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\begin{document}

\title{Quantized Black Holes and Their Radiation}

\classification{97.60.-s}
\keywords      {Black holes; entropy; adiabatic
invariance; discrete spectrum}


\author{I.B. Khriplovich}{
  address={Budker Institute of Nuclear Physics, 630090 Novosibirsk, Russia}
}

\begin{abstract}
Temperature and entropy of black holes are discussed. The maximum entropy of a quantized surface is
demonstrated to be proportional to the surface area in the classical limit. The general
structure of the quantum spectrum of a black hole horizon is
found. The discrete spectrum of thermal radiation of a black hole
fits the Wien profile. The natural widths of the lines are much
smaller than their separation.
\end{abstract}

\maketitle

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\section{Temperature of Black Holes}
Long ago, in 1971, J. Wheeler realized that the classical
description of black holes is incomplete in principle. His line of
reasoning looked as follows. Let us take a box filled with the
black body radiation at some temperature $T$. Obviously it
possesses a finite entropy as well. We drop the box into a black
hole. Then the entropy of the observable part of the universe 
decreases forever. But this is an explicit violation of the second
law of thermodynamics! To save the second law, Bekenstein
suggested \cite{bek} that the black hole itself has some entropy
which increases when the box is absorbed. Then it is only natural
to ascribe some finite temperature as well to a system with a
finite entropy. This conclusion is quite natural from a somewhat
different point of view. A black hole is an ideal absorber, an
absolutely black body, for which the temperature is a quite
natural property.

Let us try at first to estimate this temperature just by
dimensional arguments. The classical parameters at our disposal,
the Newton gravitational constant $k$, the mass $M$ of black hole,
and the speed of light $c$, are insufficient for the purpose ($Mc^2$
is too large, and does not contain $k$). But there is the Planck
constant $\hbar$. With it one can easily construct the necessary
combination: the black hole temperature is on the order of magnitude
$\hbar c^3/(k M)$.

To derive the numerical factor in the relation $T \sim \hbar
c^3/(k M)$, we consider the following problem \cite{pad}. Let a
semiclassical wave packet of a massless field propagate from a
point $r_0 = r_g + \ep$ close to the horizon of a black hole with
mass $M$ ($r_g = 2kM/c^2$ is the horizon radius) to a distant point
$r$. A straightforward (but rather
tedious) calculation demonstrates that, independently of the
initial spectrum of the wave packet, at infinity it is
completely universal:
\beq\label{spd}
|f(\om)|^2 \sim \,\exp\;(-8\pi k M \om/c^3)\,.
\eeq
If one goes over in it from the frequency $\om$ to the energy
$\hbar \om$, it corresponds to the Boltzmann distribution with the
temperature
\beq\label{T}
T=\,\hbar c/(4\pi  r_g)\,=\,\hbar c^3 /(8\pi k M)\,.
\eeq
This expression for the black hole temperature was obtained by S.
Hawking \cite{haw}.

\section{Radiation of black holes}

The inevitable result of the finite temperature $T$ of a black
hole is the conclusion that in fact it radiates. Black hole produces not only photons and neutrinos
with energies on the order of $T$, but particles of non-vanishing
rest mass $m$ as well (only if its temperature is sufficiently
high). Thus, one of the most amazing properties of black holes is
that they shine!

V.N. Gribov was the first who made this
conclusion\footnote{Gribov precisely formulated the
statement that black holes radiate in discussions taking place in
1971 or 1972. This was told to me independently by A.D. Dolgov,
D.I. Diakonov, L.B. Okun', who had been present at those
discussions. Unfortunately, Gribov did not publish this
result, perhaps he considered it self-evident. In 1974
radiation of black holes was predicted independently by S.
Hawking~\cite{haw}.}. His argument was as follows. The uncertainty
relation $\De E \De t \geq \hbar$ allows the creation of pairs of
particles from vacuum for the time $t$ that does not exceed $\hbar
/E$; here $E$ is the total energy of the pair ($E \geq 2mc^2$ for massive
particles). The gravitational field near the horizon is very
strong, so that the energy conservation by itself allows one of
the particles to be absorbed by the black hole, and the second one
to go to infinity. In quantum mechanics, due to the tunneling
effect of such a sort, the processes of particle creation become
possible. One can recall in this connection the creation of electron-positron pairs
in strong electric fields.

Contrary to na\"{\i}ve expectations, the radiation of black holes is not described by the
common Planck and Fermi distributions. Indeed, these distributions
are valid only if the size of a radiating body is much larger than
the typical wave lengths, i.e. in the semiclassical limit. Let us
recall the common model of an absolutely black body: an opening in a cavity
filled with radiation \cite{lan} (we talk here about photons, but these arguments
apply to particles of any spin). Obviously, the radiation of wave
lengths large as compared to the size of the opening is strongly
suppressed. So, the common temperature distributions apply
only under the condition $\omega a \gg 1$, where $a$ is the
typical size of a radiating body.

For a black hole, however, the typical frequencies are $\om \sim
(4\pi r_g)^{-1}$ (see (\ref{spd})), and the size of radiating body
$a$ is $r_g$. Thus, here
\beq
\om a \sim1/(4\pi) \ll 1\;.
\eeq
As a result, the black body radiation differs essentially from the
common thermal one. In particular, due to the centrifugal effect,
at $\om r_g \sim 1/(4\pi) \ll 1$, the density of a partial wave of
radiated field near the horizon, and therefore the
radiation intensity in this wave, falls down rapidly with the
growth of the total angular momentum $j$. The results of numerical
solution of wave equations for particles of different spins in the
field of a black hole \cite{page} confirm this qualitative
conclusion. The total radiation intensities for (ultrarelativistic)
electrons ($j_{\rm
min}=1/2$), photons ($j_{\rm min}=1$), and gravitons ($j_{\rm
min}=2$) are related as 22:9:1. On the other hand, the ratios of
the radiation intensities in the waves with $j = j_{\rm min}$ and
$j = j_{\rm min}+1$ (corresponding to a single value of the
projection $j_z$, i.e. without the weight $(2j +1)$) are as
follows: for electron 26:1, for photon 47:1, for graviton 95:1. It
is curious that, according to the same numerical calculations, the
total intensity of the photon radiation by a black hole is
quantitatively close to the na\"{\i}ve Planck one.

We mean here sufficiently light black holes, with mass  $\sim
10^{15} - 10^{16}$ g, and with typical temperature in the interval
$10 - 1$ MeV, correspondingly. Electrons and positrons emitted by
such black holes can be treated with good accuracy as
ultrarelativistic ones. However, for real black holes the
temperature (\ref{T}) is negligibly small: for the mass comparable
with that of the Sun it is only about $10^{-7}$ K. 

As to the stars
with masses $\sim 10^{15} - 10^{16}$ g, their gravitational field is
too weak, they cannot compress to their gravitational radii, they
cannot turn into black holes. Such light black holes in principle
could arise at the most early stages of the Universe evolution.

But could these mini-holes survive since those times? Could their
age approach the Universe life time $\tau \sim 10^{10}$ years, or
$10^{17}$ s? The problem here is the black hole thermal radiation
itself. Let us estimate its intensity $I$ by dimensional
arguments: divide $T$ by the characteristic time $r_g/c$:
\beq\label{I}
I \sim cT/r_g \sim m_p^4 c^4/(\hbar M^2).
\eeq
We have introduced here the so-called Planck mass
\beq\label{mp}
m_p =\left(\hbar c/k\right)^{\!{1/2}}= 2.2 \times 10^{-5}\,{\rm g}\,.
\eeq
Obviously, $I= - c^2 dM/dt$. Solving the
differential equation
\[
dM/dt\,=-\,m_p^4\, c^2/(\hbar M^2)\,,
\]
we find that to survive until our time a black hole should have an
initial mass
\beq\label{Mmp}
M > m_p\left(\tau/t_p\right)^{\!{1/3}}\,\sim \,10^{15}\,g\,.
\eeq
Here $t_p$ is the so-called Planck time
\beq\label{tp}
t_p=\,\hbar/(m_p c^2)\,=\,\left(\hbar
k/c^5\right)^{\!{1/2}}\,= 0.54 \times 10^{-43}\, {\rm s}\,.
\eeq

Together with the energy, a black hole loses its mass. Then,
according to relation (\ref{I}), the intensity of its radiation
grows, and the gravitational radius
of a black hole gets smaller. However a star cannot radiate more energy than it has. The
radiation stops when the black hole temperature becomes
comparable to its rest energy, at
\[
Mc^2 \sim T \sim \,m_p^2 c^2/M\,,
\]
i.e. when the mass of such a mini-hole decreases to the Planck
mass:
\[
M \sim m_p\,.
\]
Here our semiclassical consideration of quantum effects
becomes inapplicable, and a consistent quantum theory of gravity
is necessary.

It is instructive to look at relation (\ref{T}) somewhat
otherwise. It demonstrates that the energy of a black
hole, together with its mass, decreases as the temperature
increases. Thus, the heat capacity of a black hole is
negative. This unusual property is quite typical for gravitating systems in
general \cite{lan}. As to a black hole, its
negative heat capacity is directly related to the instability
caused by radiation. Let us recall, however, that the classical
instability of an electron bound in the Coulomb field,
also caused by radiation, is stabilized by
quantum effects. In the case of black holes as well, it is natural
to assume that on the Planck scale their semiclassical radiative
instability is stabilized by quantum effects.

One more fact related to the radiation of black holes. For the
typical time interval $\De t \sim r_g/c$ between the acts of
radiation, the uncertainty of the energy of a black hole is $\De E
\sim \hbar/\De t \sim \hbar c^3/kM$. The corresponding
uncertainty in the
gravitational radius is \cite{yor}
\[
\De r_g \sim \,k\, \De M/c^2\, \sim \,k \,\De E/c^4\,
\sim \,\hbar/(M c)\,.
\]
Obviously, at least due to this
uncertainty, the time of the fall of a point-like particle to the
horizon (which is logarithmically divergent in the classical
approach!) becomes finite:
\[
t \simeq r_g \ln\, r_g/\De r_g\, \simeq r_g
\ln\,M^2/m_p^2\,.
\]
The arising logarithm is huge, $\ln
M^2/m_p^2 \simeq 10^2$, but here it is not of much
importance.

\section{Entropy and Horizon Area}

Now, when the temperature of a black hole is known, its entropy is
calculated with the well-known thermodynamical formula $dE = TdS$.
In our case $T$ is given by formula (\ref{T}), and $E = Mc^2$. Solving
the differential equation
\[
dM=\,\hbar c \, dS/(8 \pi k M)
\]
with the natural boundary condition $\,S=0\,$ for $\,M = 0$,
we find $\,S=\,4\pi k M^2/(\hbar c)\,$.
It is convenient to introduce the so-called Planck length
\beq\label{lp}
l_p = (\hbar\, k/c^3)^{\!{1/2}}=1.6 \times
10^{-33}\; {\rm cm}\,.
\eeq
Then we arrive at the following remarkable relation between the
entropy of a Schwarzschild black hole and the area of its horizon
$A=4\pi r_g^2$:
\beq\label{sa}
S=\,\pi r_g^2/l_p^2\,=\,A/(4 l_p^2)\,.
\eeq

The corresponding analysis for a charged black hole is more
intricate. In the Schwarzschild case, the horizon area $A=16\pi k^2 M^2$
(from now on, we put $c=1$) depends
on the only parameter $M$, and
the adiabatic invariance of $A$ means that $M$ is also an
adiabatic invariant. But the horizon area of a charged black hole
depends not only on its mass, but on its charge $q$ as well (see, e.g., \cite{ibk}):
\beq
A_{ch}=4\pi r_{gc}^2; \quad r_{gc} = kM+\sqrt{k^2 M^2 - k q^2}\,.
\eeq
So, what happens with this black hole when a small charge $e$ is lowered
adiabatically to its horizon? What remains
constant, the horizon area or the mass (if either)?

To answer the question, we resort to a thought experiment. Let a
particle with energy $\ep$ at infinity and charge $e$ falls radially
to the horizon. The charges $e$ and $q$ are of the same sign, so that the
electrostatic repulsion could compensate for the gravitational
attraction. As a result of the capture, the black hole mass increases by
$\De M = \ep$, and its charge by $\De q =e$ (both energy and charge are
conserved). The change of the horizon area in this process is
\beq\label{ai}
\De A_{ch}=\,{8\pi r_{gc} k \over \sqrt{k^2 M^2 - k q^2}}\left(\ep\,
 - \frac{e q}{r_{gc}}\right)\,.
\eeq
If one chooses $\ep = eq/r_{gc}$, then for a non-extremal black hole
(i.e. for $q^2 < kM^2$) this expression turns to zero. Let us prove now
that at the capture the particle velocity tends to zero, i.e.
that the capture itself is adiabatic. To this end, we use the identity
$g_{\mu\nu}u^{\mu}u^{\nu}= 1$. In the present case of diagonal metric
and radial motion, this identity reduces to $g_{00}(u^0)^2 + g_{rr}(u^{r})^2 = 1\,$, or
\beq\label{ide}
g^{00}(u_0)^2 + g_{rr}(u^{r})^2 = 1\,.
\eeq
In this case
\[
g^{00} = -\,g_{rr} = a^{-2}(r) = \left(1-\,2 k M /r\,+\,kq^2
/r^2\right)^{-1}.
\]
By definition, $u_0 = (p_0 - eA_0)/m = (\ep - e q/r)/m$.
We recall here that the zeroth component of the covariant momentum is the
conserved particle energy (it gets obvious in the Hamilton-Jacobi formalism).
Again by definition, $(u^{r})^2 = (dr/d\tau)^2$, where $\tau$ is
the invariant, local time. Finally, equation (\ref{ide}) is rewritten as follows:
\beq\label{ad}
\left(\ep - eq/r\right)^2 = m^2 \left(dr/d\tau\right)^2 + m^2 a^2\,.
\eeq
At the horizon, at $r = r_{gc}$, $a$ vanishes. It gets clear now
that at the capture by a charged black hole of a particle with
energy $\ep = eq/r_{gc}$, its radial velocity
measured by a local observer,  $dr/d\tau$, tends to zero, and the duration of
this process measured by the same clock tends to infinity.
Thus, the area $A$ of the horizon of a charged black hole (but not
the mass) is the adiabatic invariant\footnote{Originally, the fact
of the adiabatic invariance of the horizon area was established by
D. Christodoulu and R. Ruffini for rotating black holes
\cite{ch,chr}. But here we confine to a more simple
case of charged black holes.}. We recall now that the entropy
remains constant under adiabatic processes. Therefore, just
$A/(4l_p^2)$ should be identified with the entropy of a charged
black hole.

\section{Quantization of Black Holes. Holographic Bound}

On the other hand, the quantization of an adiabatic invariant is
perfectly natural. And just on this argument was based the idea of
quantizing the horizon area of black holes proposed by J.
Bekenstein \cite{bek1}.

We start the discussion with eq. (\ref{ad}), rewriting it as
\beq\label{ad1}
\ep - eq/r = \sqrt{m^2 \left(dr/d\tau\right)^2 +
m^2 a^2}\,.
\eeq
It is convenient to go over here from the contravariant radial component
$p^r = m\; dr/d\tau$ of the momentum vector to the corresponding
component $\pi^r$ in the locally inertial frame: $\pi^r = \sqrt{-g_{rr}}\,p^r =
p^r/a(r)$. And in the thus arising expression
\[
\ep - eq/r = a(r)\sqrt{(\pi^r)^2 + m^2}
\]
we go over from $r$ to the coordinate $\rho$, given also
in the locally inertial frame and counted off the horizon:
\[
\rho = \int_{r_{gc}}^r dr \sqrt{-g_{rr}(r)} = \int_{r_{gc}}^r
\frac{dr}{\sqrt{a(r)}} = \frac{2\,r_{gc}\sqrt{r - r_{gc}}}
{\sqrt{r_{gc}-r_{gc}^-}}\,;
\]
here $r_{gc}^- = kM - \sqrt{k^2M^2 - k q^2}$; 
we assume that $\rho \ll r_{gc}$. As a result,
eq. (\ref{ad1}) transforms for $\,r \to r_{gc}$ to
\beq\label{ad2}
\ep - \frac{eq}{r} = \frac{\sqrt{k^2M^2 - k
q^2}}{r_{gc}}\,\rho\,\sqrt{(\pi^r)^2 + m^2}\,.
\eeq
At last, in virtue of (\ref{ai}), the change of the horizon
surface in this process is
\[
\De A_{ch}= \,{8\pi k r_{gc}\over \sqrt{k^2 M^2 - k
q^2}}\left(\ep\,
 - \frac{e\, q}{r_{gc}}\right)
\]
\beq\label{ai1}
\quad \quad = \,8\pi k \,\sqrt{\rho^2\,(\pi^r)^2 + (m\,\rho)^2}\,.
\eeq
Of course, $\;m\rho$ vanishes in the limit $\rho \to 0$.
However, in virtue of the uncertainty relation, $\rho\,\pi^r$
stays finite in this limit: $\rho\,\pi^r \gsim \;\hbar$. Thus,
the minimum change of the horizon surface is
\beq\label{qu0}
\De A_{min} \simeq 8\pi\,\hbar k\, = \,8\pi \,l_p^2\,.
\eeq

Obviously, the fact that the minimum possible change of the horizon
area is finite, makes the horizon quantization quite natural. We assume
therefore that the whole horizon area $A$ consists of patches of typical
size $\sim 8\pi\,l_p^2\,$. Each of them is characterized by a
quantum number $j$, such that the contribution $a$ of a patch to the
area depends on this quantum number, $\;a=a(j)$. Besides, a patch can possess a
quantum number $m$, such that $a$ is independent of it\footnote{In principle,
both $j$ and $m$ may refer not only to a single quantum number each, but to sets
of them: $j = (j_1,\,j_2,\,...)\,, \; m = (m_1,\,m_2\,...)$.}. Then, the horizon
area is conveniently rewritten as
\beq\label{qua}
A = 8\pi\ga\, l_p^2 \sum_{jm}\,a(j)\,\nu_{jm}\,,
\eeq
where $\nu_{jm}$ is the number of patches of given $j$ and $m$. The numerical
factor $\ga$ will be determined below for given function $a(j)$ and given
statistical weight $g(j)$ (as usual, the latter equals the number of possible values of
$m$ for given $j$). Correspondingly, the entropy of a black hole is
\beq\label{que}
S = 2\pi\ga\, \sum_{jm}\,a(j)\,\nu_{jm}\,.
\eeq

The occupation numbers $\nu_{jm}$ can be related to $a(j)$ and
$g(j)$ by using the so-called holographic bound. According to
it, the entropy $S$ of any spherical
nonrotating system confined inside a sphere of area $A$ is bounded
by relation
\beq\label{hb}
S \leq A/(4 l_p^2)\,,
\eeq
with the equality attained only if the system is a black
hole [11--13].

A simple intuitive argument confirming this bound is as
follows~\cite{sus}. Let us allow the discussed system to collapse
into a black hole. Due to the spherical symmetry, this process is
not accompanied by radiation or any other loss of matter.
During the collapse the entropy increases from $S$ to $S_{bh}$, or
at least remains constant. And the resulting horizon area $A_{bh}$
is certainly smaller than the initial confining one $A$. Now, with
the account for relation (\ref{sa}) for a black hole, we arrive,
through the obvious chain of (in)equalities
\[
S \leq S_{bh} = A_{bh}/(4 l_p^2) \leq A/(4 l_p^2)\,,
\]
at the discussed bound (\ref{hb}).

The holographic bound looks quite surprising since usually the
entropy of a body is proportional to its volume, but not to the
area of its surface. However, in regular situations limit
(\ref{hb}) is so weak quantitatively that no contradiction with
the common sense arises. In fact, at least
for spherically symmetric black holes, the holographic bound has
been checked by careful analysis of various physical situations,
and therefore its validity is firmly established.

The result (\ref{hb}) can be formulated otherwise. Among the
spherical surfaces of a given area, it is the surface of a black
hole horizon that has the largest entropy.

Let us come back now to our problem. We will consider the
``microcanonical'' entropy $S$ of a quantized surface, defined as
the logarithm of the number of states of this surface for a fixed
value $A$ of its area (instead of fixed energy in common
problems).

Obviously, this number of states $K$ depends essentially on the assumption
concerning the distinguishability of the patches. So, let us
discuss first of all which of a priori possible assumptions
is reasonable here from the physical point of view~\cite{kh1}.

We start with the possibility of complete indistinguishability of
patches. It means that for given $\nu_{jm}$ no permutation of any
patches results in new states, i.e. this is the only state at all.
Correspondingly, the entropy in this case just turns to zero.

Let us consider now the opposite assumption, that of completely
distinguishable patches. In this case the total number of states
is
\[
K = \nu !\,,\quad \nu = \sum_j \nu_j =  \sum_{jm} \nu_{jm}\,,
\]
with the microcanonical entropy\footnote{We assume that all
occupation numbers are sufficiently large, so that the simple
Stirling approximation is applicable for all factorials.}
\[
S=\nu \ln \nu\,.
\]
Obviously, here the maximum entropy for fixed $A \sim \sum_j
a(j)\; \nu_j$ is attained with all $a(j)$ being as small as
possible. Then, in the classical limit $\nu \gg 1$, the entropy of
a black hole grows faster than its area: $A \sim \nu$, but $S =
\nu \ln \nu \sim A\ln A$. Thus, the assumption of complete
distinguishability is in conflict with the holographic bound, and
therefore should be discarded.

Now the third possibility (used to be quite popular). Here the total number of
states and the entropy are
\begin{equation}\label{pro}
K=\prod_j g(j)^{\nu_j}\,,\quad \mbox{and} \quad S=\sum_j \nu_j \,\ln
g(j).
\end{equation}
This scheme corresponds in fact to the
following assumptions on the distinguishability of patches:\\

\begin{tabular}[h]{cccr}
%\vspace{3mm}
 nonequal $j$, & any $m$ & $\longrightarrow$ &
indistinguishable;\\
%\vspace{3mm}
equal $j$, & nonequal $m$ &
$\longrightarrow$ & distinguishable;\\
\vspace{3mm}
equal $j$, & equal $m$ & $\longrightarrow$ & indistinguishable.\\
\end{tabular}\\
\noindent The combination of the first two of them looks strange
and unnatural (except the special case when only a single value of
$j$ is allowed for all patches).

The only reasonable set of assumptions on the distinguishability
of patches, which may result in acceptable physical predictions
(i.e. may comply both with the relation
(\ref{sa}) between the entropy and the horizon surface,
and with the holographic bound (\ref{hb})) is as follows:\\

\begin{tabular}[h]{cccr}
 %\vspace{3mm}
 nonequal $j$, & any $m$ & $\longrightarrow$ &
distinguishable;\\
%\vspace{3mm}
equal $j$, & nonequal $m$ &
$\longrightarrow$ & distinguishable;\\
\vspace{3mm}
equal $j$, & equal $m$ & $\longrightarrow$ & indistinguishable.\\
\end{tabular}\\
\noindent Under these assumptions, the number of states of the
horizon surface, for a given number $\nu_{jm}$ of patches with
quantum numbers $j$ and $m$, is obviously \cite{kk1}
\begin{equation}\label{mk}
K = \nu\,!\, \prod_{jm}\,\frac{1}{\nu_{jm}\,!}\;, \; {\rm where}
\; \nu =\sum_j \nu_j\,, \; \nu_j = \sum_m \nu_{jm}\,,
\end{equation}
and the corresponding entropy equals
\begin{equation}\label{ms}
S=\ln K = \ln(\nu\,!)\,- \sum_{jm}\,\ln(\nu_{jm}\,!)\,.
\end{equation}
The structures of the last expression and of formula (\ref{que})
are so different that in a general case the entropy certainly
cannot be proportional to the area. However, this is the case for
the maximum entropy. We will calculate it for a fixed area, i.e.
for a fixed sum
\begin{equation}\label{N}
N\,=  \sum_{jm}^\infty a(j)\,\nu_{jm}={\rm const} \,.
\end{equation}
The problem reduces to the solution of the system of equations
\begin{equation}\label{sys}
\ln \nu  - \ln \nu_{jm} = \mu\, a(j)\,,
\end{equation}
where $\mu$ is the Lagrange multiplier for the constraining
relation (\ref{N}). These equations can be rewritten as
\begin{equation}\label{nu1}
\nu_{jm}=\nu\, e^{- \mu\, a(j)}, \quad \mbox{or} \quad \nu_j =
\,\nu\,g(j)\,e^{- \mu\, a(j)}\,.
\end{equation}
Now we sum expression (\ref{nu1}) over $j$, and with $\sum_j
\nu_j = \nu$ arrive at the equation for $\mu$:
\begin{equation}\label{equ}
\sum_j g(j)\, e^{- \mu \,a(j)} = 1.
\end{equation}

On the other hand, multiplying equation (\ref{sys}) by $\nu_{jm}$
and summing over $jm$, we arrive, with the constraint (\ref{N}),
at the following result for the maximum entropy for given $N$:
\begin{equation}\label{enf0}
S_{\rm max}= \,\mu \,N\,=\,\mu\,A/(8\pi\gamma \,l_p^2)\,.
\end{equation}

Thus, equation (\ref{qua}) for the quantized area can be written
as
\begin{equation}\label{qu1}
A = 8\pi\gamma\, l_p^2\, \nu \sum_j\,g(j)\,a(j)\, e^{- \mu
a(j)}\,,
\end{equation}
where $\gamma = \mu/(2\pi)$, and the value of $\mu$ is found from
equation~(\ref{equ}).

Let us note that, strictly speaking, the summation in
formulae (\ref{equ}), (\ref{qu1}) goes not to infinity, but to some
$j$, corresponding to the maximum contribution $a_{{\rm max}}$ to
the horizon area. The value of $a_{{\rm max}}$ follows from the
obvious condition: none of the occupation numbers $\nu_{jm}$
should be less than unity. Then equation  (\ref{nu1}) results
in the estimate
\beq\label{jm}
a_{{\rm max}} \sim \,\ln \nu/\mu\,.
\eeq

We illustrate now these general relations with an example of a concrete
model, that of loop quantum gravity (LQG) \cite{leh}.
A quantized surface in LQG looks as follows. One ascribes to it a
set of punctures (corresponding to our patches). Each puncture
is supplied with an integer or half-integer quantum number $j$:
\beq\label{j}
j= 1/2, 1, 3/2, ...\; .
\eeq
The projections $m$ of these ``angular momenta'' (unrelated to the
common ones) run as usual from $-j$ to $j$. The area of a surface is
\beq\label{Aj1}
A =8\pi\ga\, l_p^2 \sum_{jm} \sqrt{j(j+1)}\,\nu_{jm}.
\eeq
This is in fact a special case of the above general expressions
with
\beq\label{ag}
a(j)\, =\, \sqrt{j(j+1)}\,, \quad g(j) = 2j+1.
\eeq
The numerical factor $\ga$ in (\ref{Aj1}) (the so-called
Barbero--Immirzi parameter) corresponds in LQG to a family of
nonequivalent quantum theories, all of them being a priori,
without additional arguments, viable ~\cite{imm,rot}. In this
case, our ``secular'' equation (\ref{equ}) and its solution are,
respectively, \cite{kk,cor}:
\beq\label{equ1}
\sum_{j=1/2}^{\infty} (2j+1)\, e^{- \mu \sqrt{j(j+1)}} = 1,\;\;
\ga = \frac{\mu}{2\pi}= 0.274.
\eeq

\section{Quantization of Rotating Black Holes}

In the next section we will discuss the radiation spectrum of
quantized black holes. Here, generally speaking, one should take
into account the selection rules for angular momentum. Therefore,
the quantization rule for the mass of a Schwarzschild black hole
should be generalized to that of a rotating Kerr black hole.

To derive the quantization rule for Kerr black hole, we come back
to the thought experiment analyzed in~\cite{ch,chr}. Therein,
under the adiabatic capture of a particle with an angular momentum
$j$, the angular momentum $J$ of a rotating black hole changes by
a finite amount $j$, but the horizon area $A$ does not change. Of
course, under some other variation of parameters it is the angular
momentum $J$ that remains constant. In other words, we have here
two independent adiabatic invariants, $A$ and $J$, for a Kerr
black hole with a mass $M$.

Such a situation is quite common in ordinary mechanics. For
instance, the energy of a particle with mass $m$, bound in the
Coulomb field $U(r)~=~-~\alpha/r$, is
\begin{equation}\label{Ec}
E=-\,\frac{m \alpha^2}{2\,(I_r + I_{\phi})^2}\,,
\end{equation}
where $I_r$ and $I_{\phi}$ are adiabatic invariants for the radial
and angular degree of freedom, respectively. Of course, the energy
$E$ is in a sense an adiabatic invariant also, but it is invariant
only with respect to those variations of parameters under which
both $I_r$ and $I_{\phi}$ remain constant (or at least their sum).
As to quantum mechanics, in it formula (\ref{Ec}) goes over into
\begin{equation}\label{Eq}
E=-\,\frac{m \alpha^2}{2\,\hbar^2\, (n_r + 1 + l)^2}\,,
\end{equation}
where $n_r$ and $l$ are the radial and orbital quantum numbers,
respectively.

This example prompts the solution of the quantization problem for
a Kerr black hole. It is conveniently formulated in terms of the
so-called irreducible mass $M_{ir}$ of a black hole, related by
definition to its horizon radius $r_h$ and area $A$ as follows:
\begin{equation}\label{rel}
r_h= 2 k M_{ir}\,, \quad A = 16\pi k^2 M_{ir}^2\,.
\end{equation}
Together with the horizon area $A$, the irreducible mass is an
adiabatic invariant. In accordance with (\ref{qua}) and (\ref{N}),
it is quantized as follows:
\begin{equation}\label{mirq}
M_{ir}^2 =\,\frac{1}{2}\,m^2_p \gamma\,N\,,
\end{equation}
where $m^2_p=\hbar c/k$ is the Planck mass squared.

Of course, for a Schwarzschild black hole $M_{ir}$ coincides with
its ordinary mass $M$. However, for a Kerr black hole the
situation is more interesting. Here \cite{chr}
\begin{equation}\label{mik}
M^2=M_{ir}^2 + \,\frac{J^2}{r_h^2}\,=M_{ir}^2 + \,\frac{J^2}{4k^2
M_{ir}^2}\,,
\end{equation}
where $J$ is the internal angular momentum of a rotating black
hole. Let us note that equation (\ref{mik}) describes a relativistic rotator with
the rest mass $M_{ir}$ and the moment of inertia $M_{ir} r_h^2$ (defined as usual in the 
nonrelativistic limit) \cite{ibk}.

Now, with equation (\ref{mirq}), we arrive at the
following quantization rule for the mass squared $M^2$ of a
rotating black hole:
\begin{equation}\label{mqj}
M^2 =\,\frac{1}{2}\,m^2_p \left[\gamma N +\,\frac{J(J+1)}{\gamma
N}\right].
\end{equation}
Obviously, as long as a black hole is far away from an extremal
one, i.e. while $\gamma N \gg J$, one can neglect the dependence
of $M^2$ on $J$.

\section{Radiation Spectrum of Quantized Black Hole}

It follows from expression (\ref{mqj}) that for a rotating black
hole the radiation frequency $\omega$, which coincides with the
loss $\Delta M$ of the black hole mass, is
\begin{equation}\label{om}
\omega = \Delta M = T \mu \,\Delta N
+\,\frac{1}{2kM}\,\frac{J+1/2}{\gamma N}\, \Delta J\,=T\Delta
S~+~\Omega~\Delta J\,,
\end{equation}
where $\Delta N$ and $\Delta J$ are the losses of the area quantum
number $N$ and of the angular momentum $J$, respectively; $\Delta
S = \mu \,\Delta N$, and the effective angular velocity is $\Omega
= (J+1/2)/(M r_h^2)$. Obviously, equation (\ref{om}) is in fact a
common thermodynamic relation.

We will be interested mainly in the first, temperature term in
(\ref{om}), dominating for black holes far from
the extremal regime, i.e. for $J \ll \gamma N$.
Just this effect is discussed in detail below.

As to the non-temperature radiation of a black hole close to an
extremal one, as described by the term with $\De J$ in (\ref{om}),
this effect is due to the tunnelling  (see relatively recent
discussion of this problem in~\cite{khr,kk1}). Loss of the charge
by a charged extremal black hole occurs due to the Coulomb
repulsion between the black hole and emitted particles of the same
sign of the charge. For an extremal rotating black hole, the
radiation is caused by the interaction of angular momenta:
particles (massless mainly), with the total angular momenta
parallel to that of the black hole, are repelled from it.

But let as come back to the temperature radiation. The natural
assumption is that it occurs when a patch with a given value of
$j$ disappears, which means that
\begin{equation}\label{dom}
\Delta N_j = a(j)\,, \quad \omega_j = T \mu\, a(j)\,.
\end{equation}
Thus we arrive at the discrete spectrum with a finite number of
lines. Their frequencies start at $\omega_{{\rm min}}=T\mu \,
a_{{\rm min}}$, where $a_{{\rm min}}$ is the minimum value of
$a(j)$, and terminate at $\omega_{{\rm max}} \sim T\ln\nu$ (we
recall here that $a_{{\rm max}} \sim \ln \nu/\mu$). Thus, the
number of lines is not so large, $\sim 10^2$, if the mass of black
hole is comparable to that of the Sun. However, due to the
exponential decrease of the radiation intensity with $\omega$ (see
below), the existence of $\omega_{{\rm max}}$ and finite number of
lines are not of much importance.

To substantiate the made assumption, we come back to the lower
bound (\ref{qu0}) on the change of the horizon area under an
adiabatic capture of a particle. The presence of the gap
(\ref{qu0}) in this process means that this threshold capture
effectively consists in the increase by unity of a single occupation
number $\nu_{jm}$. If the capture
were accompanied by a change of few occupation numbers, some of them increasing and
some of them decreasing, the
change of the area could be made in general as small as one wishes
\footnote{Except the case when $a(j)$ is a linear function of $j$,
and, correspondingly, the area spectrum is equidistant. Generally
speaking, this case cannot be excluded.}.

It is only natural to assume that in the radiation process as
well, the change of few occupation numbers, instead of one, is at least
strongly suppressed. In this way we arrive at
equations~(\ref{dom}).

Our next assumption, at least as natural as this one, is that the
probability of radiation of a quantum with frequency $\omega_j$ is
proportional to the occupation number $\nu_j$. Correspondingly,
the radiation intensity $I_j$ at this frequency $\omega_j$ is
proportional to $\nu_j\, \omega_j$:
\begin{equation}\label{ij}
I_j \sim \nu_j\, \omega_j \sim \nu\, g(j)\, \omega_j \,
e^{-\omega_j/T}.
\end{equation}
Thus, we have arrived in a natural way at the exponential Wien
profile for $\om_j \gg T$. The conclusion that the discrete
thermal radiation spectrum of a black hole should fit the Wien
profile was made in \cite{bem} for the case of
equidistant horizon quantization

Numerical estimates demonstrate that the total radiation
intensity of photons by quantized black holes is about the same as that of
classical ones (of course, if $\mu a_{{\rm min}} \lsim 1$). It
follows also from the same estimates that the total natural widths
of the radiation lines of quantized black holes do not exceed
few percent of the line separations. Thus, the radiation
spectrum is really discrete.


\begin{theacknowledgments}
I am grateful to A.A. Pomeransky  for useful discussions. The work
was supported by the Russian Foundation for Basic Research through
Grant No. 08-02-00960-a.
\end{theacknowledgments}

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\IfFileExists{\jobname.bbl}{}
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  \typeout{******************************************}
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  \typeout{** the bibliography and then re-run LaTeX}
  \typeout{** twice to fix the references!}
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\begin{thebibliography}{99}

\bibitem{bek}
J. Bekenstein, \emph{Lett. Nuovo Cimento} \textbf{4},
  737 (1972).

\bibitem{pad}
T. Padmanabhan, \emph{Phys. Rev. D} \textbf{59},
  124012 (1999).

\bibitem{haw}
S. Hawking, \emph{Nature} \textbf{248}, 30 (1974).

\bibitem{lan}
L.D. Landau, and E.M. Lifshitz, \emph{Statistical Physics},
Pergamon Press, London - Paris, 1958, pp. 62--63, 177.

\bibitem{page}
D. Page, \emph{Phys. Rev. D} \textbf{13}, 198 (1976).

\bibitem{yor}
J. York, Jr. \emph{Phys. Rev. D} \textbf{28}, 2929 (1983).

\bibitem{ibk}
I.B. Khriplovich, \emph{General Relativity}, Springer, New York, 2005, pp. 57--59.

\bibitem{ch}
D. Christodoulu, \emph{Phys. Rev. Lett.}, \textbf{25}, 1596
(1970).

\bibitem{chr}
D. Christodoulu, R. Ruffini, \emph{Phys. Rev.}, \textbf{D4}, 3552
(1971).

\bibitem{bek1}
J. Bekenstein, \emph{Lett. Nuovo Cimento} \textbf{11},
  467 (1974).

\bibitem{bek2}
J. Bekenstein, \emph{Phys. Rev. D}, \textbf{23}, 287 (1981).

\bibitem{tho}
G. 't Hooft, in \emph{Salam Festschrift,} World Scientific, Singapore, 1993.

\bibitem{sus}
L. Susskind, \emph{J. Math. Phys.}, \textbf{36}, 6377 (1995).

\bibitem{kh1}
I.B. Khriplovich, \emph{Zh. Eksp. Teor. Fiz.}, \textbf{126}, 527
(2004)\newline $[$\emph{Sov. Phys. JETP}, \textbf{99}, 460
(2004)$]$.

\bibitem{kk}
R.V. Korkin, I.B. Khriplovich, \emph{Zh. Eksp. Teor. Fiz.},
\textbf{122}, 5 (2002) $[\,$\emph{Sov. Phys. JETP}, \textbf{95}, 1
(2002)$]$.

\bibitem{leh}
S. Frittelli, L. Lehner, C. Rovelli, \emph{Class. Quantum Grav.}
\textbf{13}, 2921 (1996).

\bibitem{imm}
G. Immirzi, \emph{Class. Quantum Grav.}, \textbf{14}, L177 (1997).

\bibitem{rot}
C. Rovelli, T. Thiemann, \emph{Phys. Rev.}, \textbf{D57}, 1009
(1998).

\bibitem{cor}
A. Corichi, J. Diaz-Polo, E. Fernandez-Borja \emph{Class. Quantum Grav.},
\textbf{24}, 1495 (2006).

\bibitem{khr}
I.B. Khriplovich, \emph{Zh. Eksp. Teor. Fiz.}, \textbf{115}, 1539
(1999)\newline $[$\emph{Sov. Phys. JETP}, \textbf{88}, 845
(1999)$]$.

\bibitem{kk1}
R.V. Korkin, I.B. Khriplovich, \emph{Zh. Eksp. Teor. Fiz.},
\textbf{121}, 531 (2002) $[$\emph{Sov. Phys. JETP}, \textbf{94},
453 (2002)$]$.

\bibitem{bem}
J. Bekenstein, V. Mukhanov, \emph{Phys. Lett. B}, \textbf{360}, 7
(1995).

\end{thebibliography}

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