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\begin{document}
\draft
\title{Anti-de Sitter black holes, perfect fluids, and holography}
\author{Saurya Das and Viqar Husain}
\address{ Dept. of Mathematics and Statistics, } 
%\address{}
\address{University of New Brunswick,}  
\address{Fredericton, New Brunswick - E3B 5A3, CANADA }
\address{ EMail: saurya,~husain@math.unb.ca   }

%\address{  }
\maketitle
\thispagestyle{empty}

\begin{abstract}

We consider asymptotically anti-de Sitter black holes in 
$d$-spacetime dimensions in the thermodynamically stable regime.  
We show that the Bekenstein-Hawking entropy and its leading
order corrections due to thermal fluctuations can be reproduced by a 
weakly interacting fluid of bosons and fermions (`holographic gas') 
in $\Delta=\alpha(d-2)+1$ spacetime dimensions, where 
the energy-momentum dispersion relation for the constituents of the 
fluid is assumed to be $\epsilon = \kappa p^\alpha$. We examine  
implications of this result for entropy bounds and the holographic 
hypothesis. 

\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\vs{1cm}

\begin{multicols}{2}

\section{Introduction} 

It is believed that black holes are thermodynamical objects 
with entropy given by one-quarter their horizon areas $A_H$ (in Planck
units) \cite{bh}
%
\be
S_{BH} = \f{A_H}{4 \ell_{Pl}^{d-2}}~,
\la{bekhaw1}
\ee
%
where $\ell_{Pl} = (G\hbar/c^3)^{1/(d-2)}$ is the 
Planck length and $G_d$ the Newton's constant in 
$d$-spacetime dimensions. They satisfy the two laws of 
`black hole thermodynamics'  
%
\ba
d(Mc^2) &=& T_H~dS + \mbox{work terms} \\
\D S_{BH} &\geq& 0~ 
\ea
where $T_H$ is the Hawking temperature. These laws are semi-classical, 
and apply to large black holes with  $A_H \gg \ell_{Pl}^{d-2}$.  

The idea that black holes are thermodynamic systems in equilibrium 
has led to two related areas of study. 
%
The first constitutes suggestions for the underlying microscopic degrees 
of freedom. These are postulated based on one's views of what are the 
fundamental degrees of freedom arising from a quantum theory of gravity. 
The goal is to reproduce the entropy formula (\ref{bekhaw1}) by tracing 
a density matrix describing a black hole over unobserved quantum degrees 
of freedom. A related approach involves calculating the number of microstates
associated with macroscopic parameters of a black hole, such as its mass
and charge \cite{vs,ash,anydim}. 

The second area of study arises from the observation that the entropy of 
a confined non-black hole system is proportional to spatial volume, rather 
than  bounding area. Reconciling this with the fact that black holes are 
believed to be the most entropic systems leads to a viewpoint known as the 
holographic hypothesis. This hypothesis states that a $d-$dimensional theory 
may be encoded exactly in a $(d-1)-$dimensional theory. What remains is the 
significant task of providing the details of the encoding, the ``holographic 
map.'' A part of this map consists in matching the thermodynamics of 
black holes with a suitable system in one lower dimension.  

In this paper we consider the second question regarding holography in detail. 
Specifically we focus on the AdS-Schwarzschild black hole because 
it is thermodynamically stable, and because of its role in the AdS/CFT 
correspondence. We show that thermodynamic properties of this black hole can be
encoded in a holographic gas of free bosons and fermions.    
The dimension of the gas depends on the dimension of the black hole, as well
as the dispersion relation of its constituents. 

In the next section, we review recent work on thermal fluctuation corrections 
to entropy and its application to black holes. In section (\ref{aads}) 
we discuss the thermodynamics of asymptotically anti-de Sitter 
black holes and thermal fluctuation corrections to entropy. In section 
(\ref{holodual}), we show that a perfect fluid of bosons and fermions captures 
the entire thermodynamics of these black holes, including thermal fluctuation 
corrections. We examine the implications of our results for entropy 
bounds \cite{thooft,susskind,bekholo,bousso} 
in section (\ref{holography}), and finally conclude in 
section (\ref{discussions}) with a summary and list of open questions.  

\section{Thermal fluctuations and entropy}

Before we give a systematic derivation of leading order corrections to
entropy of a system due to small thermal fluctuations, we present a
heuristic argument showing how logarithmic corrections to entropy 
arise for a black hole. 

Consider Wheeler's ``it from bit'' idea, which ascribes one bit 
of information (equivalently a spin 1/2) to each Planck area on the 
black hole's horizon \cite{thooft,wheeler}. This idea also incorporates 
``holography'' in that degrees of freedom are associated with area rather 
than volume. If the area of the black hole
horizon is $A$, the number of microstates is $\Omega = 2^n$, with $n=A/l_p^2$,
so that entropy $S_{BH} =\ln \Omega$ is proportional to horizon area. 
This model may be obviously generalised to associate a spin $s$ with each 
Planck area without changing the basic result. The number of microstates is 
now $\Omega = (2s+1)^n$, which gives entropy  proportional to area with a 
different coefficient.

Corrections to entropy arise in this picture if the component
of total spin along a given axis is fixed. Let $x^i$ of the $n$ spins have
a state $s^i$ chosen from the $(2s+1)$ projections along a fixed axis.
(The total spin component along that axis is then 
$ \sum_i [x_i (2s/(2s+1))s_i-s ]$.)
The number of possible states ${\cal N}$ subject to
$\sum_i x_i = n$ is given by the multinomial distribution function
(assuming that the probability for finding
any spin in state $i$ is independent of $i$, and equal to $1/(2s+1)$ ):
%
\be
{\cal N} = \f{n!}{x_1! x_2! \cdots x_{2s+1}!}~~.
\la{deg1}
\ee
%
The entropy $S=\ln {\cal N}$ may be  evaluated using Stirling's formula,
assuming $x^i >> 1$ to give
%
\be
{\cal S}   = a n - s \ln n
= b S_{BH} - s \ln \le( S_{BH} \ri) 
\la{it1}
\ee
%
where $S_{BH}= A\ {\rm ln}(2s+1)/l_p^2$ is the entropy with no restriction on 
total spin component, $a \equiv \sum_i k_i \ln k_i^{-1}$ and $b=a/\ln(2s+1)$ 
are of order unity, and $k_i \equiv x_i/n$. This is a generalisation of the 
$s=1/2$ case considered in \cite{dmk}. Thus, the leading corrections 
are  logarithmic, with the exact coefficient depending on the 
spin of the building blocks of the black hole. 

We now discuss some generalities of thermodynamic systems in equilibrium and 
their thermal fluctuations, before proceeding to AdS black holes and 
the free boson/fermion gas. This is a review of work \cite{bm,rkb,dmb,cap} 
which shows that logarithmic corrections to entropy arise from quite general 
considerations. 
 
 
The canonical partition function for any thermodynamic system in equilibrium
is 
%
\be
Z(\b)
= \int_0^\infty \r(E)~e^{-\b E} dE~~.
\label{part1}
\ee
%
The density of states $\r(E)$ can be written as an inverse 
Laplace transform of the partition function,  
%
\be
\r(E) = \f{1}{2\p i} \int_{-i\infty}^{i\infty} 
{Z(\b) e^{\b E}} d\b
= \f{1}{2\p i} \int_{c-i\infty}^{c+i\infty} e^{S(\b)}d\b~~,
\la{density1}
\ee
%
where 
%
\be
S (\b) =\ln Z + \b E 
\la{efn1}
\ee
is the `entropy function'.  A saddle-point approximation of $S(\beta)$ 
around the equilibrium temperature $\b_0^{-1}$ gives 
%
\be
\r(E) = \f{e^{S_0}}{2\p i} \int_{c-i\infty}^{c+i\infty}
e^{1/2 (\b - \b_0)^2 S_0''} d\b ~,
\la{saddle}
\ee
%
where $S_0 \equiv S(\b_0)$ and $S_0'' = S_{,\b\b|\b_0}$. 
By substituting $\b-\b_0=ix$, and choosing an appropriate contour, the 
integral can be evaluated exactly to give 
%
%
\be
\r(E) = \f{e^{S_0}}{\sqrt{2\p S_0''}} ~~.
\la{corr0}
\ee
%
The corrected entropy is  
%
\be
{\cal S} := \ln \r(E) = S_0 - \f{1}{2} \ln S_0'' + ~\mbox{(smaller terms)}.
\la{entcorr2}
\ee
%
That the quantity $S_0''$ is indeed a measure of fluctuations of the system 
can be seen from the  relation  
%
\ba
S''(\b) &=& \f{1}{Z} \le( \f{\pa^2 Z(\b) }{\pa\b^2} \ri) 
- \f{1}{Z^2} \le( \f{\pa Z}{\pa\b}  \ri)^2 \\
&=& <E^2> - <E>^2~,
\ea
%
where we have used the definitions  
%
\be
<E> = - \le( \f{\pa \ln Z}{\pa \b}\ri)_{\b=\b_0} 
% = - \f{1}{Z} \le(  \f{\pa Z}{\pa \b} \ri)_{\b=\b_0},  
<E^2> = \f{1}{Z} \le( \f{\pa^2 Z}{\pa \b^2} \ri)_{\b=\b_0}~.
\ee
%
Using the fact that the specific heat of a thermodynamic system in 
equilibrium can be written as 
%
\ba
C \equiv \le( \f{\pa E}{\pa T}\ri)_{T_0}~ 
&=& \f{1}{T^2}  \le[ \f{1}{Z} \le( \f{\pa^2 Z}{\pa\b^2} \ri)_{\b=\b_0}  - 
\f{1}{Z^2} \le( \f{\pa Z}{\pa \b} \ri)^2_{\b=\b_0} \ri] \nn \\ 
&=& \f{S_0''}{T^2}
\ea
%
gives 
%
\be
S_0''= C T^2~, 
\ee
%
and hence from (\ref{entcorr2}) that 
%
\be
{\cal S } = S_0 - \f{1}{2} \ln \le( CT^2  \ri) + \cdots~~. 
\la{master1}
\ee
%
This formula is applicable to all thermodynamic systems. In particular 
it may be applied to black holes by setting $S_0 = S_{BH}$, $T=T_{BH}$ and  
$C=C_{BH}$ for the specific black hole under consideration. 
It is understood that the quantity within the logarithm is divided
by $k_B^2$, the square of the Boltzmann constant. 

The application of (\ref{master1}) to black holes was considered in detail 
in  \cite{dmb,cap}, where it was shown that it provides a general approach
to understanding corrections to black hole entropy computed in various 
models in the literature. For related and other approaches, see  
\cite{fursaev,solo1,other,qg,obregon,carlip,jy,qg1,bss,gks,k,ms,medved,k2,medved2,gg2}.  
For other applications of (\ref{master1}), see \cite{appln}. 

 
\section{Asymptotically Anti-de Sitter black holes} 
\la{aads}

We review thermodynamic properties of BTZ and 
AdS-Schwarzschild black holes, 
and compute entropy corrections using the method discussed above. Note that this
method does not apply to Schwarzschild black holes which have negative specific 
heat. 

For the BTZ black holes 
%
\ba
ds^2 &=& - \le( \f{r^2}{\ell^2} - \f{8 G_3 M}{c^2} \ri)~c^2 dt^2  \nn \\
&+& \le( \f{r^2}{\ell^2} - \f{8 G_3 M}{c^2} \ri)^{-1}~dr^2   + r^2 d\theta^2 ~,
\ea
%
the entropy, temperature and specific heat are 
%
\ba
S_{BH} &=& \f{2\p r_+ }{4\ell_{Pl}}  
= \le( \f{ \ell \p c^2 \sqrt{2}}{\hbar\sqrt{G_3}} \ri) 
{\sq{M}}  \\
T_H &=& \f{\hbar c r_+}{2\p \ell^2} = 
\f{ \hbar \sqrt{2G_3 M}}{\p \ell}  
= {\le( \f{\hbar \sqrt{G_3}}{\p \ell c }  \ri)^2}~{S_{BH}}
\la{btztemp}  \\
%\f{ \hbar \sqrt{2G_3 M}}{\p \ell}  
%
C_{BH} &=& \f{dMc^2}{dT_H} = {S_{BH}}~~. 
\ea
%
where $r_+ = \ell\sqrt{8 G_3 M}/c$ is the horizon radius. 
Substituting in (\ref{master1}) gives 
%
\ba
{\cal S} &=& S_{BH} - \f12 \ln \le(S_{BH}  S_{BH}^2\ri) + \cdots \nn \\
&=& S_{BH} - \f{3}{2}\ln \le( S_{BH} \ri) + \cdots ~~.
\la{btzcorr}
\ea
%
This agrees with correction computed using conformal field theory by Carlip 
\cite{carlip},  including the coefficient $-3/2$ in front of the logarithm. 

Similarly, for AdS-Schwarzschild black holes in $d$-dimensions, 
with a  cosmological constant  
$$
\Lambda = - (d-1)(d-2)/2 \ell^2, 
$$ 
the metric is  
%
\ba 
&ds^2& = -
 \le(1 - \f{16\p  G_d M}{(d-2)\O_{d-2}c^2 r^{d-3}} 
+ \f{r^2}{\ell^2} \ri) c^2 dt^2  \nn \\  
&+& \le(1 - \f{16\p  G_d M}{(d-2)\O_{d-2}c^2 r^{d-3}} 
+ \f{r^2}{\ell^2} \ri)^{-1} dr^2 \nn \\
&+& r^2 d\O_{d-2}^2 ~.
\la{adsscmetric}
\ea
%
The entropy, temperature and specific heat are given by 
%
\ba
S_{BH} &=& \f{ A_{d-2}}{4\ell_{Pl}^{d-2}} 
= \f{\O_{d-2} r_+^{d-2}}{4\ell_{Pl}^{d-2}} \la{adsscent}\\
T_H &=& {\hbar c}{}~\f{(d-1) r_+^2 + (d-3)\ell^2}{4\p \ell^2 r_+} 
\la{adssctemp} \\
C &=& (d-2)\le[ \f{(d-1)r_+^2/\ell^2 
+ (d-3)}{(d-1)r_+^2/\ell^2-(d-3)} \ri] S_{BH} ~, \la{adsscspht}
\ea
%
where $\O_{d-2}$ is the area of an unit $S^{d-2}$. 
In the so-called `high-temperature limit' $r_+ >> \ell$, a regime in 
which the specific heat is positive, the corrected entropy using 
({\ref{master1}}) is  
%
\ba
{\cal S} &=& 
S_{BH} - \f{1}{2} \ln \le( {S_{BH} S_{BH}^{2/(d-2)}} \ri) + \cdots \nn \\
&=&  S_{BH} - \f{d}{2(d-2)} \ln \le( {S_{BH}}{} \ri) + \cdots ~~~.
\la{adssccorr}
\ea
%
It is curious to note that on substituting $d=3$ in the above, the 
BTZ result (\ref{btzcorr}) is recovered, although the metric of the latter 
is not a special case of (\ref{adsscmetric}). 

The free energy of the black hole in the limit $r_+>>\ell$ may be computed 
from Eq.(\ref{adsscmetric}-\ref{adssctemp}):
%
\be
F_{BH} = Mc^2  - T_{H}S_0 = -c_1 T_H^{d-1}~,
\la{fbh}
\ee
%
where
%
\be
c_1 =  \f{\O_{d-2}}{4(d-1) \ell_{Pl}^{d-2}} 
\le(\f{4\p \ell^2}{\hbar c (d-1)} \ri)^{d-2}~~.
\ee
%
The entropy may be expressed as 
\be
S_{BH} = -\f{\pa F_{BH}} {\pa T_{BH}} = c_1(d-1)T_H^{d-2}.
\la{sbh}
\ee
\section{Holographic gas}
\la{holodual}

We now attempt to model the thermodynamic properties of asymptotically AdS black holes 
using a perfect fluid of free bosons and fermions at temperature $T$ in spacetime 
dimension  $\D$, and with dispersion relation 
%
\be
\e = \k p^\a~.
\la{dispersion}
\ee
%
We compute the equation of state, free energy and entropy using 
standard methods. 

The thermodynamic potential $\O$ is \cite{landau}
%
\be
\O = \pm  T \sum_i \ln \le[ 1  \mp e^{(\m-\e_i)/T} \ri] 
\ee
%
where $+$ and $-$ signatures refer to bosons and fermions respectively,
and $\mu$ is the chemical potential.
In the continuum limit, partial integration gives  
%
\ba
\O &=&  \pm \f{ V_{\D-1} \O_{\D-2} T}{(2\p \hbar)^{\D-1}} \int_0^\infty
\ln \le[1 \mp e^{(\m-\e)/T }  \ri] p^{\D-2} dp \nn \\ 
%
&=&  - \f{ V_{\D-1} \O_{\D-2}}{(\D-1) (2\p \hbar)^{\D-1} k^{(\D-1)/\a}} 
\int_0^\infty
\f{\e^{({\D-1})/{\a}}}{e^{({\m-\e})/{T}} \pm 1} d\e~.
\la{omega}
\ea
%
The energy of the gas is 
%
\ba
E &=& \int_0^\infty \e~d\Gamma (\e)  \nn \\
%
&=& \f{V_{\D-1} \O_{d-2} }{\a (2\p\hbar)^{\D-1} k^{(\D-1)/\a}}
\int_0^\infty \f{\e^{(\D-1)/\D} d\e}{e^{(\m-\e)/T} \pm 1}~. 
\ea
%
Comparing with (\ref{omega}) and using the relation 
$\O = -PV_{\D-1}$ gives  
\be
E = \le(\f{\D-1}{\a}\ri)~P~V_{\D-1}~.  
\la{state1}
\ee
%
This can be written in terms of the energy density 
$\r = E/V_{\D-1}$ as
%
\be
P = k \r~,
\ee
%
where 
\be
k = \f{\a}{\D-1}~. 
\label{k}
\ee
% 
This relates spacetime dimension $\D$ and dispersion relation power $\a$ 
to the coefficient $k$ in the equation of state.  
 
The free energy of the gas is 
%
\be
F_{gas} = -  T \ln z = \pm T \sum_{i} \ln (1 \mp e^{-\b \e_i} ) ~.
\ee
%
The temperature dependence is obtained by partial integration 
after taking the continuum limit: 
%
\ba
F_{gas}  &=& \pm \f{ T V_{\D-1} \O_{\D-2}}{(2\p \hbar)^{\D-1}} \int_0^\infty 
\ln ( 1 \mp e^{-\b \e}) dp~p^{\D-2} ~ \nn \\
&=& -~c_2 V_{\D-1}~T^{\f{\D-1}{\a}+1} ~,
\la{freegas}
\ea
%
where 
$V_{\D-1}$ is the volume of the gas, and  
%
\ba
c_2 &=& \f{\O_{\D-2}}{(\D-1)\k^{(\D-1)/\a} (2\p \hbar)^{(\D-1)}} 
\times \nn \\
&~&\zeta \le(\f{\D-1}{\a} + 1 \ri) 
\C \le( \f{\D-1}{\a} + 1 \ri) \times \nn \\
&~& (n_B + n_F  - \f{n_F}{2^{(\D-1)/\a}} )~.
\ea
% 
Here $n_B (n_F)$ is the total number of bosonic (fermionic) degrees of 
freedom for the fluid. The entropy is 
%
\be
S_{gas} = - \f{\pa F_{gas}}{\pa T} = c_2 
\le(\f{\D-1}{\a}+1 \ri) V_{\D-1} T^{\f{\D-1}{\a} }~~.
\la{sgas}
\ee
% 

To compare thermodynamic quantities of the gas with those of the AdS 
black hole, we derive the consequences of imposing 
%
\be
S_{BH} = S_{gas}~.
\la{sbhgas}
\ee
%
Matching powers of temperature in the entropy formulas 
(\ref{sbh}) and (\ref{sgas}) gives our first result 
relating spacetime dimensions and $\a$:
% 
\be
\D =\a (d-2) + 1.
\la{match1} 
\ee
%
Thus, given an anti-de Sitter Schwarzschild black hole in $d$-spacetime
dimensions, there will be a holographic gas in $\D$-spacetime dimensions
capturing all thermodynamic information of the black hole.

Using (\ref{k}), this may be written as 
\be
k = \f{1}{d-2}~.
\ee
The last equation shows that the fluid is physical, in the 
sense of causality, for all cases: the dominant energy condition 
(positive energy, and null or lightlike fluxes)
gives the condition $0\leq k \leq 1$, which is satisfied for  
$d \geq 3$ (the cases of interest here). 

Next we compare the coefficients of the power of temperature
in (\ref{sbhgas}). There are two ways to do this depending on the point of 
view taken on holography: (i) the holographic degrees of freedom associated 
with a black hole reside on a surface $r=r_0$ in the black hole spacetime, or 
(ii)  the holographic degrees of freedom do not reside on any bounding 
surface in the black hole background, but rather are defined on 
their own flat background spacetime. 

According to (i), the temperature $T$ of the gas must be 
taken as the red-shifted black hole temperature
\be 
 T = \f{T_H}{\sqrt{-g_{00}}} = \f{\ell~T_H}{r_0}
\ee
The black hole entropy in terms of $T$ is 
\be 
S_{BH} = c_1(d-1)\left( {r_0\over l}\right)^{d-2} T^{d-2}.
\ee
Matching coefficients of powers of $T$ in (\ref{sbhgas}) now gives
%
\be 
c_1 = c_2~\Omega_{\D - 1}~\ell^{d-2}~r_0^{(\a-1)(d-2)}~,
\la{match2}
\ee
%
where we have used the fact that the gas lives in a subspace of the 
surface $r=r_0$ so that its volume is $V_{\D-1} = \Omega_{\D-1}r_0^{\D-1}$. 
%

We point out that a special case of the relation (\ref{match2}) arises in 
the context of the AdS/CFT conjecture. For $d=5$, $\a=1$, $\D=4$ and 
$n_B=n_F=8(N^2-1)$, it was shown \cite{gubser,berman}, with 
the additional AdS/CFT relation 
$$
\p \ell^3/G_5 = 2N^2,
$$ 
that Eq.(\ref{match2}) is satisfied up to a factor of $4/3$. 

According to the point of view (ii), the temperature of the gas is equated to the 
black hole temperature without any red-shift factors. This leads to the following 
relation between coefficients:
%
\be
c_1 = c_2 V_{\D - 1}\ell^{d-2}  
\ee
%
Note that in either case (i) or (ii), the holographic
dimension $\D$ is given by Eq.(\ref{match1}).   

From (\ref{match1}), it follows that for the special case $\a=1$, 
%
\be
\D = d -1 ~,
\la{minusone}
\ee
%
which is normally assumed in the context of holography 
\cite{thooft,susskind}. Another indication that $\a=1$ may be 
`preferred' lies in the fact that the $r_0$-dependence drops out of 
the relation (\ref{match2}) only for this value of $\a$.
 

It is interesting to note that for near-extremal stringy black holes,
the near horizon geometry is BTZ \cite{dadg}. Thus, to describe the  
thermodynamics of these black holes, one substitutes $d=3$ in (\ref{minusone}). 
This gives a $(1+1)-$dimensional holographic gas. Such a gas has been 
shown to reproduce the entropy and Hawking radiation rates of near extremal 
black holes in string theory\cite{dma}. 
%

We now compare the leading order entropy corrections of the 
black hole and gas. Using $C_{gas} = d(F_{gas} + T S_{gas})/dT$, 
the dimension matching equation (\ref{match1}) and
Eq.(\ref{master1}), the corrected entropy of the gas is  
%
\be
\cals^{corr}_{gas} 
= S_{gas} - \f{d}{2(d-2)} \ln S_{gas} 
- \f{1}{d-2} \ln \le( c_2 V_{\D-1} \ri).  
\la{gascorr} 
\ee

Matching the uncorrected entropies $S_{BH} = S_{gas}$ 
in this equation,  we see that the leading logarithmic term agrees precisely 
with that on the black hole side, Eq.(\ref{adssccorr}),  for all spacetime 
dimensions, and for any value of $\a$ in the dispersion relation (\ref{dispersion}).  
The last term in (\ref{gascorr}) depends on the volume of the gas, as well the 
number of species, and can be interpreted as a finite size effect. These 
are considered sub-dominant so long as the entropy $S_{BH}$ remains large. Thus, 
the exact details of the boundary theory are seen to be irrelevant for all 
for the leading order corrections to match. (Finite size effects were also considered 
in \cite{finite1}.)

The issue of log corrections for $d=5$ and 
$\a=1$ were first analysed in \cite{mukherji}. 
However, there the authors had ignored the factor of
$T_H^2$ in (\ref{master1}), by setting the `scale' of the logarithm to 
$T_H$ itself. This scale is actually the Boltzmann constant $k_B$,
which has been set to unity here. Consequently, the coefficient in 
front of the log term was incorrect there, both for the black hole and
the gas. In addition, the finite-size effect terms were missed.

For $AdS_5/CFT_4$ correspondence, when $d=5,~\a=1,~\D=4,~n_b=n_F=8(N^2-1)$, the 
coefficient of the log corrections is $5/6$, agreeing perfectly with its black 
hole counterpart, Eq.(\ref{adssccorr}). 


\section{Entropy bounds with AdS black holes}
\la{holography} 

In this section we consider the Bekenstein and spherical bounds 
for asymptotically anti-de Sitter black holes, and examine the 
effects of entropy corrections on these bounds. 
 
The Bekenstein bound states that the entropy of matter in a closed
region of linear dimension $R$ and energy $E$ is bounded
above by the inequality \cite{bekholo,bousso}
%
\be
S_{matter} \leq \f{2\p ER}{\hbar c}~.
\ee
%

One way to arrive at this result is to consider a `Geroch process,' in 
which a matter system in a box is lowered slowly from an asymptotically flat 
region, and then dropped into the black hole where the box just touches 
the horizon. The argument assumes that the energy of a floating box 
near the horizon is added to the black hole, which increases its entropy. 
The generalised second law $\Delta S_{Tot} = S^{final}_{BH} - (S^{initial}_{BH} 
+ S_{matter}) \ge 0$ then leads to the desired inequality. The drop off point 
occurs where the center of mass of the matter system is a distance $R$ from the 
horizon, where the energy of the matter acquires a factor $\sqrt{-g_{00}}$ 
relative to its energy at infinity. 

This argument may be applied to an AdS-Schwarzschild black hole
background, where the 
energy of the system 
is assumed to be $E$ at a large radius $r>>r_+$.  The energy 
of the matter a proper distance $R$ from the horizon is 
\be 
{\cal E} = \sqrt{-{g_{00}}|_{R}}~E = \le[ \f{(d-1) r_+}{2\ell^2} \ri] ER~.  
\ee
The energy gain of the black hole is 
%
\be
\d E \leq 
\le[ \f{(d-1) r_+}{2\ell^2} \ri] ER~,  
\ee
%
and the corresponding entropy gain is 
%
\be
\d S_{BH} = \f{\pa  S_{BH}}{\pa E}~\d E~.
\la{geroch1}
\ee
%
Now, in order to find the corrected Bekenstein bound, we use 
(\ref{adssccorr}) to compute the entropy derivative. This gives
a `corrected Hawking temperature' $T_H' = \pa \cals/\pa M_{BH}$ 
given by 
%
\be
T_H' = T_H + \f{d(d-1)\hbar c \ell_{Pl}^{d-2}}{\p (d-2) \O_{d-2} \ell^2
r_+^{d-3}}~.
\ee
%
Substituting this in (\ref{geroch1}) gives 
%
\be
\d S_{BH} = \le[ 1 - \f{2\ell_{Pl}^{d-2}}{\O_{d-2}(d-2)r_+^{d-2}} \ri]~
\f{2\p ER}{\hbar c}~.
\ee
%
Imposing the generalised 2nd. law gives    
%
\be
S_{matter} \leq  \le[ 1 - \f{2\ell_{Pl}^{d-2}}{\O_{d-2}(d-2)r_+^{d-2}} \ri]~
\f{2\p ER}{\hbar c}~.
\ee
%
Notice that the leading term is identical to the Schwarzschild case
\cite{bousso}. The correction,  which depends on
the horizon radius, reduces the bound. The correction
term  attains its maximum when the radius is of the order of Planck
length, which gives
%
\be
S_{matter} \leq  \le[ 1 - \f{2}{\O_{d-2}(d-2)} \ri]~
\f{2\p ER}{\hbar c}~.
\ee
%
This equation may be regarded as the modified Bekenstein bound. 
(Corrections to the Bekenstein bound from 
finite volume corrections were analysed in \cite{kutlar}.)

It is interesting to see what happens to this bound at the 
expected saturation point where the system $(E,R)$ is 
an AdS-Schwarzschild black hole. The entropy bound becomes
%
\be
S_{matter}  \leq \le( \f{d-2}{2} \ri) 
\le[ 1 - \f{2}{(d-2)\O_{d-2}} \ri] 
\le( \f{r_+^2}{\ell^2}\ri)~ S_{BH}~.
\la{bound2}
\ee
%
Since $r_+/\ell \gg 1$,  the right side is much larger than the black hole
entropy. That is, the black hole entropy is far from saturating the Bekenstein
bound, implying that the latter is 
rather weak in this case. This is a significant departure 
from the asymptotically flat case where the upper limit is provided by 
the black hole entropy. 

The spherical entropy bound similarly undergoes the modification
%
\be
S_{matter} \leq \f{A_H}{4\ell_{Pl}^{d-2}} 
- \f{d}{2(d-2)} \ln \f{A_H}{4\ell_{Pl}^{d-2}} ~.
\la{bound3}
\ee
%
Of course in this case, the black hole saturates this bound, by 
definition. From (\ref{bound2}) and (\ref{bound3}) it follows that 
the two bounds are {\it not} of the same order for asymptotically 
anti-de Sitter black holes, and that the Bekenstein bound is much 
weaker than the spherical bound. (After this work was done, we became aware
of a recent paper where corrections to Bekenstein and spherical 
entropy bounds have been proposed \cite{ggour}. The 
correction to the spherical bound proposed in that paper agrees with our's, 
while that to the Bekenstein bound does not).   

Finally we point out the AdS black hole analogues of two related   
entropy bounds previously discussed for Schwarzschild 
black holes \cite{thooft}. These are bounds on the entropy 
of a collection of black holes, and on the entropy of matter 
contained in a closed spatial region. 

We ask if the spherical entropy bound is satisfied 
for a collection of AdS-Schwarzschild black holes with 
masses $M_i$ in a bounded region of area $A$. To do so
let us first compare the sum of the entropies 
$S_{total}=\Sigma S_i$ of a collection of black holes with 
the entropy $S_M$ of a black hole of mass $M=\Sigma M_i$:  
%
\ba
S_{total} &=& \f{\O_{d-2}} {4\ell_{Pl}^{d-2}}
\le[\f{16\p G_d\ell^2}{(d-2) \O_{d-2} c^2} \ri]^{(d-2)/(d-1)}~
\nn \\
&& \times  \sum_i M_i^{(d-2)/(d-1)}~,
\ea
%
and  
%
\ba
S_M &=& \f{\O_{d-2}}{4\ell_{Pl}^{d-2}}
~\le[ \f{16\p G_d\ell^2}{(d-2) \O_{d-2} c^2} \ri]^{(d-2)/(d-1)}~
\nn \\
&& \times  \left( \sum_i M_i \right)^{(d-2)/(d-1)}~,
\ea
%
The relation   
%
\be
\sum_i M_i^{(d-2)/(d-1)} > 
\le(  \sum_i M_i\ri)^{(d-2)/(d-1)}~
\ee
%
for powers $(d-2)/(d-1)<1$ gives 
%
\be
S_{total} >  S_M 
\ee
%
This is the opposite of the Schwarzschild black hole result,
and indicates that if $A$ is taken to be the horizon area of 
the composite black hole, then $S_{total} > A/4\ell_{Pl}^{d-2}$.  
Thus an upper bound on the entropy of a collection of AdS black 
holes cannot be derived form this argument. 

To obtain a specific example of an entropy bound on a matter 
system derived using AdS black holes, we use the gas 
discussed in the last section. The energy and entropy of the gas, consisting 
of $Z$ species, and with the dispersion relation (\ref{dispersion}), 
follows from Eq.(\ref{freegas}), with the replacement 
$\D \rightarrow d$  (also see \cite{thooft,bousso,wald}):
%
\ba
E &=& Z c_2 \le( \f{d-1}{\a} \ri) V_{d-1} T^{(d-1)/\a+1} ~, \\
S &=& Z c_2 \le( \f{d-1}{\a} +1 \ri) V_{d-1} T^{(d-1)/\a}  ~.
\ea
%
With $V_{d-1}= \O_{d-1} R^{d-1}$ we can write the entropy 
$S(Z,E,R)$ as 
%
\be
S = c_3 Z^{\a/(d-1+\a)} R^{(\a(d-1))/(d-1+\a)}
~E^{(d-1)/(d-1+\a)}~, 
\la{gasentropy}
\ee
%
where 
$$
c_3 = c_2^{\a/(d-1+\a)} \O_{d-1}^{\a/(d-1+\a)} 
\le(  \f{\a}{d-1}  \ri)^{(d-1)/(d-1+\a)}~.
$$
An entropy bound arises by requiring that the gas 
is outside its AdS-Schwarzschild radius. This is obtained 
from  the expression for the mass of the Schwarzschild-AdS black hole
in the $r\gg \ell$ limit: 
%
\be
M = \le[ \f{(d-2) \O_{d-2} c^2}{16\p G_d \ell^2} \ri]
r_+^{d-1}~.
\ee
As a result the entropy (\ref{gasentropy}) satisfies the inequality:
%
\be
S < c_4 Z^{\a/(d-1+\a)} \f{A^{(d-1)/(d-2)}}{\ell^{2(d-1)/(d-1+\a)}}
\la{gasbound1}
\ee
%
where 
$$
c_4 = c_3 \le[ \f{(d-2)\O_{d-2} c^4}{16\p G_d} \ri]^{(d-1)/(d-1+\a)}
~~. 
$$
Note that the exponent of $A$ in (\ref{gasbound1}) is greater than unity, 
unlike the asymptotically flat case, again resulting in a weaker bound. 
Eq.(\ref{gasbound1}) can be written as: 
%
\be 
\cals < c_5 \le( \f{\ell_{Pl}}{\l} \ri)^{\f{(\a-1)(d-1)}{(d-1+\a)}} 
\le( \f{\ell_{Pl}}{\ell}  \ri)^{ \f{(d-1-\a)}{(d-1+\a)}}~
\f{r_+}{\ell}~S_{BH}~,
\ee 
%
where $S_{BH}$ is the black hole entropy for the same horizon area, 
and
%
\ba
\l^{\a-1} &\equiv& \f{\e \hbar^{\a-1}}{cp^\a} \\
c_5 &=& 4 \a^{\f{d-1}{d-1+\a}} \O_{d-1}^{\f{\a}{d-1}} 
%\nn \\ &\times& 
\le( \f{d-2}{16\p} \ri)^{\f{d-1}{d-1+\a}} \nn \\
&\times& 
\le[\zeta\le(\f{d-1}{\a} +1 \ri) 
\Gamma\le(\f{d-1}{\a} +1 \ri) \ri] 
\nn \\ &\times& 
\le[ \le(n_B + n_F - \f{n_F}{2^{(d-1)/\a}}\ri)  \ri]^{\f{\a}{(d-1+\a)}}~.
\ea
%
Given $r_+/\ell \gg1$, the magnitude of the proportionality
factor multiplying $S_{BH}$ on the RHS depend on the two ratios 
$\ell_{Pl}/\l$ and $\ell_{Pl}/\ell$. 



\section{Discussions} 
\la{discussions}

We have shown that the thermodynamics of  
black holes can be reproduced by a holographic gas with a generalised 
dispersion 
relation. Specifically we have the general result that for 
AdS-Schwarzschild black 
holes in $d$-spacetime dimensions, the thermodynamics can be encoded  
in a gas of 
free bosons and fermions in $\D = \a (d-1) + 1$ spacetime dimensions. 
Thus for a 
given $d$, a variety of $\D$ can serve our purpose, depending on $\a$. 
We have also 
seen that some results in the AdS/CFT context arise as special cases of the 
thermodynamic matching we have used. 

We have also derived corrections to entropy bounds, and discussed specific 
examples of the bounds, using Schwarzschild-AdS black holes. 
An interesting result 
here is the lack of a spherical bound for a collection of these black holes.

Particles with $\a=1$ do not seem to be necessary in the holographic mapping, 
although they are sufficient. It is interesting to note however, 
that for $\a=1$, the 
relations that map the entropies  of the black hole and gas 
become independent of $r_0$, the `location' of the holographic gas 
in the black hole spacetime. 
Thus, this may be an additional reason  to attach a preferred status to 
particles satisfying the  relativistic dispersion relation. 

There are several open questions which would be interesting to pursue.
From the integral in (\ref{saddle}), it is clear that the next-to-leading-order
corrections are difficult, if not impossible to compute
analytically. It may be possible to do this numerically to find 
the dependence of the corrections on horizon area. As noted earlier, 
the corrected entropy is always less than the uncorrected one, signifying
a reduction in the number of accessible states, when fluctuations are taken 
into account. It may be useful to find an interpretation of this result from
the point of view of information theory, in which decrease of entropy is
associated with an increase in information \cite{khinchin,shannon}.  
%

An important generalisation of the current
formalism would be to calculate entropy corrections for asymptotically
flat black holes. One way would be to consider these black holes in
a finite box, whose size can be adjusted such that the specific heat of the 
resultant system is positive \cite{ad}. It would also be interesting 
to compare these corrections with those coming from other sources, such
as quantum spacetime fluctuations \cite{pm2}.

A related problem concerns  de-Sitter black holes. 
Here the temperature and specific heat can be obtained from the 
corresponding expressions for asymptotically anti-de Sitter 
black holes [Eqs.(\ref{adssctemp}-\ref{adsscspht})],
by substituting $\ell^2 \rightarrow -\ell^2$:
\ba
T_H &=& {\hbar c}{}~\f{- (d-1) r_+^2 + (d-3)\ell^2}{4\p \ell^2 r_+} 
\la{dssctemp} \\
C &=& (d-2)\le[ \f{(d-1)r_+^2/\ell^2 
- (d-3)}{(d-1)r_+^2/\ell^2-(d-3)} \ri] S_{BH} ~. \la{dsscspht}
\ea
%
Here in the regime in which Hawking temperature is positive, 
the specific heat is negative, again signaling an apparent 
breakdown of the approach. However, since the energy of the black holes 
is negative,  one may use a modified definition of specific heat, 
namely $C = d(-E)/dT$. This gives a positive $C$  and permits 
some calculations \cite{spradlin,balasub,shanki2}. 
Corrections to entropy would be  identical to that 
for the AdS case, Eq.(\ref{adssccorr}), and it appears that the 
mapping of thermodynamics can also be done. 


\vs{.4cm}
\no
{\bf Acknowledgements}

The authors would like to thank J. Gegenberg for useful discussions. 
S.D. thanks R. K. Bhaduri, P. Majumdar and 
S. Shankaranarayanan for useful correspondence, 
and M. M. Akbar, D. S. Berman, M. Cavaglia, A. Dasgupta, 
W. R. Knight, R. Maartens
and O. Obregon
for helpful comments and discussions. This work was supported in part by 
the Natural Sciences and Engineering Research Council of Canada.  

 
 
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\end{document}






