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UB-ECM-PF-03/10
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\begin{centering}
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\vspace{.3in}
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{\Large{\bf Cardy-Verlinde Formula and  Ach\'ucarro-Ortiz Black
Hole}}
\\

\vspace{.5in} {\bf Mohammad R. Setare$^{1}$ and Elias C.
Vagenas$^{2}$ }\\
\vspace{.3in} $^{1}$\, Department of Physics, Sharif University of
Technology, Tehran, Iran\\and\\Institute for Theoretical Physics
and Mathematics, Tehran, Iran\\and\\Department of Science, Physics
group, Kordestan University, Sanandeg, Iran
\\rezakord@yahoo.com\\
\vspace{0.4in}

$^{2}$\, Departament d'Estructura i Constituents de la Mat\`{e}ria\\
and\\ CER for Astrophysics, Particle Physics and Cosmology\\
Universitat de Barcelona\\
Av. Diagonal 647, E-08028 Barcelona\\
Spain\\
evagenas@ecm.ub.es\\
\end{centering}

\vspace{0.6in}
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%%%%%%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}


In this paper it is shown that the entropy of the black hole
horizon in the Ach\'ucarro-Ortiz  spacetime, which is the most
general two-dimensional black hole derived from the
three-dimensional rotating BTZ black hole, can be described by the
Cardy-Verlinde formula. The latter is supposed to be an entropy
formula of conformal field theory in any dimension.


\end{abstract}
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\end{titlepage}

\newpage

\baselineskip=18pt
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%%%%%%%%%%%%%%%%%%%%%%%%%%% INTRODUCTION %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}
Holography is believed to be one of the fundamental principles of
the true quantum theory of gravity \cite{{HOL},{RAP}}. An
explicitly calculable example of holography is the much--studied
anti-de Sitter (AdS)/Conformal Field Theory (CFT) correspondence.
More recently, it has been proposed that defined in a manner
analogous to the AdS$_{d}$/CFT$_{d-1}$ correspondence, quantum
gravity in a de Sitter (dS) space is dual to a certain
 Euclidean  CFT living on a spacelike boundary of the
dS space~\cite{Strom} (see also earlier works
\cite{Hull,Mazu,Bala}). Following the proposal, some
investigations on the dS space have been carried out recently
\cite{Mazu,Bala,mu1,Li,Noji2,Klem3,Gao,Bros,Haly,
Toll,Shir,McIn,Strom1,BBM,myung1,Carn,CMZ,Dani,gez1,deh,Ogus,set,set1,set3}.\\
The Cardy-Verlinde formula recently proposed by E. Verlinde
\cite{Verl}, relates the entropy of a  certain CFT to its total
energy and Casimir energy in arbitrary dimensions. In the spirit
of AdS$_{d}$/CFT$_{d-1}$ and dS$_{d}$/CFT$_{d-1}$ correspondence,
this formula has been shown to hold exactly for the cases of dS
Schwarzschild, dS topological, dS Reissner-Nordstr\"om , dS Kerr,
and dS Kerr-Newman black holes. In this paper we have further
checked the Cardy-Verlinde formula with the two-dimensional
Ach\'ucarro-Ortiz black hole which is derived from the
three-dimensional rotating BTZ black hole. In 1992 Ba\~nados,
Teitelboim and Zanelli (BTZ) \cite{banados1,banados2} showed that
$(2+1)$-dimensional gravity has a black hole
 solution. This black hole is described by two parameters,
its mass $M$ and its angular momentum (spin) $J$. It is locally
anti-de-Sitter space and thus it differs from Schwarzschild and
Kerr solutions in that it is asymptotically anti-de-Sitter instead
of flat. Additionally, it has no curvature singularity at the
origin.
 AdS black holes, are members of the two-parametric family
of BTZ black holes, Specifically AdS(2) black hole is most
interesting in the context of string theory and black hole physics
\cite{strominger1,strominger2}.\\
For two-dimensional (2D) gravitational systems (more in general
systems that admit 2D CFTs as duals) one can make use, directly,
of the Cardy formula \cite{cardy} that gives the entropy of  a CFT
in terms of the central charge $c$ and the eigenvalue of the
Virasoro operator $l_{0}$. However, this is possible only for 2D
systems for which one can explicitly show (e.g using the
AdS$_{d}$/CFT$_{d-1}$ correspondence) that they are in
correspondence with a 2D CFT \cite{CM99,andy}. Even in this most
favorable  case the use of the Cardy formula for the computation
of the entropy of the gravitational system is far from trivial.
The central charge $c$ and the eigenvalue $l_{0}$ of the Virasoro
operator  have to be expressed in terms of the gravitational
parameters, an operation that sometimes turns out to be very hard
\cite{CCCM}. The two-dimensional (2D) limit of the Cardy-Verlinde
proposal is interesting for various reasons. From investigations
of the AdS$_{d}$/CFT$_{d-1}$ correspondence, we know that there
are 2D gravitational systems that admit 2D CFTs as duals
\cite{CM99,cadcav}. In this case one can make direct use of the
original Cardy formula \cite{cardy} to compute the entropy
\cite{CM99,cadcav}. A comparison of these results with a 2D
generalization of the Cardy-Verlinde formula could be very useful
in particular for the understanding of the puzzling features of
the AdS$_{d}$/CFT$_{d-1}$ correspondence in two dimensions
\cite{strominger}. Another point of interest in extending the
Cardy-Verlinde formula to $d=2$ is the clarification of the
meaning of the holographic principle for 2D spacetimes. The
boundaries of spacelike regions of 2D spacetimes are points, so
that even the notion  of holographic bound is far from trivial. A
generalization of the work of Verlinde to two spacetime dimensions
presents several difficulties, essentially for dimensional
reasons. First of all, in two dimensions one cannot establish a
area law, since black hole horizons are isolated points. Moreover,
the spatial coordinate is not a ``radial'' coordinate and hence
one cannot
impose a natural normalization on it.\\

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Ach\'ucarro-Ortiz  Black Hole}
The black hole solutions of Ba\~nados, Teitelboim and Zanelli
\cite{banados1,banados2} in $(2+1)$ spacetime dimensions are
derived from a three dimensional theory of gravity \be S=\int
dx^{3} \sqrt{-g}\,({}^{{\small(3)}} R+2\Lambda) \ee with a
negative cosmological constant ($\Lambda>0$).
\par\noindent
The corresponding line element is \be ds^2 =-\left(-M+
\frac{r^2}{l^2} +\frac{J^2}{4 r^2} \right)dt^2
+\frac{dr^2}{\left(-M+ \displaystyle{\frac{r^2}{l^2} +\frac{J^2}{4
r^2}} \right)} +r^2\left(d\theta -\frac{J}{2r^2}dt\right)^2
\label{metric}\ee There are many  ways to reduce the three
dimensional BTZ black hole solutions to the two dimensional
charged and uncharged dilatonic black holes \cite{ortiz,lowe}. The
Kaluza-Klein reduction of the $(2+1)$-dimensional metric
(\ref{metric}) yields a two-dimensional line element:
 \be ds^2 =-g(r)dt^2 +g(r)^{-1}dr^2
\label{metric1}\ee where \be g(r)=\left(-M+\Lambda r^2
+\frac{J^2}{4 r^2}\right)\label{metric2}
 \ee

with $M$ the ADM mass, $J$ the angular momentum (spin)
 of the BTZ black hole and $-\infty<t<+\infty$, $0\leq r<+\infty$, $0\leq \theta <2\pi$.
\par \noindent
The outer and inner horizons, i.e. $r_{+}$ (henceforth simply
black hole horizon) and $r_{-}$ respectively, concerning the
positive mass black hole spectrum with spin ($J\neq 0$) of the
line element (\ref{metric}) are given as  \be
r^{2}_{\pm}=\frac{l^2}{2}\left(M\pm\sqrt{M^2 -
\displaystyle{\frac{J^2}{l^2}} }\right) \label{horizon1} \ee and
therefore, in terms of the inner and outer horizons, the black
hole mass and the angular momentum are given, respectively, by \be
M=\frac{r^{2}_{+}}{l^{2}}+\frac{J^{2}}{4r^{2}_{+}}\label{mass}\ee
and \be J=\frac{2r_{+}r_{-}}{l^2}\label{ang}\ee with the
corresponding to the angular momentum angular velocity to be
\be\Omega=\frac{J}{2 r^{2}}\label{angvel}\hspace{1ex}.\ee
\par\noindent The Hawking temperature $T_H$ of the black hole
horizon is \cite{kumar1} \bea T_H &=&\frac{1}{2\pi
r_{+}}\sqrt{\left(\displaystyle{\frac{
r_{+}^2}{l^2}+\frac{J^2}{4r_{+}^2}}\right)^2-\displaystyle{\frac{J^2}{l^2}}}\nn\\
&=&\frac{1}{2\pi r_{+}}\left(\displaystyle{\frac{
r_{+}^2}{l^2}-\frac{J^2}{4r_{+}^2}}\right)\label{temp1}
\hspace{1ex}.\eea \par\noindent The area $\mathcal{A}_H$ of the
black hole horizon is \bea \mathcal{A}_H &=&2\pi l
\left(\frac{M+\sqrt{M^2 -\frac{J^2}{l^2} }}{2}\right)^{1/2}\label{area1}\\
&=&2\pi r_{+} \label{area2}\eea thus the entropy of the
two-dimensional Ach\'ucarro-Ortiz black hole, if we employ the
well-known Bekenstein-Hawking area formula ($S_{BH}$) for the
entropy \cite{bekenstein1,bekenstein2,hawking3}, is given as  \be
S_{bh}=\frac{1}{4\hbar G} \mathcal{A}_H =S_{BH} \ee and using the
BTZ units where $8\hbar G =1 $ takes the form \be S_{bh}=4\pi
r_{+} \label{entr1}\hspace{1ex}.\ee
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Cardy-Verlinde Formula}
In a recent paper, E. Verlinde \cite{Verl} propound a
generalization of the Cardy formula which holds for ($1+1$)
dimensional Conformal Field Theory (CFT) to $(n+1)$-dimensional
spacetime described by the metric \be
ds^{2}=-dt^{2}+R^{2}d\Omega_{n}\ee where $R$ is the radius of a
$n$-dimensional sphere.
\par\noindent The generalized
Cardy formula (hereafter named Cardy-Verlinde formula)  is given
by \be S_{CFT}=\frac{2\pi
R}{\sqrt{ab}}\sqrt{E_{C}\left(2E-E_{C}\right)} \label{cvf}\ee
where $E$ is the total energy and $E_{C}$ is the Casimir energy.
The definition of the Casimir energy is derived by the violation
of the Euler relation as \be E_{C}\equiv n\left(E+pV-TS-\Phi
Q\right)\label{casimir1}\ee where the pressure of the CFT is
defined $p=E/nV$. The total energy may be written as the sum of
two terms \be E(S, V)=E_{E}(S, V)+\frac{1}{2}E_{C}(S,
V)\label{ext}\ee where $E_{E}$ is the purely extensive part of the
total energy $E$. The Casimir energy $E_{C}$ as well as the purely
extensive part of energy $E_{E}$ expressed in terms of the radius
$R$ and the entropy $S$ are written as \bea
E_{C}&=&\frac{b}{2\pi R}S^{1-\frac{1}{n}}\label{cftcas1}\\
E_{E}&=&\frac{a}{4\pi
R}S^{1+\frac{1}{n}}\label{exten1}\hspace{1ex}.\eea After the work
of Witten on AdS$_{d}$/CFT$_{d-1}$ correspondence,  E. Verlinde
proved that the Cardy-Verlinde formula  (\ref{cvf}) can be derived
using the thermodynamics of an AdS Schwarzschild black holes in
arbitrary dimension.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Entropy of Ach\'ucarro-Ortiz black hole  in Cardy-Verlinde Formula}
We would like to derive the entropy of the two-dimensional
Ach\'ucarro-Ortiz black hole (\ref{entr1}) from the Cardy-Verlinde
formula (\ref{cvf}).
 First, we evaluate the Casimir energy $E_{C}$ using (\ref{casimir1}).
 It is easily seen from (\ref{temp1}) and (\ref{entr1}) that
 \be
 T_{H}S_{bh}=2\left(\displaystyle{\frac{
r_{+}^2}{l^2}-\frac{J^2}{4r_{+}^2}}\right)\label{ts}
 \ee
while from (\ref{ang}) and (\ref{angvel}) we have \be
\Omega_{+}J=\frac{J^2}{2r^{2}_{+}}\label{jo}\hspace{1ex}.\ee Since
the two-dimensional Ach\'ucarro-Ortiz black hole
 is asymptotically anti-de-Sitter, the total energy is $E=M$ and
thus the Casimir energy, substituting (\ref{mass}), (\ref{ts}) and
(\ref{jo}) in (\ref{casimir1}), is given as \be E_{C}=
\frac{J^2}{2r^{2}_{+}}\label{casimir2}\ee where in our analysis
the charge $Q$ is the angular momentum $J$ of the two-dimensional
Ach\'ucarro-Ortiz black hole, the corresponding electric potential
$\Phi$ is the angular velocity $\Omega$ and $n=1$. Making use of
expression (\ref{cftcas1}), Casimir energy $E_{C}$ can also be
written as \be E_{C}=\frac{b}{2\pi
R}\label{cftcas2}\hspace{1ex}.\ee Additionally, it is obvious that
the quantity $2E-E_{C}$ is given, by substituting (\ref{ts}) and
(\ref{jo}) in (\ref{casimir1}), as \be
2E-E_{C}=2\frac{r^{2}_{+}}{l^2}\label{e-ec}\hspace{1ex}.\ee The
purely extensive part of the total energy $E_{E}$ by substituting
(\ref{e-ec}) in (\ref{ext}) is \be
E_{E}=\frac{r^{2}_{+}}{l^2}\label{exten2}\ee whilst by
substituting (\ref{entr1}) in (\ref{exten1}) is \be
E_{E}=\frac{4\pi a }{R}r^{2}_{+}\label{exten3}\hspace{1ex}.\ee At
this point it is useful to evaluate the radius $R$. By equating
the right hand sides of  (\ref{casimir2}) and (\ref{cftcas2}) the
radius is written as \be R=\frac{b r^{2}_{+}}{\pi
J^2}\label{radius1}\ee while by equating the right hand sides
(\ref{exten2}) and (\ref{exten3}) it can also be written as \be
R=4\pi a l^{2}\label{radius2}\hspace{1ex}.\ee Therefore, the
radius expressed in terms of the arbitrary positive coefficients
$a$ and $b$ is \be R=
2r_{+}\left(\frac{l}{J}\right)\sqrt{ab}\label{radius3}\hspace{1ex}.\ee
Finally, we can substitute the expressions (\ref{casimir2}),
(\ref{e-ec}) and (\ref{radius3}) which were derived in the context
of thermodynamics of the two-dimensional Ach\'ucarro-Ortiz black
hole black hole, in the Cardy-Verlinde formula (\ref{cvf}) which
in turn was derived in the context of CFT  \be
S_{CFT}=\frac{2\pi}{\sqrt{ab}}2r_{+}\left(\frac{l}{J}\sqrt{ab}\right)
\sqrt{\frac{J^2}{2r^{2}_{+}}2\frac{r^{2}_{+}}{l^2}}\ee and we get
\be S_{CFT}=S_{bh}\hspace{1ex}.\ee It has been proven that the
entropy of the the two-dimensional Ach\'ucarro-Ortiz black hole
can be expressed in the form of Cardy-Verlinde formula.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
Among the family of $AdS_{d}/CFT_{d-1}$ dualities, the pure
gravity case $AdS_3/CFT_2$ is the best understood. In contrast,
the AdS/CFT correspondence in two space-time dimensions is quite
enigmatic. Some progress has been made in
\cite{{CM99},{strominger}}. The aim of this paper is to further
investigate the $AdS_2/CFT_1$ correspondence in terms of
Cardy-Verlinde entropy formula. Naively, one might expect that
holographic dualities in a two-dimensional bulk context would be
the simplest cases of all. This may certainly be true on a
calculational level; however, one finds such two-dimensional
dualities to be plagued by conceptually ambiguous features. One of
the remarkable outcomes of the AdS/CFT and dS/CFT correspondence
has been the generalization of Cardy's formula (Cardy-Verlinde
formula) for arbitrary dimensionality as well as for a variety of
AdS and dS backgrounds. In this paper, we have shown that the
entropy of the black hole horizon of Ach\'ucarro-Ortiz spacetime
can also be rewritten in the form of Cardy-Verlinde formula.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Acknowledgments}
The work of Elias C. Vagenas has been supported by the European
Research and Training Network ``EUROGRID-Discrete Random
Geometries: from Solid State Physics to Quantum Gravity"
(HPRN-CT-1999-00161).
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\begin{thebibliography}{99}
\bibitem {HOL}G. 't Hooft, {\it Dimensional Reduction in Quantum
Gravity}, gr-qc/9310026;  L. Susskind, J. Math. Phys. {\bf36}
(1995) 6377, hep-th/9409089.

\bibitem{RAP}R. Bousso, JHEP {\bf9907} (1999) 004, hep-th/9905177; ibid. {\bf9906} (1999)  028, hep-th/9906022; ibid.
  {\bf0104} (2001) 035, hep-th/0012052.

\bibitem{Strom}A. Strominger, JHEP {\bf0110} (2001) 034,
hep-th/0106113; M. Spradlin, A. Strominger and A. Volovich, {\it
Les Houches Lectures on De Sitter Space}, hep-th/0110007.

\bibitem{Hull}C.M. Hull, JHEP {\bf9807} (1998) 021,
hep-th/9806146; ibid. {\bf9811} (1998)  017, hep-th/9807127;
 C.M. Hull and R.R. Khuri, Nucl. Phys. B {\bf536} (1998)  219,
 hep-th/9808069; ibid. Nucl. Phys. B {\bf575} (2000) 231, hep-th/9911082.

\bibitem{Mazu}P.O. Mazur and E. Mottola,
Phys. Rev. D {\bf64} (2001) 104022, hep-th/0106151.

\bibitem{Bala}V. Balasubramanian, P. Horava and D. Minic,
 JHEP {\bf0105}  (2001)  043, hep-th/0103171.

\bibitem{mu1}M.-I. Park, Nucl. Phys. B {\bf544}  (1999) 377, hep-th/9811033.

\bibitem{Li}M. Li, JHEP {\bf0204} (2002)  005, hep-th/0106184.

\bibitem{Noji2}S. Nojiri and S.D. Odintsov,
Phys. Lett. B {\bf519} (2001) 145, hep-th/0106191; S. Nojiri and
S.D. Odintsov, JHEP {\bf0112} (2001)  033, hep-th/0107134; S.
Nojiri, S.D. Odintsov and S. Ogushi, Phys. Rev. D {\bf65} (2002)
023521, hep-th/0108172; S. Nojiri, S.D. Odintsov and S. Ogushi,
Int. J. Mod. Phys. A {\bf17} (2002) 4809, hep-th/0205187.

\bibitem{Klem3}D. Klemm, A.C. Petkou and G. Siopsis, Nucl. Phys. B {\bf601} (2001)  380, hep-th/0101076;
 D. Klemm, Nucl. Phys. B {\bf625} (2002)  295, hep-th/0106247; S.
 Cacciatori and D. Klemm, Class. Quant. Grav. 19 (2002)  579, hep-th/0110031.

\bibitem{Gao}Y.-h. Gao, {\it Symmetries, Matrices, and de Sitter
Gravity}, hep-th/0107067.

\bibitem{Bros}J. Bros, H. Epstein and U. Moschella, Phys. Rev. D
{\bf65} (2002) 84012, hep-th/0107091.

\bibitem{Haly}E. Halyo, {\it De Sitter Entropy and Strings}, hep-th/0107169.

\bibitem{Toll}A.J. Tolley and N. Turok,
{\it Quantization of the massless minimally coupled scalar field
and the dS/CFT correspondence}, hep-th/0108119.

\bibitem{Shir}T. Shiromizu, D. Ida and T. Torii, JHEP {\bf0111}  (2001)  010, hep-th/0109057.

\bibitem{McIn}B. McInnes, Nucl. Phys. B {\bf627} (2002)  311, hep-th/0110061.

\bibitem{Strom1}A. Strominger, JHEP {\bf0111} (2001)  049, hep-th/0110087.

\bibitem{BBM}V. Balasubramanian, J. de Boer and D. Minic, Phys. Rev. D
{\bf65}(2002)  123508, hep-th/0110108.

\bibitem{myung1}Y. S. Myung, Mod. Phys. Lett. A {\bf16} (2001)  2353, hep-th/0110123.

\bibitem{Carn}B. Carneiro da Cunha, Phys. Rev. D {\bf65}
(2002)  104025, hep-th/0110169.

\bibitem{CMZ}R.G. Cai, Y.S. Myung and Y.Z. Zhang, Phys. Rev. D {\bf65}
(2002)  084019, hep-th/0110234.

\bibitem{Dani}U.H. Danielsson, JHEP {\bf0203}  (2002) 020, hep-th/0110265.

\bibitem{gez1}A.M. Ghezelbash and R.B. Mann, JHEP {\bf0201}
(2002) 005, hep-th/0111217; A.M. Ghezelbash, D. Ida, R.B. Mann and
T. Shiromizu, Phys. Lett. B {\bf535} (2002)  315, hep-th/0201004.

\bibitem{deh} M.H. Dehghani, Phys. Rev. D {\bf66}  (2002) 044006, hep-th/0205129;
ibid. D {\bf65} (2002)  104030, hep-th/0201128; ibid.  D {\bf65}
 (2002) 104003, hep-th/0112002.

\bibitem{Ogus}S. Ogushi, Mod. Phys. Lett. A {\bf17}(2002)  51, hep-th/0111008.

\bibitem{set}M.R. Setare, Mod. Phys. Lett. A {\bf17} (2002)  2089, hep-th/0210187.

\bibitem{set1} M. R. Setare and R. Mansouri, {\it Holographic Thermodynamic
on the Brane in Topological Reissner-Nordstr\"om  de Sitter
Space},  to appear in Int. J. Mod. Phys. A, hep-th/0210252.

\bibitem{set3}M.R. Setare and M.B. Altaie, {\it The Cardy-Verlinde formula and
 entropy of Topological Kerr-Newman black holes in de Sitter
 spaces}, submitted to Eur. Phys. J. C .

\bibitem{Verl}E. Verlinde, {\it On the Holographic Principle in a Radiation Dominated Universe}, hep-th/0008140.

\bibitem{banados1} M. Ba\~nados, C. Teitelboim and J.
Zanelli, Phys. Rev. Lett. {\bf69} (1992) 1849, hep-th/9204099.

\bibitem{banados2} M. Ba\~nados, M. Henneaux, C. Teitelboim and J.
Zanelli, Phys. Rev. D {\bf48} (1993) 1506, gr-qc/9302012.

\bibitem{strominger1} J. Maldacena, J. Michelson and A.
Strominger, JHEP {\bf9902} (1999) 011, hep-th/9812073.

\bibitem{strominger2} M. Spradlin and A. Strominger,
JHEP {\bf9911} (1999) 021, hep-th/9904143.

\bibitem{cardy} J.A. Cardy, Nucl. Phys. B {\bf270} (1986) 186.

\bibitem{CM99} M. Cadoni and S. Mignemi,
 Phys. Rev. D {\bf59} (1999) 081501, hep-th/9810251;
 M. Cadoni and S. Mignemi, Nucl. Phys. B {\bf557} (1999) 165, hep-th/9902040.

\bibitem{andy}
A. Strominger, JHEP {\bf9802} (1998) 009, hep-th/9712251.

\bibitem{CCCM}
M. Cadoni, P. Carta, M. Cavagli\`a  and  S. Mignemi, Phys. Rev. D
{\bf65} (2002) 024002, hep-th/0105113.

\bibitem{cadcav}
M. Cadoni and M. Cavagli\`a, Phys. Rev. D {\bf63} (2001) 084024,
hep-th/0008084;  M. Cadoni and M. Cavagli\`a, Phys. Lett. B
{\bf499} (2001) 315, hep-th/0005179.

\bibitem{strominger} A. Strominger, JHEP {\bf9901} (1999) 007, hep-th/9809027;
 S. Cacciatori, D. Klemm and D. Zanon, Class. Quant. Grav. {\bf17} (2000) 1731, hep-th/9910065.

\bibitem{ortiz} A. Ach\'ucarro and M.E. Ortiz, Phys. Rev.
D {\bf48} (1993) 3600, hep-th/9304068.

\bibitem{lowe} D.A. Lowe and A. Strominger, Phys. Rev. Lett.
{\bf73} (1994) 1468, hep-th/9403186.

\bibitem{kumar1} A. Kumar and  K. Ray, Phys. Lett. B {\bf351} (1995) 431, hep-th/9410068.

\bibitem{bekenstein1} J.D. Bekenstein, Phys. Rev. D {\bf7} (1973)
2333.

\bibitem{bekenstein2} J.D. Bekenstein, Phys. Rev. D{\bf9} (1974)
3292.

\bibitem{hawking3} S.W. Hawking, Phys. Rev. D {\bf13} (1976) 191.



\end{thebibliography}
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