\,x\Bigg)\,.\eea  There exists a solution that is
analytic in the neighborhood of the origin which can be normalized
to unity. A stability check may be needed for the both limits:
$x\to 0$ and $x\to \infty$. This is the case extensively studied
by Gibbons and Hartnoll in~\cite{Gibbons02a}. For a consideration
of the $\alpha>0$ case in next section, we briefly discuss first
some important features of the $\alpha=0$ solution.

To study the asymptotic behavior, one uses the following relation
for hypergeometric functions~\cite{Askey} \bea
&&_2F_1(a,b;c;x)=h_1\,(-x)^{-a}\times \nn \\
&&{}~~~~~~~~~~
_2F_1\Big(a,\,a-c+1;\,a-b+1;\frac{1}{x}\Big)\nn \\
&&+\,h_2\,(-x)^{-b}\,{}_2F_1\left(b,\,b-c+1; b-a+1;
\frac{1}{x}\right), \label{infinite} \eea where \be
h_1=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(c-a)\Gamma(b)}\,, \quad
h_2=\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(c-a)\Gamma(a)}\,. \ee

As $x\to \infty$, the asymptotic solution is \bea &&
{}_2F_1(a,\,b;\,c;\,x)=h_1\,(-x)^{-a}
\left(1+\frac{a(a-c+1)}{a-b+1}\,\frac{1}{x}+\cdots\right)\nn \\
&&~~~~~~~~~+\,h_2\,(-x)^{-b}
\left(1+\frac{b(b-c+1)}{b-a+1}\,\frac{1}{x}+\cdots\right)\,.\eea
The boundedness requires that as $r\to\infty$ the function
$\Phi(r)$ must go as ${\cal O}\left(r^{(n-1)/2}\right)$ or lower
power of $r$. The solution which has this behavior is given by the
linear combination of $\Phi_{\pm}$ such that the term $(x)^{-b}$
of two solutions cancel. Hence \bea \label{thirdsol} \Phi_3 (x)&=&
x^{-(n+1)/4+i\omega\,l/2}\,\left(1+x\right)^{-\,i\omega\,l/2} \nn
\\&\times & {}_2F_1\Bigg(\frac{2\nu+n+2}{4}-\frac{i\omega\,l}{2},
\, \frac{-\, 2\nu+n+2}{4}-\frac{i\omega\,l}{2};\nn \\
&{}&~~~~~~~~~~~~~~~~~ \,\frac{n+2}{2};\,-\,\frac{1}{x}
\Bigg)\,.\eea To evaluate the behavior as $r\to 0$, one can
apply~(\ref{infinite}) to~(\ref{thirdsol}), replacing $1/x$ by $x$
for ${}_2F_1$. Then the leading behavior of the solution as $x\to
0$ is given by ${\cal O}\left(x^{-(n+1+2\nu-n-2)/4}\right) \sim
{\cal O}\left(r^{(1-2\nu)/2}\right)$. The range of convergence for
$c>a+b$ is $|x|<1$ and $x=1$, which implies that
$0>-\,i\omega\,l$. Therefore, a situation that the hypergeometric
series terminates for some special (positive) values of $\omega$
simply does not arise here~\cite{Gibbons02a}. As a consequence,
the massless configurations where ${\cal M}^{n-1}$ has constant
positive ($k=+1$) curvature are stable under tensor perturbations.
Some further discussion on the asymptotic behavior of the
solutions will also be given in next section while discussing the
$\alpha>0$ case.

\subsection{Instability of hyperbolic black holes with $\mu=0$}

The instability of $k=-1$ massless ($\mu=0$) black hole spacetimes
in Einstein gravity ($\alpha^\p=0$), under tensor perturbations,
was studied in~\cite{Gibbons02a} with some details. So we will be
brief in our analysis of this particular case, but we shall review
some results reported earlier as there arise some quantitative
differences. For $\alpha=0$, the metric solution $f(r)$ takes the
form \be \label{bhmetric}
f(r)=k+\frac{r^2}{l^2}-\frac{\mu}{r^{n-2}} \,.\ee With $k=-1$,
$\mu=0$, the above metric, although it possesses a horizon ar
$r_+=l$, may not describe a black
hole~\cite{Vanzo,Mann96a,Emparan99b}. This spacetime is simply a
topological space, which requires suitable identifications at
conformal infinity. The gravitational potential takes the form
\bea
V(x)&=&\frac{1}{l^2}\Bigg(\frac{4(n-1+\lambda)-(n-3)(n-5)}{4\,x}\nn
\\ &{}&~~~~~~ +\,\frac{(n-1)(n+1)}{4}\Bigg)\left(x-1\right) \,, \eea where
$x\equiv r^2/l^2$. One can easily show that the above potential
can be positive for all values of $x$ only if \be
\lambda=5-3n\equiv \tilde{\lambda}\,.\ee This is a special
situation. One might note that for a quotient space, the
Lichnerowicz spectrum should be bounded below by
$\lambda_{min}=-\,2(n-1)$. The spacetime dimensions $n+1=4$ is in
borderline for which $\tilde{\lambda}=\lambda_{min}$. The issue of
instability, however, arises only for $n+1>4$, because there are
no tensor harmonics on $S^2$ or $H^2$. For $n+1\geq 5$,
$\tilde{\lambda}$ is smaller than the minimum eigenvalue
$\lambda_{min}$, and in this case the potential can be negative.
More precisely, as the plots in Fig~(\ref{figure9}) show, the
potential is negative but bounded from below when \be
\label{critical}
\lambda<\lambda_{crit}=\frac{(n-2)(n-10)-1}{4}\,,\ee and unbounded
when $\lambda \geq \lambda_{crit}$, the latter might signal
instability of a massless black hole, when $k=-1$.

To gather more information, we need to solve the Schr\"odinger
equation. In terms of the dimensionless parameter $x\equiv
r^2/l^2$, the Schrodinger equation, under the above potential, can
easily be expressed as a hypergeometric equation \bea
\label{HyperGk=-1} &&
x(x-1)\,\Phi^{\p\p}(x)+\frac{x+1}{2}\,\Phi^\p(x)-\frac{1}{4}\,
\Bigg(\frac{(n-1)(n+1)}{4}\nn \\
&& +\frac{4\lambda+1-(n-2)(n-10)}{4x}+\frac{\omega^2\,l^2}{4}\,
\frac{1}{x-1}\Bigg)\Phi(x)=0\,. \nn \\
&&~~~~~~~~~~~~~~~~~~ \eea Then one can make the following
substitutions \bea \label{substitute} 2\tnu
&=&\sqrt{4(2-\lambda)+(n-2)(n-10)}\,,
\quad \tsigma=-\,\frac{\omega\,l}{2}\nn \\
a&=&\tsigma+\frac{2\tnu+(n+2)}{4}\,, \quad
b=\tsigma+\frac{2\tnu-(n-2)}{4}\,, \nn \\
&{}&~~~~~~~~~~~~~~~~~ c=\tnu+1\,,\eea and decompose the harmonic
function as $\Phi(x)=x^{(3+2\tnu)/4}\,
{(x-1)}^{\tsigma}\,\varphi(x)$. Then two independent solutions
of~(\ref{HyperGk=-1}) are  \bea &{}& \varphi(x)= {}_2F_1(a,\, b;\,
c;\, x)\,\quad \mbox{and}\nn \\ &{}&~~x^{1-c}\,~{}_2F_1(a-c+1,\,
b-c+1;\, 2-c;\, x)\,. \eea

A general solution of~(\ref{HyperGk=-1}) is therefore given by the
linear combination of the following two solutions~\footnote{We
believe that the expression given in Ref.~\cite{Gibbons02a} (c.f.,
Eq.(43)) need to be presented as appeared in the
expression(~\ref{k-1sol}) with some change of notations $2\tnu=C$,
$n=d+1$, and $l=1/L$.} \bea \label{k-1sol} &{}& \Phi_\pm
(x)=x^{(3\pm
2\tnu)/4}\,\left(x-1\right)^{-\,\omega\,l/2}\times \nn \\
&{}& {}_2F_1\Bigg(\frac{\pm\,
2\tnu+(n+2)}{4}-\frac{\omega\,l}{2},\, \frac{\pm\,
2\tnu-(n-2)}{4}-\frac{\omega\,l}{2};\nn \\
&{}&~~~~~~~~~~~~~~~~~~~~~~~~\, \pm\,\tnu+1;\,\,x\Bigg)\,.\eea The
boundedness requires that as $r\to\infty$ the function $\Phi(r)$
must go as ${\cal O}\left(r^{(n-1)/2}\right)$ or lower power of
$r$. The solution which has this behavior as $r\to \infty$ is
given by the linear combination of $\Phi_{\pm}$ such that the term
$(-x)^{-b}$ of two solutions cancel, and hence the asymptotic
behavior goes like ${\cal O}\left(x^{(3\pm 2\tnu\mp 2\tnu
-n-2)/4}\right)\sim {\cal O}\left(r^{-(n-1)/2}\right)$. The
solution that is well behaved as $r\to \infty$ therefore takes the
form \bea &&\Phi_3 (x)=x^{(2\omega l
-n+1)/4}\,\left(x-1\right)^{-\,\omega\,l/2}\times ~~~~~~~~~~~~\nn
\\
&&{}_2F_1\left(\frac{2\tnu+n+2}{4}-\frac{\omega\,l}{2},\,
\frac{-2\tnu+n+2}{4}-\frac{\omega\,l}{2};\,
\frac{n+2}{2};\,\,\frac{1}{x}\right)\nn \\
&& ~~~~~~~~~~~~~~~~~~~~~~  \,.\eea

We can find the new solutions of the hypergeometric equation by
using the relation \bea &{}&{}_2F_1\left(a,\, b;\,c;\,x\right)=
k_1\,{}_2F_1\left(a,\,b;\,a+b+1-c;\,1-x\right)\nn \\
&{}&~~~~~~~~~~~~~~~~~~~~~~~
+\,k_2\,\left(1-x\right)^{c-a-b}\,\times\nn \\
&&~~~~~~{}_2F_1 \left(c-b,\,c-a;\,1+c-a-b;\,1-x\right)\,,\eea
where \be
k_1=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\,,\quad
k_2=\frac{\Gamma(c)\Gamma(-c+a+b)}{\Gamma(a)\Gamma(b)}\,. \ee For
a generic solution with $k_1,\,k_2\neq 0$, the dominant term as
$x\to 1$ is ${\cal O}\left((x-1)^{-\omega\,l/2}\right)$. This
solution is not square integrable at $x=1$. In the special case,
$k_1=0$, which is accomplished by choosing $c=a$ or $c=b$, one
finds the solution with better behavior at $x=1$. In this case,
the leading term as $x\to 1$ is \be
(x-1)^{-\omega\,l/2}\,\left(\frac{x-1}{x}\right)^{\omega\,l}=
{\cal O}\left((x-1)^{\omega\,l/2}\right)\,.\ee This solution is
normalizable and bounded at the horizon for $\omega>0$. The
condition that either $(c-a)$ or $(c-b)$ is zero would be that \be
\omega =\left(\frac{2\tnu-(n+2)}{2\,l}\right)>0
%% \quad \mbox{or}\quad
%%\omega=\left(\frac{-2\tnu-n-2}{4 l}\right)>0
\,.\ee This holds if $2\tnu>n+2$, which requires $\lambda_{min}
<(6-4n)$ (c.f., the first expression in Eq.~(\ref{substitute})).
As we noted above, for $H^{n-1}$ space and its quotient
$H^{n-1}/\Gamma$, this eigenvalue may not be allowed, because the
Lichnerowicz spectrum is bounded below by $\lambda_{min}=
-\,2(n-1)$. This should not mean that a massless topological
spacetime is stable. In fact, in the following range of the
eigenvalues, which are allowed by the theory such that $2\tnu
> 1$, \be \label{mainconstr} \frac{(n-2)(n-10)-1}{4}\leq \lambda <
\frac{(n-2)(n-10)+8}{4}\,, \ee one still has
$\lambda>\lambda_{min}$, but the gravitational potential is always
unbounded from below. Though a negative potential does not
necessarily imply instability of the background, an unbounded
potential certainly signals instability of a massless topological
black hole spacetime.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=5dpoten.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=7dpoten.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=10dpoten.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The potential in function of horizon $r_+$ at fixed
$l=1$; (a) $n=4$ and $\lambda=-\,5,~-\,14/4,~-\,3$ (top to
bottom); (b) $n=6$ and $\lambda=-\,6,~-\,17/4,~-\,4$ (top to
bottom); (c) $n=9$ and $\lambda=-\,3,~-\,9/4,~-\,1$ (top to
bottom). The above potential will be bounded and positive only for
$\lambda=5-3n$, and it is always unbounded for $\lambda$
satisfying~(\ref{mainconstr}).} \label{figure9}
\end{figure}

As some specific examples, instability of a massless topological
black hole can arise in the following range: \bea \bullet &{}&
n=4: \quad -\,\frac{13}{4}\leq \lambda
<-\,1 \nn \\
\bullet &{}& n=6: \quad -\,\frac{17}{4}\leq \lambda <-\,2\nn \\
\bullet &{}& n=9: \quad -\,2\leq \lambda <\frac{1}{4}\,.\nn \eea

In four dimensions, the Schwarzschild AdS black hole is shown to
be stable against small electromagnetic and gravitational
perturbations~\cite{Cardoso01a}. In~\cite{Wang01a}, using a
numerical method, a classical instability for the evolution of a
massive scalar field coupled to curvature in four-dimensional
topological spacetime is explored. Recently, an analytic
expression for the quasi-normal modes of a scalar perturbation is
presented in~\cite{Aros02a} for $n\geq 3$, where a massless
topological black hole itself is taken as the ground state. Here
we see that, for $k=-1$, a massless background can be unstable
under metric perturbations.

As for tensor perturbations, one can check instability by
evaluating the (quasi-)normal modes at the horizon. To this aim,
one defines $\omega\,l=-\,i\bar{\omega}$, then the quasinormal
frequencies are determined by the conditions
$a\vert_{\tsigma}=-\,N$ or $b\vert_{\tsigma}=-\,N$, with
$N=0,1,2,\cdots $, so \be
\bar{\omega}=i\left[2N+\frac{2(\tnu+1)\pm n}{2}\right]\,. \ee The
quantity in the square bracket is generally positive, even if we
take the negative sign. To make it negative we require
$2\tnu<n-2$, even if $N=0$. This further implies that
$\lambda>2n+5$, but for such value of $\lambda$ the potential is
unbounded from below, thereby signalling a gravitational
instability of a massless topological black hole.

\subsection{Stability of negative mass extremal state}

We discuss first the extremal solution in Einstein gravity. The
metric background with a negative extremal mass $\mu_{extr}<0$ is
given by \be \label{extremalMetric} f(r)=-1+\frac{r^2}{l^2}+
B\left(\frac{l}{r}\right)^{n-2}\,, \quad
B=2\sqrt{\frac{(n-2)^{n-2}}{n^n}}\,. \ee The above metric
background has constant negative curvature, namely,
$R_{ab}=-\,(n/l^2)\,g_{ab}$, and it is a candidate background
metric for hyperbolic black
holes~\cite{Vanzo,Birmingham98a,Emparan99a}. The gravitational
potential therefore reads \bea \label{extremal1+n}
V(r)&=&\Bigg[\frac{4(n-1+\lambda)-(n-3)(n-5)}{4\,r^2}\nn \\
&{}&~~~~~+\, \frac{(n-1)(n+1)}{4\,l^2}-\frac{B(n-1)^2 l^{n-2}}{4\,
r^n}\Bigg]\nn \\
&{}&\times\, \left(-1+\frac{r^2}{l^2}
+B\left(\frac{l}{r}\right)^{n-2}\right) \,. \eea The spacetime one
is allowed to take is $r>r_{extr}=l\,\sqrt{(n-2)/n}$. As expected
the potential vanishes at the extremal horizon $r=r_{extr}$. This
potential does not involve any negative energy eigen states for
$\lambda>\lambda_{crit}$ and $r_+\geq r_{extr}$, and there is a
bound state at the extremal horizon with $E=0$.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=potential4.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=potential5.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=potential6.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The potential defined for a negative mass extremal
background with $\alpha=0$. The values are chosen, from top to
bottom, at (a) $n=4$ and $\lambda=0,\,-1$ and $-2$; (b) $n=5$ and
$\lambda=-1,\,-2$ and $-3$; (c) $n=6$ and $\lambda=-2,\,-3$ and
$-4$.} \label{figure10}
\end{figure}

\subsection{Extremal solution in $n=4$}

We are interested to check stability for the $k=-1$,
$\mu_{extr}<0$ solution. For $n=4$, the metric
function~(\ref{extremalMetric}) takes the form  \be
\label{4dextremalM} f(x)=\frac{(2x-1)^2}{4x}\,, \ee where $x\equiv
r^2/l^2$ is a dimensionless scale. There is no cosmological
horizon for an extremal solution, so the perturbation extends to
$x\to \infty$. The extremal horizon is at $2x=1$. The
potential~(\ref{extremal1+n}) reads as \be
V(x)=\frac{1}{l^2}\,\left(\frac{4(3+\lambda)+1}{4\,x}+\frac{15}{4}
-\frac{9}{16\,x^2}\right) \left(\frac{(2x-1)^2}{4\,x}\right)\,.
\ee This potential vanishes at the extremal horizon $2x=1$, but it
is everywhere positive outside the extremal horizon. The
Schrodinger equation under the potential takes the form \bea
&&\frac{(2x-1)^2}{8x^2}\Bigg[2x(2x-1)^2\,\Phi^{\p\p}(x)+(4x^2-1)\,\Phi^\p(x)
-2x\times \nn\\
&&\left(\frac{4(3+\lambda)+1}{4\,x}+\frac{15}{4}-\frac{9}{16\,x^2}\right)
\Phi(x)\Bigg] -\omega^2 l^2\,\Phi(x)=0\,.\nn \\
&& ~~~~~~~~~ \eea The condition of boundedness is automatic,
because it simply requires that $\Phi(x)$ is bounded at $2x=1$,
where $f(x)=0$, as seen in the above plots for the potential. The
finite energy condition will be that $\Phi(x)$ goes to zero on the
extremal horizon, because the zero of $f(x)$ is simple. We can get
some insights about stability of the solutions by inspecting the
graphics for the gravitational potential.

The above equation may be solved exactly only for $\omega=0$ modes
\bea &{}&
\Phi(x)=c_1\,(2x-1)^{-\,\gamma/4}\,x^{3/4}\,{}_2F_1(a,b;1;2x)\nn
\\
&{}& ~~~~ +\,c_2\, (2x-1)^{-\,\gamma/4}\,
x^{3/4}\,{}_2F_1(a,b;1,2x)\nn \\
&{}& ~~~~~~~\times \, \int
\frac{(2x-1)^{-1+\gamma/2}}{x\,\left({}_2F_1(a,b;1;2x)\right)^2}\,dx
\,,\eea where \be \gamma=\sqrt{2\lambda+32}\,,\quad
a=\frac{6-\gamma}{4}\,,\quad b=\frac{-\,(2+\gamma)}{4}\,.\ee This
solution converges at $2x=1$ for any eigenvalue
$\lambda>\lambda_{crit}$, and as well in the
range~(\ref{mainconstr}), and it is also normalizable there. The
situation is similar in the $n>4$ case. Now for the eigenvalues
$\lambda$ in the range given by~(\ref{mainconstr}), where one
might expect instability to arise for a massless background, the
potential is bounded and positive. Therefore, in the spacetime
region $r\geq r_{extr}$, the potential is always positive, tending
to zero at the extremal state. This means the extremal ground
state could be stable under tensor perturbation.

\subsection{Extremal solution in $n=6$}

For $n=6$, the metric function~(\ref{extremalMetric}) takes the
form \be f(x)=\frac{(3x+1)(3x-2)^2}{27\,x^2}\,, \ee and the
corresponding potential is \bea
V(x)&=&\frac{1}{l^2}\,\left(\frac{4\lambda+17}{4x}+\frac{35}{4}
-\frac{25}{27\,x^2}\right)\nn \\
&{}&~~~~~~\times \left( \frac{(3x+1)(3x-2)^2}{27\,x^2}\right)\,.
\eea This potential is plotted in Fig.~(\ref{figure10}), and there
is no need to repeat the above calculation for $n=6$, because this
potential does not cover the whole spacetime region. For example,
when $\lambda=-13/4$, the potential is not well behaved when
$x<0.27$.

More precisely, for the $\alpha=0$ solution, the spacetime region
$r<r_{e}$ can have an internal infinity~\cite{Mann96a}, where the
potential is not well defined. This might force the reference
spacetime to be incomplete in Einstein gravity, specially for the
$k=-1$ case. This is clearly seen in the plots of gravitational
potential $V(r)$ vs horizon position, as shown in
Fig.~(\ref{figure10}) for different spatial dimensions ($n$) and
eigenvalues ($\lambda$). This problem may be resolved once the
background metric and hence the potential receives higher
derivative curvature corrections.


\section{Gauss-Bonnet black holes and Gravitational Stability}

For $\mu=0$, the AdS vacuum solution is given by \be \label{mu=0}
f(r)=k+\frac{r^2}{2\alpha}\left(1\mp \sqrt{1-
\frac{4\alpha}{l^2}}\,\right)\equiv k+C\,r^2\,. \ee That is, the
$AdS_{n+1}$ metric spacetime has constant curvature $ R_{ab}=- n
C\, g_{ab}$.

For $k=-1$, we take a negative mass extremal state as the ground
state. In this case, the background metric in five dimensions
takes the form \be \label{GBExtremal}
f(r)=-1+\frac{r^2}{2\alpha}\left[1 - \sqrt{\left(1 -
\frac{4\alpha}{l^2}\right)\left(1-\frac{\alpha
l^2}{r^4}\right)}\right]\,.\ee In the $\alpha\to 0$ limit, an
anticipated, this reduces to~(\ref{extremalMetric}), which
satisfies $R_{ab}=-\,(n/l^2)\,g_{ab}$. For a finite coupling
$\alpha$, the metric background~(\ref{GBExtremal}) is only
asymptotically locally AdS, which can be easily seen by allowing a
particular coupling. A requirement that $4\alpha < l^2$ is
complementary to the condition $r^2 \geq l^2/2$. The above
background, unlike the AdS vacuum used in the spherical case,
represents an extremal black hole which itself receives
corrections from the coupling $\alpha$, similar situation arises
for stringy corrections of the form $\left(\alpha^\p\right)^3
R^4$~\cite{Klemm99a}.

Here we shall be interested in the following
two backgrounds: \\
\noindent (i) an AdS space with zero mass ($\mu= 0$), which is
the background for spherical black holes, and\\
\noindent (ii) an extremal metric with $\mu_{extr}<0$, which is
taken as the background for $k=-1$ hyperbolic black holes.

\subsection{Stability of a massless Gauss-Bonnet black hole}

As shown in the previous section, for $\alpha=0$, the
Schwarzschild AdS black hole with $\mu=0$ and $k=+1$ horizon is
stable under tensor perturbations. Here we extend this analysis
for a non-trivial GB coupling. The background metric function,
with $\alpha>0$, therefore reads \be \label{k=1bg} f(r)=1+C\,r^2
\,, \quad \frac{1}{C}\equiv\frac{2\alpha}{1\mp
\sqrt{1+\frac{8\alpha\Lambda}{n(n-1)}}}
>0\,.\ee
For generality, we may allow both sign for $C$, but we will
ultimately find that, for $n+1\leq 8$, only the negative sign in
front of the square root in~(\ref{k=1bg}) can give stable solution
under metric perturbations. In maximally symmetric spaces, as
explicitly shown in the Appendix (sub-section B), one may take
$C\equiv 1/l^2$, which corresponds to the negative sign
in~(\ref{k=1bg}). In this case, the Einstein vacuum is also the
vacuum (ground state) of the Einstein-Gauss-Bonnet theory (see
Appendix for further details).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{figure}[ht]
\begin{center}
\epsfig{figure=potential7.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=potential8.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=potential9.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The potential vs horizon radius for $k=+1$. The values
are fixed at $n=7$, $\lambda=17/4$, $\alpha=2/11$ (upper plot);
$n=8$, $\lambda=14/4$, $\alpha=2/9$ (middle plot); $n=9$,
$\lambda=9/4$, $\alpha=1/4$ (lower plot). The upper most curve
corresponds to the $\alpha=0$ case, and two other curves
correspond to $\mp$ signs in~(\ref{k=1bg}): $-$ and $+$ (up to
down) in the upper most plot, and reverse in the lower two
plots).} \label{figure11}
\end{figure}

Let us introduce a dimensionless scale $x=Cr^2$, then the
corresponding Schr\"odinger equation may be expressed as
hypergeometric equation \bea \label{GBSchwarz} &{}&
x(x+1)^2\,\Phi^{\p\p}(x)+
\frac{(x+1)(3x+1)}{2}\,\Phi^\p(x)\nn \\
&{}& -\, \frac{x+1}{4}
\Bigg(\frac{4\lambda+(n-2)(n-10)-1}{4x}\nn \\
&{}&+\, \frac{n^2-8n-1+d}{4}+
\frac{\omega^2}{(x+1)C}\Bigg)\Phi(x)=0\,, \eea where \bea
\label{constant=d} d &=&
\frac{8nC}{(1-2C\alpha)}\,\frac{2\alpha}{1\mp \left(1-2C\alpha\right)}\nn \\
&{}&\times
\,\left[1-\frac{C\alpha(3n^2-3n+2)}{n(n-3)}\right]\,.\eea  We
shall choose the minus sign out of $\mp$ in Eq.~(\ref{k=1bg}) or
Eq.~(\ref{constant=d}) because this branch gives stable solutions
in all dimensions. Another reason for choosing the negative sign
is that, for this branch, in the limit $\alpha\to 0$, the black
hole solutions reduce to that of Einstein gravity.

The hypergeometric equation~(\ref{GBSchwarz}) may be solved
exactly, and two independent solutions are \bea \Phi_\pm
(x)&=&x^{(1\pm
2\nu)/4}\,\left(1+x\right)^{-\,i\omega /{2\sqrt{C}}}\nn \\
&{}& {}_2F_1\Bigg(\frac{\pm\, 2\nu \mp 2\beta +2}{4}
-\frac{i\omega}{2\sqrt{C}},\, \frac{\pm\, 2\nu \pm 2\beta +2}{4}\nn \\
&{}& ~~-\,
\frac{i\omega}{2\sqrt{C}};\,\pm\,\nu+1;\,-\,x\Bigg)\,,\eea where
$\nu=\frac{1}{2}\,\sqrt{4\lambda+(n-2)(n-10)}$ and
$\beta=\frac{1}{2}\,\sqrt{n^2-8n+d}$. To determine the conditions
for $\nu$ and $\beta$ to be real we need to satisfy \be \lambda
\geq \lambda_{crit} = \frac{1+(n-2)(10-n)}{4}\,,\quad
\mbox{and}\quad d\geq 8n-n^2+1 \,.\ee The first constraint is the
same as in the $\alpha=0$ case. For $\alpha= 0$, one has $d=8n$,
so $2\beta=n$. As we noted above, one of the roots of $C$ reduces
to the solution in Einstein gravity, namely, $C=1/\ell^2$. We may
derive a more useful constraint for $C\alpha$ from the condition
$d\geq (8n-n^2+1)$, so $2\beta\geq 1$, which implies that \be
\label{bound} C\alpha \leq \frac{(n-3)(n-1)}{2(n^2+11)}\,.\ee
Using~(\ref{k=1bg}), along with $\Lambda=-\,n(n-1)/(2l^2)$, we
find the constraint \be \label{newconstr} \frac{\alpha}{l^2}\leq
\frac{(n-3)(n-1)(n^2+4n+19)}{4(n^2+11)^2}\,. \ee So, for example,
when $n+1=5$, we find $\alpha/l^2\leq 17/324$. This bound may be
slightly enhanced by minimizing the eigenvalue $\lambda$, but
satisfying $\nu \geq 1/2$, $\beta\geq (1-\nu)$. At any rate, the
bound $\alpha/l^2\leq 17/324$, which is required here to make the
potential non-negative at the linearized level, already puts a
stronger bound to $\alpha/l^2$ than required for the thermodynamic
stability, namely, $\alpha/l^2\leq 1/12$. Therefore, the
quasi-local energy of the $k=1$ Schwarzschild AdS spacetimes and
the $k=-1$ extremal black hole entropy reported in~\cite{IPN02b}
are non-negative, given that a classical background is stable.

By the same arguments as for the pure AdS case with $\alpha=0$,
the solution that is well behaved as $x\to \infty$ is given by the
linear combination of $\Phi_\pm$, which reads \bea \Phi_3 (x)&=&
x^{-(2\beta+1)/4+i\omega/{2\sqrt{C}}}\,\left(1+x\right)^{-\,i\omega
/{2\sqrt{C}}} \nn \\
&{}& {}_2F_1\Bigg(\frac{2\beta + 2\nu +2}{4}-\frac{i\omega
}{2\sqrt{C}},\, \frac{2\beta - 2\nu +2}{4}\nn \\ &{}& ~~ -\,
\frac{i\omega}{2\sqrt{C}};\,\beta+1;\,-\,\frac{1}{x}\Bigg)\,.\eea
In the $\alpha=0$ case, the leading behavior as $r\to \infty$ goes
like ${\cal O}\left(r^{-(n+1)/2}\right)$, while it goes as $r\to
0$ like ${\cal O}\left(r^{1-2\nu)/2}\right)$. Both solutions
converge, and are normalizable only if $2\nu < 1$, that is, if
$4\lambda< 1+(n-2)(10-n)$. However, in this limit, there may be a
continuum of negative energy bound state due to a negative (or an
unbounded) potential. Fortunately, these solutions do not satisfy
the usual boundedness condition, \be \int
\phi^2\,\frac{r^{n-1}}{f(r)}\,dr=1 \ee because
$\phi=\Phi\,r^{-(n-1)/2}\sim r^{-n}$ diverges at the origin. Thus,
for the $k=+1$ case, a zero mass background is stable under tensor
perturbations~\cite{Gibbons02a}, and this is true also for the
$\alpha>0$ solution, in satisfying~(\ref{newconstr}). In the
latter case, the leading behavior of the solution as $x\to 0$ is
given by ${\cal O}\left(x^{-(2\beta+1+2\nu-2\beta-2)/4}\right)
\sim {\cal O}\left(r^{(1-2\nu)/2}\right)$, which is same as in the
$\alpha=0$ case.

For the positive root in~(\ref{k=1bg}), the gravitational
potential is unbounded when $n\leq 7$, thus a metric background
with positive root is unstable~\cite{Deser85a}. In the bottom plot
of Fig.~(\ref{figure11}), the positions of the curves are
reversed. That is, in spacetime dimensions $n+1> 8 $, a black hole
solution with the plus root in~(\ref{k=1bg}) can also be stable.
We have taken here a minimal $\lambda$, and $\alpha$ is allowed to
take its value towards its upper bound.

One also notes that if the base manifold is a round sphere
$S^{n-1}$, the spectrum of the Lichnerowicz operator is bounded
from up by $2(n-1)$. Interestingly, for a small GB coupling, in
(\ref{newconstr}), the potentials are bounded from below. So a
massless AdS metric with $k=1$ is stable only for a small
Gauss-Bonnet coupling $\alpha$. In other words, since
$\alpha=16\pi G_{n+1}\,(n-2)(n-3)\,\alpha^\p\sim M^{n-1}$, when
the AdS length scale is much larger than the corresponding
$(n+1)$-dimensional Planck scale $M$, the potential is bounded
from below and stable under gravitational perturbations.

A remark is in order. In the strong coupling regime, such as
$\alpha\sim l^2$, additional higher order curvature corrections
are important. Moreover, when $n+1>7$, a Gauss-Bonnet type
quadratic corrections to Einstein gravity may be supplemented by
the higher order Euler invariants, such as cubic order in Riemann
tensors or/and additional higher order corrections, like that of
$R^4$ terms.

\subsection{Potential for extremal background}

For $\alpha>0$, the $\mu_{extr}<0$, $k=-1$ (extremal) background
does not have constant curvature. So in this case, the
perturbation equations would involve terms like $a(l^6/r^8)$ and
higher powers of $(l/r)$, with some constant $a$, namely $a<<1$
when $\alpha<<l^2$. This complicates the process for finding exact
solution to the Schrodinger equation. But, in the limit
$\alpha<<l^2$, as well as $r^2>2\alpha$, so that $R_{ab}\simeq
-\,(n/l^2) \,g_{ab}$, we may get some approximate ideas about
stability of an extremal background by inspecting the
gravitational potential.

\begin{figure}[ht]
\begin{center}
\epsfig{figure=potential10.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%
\begin{center}
\epsfig{figure=potential11.eps,height=4.0cm,width=7.0cm}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\caption{The potential vs horizon radius, with an extremal state
as the ground state. The values are fixed at $n=4$, $l^2=1$,
$\lambda=-1,\,-1.6,\,-2,\,-3$ (from top to bottom); and
$\alpha=1/17$ (upper plot) and $\alpha=1/20$ (lower plot). }
\label{figure12}
\end{figure}

For $k=-1$, the gravitational potential, for small $\alpha/l^2$
or/and $r^2>2\alpha$, may be given by \bea V(r) &\simeq &
\Bigg[\frac{4\lambda+13}{4r^2}-\frac{17}{8\alpha}
+\sqrt{\frac{(l^2-4\alpha)\,r^4}{(r^4-\alpha l^2)\,l^2}}\nn
\\&\times& \left(\frac{17}{8\alpha}
-\frac{13\,l^4}{8(l^2-\alpha)r^4}\right)+\frac{4(2l^2-19\alpha)}
{l^2(l^2-2\alpha)}\Bigg] f(r)\, \nn \\
\eea where $f(r)$ is given by~(\ref{GBExtremal}). For a small
coupling $\alpha<<l^2$, the gravitational potential can be bounded
from below for all eigenvalues satisfying
$\lambda>\lambda_{crit}=-\,13/4$. The coupling $\alpha/l^2$ is
equally important parameter for a classical stability of the
background. As the plots in Fig.~(\ref{figure12}) show, when
$\alpha/l^2\leq 1/18$ holds, the gravitational potential is always
bounded from below and it is positive for large $r$, but it can be
negative and unbounded for $\alpha/l^2> 1/18$. It is quite
interesting that the above constraint for $\alpha/l^2$, which we
actually derived for $k=+1$, is effective to the $k=-1$ case.

It is possible that the plots for gravitational potentials would
be modified at small scales, namely, $r^2<l^2/2$, due to the terms
like $l^6/r^8$, and also due to additional higher derivative
curvature corrections, like $R^4$ terms. However, any corrections
to the potential coming from them would only contribute in the
order of $1/r^8$, so they will not destabilize the potential for
large $r$, as well as, for small $r$ but in the limit
$\alpha<<l^2$ and $r^2>2\alpha$.

For $k=-1$, the base manifold has negative curvature, so a
negative potential does not imply instability of the background,
if it can be bounded from below. We can be little more precise
here. The Breitenloher-Freedman bound $\lambda>-\,n^2/4$, which
may be required for the positivity of energy (or unitarity) in
$AdS_{n+1}$ spacetime, is $\lambda>-\,4$, when $n+1=5$. In our
case, we must satisfy $\lambda> -\,13/4$, such that the potential
is bounded from below and positive for large $r$, which puts a
stronger bound for stability of an extremal background than the
Breitenloher-Freedman bound.

A bounded and positive potential certainly implies that there are
no unstable modes. Therefore, we see that, for a small but
non-zero Gauss-Bonnet coupling, the hyperbolic black hole, whose
ground state is the extremal black hole spacetime with a negative
mass, may be stable under the tensor perturbation. We have shown
that the AdS extremal solution has a minimal energy under small
metric perturbations.

\section{Discussion and Outlook}

In this paper, by taking Gauss-Bonnet type curvature corrections
into account, we have presented three important ideas
together: \\
$\bullet$ choice of backgrounds for AdS black holes with
spherical, flat or hyperbolic event
horizons \\
$\bullet$ thermodynamic (in)stability of AdS black
holes with hyperbolic and spherical event horizons, and \\
$\bullet$ classical (gravitational) instability of background
spacetimes of dimensions $n+1>4$.\\

Having made it clear that what are the backgrounds to be used for
black hole spacetimes with spherical ($k=+1$), flat ($k=0$) or
hyperbolic ($k=-1$) event horizons, we have computed the
Gauss-Bonnet curvature corrections to the AdS black hole
thermodynamics by allowing $k=0,\,\pm 1$. It is shown that there
exists a unique $k=-1$ extremal black hole solution which has the
lowest energy for all spacetimes in its asymptotic class.

The $AdS_5$ hyperbolic black hole presents some interesting
features, such as, the free energy and the entropy are dependent
of the Gauss-Bonnet coupling $\alpha$, but the total energy is
not, thereby indicating a possibility that entropy and free energy
scale with the coupling of $\alpha^\p$-corrections, when going
from strong to weak coupling~\cite{Emparan99b,Gubser98a,Klemm99a}.

We have shown that for a thermodynamic stability of hyperbolic
black holes the extremal entropy and the specific have to be
non-negative at the background. By considering the free energy,
the corresponding thermal phase diagrams are obtained for $n=4$
and $n=6$ in the limit of sufficiently small and large
$\alpha^\p$-coupling, allowed in the theory. The Gauss-Bonnet
corrections to the black hole thermodynamics do not give rise to
Hawking-Page transition as a function of temperature for flat and
hyperbolic horizons. It is plausible that, for $k=-1$, when the
coupling $\alpha$ is large, there appears a new phase transition.
For example, in five dimensions, such behavior is seen for the
coupling $0.25\, l^2<\alpha<0.1305\,l^2$. (We did not elaborate
more upon this possibility, because it is not clear yet whether a
dual description should exist with a GB term, we will briefly
comment on it below.)

As for spherical black holes, there would appear a stable branch
of small black holes when the coupling $\alpha$ is small. It is
observed that, in five dimensions, there might occur a first-order
thermal phase transition only if $\alpha<0.0278\,l^2$. Moreover,
the Hawking-Page transition temperature decreases with increasing
the coupling of $\alpha^\p$-corrections (i.e. increasing string
scale), and this holds for all spacetime dimensions $n+1\geq 5$.

We have explored a classical (in)stability of AdS spacetimes of
dimensions $n+1>4$, considering the curvature $k=+1$ and $k=-1$,
applicable, respectively, to spherical and hyperbolic event
horizons. A base manifold with a negative curvature may be stable
under tensor perturbation for an extremal background but in the
Einstein gravity the gravitational potential is not well behaved
in the small distance scale $r<r_{extr}$. For solutions of the
Einstein-Gauss-Bonnet theory we need a small GB coupling,
$\alpha<<l^2$, in order to obtain a bounded and positive
potential. This is perhaps one of the important achievements of
the present paper.

\medskip
We end with few remarks and future problems.

We have computed curvature corrections to the $(n+1)$-dimensional
AdS black hole thermodynamics using the conventional
regularization scheme which follow by the subtraction of
divergences from a reference state to which the black hole
solution is asymptotically matched. It might be more interesting
to recover the obtained results in the Gauss-Bonnet theory using
the regularization where divergences are removed by subtraction of
local counter-terms at the boundary in the spirit of AdS/CFT
correspondence, as proposed in~\cite{Vijay99a} and implemented for
the Einstein gravity in~\cite{Emparan99a}.

With some further efforts, it might be possible to apply the black
hole thermodynamics with a GB term to the AdS/CFT correspondence.
One has AdS black holes as exact solutions when there are
interactions of the form $R+a R^2$, $R+b R_{ab}R^{ab}$ or
$R+\alpha_1 {\cal R}_{GB}^2$. So, one would find it convenient to
modify the Lagrangian of gravity \bea {\cal L}&\sim&
R+\alpha_1\,{\cal R}_{GB}^2+ a\,R^2+b\,R_{ab}R^{ab}= R+
(a+\alpha_1)R^2\nn
\\&{}&+\, (b-4\alpha_1)R_{ab}R^{ab}
+\alpha_1\,R_{abcd}R^{abcd}\,.\nn \eea To this Lagrangian one can
construct AdS black holes exactly and study its thermodynamics. As
discussed in~\cite{Odintsov99a}, the $\alpha_1(\mbox{Riemann})^2$
term arises as next to the leading order terms in AdS/CFT
correspondence for ${\cal N}=4$ and ${\cal N}=2$ super Yang-Mills
theories. One can fix the coefficients of higher derivative terms
to be the supergravity prediction of ${\cal O}\left(N^2\right)$
correction to the trace anomaly and study the Witten-Hawking-Page
phase transitions. This might help to further explore the
confining (deconfining) phase transitions in dual gauge
theories~\footnote{We thank Sergei Odintsov for this interesting
remark.}.

We have studied (in)stability of the black hole spacetimes with
different event horizons, considering only the AdS part on
$AdS_{n+1}\times Y$ for some $Y$. Out of many possible backgrounds
in $D=11$ supergravity~\cite{Page84a}, two of them are (i)
$AdS_2\times H^2\times S^7$ (ii) $AdS_5\times H_2\times S^4$; both
these backgrounds are
non-supersymmetric~\cite{Emparan98a,Kehagias00a}. It would be
interesting to know stability of the solutions by considering the
eigenvalues of the Lichnerowicz operator in the transverse space
$Y$ acting on symmetric tensors.

One could ask whether it is possible to build a stable
(meaningful) field theory on the AdS asymptotic (conformal
boundary in the sense of Penrose) having negative scalar
curvature, namely, $I\!\!R^{1}\times H^3$. Witten and Yau have
advocated~\cite{Edward99a} that a massless scalar coupled to a BPS
brane action, defined for a hypersurface $I\!\!R^{1}\times H^3$,
which has constant negative curvature, has a potential (or brane
action) unbounded from below. The instability of a massless
topological black hole as the background discussed in section V is
consistent with those results.

The calculations in~\cite{Edward99a} show that stable solutions on
$I\!\!R^{1}\times H^3$ cannot be obtained in the effective field
theory by using only the lowest order terms in the $10d$ or $11d$
supergravity. One possibility to improve the situation is to allow
higher derivative curvature corrections to the bulk action and to
take a non-supersymmetric bulk background, for example, an
extremal black hole metric with a negative mass or a non-zero
temperature background. With these modifications, it might be
possible that the potential is bounded from below. It would be
interesting to study these issues in some general framework.

\medskip {\it Acknowledgements}:

The author would like to thank Danny Birmingham, Rong-Gen Cai,
Chiang-Mei Chen, Roberto Emparan, Pei-Ming Ho, Miao Li, Shin'ichi
Nojiri, Sergi Odintsov, Sumati Surya and John Wang for helpful
discussions on the subjects. He has enjoyed many fruitful
conversations with Sean Hartnoll. This work is supported by the
National Science Council, the center for Theoretical Physics at
NTU, Taiwan, R.O.C..

\section*{Appendix : Linearized Curvature Terms for $R^2$-Gravity}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}

Our convention of metric is $(-,+,+,\cdots,+)$, and of curvatures
is \bea R_{abc}\,^d
&=&\partial_b\Gamma_{ac}^d-\partial_a\Gamma_{bc}+
\Gamma_{eb}^d\Gamma_{ac}^e - \Gamma_{ea}^d\Gamma_{bc}^e\nn \\
R_{acb}\,^c &=&R_{ab}\,, ~~~ R_{ab}g^{ab}=R \nn \\
\Gamma_{ab}^c &=&\frac{1}{2}\,g^{cd}\left(\partial_b g_{ad}
+\partial_a g_{bd} -\partial_d g_{ab}\right)\,. \eea

Consider a small perturbation to the metric \be \bar{g}_{ab}\to
g_{ab}= \bar{g}_{ab}+h_{ab}\,, \quad g^{ab}=\bar{g}^{ab}-h^{ab}\,,
\quad \vert h^a\,_b\vert <<1\,. \ee Under this perturbation the
curvatures transform as~\cite{IPN01d}) \bea R_{abc}\,^d &\to &
R_{abc}\,^d
 -\bar{\nabla}_{[a}\bar{\nabla}_{|c|} h_{\,b]}\,^d +\bar{\nabla}_{[a}\bar{\nabla}^d
h_{b]\,c} \nn \\
&{}&+\,\half\left(\bar{R}_{abe}\,^d h_c\,^e- \bar{R}_{abc}\,^e
h_e\,^d\right)\label{curvature}\nn \\
R_{ab}&\to & R_{ab}+\frac{1}{2}\,\left(\Delta_L\right) h_{ab}
-\frac{1}{2}\,\bar{\nabla}_a\bar{\nabla}_b h
+\bar{\nabla}_{(a}\bar{\nabla}^c
h_{b)\,c}\label{Riccitensor} \nn \\
R &\to & R -\bar{R}_{ab} h^{ab}- \bar{\nabla}^2 h  +
\bar{\nabla}_a\bar{\nabla}_b h^{ab}\label{Ricciscalar} \eea where
$h=h_p^p$. The Lichnerowicz operator $\Delta_L$ acting on a
symmetric second rank tensor $h_{ab}$ reads as \be \left(\Delta_L
h\right)_{ab}=-\bar{\nabla}^c\bar{\nabla}_c h_{ab}-2\bar{R}_{apbq}
h^{pq}+ 2\bar{R}_{(a}\,^c \,h_{b)\,c} \,.\ee

\subsection{Gauge freedom}
The diffeomorphism invariance of the gravitational theory, under
\be h_{ab}\to h_{ab}+\bar{\nabla}_a\xi_b+\bar{\nabla}_b\xi_a
\,,\ee implies gauge invariance of the linearized theory. To this
effect, one of the physical gauges is the transverse (or harmonic)
gauge \be \bar{\nabla}^a \hat{h}_{ab}\equiv
\bar{\nabla}^a\left(h_{ab}-\frac{1}{2}\,g_{ab}\,h_c^c\right)=0\,,
\ee which may always be imposed. Under this the Lichnerowicz
operator is compatible with the transverse tracefree condition,
$h_c^c=\bar{\nabla} ^a h_{ab}=0$. This gauge does not eliminate
all of the gauge freedom, but does simplify the perturbation
equations \be \delta
R_{ab}=\frac{1}{2}\,\left(\Delta_L\right)_{ab} \,.\ee


\subsection{Background solutions}

The Einstein field equations modified by a Gauss-Bonnet term read
\be R_{ab}-\frac{2\Lambda}{n-1}\, g_{ab}=16\pi G
\left[\frac{2\alpha^\p}{n-1}\,g_{ab}H-2\alpha^\p\,H_{ab}\right]\,,
\label{A1} \ee where $H_{ab}\equiv R
R_{ab}-2R_{acbd}R^{cd}+R_{acde}R_b\,^{cde}-2R_{ac} R_b^c$ and $H$
is its trace. For $\alpha^\p=0$, the field equations~(\ref{A1})
admit black holes as the exact vacuum solution, whose $g_{00}$
component in $(n+1)$ dimensions is \be \label{g00}
g_{00}=f(r)=k-\frac{2\Lambda
r^2}{n(n-1)}-\frac{\mu}{r^{n-2}}\,,\ee where $\mu$ is an
integration constant, and in suitable co-ordinates, $k=0,~\mp1$.
There are two independent constants $\mu$ and $\Lambda$, where, of
course, the parameter $\mu$ is related to the ADM mass of a black
hole. The base manifold ${\cal M}^{n-1}$ (i.e., the hypersurface
at the event horizon) will have constant negative (if $k=-1$) or
zero (if $k=0$) or positive ($k=1$) scalar curvature.

The metric solution~(\ref{g00}) satisfies \be
R_{abcd}=\frac{2\Lambda}{n(n-1)}\left(g_{ac}g_{bd}-g_{ad}g_{bd}\right)\,,
\quad R_{ab}=\frac{2\Lambda}{(n-1)}\,g_{ab}\,,\ee which is
maximally symmetric. While $\Lambda$ is fixed by the curvature of
the spacetime, namely \be \Lambda=\mp
\,\frac{n(n-1)}{2{\ell}^2}\,.\ee For $\Lambda<0$, one chooses the
negative sign so that ${\ell}$ is the curvature radius of the AdS
space.

For a non-trivial $\alpha$, the cosmological constant is fixed \be
\Lambda=-\,\frac{n(n-1)}{2 l^2}\,.\ee The new length scale $l^2$
is not the same as ${\ell}^2$, but there must be involved some
relation between them. On physical ground, the form of $\Lambda$
should be consistent with the exact (vacuum) solution of the field
equations~(\ref{A1}). For $\alpha^\p>0$, the metric solution is
known to be \be \label{muneq0}
f(r)=k+\frac{r^2}{2\alpha}\left(1\mp \sqrt{1+
\frac{8\alpha\Lambda}{n(n-1)}+\frac{4\alpha\mu}{r^n}}\right)\,,
\ee where $\alpha=16\pi G\, (n-2)(n-3)\alpha^\prime$. For a
spherically symmetric black hole (i.e., $k=1$), the background
(AdS vacuum solution) is given by the setting $\mu=0$. The metric
function then takes the form \be f(r)=1+C\,r^2\,, \quad C\equiv
\frac{1\mp \sqrt{1+ \frac{8\alpha\Lambda}{n(n-1)}}}{2\alpha}\,,
\ee where $C$ is a constant of dimension $(\mbox{length})^2$. Our
convention is that $C>0$ for $\Lambda<0$ and $C<0$ for
$\Lambda>0$. The above metric has constant curvature, namely, \be
\label{symmetricGB}
R_{abcd}=-\,C\,\left(g_{ac}g_{bd}-g_{ad}g_{bd}\right)\,, \quad
R_{ab}=-\,n C\,g_{ab}\,.\ee In these backgrounds, the field
equations~(\ref{A1}) amount to fix the cosmological constant \be
\Lambda=-\,\frac{n(n-1)C}{2}\,\left(1-\alpha C\right)\,.\ee

Finally, the background metric for a spherically symmetric
solution takes the form \bea f(r)&=&
1+\frac{r^2}{2\alpha}\left[1\mp
\sqrt{1-4\alpha C(1-\alpha C)}\,\right]\nn \\
&=&1+C\,r^2\,,\quad 1+\frac{r^2}{\alpha}-C r^2 \,. \eea In
identifying $C$ by $1/{\ell}^2$, one has $f(r)=1+r^2/{\ell}^2$
which is the same as in Einstein gravity. The message is clear,
that is, for a spherically symmetric solution, the AdS vacuum of
the Einstein theory is also the vacuum of the
Einstein-Gauss-Bonnet theory, although the black hole solutions,
which are given by $\mu\neq 0$, with $\alpha>0$ are very different
from those for $\alpha=0$.

\subsection{Linearized equations}

For the backgrounds~(\ref{symmetricGB}), the variations of the
curvatures are given by \bea \delta\left(RR_{ab}\right)&=&
-\,n(n+1)\,C\,\delta R_{ab}-n C\,\delta R\nn \\
\delta\left(R_{acde}R_b\,^{cde}\right)&=& -\,4 C\,\delta
R_{ab}+2C^2\left(h_{ab}-\bar{g}_{ab}\,h\right)\nn \\
\delta\left(R_a^c\,R_{bc}\right)&=&-\,2n C\,\delta R_{ab}-n^2
C^2\,h_{ab}\nn
\\
\delta\left(R_{acbd}R^{cd}\right)&=&
\frac{n C^2}{2}\left((n+3)h_{ab}-\bar{g}_{ab} h\right)-\nn \\
&{}& (n-1)C\,\delta R_{ab}-\bar{g}_{ab}\,C\,\delta R \,.\eea And
\bea &{}&
\delta\Bigg(g_{ab}\Big(R^2-4R_{ab}R^{ab}+R_{abcd}R^{abcd}\Big)\Bigg)\nn
\\ &{}& ~~~~~~~=\, n(n-2)(n-1)(n+1)C^2\,h_{ab}\nn \\
&{}& ~~~~~~~~~~~~- \, 2(n-1)(n-2)\,C\,\delta R \,.\eea Therefore,
the field equations~(\ref{A1}) reduce to the form \bea
\label{finallinear} \big(1-2C \alpha \big)\delta R_{ab}&=& -\,n
C(1-C\alpha)\,h_{ab}+
\frac{2(n^2+1)\alpha}{n-3}\,C^2\,h_{ab}\nn \\
&{}& -\, \frac{2\alpha\, C}{n-3}\,\Big(\delta R
+C\,\bar{g}_{ab}\,h\Big)\,.\eea

Note that a massless $(\mu=0)$ topological $(k=-1)$ black hole
solution \be f(r)=-1+C\,r^2 \,\ee is locally AdS and maximally
symmetric. In this case, the above linearized expressions are
fully effective.

For an extremal metric background, however, (\ref{finallinear})
may be used only for a large $r$. For a small coupling
$\alpha/\ell^2$, under a re-scaling $l^2\to
\ell^2/(1-\alpha/\ell^2)$, in five dimensions, the extremal
solution is \be f(r)=
-1+\frac{r^2}{2\alpha}-\frac{{\ell}^2-2\alpha}{2\alpha}
\sqrt{\left(\frac{r^4}{{\ell}^4}-\frac{\alpha}{{\ell}^2-\alpha}\right)}\,.\ee
Therefore, in the limit $\alpha\to 0$, one has \be f(r)\approx
-1+\frac{r^2}{\ell^2}+\frac{\ell^2}{4
r^2}=\frac{r^2}{\ell^2}\left(1-\frac{\ell^2}{2r^2}\right)^2\,.\ee
However, for $\alpha>0$, the metric spacetime is only
asymptotically locally AdS. As a simple choice, when
$\alpha={\ell}^2/5$, the curvatures are \bea R_0^0&=&
-\,\frac{4}{{\ell}^2}- \frac{9}{64}\,\frac{{\ell}^6}{r^8}+ {\cal
O}\left(\frac{1}{r^{12}}\right)=R_1^1\nn\\
R_{i}^{j}&=&
\left(-\,\frac{4}{{\ell}^2}+\frac{3}{64}\,\frac{{\ell}^6}{r^8}+
{\cal O}\left(\frac{1}{r^{12}}\right)\right)\delta_{i}^{j}\,. \eea
That is, one has $R_{ab}\simeq -\,\frac{n}{{\ell}^2}\, g_{ab}$,
only for large $r~(>>{\ell})$ or/and $\alpha<<\ell^2$. Though the
sub-leading terms are much suppressed for $r>{\ell}$, for an
extremal background, there would be some extra contributions on
the right hand side of~(\ref{finallinear}).


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\end{references}

%%\end{thebibliography}

%\newpage

\end{document}

%%% Comments welcome

