# Dynamical Origin of the Entropy of a Black Hole

Valery Frolov
Theoretical Physics Institute
University of Alberta, Edmonton, Canada T6G 2J1;
P.N. Lebedev Physical Institute,
Leninsky prospect 53, Moscow, 117924, Russia
Electronic address<EMAIL_ADDRESS>Igor Novikov
The Copenhagen University Observatory
Øster Voldgade 3, DK-1350 Copenhagen K, Denmark;
Nordita, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark;
Astro Space Centre of P.N. Lebedev Physical Institute,
Profsoyuznaya 84/32, Moscow, 117810, Russia
Electronic address<EMAIL_ADDRESS>
(PACS numbers: 04.20.Cv; 05.30.-D; 97.60.Lf)

###### Abstract

Modes of physical fields which are located inside a horizon and which cannot
be observed by a distant observer are identified with dynamical degrees of
freedom of a black hole. A new invariant statistical mechanical definition of
a black-hole’s entropy is proposed. It is shown that the main contribution to
the entropy is given by thermally excited ‘invisible’ modes propagating in the
close vicinity of the horizon. A calculation based on the proposed definition
yields a value of the entropy which is in good agreement with the usually
adopted value $A^{H}/(4l^{2}_{\mbox{\scriptsize{Pl}}})$, where $A^{H}$ is a
black hole surface area and $l_{\mbox{\scriptsize{Pl}}}$ is the Planck’s
length.

According to the thermodynamical analogy in black hole physics, the entropy of
a black hole is defined as $S^{H}=A^{H}/(4l_{\mbox{\scriptsize{Pl}}}^{2})$,
where $A^{H}$ is the area of a black hole surface and
$l_{\mbox{\scriptsize{Pl}}}=(\hbar G/c^{3}~{})^{1/2}$ is the Planck’s length
[1, 2]. Hawking’s theoretical discovery [3] of quantum black hole evaporation
proved the reality of the black hole temperature and fixed the coefficient
relating entropy of a black hole with its surface area. The generalized second
law (i.e. the statement that the sum $S=S^{H}+S^{m}$ of a black hole’s entropy
and the entropy $S^{m}$ of the outside matter cannot decrease) implies that
the black hole’s entropy plays the same role as the usual entropy and shows up
to which extend the energy contained in a black hole can be used to produce
work [1, 2, 4]. Four laws of black hole physics which form the basis in the
thermodynamical analogy were formulated in [5].

Despite some promising attempts [2, 6, 7, 8, 9], the dynamical (statistical
mechanical) origin of a black hole entropy has not been well understood.
Bekenstein [2] who introduced the notion of the black hole entropy related it
with ‘the measure of the inaccessibility of information (to an exterior
observer) as to which particular internal configuration of the black hole is
actually realized in a given state’ (i.e. for given values of black hole
parameters: mass, charge, and angular momentum.) According to the ‘standard’
interpretation these different internal states of a black hole are related
with different possible initial conditions which may result in the creation of
a stationary black hole with the same parameters [2]. In this approach the
entropy of a black hole is considered as the logarithm of the number of
distinct ways that the hole might have been made [8, 9].

This definition of the black-hole entropy resembles up to some extend the
usual definition of the entropy of matter. But there is a big difference which
makes the above ‘standard’ interpretation not completely satisfactory. The
entropy of matter is connected with its real internal dynamical degrees of
freedom which exist at given moment and which can be affected by an external
force. By getting information about the states of some of the internal degrees
of freedom one can reduce the entropy of matter. (The total entropy of matter
and its environment never decreases.) The loophole of the above described
‘standard’ interpretation is that one cannot indicate dynamical degrees of
freedom of a black hole which are responsible for its entropy. Even if an
observer is falling down into a black hole long time after its formation he
cannot receive more information about the initial state than the exterior
observer. The information concerning the initial conditions is lost for the
interior observer for the same reason as it is lost for an exterior observer
[10]. The collapsing body and its structure become invisible for the late-time
observer and only a few macroscopic parameters (mass, angular momentum, and
charge) remains measurable. This situation reminds the famous grin of the
Cheshire Cat remaining after the Cat himself had disappeared. In this sense
different possible initial states for a black hole are ‘Cheshire-Cat’s’
variables [11].

This difficulty of the ‘standard’ interpretation becomes especially vivid if
one considers a recently proposed gedanken experiment in which a traversable
wormhole is used to get information from black-hole’s interior [12]. Namely it
was shown that the area of the surface of a black hole decreases if one of the
mouths of a traversable wormhole is falling into a black hole while the other
mouth remains outside it. This decrease continues until the other mouth
crosses the horizon or the wormhole is destroyed. After this the surface area
of a black hole returns back to its initial value. There is an evident mystery
in such a behavior if in accordance with the usual rules one identifies the
entropy of a black hole with the loss of information about the possible origin
of a black hole, because the wormhole does not get any new information about
it. One can escape a contradiction with the generalized second law only if one
assumes that there exist some real internal degrees of freedom hidden inside
the black hole so that additional new information about the internal states of
a black hole can be obtained by using a wormhole during this process.

York [7] tried to solve the problem of ‘Cheshire-Cat’s’ variables by proposing
that quasi-normal modes of a black hole can play the role of its dynamical
degrees of freedom. This attempt also cannot be considered as completely
satisfactory. The entropy of the quasi-normal modes which are thermally
excited at the given moment of time is much smaller than
$A^{H}/(4l_{\mbox{\scriptsize{Pl}}}^{2})$ (the adopted value for the entropy
of a black hole). In order to obtain the black hole’s entropy York [7] makes
an additional assumption that the entropy “results when we add up all the
excited (normal modes) states that disappear if we allow the hole to evaporate
down to a final mass zero”. In other words the black hole’s entropy is again
related not with dynamical degrees of freedom which are really excited at the
given moment of time but with a number of different possibilities to excite
them during the life-time of a black hole.

In this paper we propose a new approach to the problem of statistical
mechanical calculation of the entropy of a black hole. The main idea of this
approach is to identify the dynamical degrees of freedom of a black hole with
those states of all physical (quantum) fields which are located inside a black
hole and cannot be seen by a distant observer. An excitation of these states
does not change the external parameters of the black hole. For fixed external
(macroscopic) parameters there exist many microscopically different (internal)
states which cannot be distinguished by observations made in the exterior
region. Due to quantum effects these internal states become thermally excited
and give contribution to the black hole entropy. This makes the definition of
the entropy and other thermodynamical characteristics of black holes quite
similar to those adopted in the usual statistical mechanical description of
matter. In the absence of mutual interactions one can consider the
contribution to the entropy of each of physical fields (including
gravitational perturbations) independently. It will be shown that the
resulting entropy of a black hole does not depend on the number of fields.

In order to make the definition of the black hole entropy more concrete we
assume that there exists a stationary black hole and denote by
$\hat{\rho}^{\mbox{\scriptsize{init}}}$ the density matrix describing in the
Heisenberg representation the initial state of quantum fields propagating in
its background. One may consider e.g. the in-vacuum state for a black hole
evaporating in the vacuum, or the Hartle-Hawking state for a black hole in
equilibrium with thermal radiation. For an exterior observer the system under
consideration consists of two parts: a black hole and radiation outside of it.
The state of radiation outside the black hole is described by the density
matrix which is obtained from $\hat{\rho}^{\mbox{\scriptsize{init}}}$ by
averaging it over the states which are located inside the black hole and are
invisible in its exterior

$\hat{\rho}^{\mbox{\scriptsize{rad}}}=\mbox{Tr}^{\mbox{\scriptsize{inv}}}\hat{\rho}^{\mbox{\scriptsize{init}}}.$
(1)

For an isolated black hole this density matrix
$\hat{\rho}^{\mbox{\scriptsize{rad}}}$ in particular describes its Hawking
radiation at infinity.

Analogously we define the density matrix describing the state of a black hole
as

$\hat{\rho}^{H}=\mbox{Tr}^{\mbox{\scriptsize{vis}}}\hat{\rho}^{\mbox{\scriptsize{init}}}.$
(2)

The trace-operators $\mbox{Tr}^{\mbox{\scriptsize{vis}}}$ and
$\mbox{Tr}^{\mbox{\scriptsize{inv}}}$ in these relations mean that the trace
is taken over the states located either outside (‘visible’) or inside
(‘invisible’) the event horizon correspondingly. We define the entropy of a
black hole as

$S^{H}=-\mbox{Tr}^{\mbox{\scriptsize{inv}}}(\hat{\rho}^{H}\ln\hat{\rho}^{H}).$
(3)

The proposed definition of the entropy of a black hole is invariant in the
following sense. Independent changes of vacuum definitions for ‘visible’ and
‘invisible’ states do not change the value of $S^{H}$. Bogolubov’s
transformations describing an independent changes of the vacuum states inside
and outside the black hole can be represented by the unitary operator
$\hat{U}=\hat{U}^{\mbox{\scriptsize{vis}}}\otimes\hat{U}^{\mbox{\scriptsize{inv}}}$
where $\hat{U}^{\mbox{\scriptsize{vis}}}$ and
$\hat{U}^{\mbox{\scriptsize{inv}}}$ are unitary operators in the Hilbert
spaces of ‘visible’ and ‘invisible’ particles, correspondingly. The above used
trace operators are invariant under such transformations.

In order to define the states one usually use modes expansion. The modes are
characterized by a complete set of quantum numbers. Due to the symmetry
properties one can choose such a subset $J$ of quantum numbers connected with
conservation laws (such as orbital and azimuthal angular momenta, helicity and
so on) that guarantees the factorization of the density matrices. In the
absence of mutual interaction of different fields the subset $J$ necessarily
includes also the parameters identifying the type of the field (e.g. mass,
spin, and charge). The factorization in particular means that

$\hat{\rho}^{\mbox{\scriptsize{init}}}=\otimes_{J}\hat{\rho}^{\mbox{\scriptsize{init}}}_{J},$
(4)

where $\hat{\rho}^{\mbox{\scriptsize{init}}}_{J}$ is acting in the Hilbert
space ${\cal H}_{J}$ of states with the chosen quantum numbers $J$, while the
complete Hilbert space is ${\cal H}=\otimes_{J}{\cal H}_{J}$. The
factorization also means that the separation into ‘visible’ and ‘invisible’
states can be done independently in each subset of modes with a fixed $J$ so
that

$\hat{\rho}^{H}=\otimes_{J}\hat{\rho}^{H}_{J}\hskip
28.45274pt\hat{\rho}^{H}_{J}=\mbox{Tr}^{\mbox{\scriptsize{vis}}}_{J}\hat{\rho}^{\mbox{\scriptsize{init}}}_{J},$
(5) $S^{H}=\sum_{J}S_{J}^{H},\hskip
28.45274ptS_{J}^{H}=-\mbox{Tr}^{\mbox{\scriptsize{inv}}}_{J}(\hat{\rho}^{H}_{J}\ln\hat{\rho}^{H}_{J}).$
(6)

where all the operators with subscript $J$ are acting in the Hilbert space
${\cal H}_{J}$.

We begin by calculating $S^{H}$ for a non-rotating uncharged black hole. We
suppose that the black hole is contained inside a spherical cavity $B$ of
radius $r_{0}$ with mirror-like boundary. We choose $r_{0}$ small enough to
guarantee the stable equilibrium of a black hole with thermal radiation inside
the cavity. Instead of physical metric $ds_{\mbox{\scriptsize{phys}}}^{2}$ it
is convenient to use its dimensionless form
$ds^{2}=r_{g}^{-2}ds_{\mbox{\scriptsize{phys}}}^{2}$ ,

where $r_{g}=2M$ and $M$ is the mass of the black hole. In the Kruskal
coordinates $(U,V)$ the metric $ds^{2}$ reads

$ds^{2}=-2BdUdV+x^{2}d\omega^{2},\hskip
28.45274ptd\omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2},$
$B=2x^{-1}\exp(1-x),\hskip 28.45274ptUV=(1-x)\exp(x-1),$ (7)

where $x=r/r_{g}$. The Killing vector $\xi$ normalized to unit at infinity in
the metric $ds^{2}$ is

$\xi^{\mu}\partial_{\mu}=1/2(V\partial_{V}-U\partial_{U}).$ (8)

For simplicity we consider a conformal massless scalar field $\hat{\varphi}$
obeying the equation

$\Box\hat{\varphi}-1/6R\hat{\varphi}=0.$ (9)

The following two sets $f^{UP}$ and $f^{SIDE}$ of classical complex solutions
of this equation which we call $UP$ and $SIDE$ modes

$f^{UP,SIDE}_{\nu lm}=x^{-1}F^{UP,SIDE}_{\nu l}(U,V)Y_{lm}(\theta,\phi),$ (10)
$\xi^{\mu}\partial_{\mu}F^{UP}_{\nu l}=-i(\nu/2)F^{UP}_{\nu l},\hskip
28.45274pt\xi^{\mu}\partial_{\mu}F^{SIDE}_{\nu l}=i(\nu/2)F^{SIDE}_{\nu l},$

will be used in our consideration. The functions $F^{UP}_{\nu l}$ and
$F^{SIDE}_{\nu l}$ obey the equation

$-\partial_{U}\partial_{V}F_{\nu l}-W_{l}F_{\nu l}=0,$ (11) $W_{l}\equiv
x^{-3}\exp(1-x)[l(l+1)+1/x].$ (12)

The $UP$ modes are radiated in the exterior space from horizon and vanish at
the past null infinity $\cal{J}^{-}$, while the $SIDE$ modes are radiated by
the inner part ($U>0$) of the horizon into the black hole’s interior [13]. We
denote by $\hat{\alpha}$ and $\hat{\alpha}^{*}$ with the corresponding
superscripts the operators of annihilation and creation of particles in these
modes. The normalized wave-packet-type solutions $f_{J}$ $(J=jnlm)$ are
constructed from modes $f_{\nu lm}$ as follows ($\delta>0$)

$f_{jnlm}=\delta^{-1/2}\int^{(j+1)\delta}_{j\delta}\exp(2\pi
in\nu/\delta)f_{\nu lm}d\nu.$ (13)

We denote by $\nu_{J}$ the average frequency of the wave packet $J$ and
introduce two sets of new modes $f_{J}^{B}$ and $f_{J}^{W}$ by the relations

$f^{B}_{J}=[1-w_{J}^{2}]^{-1/2}[f^{SIDE}_{J}+w_{J}\bar{f}^{UP}_{J}],$
$f^{W}_{J}=[1-w_{J}^{2}]^{-1/2}[f^{UP}_{J}+w_{J}\bar{f}^{SIDE}_{J}],$ (14)

where $w_{J}=\exp(-\pi\nu_{J})$. These modes are of positive frequencies with
respect to the affine parameter along the horizon $H^{-}$. In the presence of
mirror-like boundary surrounding the black hole the vacuum state with respect
to $B$ and $W$ modes coincides with the Hartle-Hawking state. The density
matrix corresponding to the Hartle-Hawking state takes the form (4) with

$\hat{\rho}^{\mbox{\scriptsize{init}}}_{J}=\rho^{0}_{J}:\exp\left[-\hat{\alpha}_{J}^{*B}\hat{\alpha}_{J}^{B}-\hat{\alpha}_{J}^{*W}\hat{\alpha}_{J}^{W}\right]:,$
(15)

where $\rho^{0}_{J}$ is the normalization constant, $:(\ldots):$ means normal
ordering with respect to operators $\hat{\alpha}^{*}$ and $\hat{\alpha}$ which
enter the expression (15). For the Hartle-Hawking state the $UP$-modes (as
well as $SIDE$-modes) are thermally excited. That is why for an eternal black
hole the density matrix (15) describes the equilibrium state of a black hole
with thermal radiation in its exterior. If the black hole is not eternal but
is formed in the process of the gravitational collapse the expression for
$\hat{\rho}^{\mbox{\scriptsize{init}}}$ should be modified. This modification
is important for the modes emitted at the time close to the moment of black-
hole’s formation. For late-time modes the expression (15) always provides
almost exact description . That is why instead of calculating the entropy of a
stationary black hole long after its formation it is possible to calculate the
entropy of an eternal black hole with the same parameters and for the initial
state decribed by the density matrix (15). We use this (technically more
simple) approach.

Consider a null surface $V=v$. Its intersection with the horizon represents
the surface of the black hole at a given moment of time. Denote by
$\Sigma_{v}$ a part of this null surface lying between the horizon and the
boundary of the cavity B. It is evident that $SIDE$-modes being propagated
inside the black hole never cross $\Sigma_{v}$ and hence according to our
definition they are ‘invisible’. Those $UP$-modes which reach $\Sigma_{v}$ are
to be considered as ‘visible’.

We introduce new coordinates $\eta,\xi$ in the exterior ($U<0$, $V>0$) region,
where $UP$-modes are propagating

$\eta=-\frac{1}{2}\ln(-V/U),\hskip 28.45274pt\xi=\frac{1}{2}q(x),\hskip
28.45274ptq(x)\equiv x-1+\ln(x-1).$ (16)

Then the wave equation (11) for modes $F^{UP}_{\nu l}$ takes the form

$[-\partial_{\eta}^{2}+\partial_{\xi}^{2}-V_{l}]F^{UP}_{\nu l}=0,$ (17)

where $V_{l}=4(x-1)x^{-3}[\mu_{l}^{2}-1+x^{-1}]$, $\mu_{l}^{2}=l(l+1)+1$. For
$l\gg 1$ the potential $V_{l}$ has maximum $\approx 16\mu_{l}^{2}/27$ at
$x=x_{m}\approx 3/2$. We call ‘escaping’ (or briefly E-modes) the $UP$-modes
with $\nu>4\mu_{l}/\sqrt{27}$ which (in the absence of the boundary $B$) are
propagating almost freely to infinity. We call ‘trapped’ (or briefly T-modes)
the $UP$-modes with $\nu<4\mu_{l}/\sqrt{27}$ which are mainly reflected by the
potential barrier and returned to the black hole’s horizon.

Let $f_{J}^{UP}$ be a ‘visible’ (either E- or T-) mode, i.e. a mode which
crosses $\Sigma_{v}$. Then the operation (5) of averaging over the states
corresponding to this mode gives the thermal density matrix

$\hat{\rho}^{H}_{J}={\normalsize
Tr}^{\mbox{\scriptsize{vis}}}_{J}\hat{\rho}^{\mbox{\scriptsize{init}}}_{J}=\rho^{0}_{J}\exp\left[-2\pi\nu_{J}\hat{\alpha}_{J}^{*SIDE}\hat{\alpha}_{J}^{SIDE}\right],$
(18)

$\nu_{J}$ being the frequency $\nu$ corresponding to the wave-packet $J$. The
standard calculation shows that the contribution of this mode $J$ to the
entropy of a black hole is [14]

$S_{J}=s(2\pi\nu_{J}),\hskip 28.45274pts(z)=\frac{z}{\exp
z-1}-\ln[1-\exp(-z)].$ (19)

It is convenient to calculate the separate contributions $S^{E}$ and $S^{T}$
of $E$ and $T$ modes and to write

$S^{H}=S^{E}+S^{T},$ (20)

where

$S^{E,T}\approx\sum_{jn}\\!^{{}^{\prime}}\sum_{l=0}^{\infty}(2l+1)\Theta[\pm(\nu_{j}^{2}-16\mu_{l}^{2}/27)]s(2\pi\nu_{j}).$
(21)

The sign $+$ in the step function $\Theta$ stands for $E$-modes, the sign $-$
stands for $T$-modes, and the prime in $\sum_{jn}^{\prime}$ means that the
summation is taken only over the modes which cross $\Sigma_{v}$.

In order to get $S^{E}$ we change the summation over $l$ by integration over
$\mu_{l}^{2}$ [15]

$\sum_{l=0}^{\infty}(2l+1)\Theta(\nu_{j}^{2}-16\mu_{l}^{2}/27)\approx\int_{0}^{27\nu_{j}^{2}/16}d\mu_{l}^{2}=\frac{27}{16}\nu_{j}^{2}.$
(22)

The expression standing under the sum for $S^{E}$ does not depend explicitly
on $n$. Denote by $N_{j}^{E}$ the total number of different values of the
index $n$ for which a mode with a given value of $j$ crosses $\Sigma_{v}$.
Then we have

$S^{E}\approx(27/16)\sum_{j}N_{j}^{E}\nu_{j}^{2}s(2\pi\nu_{j}).$ (23)

In the DeWitt’s approximation one can omit $V_{l}$ in Eq.(17) and to get that
$N_{j}^{E}=(u_{2}-u_{1})\delta/2\pi$, where $u_{1,2}$ are the values of the
retarded time $u=\eta-\xi$ corresponding to the boundary points of the surface
$\Sigma_{v}$.

The value $u_{2}$ is formally divergent. The necessary cut-off arises due to
the quantum fluctuation of the horizon. Zero-point fluctuations of the horizon
result in its spreading so that due to the quantum noise events happening
closer than $x_{\lambda}=1+\lambda$ to the horizon cannot be seen from
outside. One can show that

$\lambda=\alpha l_{\mbox{\scriptsize{Pl}}}^{2}/r_{g}^{2},$ (24)

where $\alpha$ is a dimensionless coefficient [7, 16, 17]. In our context it
means that we are to consider as ‘visible’ only those particles which cross
$\Sigma^{\lambda}_{v}$ which is the part of $\Sigma_{v}$ lying outside
$x_{\lambda}$. It should be stressed that for black holes with $M\gg
m_{\mbox{\scriptsize{Pl}}}$ (which we consider here) $\lambda\ll 1$.

By using this cut-off after changing the summation over $j$ by integration
($\delta\sum_{j=0}^{\infty}=\int_{0}^{\infty}d\nu$) we get

$S^{E}\approx\frac{27}{32\pi}\Delta u\int_{0}^{\infty}d\nu\nu^{2}s(2\pi\nu),$
(25)

where $\Delta u=q(x_{0})-\ln\lambda$ and the function $q(x)$ is defined in
Eq.(16).

Now we turn to the calculation of the contribution $S^{T}$ of $T$-modes to the
black-hole’s entropy. We write $S^{T}$ in the form

$S^{T}\approx\int_{0}^{\infty}d\nu J(\nu)s(2\pi\nu),$ (26)

where

$J(\nu_{j})=\delta^{-1}\sum_{l=0}^{\infty}(2l+1)N_{jl}^{T}\Theta(\frac{16}{27}\mu_{l}^{2}-\nu_{j}^{2}),$
(27)

and $N_{jl}^{T}$ is a number of $T$-modes with given quantum numbers $j$ and
$l$ which cross $\Sigma^{\lambda}_{v}$.

Note that due to the exponential decrease of $s(z)$ the frequencies $\nu\gg
1/2\pi$ do not contribute to $S^{T}$. For example, to provide the accuracy
$\sim 10^{-6}$ it is sufficient to consider the frequencies $\nu<2.2$. Denote
by $x(\nu)$ a turning point of a mode of frequency $\nu$
($V_{l}(x(\nu))=\nu^{2}$). For large $l$ one has
$x(\nu)-1\approx\nu^{2}/4\mu_{l}^{2}\ll 1$. In the region $|x-1|\ll 1$ one can
use the homogeneous-gravitational-field (HGF) approximation and put
$W_{l}=\mu_{l}^{2}$ in Eq.(11) and $B=x=1$ in the metric (7). This approximate
metric can be written in the Rindler-like form

$ds^{2}=-\rho^{2}d\eta^{2}+d\rho^{2}+d\omega^{2},$ (28)

where $\rho=2(x-1)^{1/2}$.

In the HGF-approximation normalized solutions $F_{\nu l}^{UP}$ of Eq.(11)
which take the value $(4\pi\nu)^{-1/2}\exp(-i\nu u)$ at the horizon $H^{-}$
are

$F_{\nu l}=A_{\nu
l}\pi^{-1}\sinh^{1/2}(\pi\nu)\exp(-i\nu\eta)K_{i\nu}(\mu_{l}\rho),$ (29)

where

$A_{\nu l}=-i\exp[-i(\nu\ln(\mu_{l}/2)-\phi_{\nu}],\hskip
28.45274pt\exp(2i\phi_{\nu)}=\Gamma(1+i\nu)/\Gamma(1-i\nu),$ (30)

and $K_{p}(z)$ is a Macdonald function.

In order to find $N_{jl}^{T}$ we note that each of the modes which crosses
$\Sigma^{\lambda}_{v}$ crosses also a space-like surface
$\Sigma^{\lambda}_{t}$ located between $p_{2}$ and mirror-like boundary $B$
and described by the equation $\eta=$const . It is possible to show that

$N_{jl}^{T}\approx
2\pi^{-2}\nu_{j}\delta\sinh(\pi\nu_{j})\int_{2\lambda^{1/2}}^{\infty}d\rho\rho^{-1}K_{i\nu}^{2}(\mu_{l}\rho).$
(31)

After substituting this relation into Eq.(27) and changing summation over $l$
by integration over $\mu_{l}^{2}$ we can rewrite this expression in the form

$J(\nu)\equiv
J_{a}(\nu)=\frac{\nu\sinh(\pi\nu)}{\pi^{2}\lambda}\int_{1}^{\infty}\frac{dy}{y}\int_{a}^{\infty}dzzK_{i\nu}^{2}(zy),$
(32)

where $a=(27\lambda)^{1/2}\nu/2$. For the frequencies $\nu$ which contribute
to the black hole’s entropy the parameter $a$ is extremely small $a\ll 1$. We
write

$J(\nu)=J_{a=0}(\nu)+\Delta J(\nu),\hskip 28.45274pt\Delta
J(\nu)=J_{a}(\nu)-J_{a=0}(\nu).$ (33)

The integrals which enter $J_{a=0}(\nu)$ can be taken exactly and we get

$J_{a=0}(\nu)=\frac{\nu^{2}}{4\pi\lambda},$ (34)

while for $\Delta J(\nu)$ one can obtain the following approximate expression

$\Delta
J(\nu)\approx\frac{27\nu^{2}}{32\pi}[\ln\lambda+\ln(27\nu^{2}/16)-1-\psi(1+i\nu)-\psi(1-i\nu)].$
(35)

Here $\psi(z)$ is the logarithmic derivative of $\Gamma$-function.

By adding this expression with the expression (25) for $S^{E}$ we finally get

$S^{H}=S^{H}_{0}+\Delta S,$ (36)

where

$S^{H}_{0}=\frac{1}{4\pi\lambda}\int_{0}^{\infty}d\nu\nu^{2}s(2\pi\nu)=\frac{1}{360\lambda},$
(37) $\Delta S=\frac{27}{32\pi}\int_{0}^{\infty}d\nu\nu^{2}Q(\nu)s(2\pi\nu),$
(38)

and $Q(\nu)=x_{0}-2+\ln(27\nu^{2}/16)-\psi(1+i\nu)-\psi(1-i\nu)$. Numerical
calculations give $\Delta S\approx(9x_{0}-23)\times 10^{-3}$.

One cannot expect to determine the entropy with accuracy higher than it is
allowed by uncertainties related with its thermodynamical fluctuations. That
is why the term $\Delta S$ in Eq.(36) which is much smaller than unity can be
neglected. Thus one can identify the black hole’s entropy with $S^{H}_{0}$. By
using Eq.(24) we get

$S^{H}_{0}=\gamma\frac{A_{H}}{4\l_{\mbox{\scriptsize{Pl}}}^{2}},\hskip
28.45274pt\gamma=\frac{1}{360\pi\alpha},$ (39)

where $A_{H}=4\pi r_{g}^{2}$ is the surface area of a black hole. It is
important to stress that $S^{H}_{0}$ does not depend on $r_{0}$ (the raduis of
the mirror-like boundary). It also does not depend on the particular choice of
the surface $\Sigma_{v}$ we introduced in order to define the separation into
’visible’ and ’invisible’ modes. For a stationary black hole the obtained
result is evidently invariant under the shift of the advanced time parameter
$v$. Instead of a null surface one can also use (without changing the result)
any space-like surface crossing the horizon. The result (39) reproduces the
standard expression for the black-hole’s entropy
$A_{H}/4l_{\mbox{\scriptsize{Pl}}^{2}}$ for the value of the parameter
$\alpha=(360\pi)^{-1}$.

The calculation of the coefficient $\alpha$ which enters the expression (24)
for the quantum fluctuations of the horizon is a delicate problem which
requires quantization of gravity. The following arguments allow one to
estimate its value. By direct measurements of black hole’s mass $M$ during
time interval $\tau$ one expects a measurement uncertainty $\Delta
M\approx\xi/\tau$ where $\xi\geq 1/2$ [17]. Spontaneous quantum emission (or
absorption) of particles by a black hole results in the jumps $\sim(8\pi
M)^{-1}$ of the mass $M$. These jumps do not allow one to take the interval
$\tau$ as long as one wishes without generating new additional uncertainties.
The best accuracy in a single measurement can be obtained if $\tau$ coincides
with the time interval $t_{1}$ between two subsequent events of emission or
absorption of quanta. The accuracy of defining of $M$ can be improved if
instead of a single measurement one makes a sequence of measurements. Let $N$
be a number of independent measurements and $t_{i}$ $(i=1,\ldots N)$ be the
time intervals of these measurements (i.e. the time intervals between the
emission or absorption of quanta). The probability of emission of a next
quantum at the time interval ($t,t+dt$) after the last previous emission is
$p(t)dt=\exp(-\bar{n}t)\bar{n}dt$, where $\bar{n}$ is the average number of
quanta radiated in a unit of time. An accuracy of the black hole’s mass
definition after large number $N$ of above described independent measurements
is [18]

$\Delta_{N}M\approx N^{-1/2}(\overline{t^{2}})^{-1/2}=(2N)^{-1/2}\bar{n},$
(40)

where $(\overline{t^{2}})=N^{-1}\sum_{i=1}^{N}t_{i}^{2}$.

In the above consideration it was inexplicitly assumed that the only origin of
the uncertainty of the measurement of the mass $M$ was connected with the
measurement procedure itself. Now we note that the quantum fluctuations of the
horizon do not allow to determine the mass $M$ of a black hole absolutely
exactly even if there would be no uncertainties in the measurement procedure
itself. The sequence of $N$ exact measurements of $M$ (in case if they were
possible) would give an accuracy $\bar{\Delta}_{N}M=N^{-1/2}\delta M$ where
$\delta M$ is the characteristic value of fluctuation of $M$ (its dispersion)
connected with quantum fluctuation of the horizon. It is natural to assume
that both accuracies (one ($\Delta_{N}M$) connected with the quantum
uncertainty of the measurement procedure and the other ($\bar{\Delta}_{N}M$)
connected with quantum fluctuations of the horizon) are of the same order of
magnitude $\Delta_{N}M\approx\bar{\Delta}_{N}M$. Hence one can write

$\delta M\approx 2^{-1/2}\xi\bar{n}.$ (41)

The value of the average particles number rate of emission $\bar{n}$ for a
scalar massless field was numerically calculated by Simkins [19] who found

$\bar{n}=6.644\times 10^{-4}M^{-1}.$ (42)

(DeWitt’s approximation gives very close result
$\bar{n}_{DeWitt}=3^{3}\zeta(3)/(2^{9}\pi^{4}M)\approx 6.5\times
10^{-4}M^{-1}$.) The value of the parameter $\alpha$ $(\delta M=\alpha/4M)$
corresponding to the expression (42) is

$\alpha\approx 1.88\times 10^{-3}\xi.$ (43)

This result is quite close to the York’s estimation of the quantum
fluctuations of the event horizon based on the deflection of the apparent
horizon from the event horizon for an evaporating black hole [16].

By using Eq.(43) one gets for the coefficient $\gamma$ which enters the
expression (39) $\gamma\approx 0.47\xi^{-1}$. For the oft-quoted minimal value
$\xi=1/2$ one has $\gamma\approx 0.94$. This estimation is in a good
accordance with the exact thermodynamical value of the black-hole’s entropy
for which $\gamma=1$.

We make now some general remarks concerning the obtained result. The entropy
of a black hole is of pure quantum nature. The gravitational field of a black
hole continuously creates pairs of particles. For alone static uncharged black
hole one of the components of a created pair is always located inside the
horizon while the other can escape to infinity and contribute to the Hawking
radiation. For low frequencies the probability of escape is exponentially
small so that almost all of the components of such pairs created outside the
horizon are reflected by the potential barrier and finally fall down into the
black hole. The existence of ‘invisible’ (hidden inside the horizon) modes
results in the entropy of the black hole. Eq.(2) shows that only those
‘invisible’ components of pairs contribute to the entropy of the black hole
for which the other component is ‘visible’. The life-time of ‘visible’
‘trapped’ modes is of order $\tau_{\nu l}\sim\kappa^{-1}\ln(l^{2}/\nu^{2})$.
For modes with large $l\sim\lambda^{-1/2}$ which give the main contribution to
the entropy this ‘life-time’ is $\sim
r_{g}\ln(r_{g}/\lambda_{\mbox{\scriptsize{Pl}}}$). The main contribution to
the entropy of a black hole is given by ‘invisible’ modes which are propagated
in the narrow $\sim\lambda_{\mbox{\scriptsize{Pl}}}$ shell region near the
horizon.

We estimated the contribution to entropy of a non-rotating uncharged black
hole by a conformal massless field. The generalization to the case of
stationary (rotating and charged) black holes and different physical fields is
straightforward. By using the above arguments one can expect that the number
of thermally exited trapped modes which contribute to the entropy will be
always proportional to the surface area of a black hole. If there are present
more than one fields each of them contributes to the entropy additively. On
the other hand the average rate $\bar{n}$ of the emission of the particles
grows as the number of fields $N$. That is why the parameter $\alpha$ which
enters the expression (39) for the entropy of a black hole and which
characterizes the quantum fluctuations of the horizon also grow as $N$. For a
large number $N$ of fields these two effects cancel each other so that the
entropy does not depend on the number of fields.

In conclusion we return to the gedanken experiment with a wormhole [12]
discussed at the beginning. The entropy of a black hole (identified with its
surface area) decreases when one of the wormhole’s mouths is falling inside a
black hole and returns to its initial value after another mouth crosses the
horizon or the wormhole is destroyed. The proposed interpretation of
‘invisible’ modes as dynamical degrees of freedom of a black hole allows one
to understand this ‘mysterious’ behavior of the entropy. In priciple byy using
a wormhole one can obtain an additional information concerning the quantum
states inside the black hole. The total number of the originally ‘invisible’
modes propagating near the gravitational radius which become ‘visible’ via the
wormhole is proportional to the change of the black hole surface area. The
decrease of number of ‘invisible’ states results in the decrease of the
entropy of the black hole. In principle by using a wormhole one can change the
states of some of the originally ‘invisible’ modes without changing external
parameters of the black hole [22]. This possibility is lost after the second
mouth of the wormhole crosses the horizon or the wormhole is destroyed. The
‘visible’ components of such excited modes continue living outside the horizon
only short time comparable with their ‘life-time’. After this they fall down
into the black hole and the corresponding pair does not contribute to the
entropy of the black hole. The system ‘forgets’ an intervention (if only it
did not change the black hole parameters) and the entropy of the black hole
returns to its initial value.

Note added: – After this paper was submitted for publication the paper by
Srednicki [20] appeared. For the massless scalar field in a flat spacetime he
showed that the entropy arising after tracing the degrees of freedom of the
field in the vacuum state residing inside a sphere is proportional to the area
of this sphere. The analogous result was obtained earlier in Ref.[21]. In the
present paper we have shown that the main contribution to the black-hole’s
entropy is given by modes, propagating near its surface, and hence the entropy
can also be considered as the ’surface effect’. Despite the formal similarity
of the results of the flat spacetime and of the black hole calculations there
is a big difference between them. The black hole horizon is a null surface.
Its geometry differs from the geometry of a timelike surface in a flat
spacetime. The physical meaning and the mathematical description of modes
which contribute to the entropy are quite different in both cases. That is why
it is not clear how far interesting arguments of Refs.[20, 21] based on the
flat spacetime calculations can be directly applied to black holes.

Acknowledgements: The authors are grateful to Werner Israel for stimulating
discussions. They also thank Don Page for helpful comments and his help in
finding the Ref.[19]. This work was supported in part by the Danish Natural
Sciences Research Council, grants N11-9640-1 and 11-9524-1SE and also by the
Natural Sciences and Engineering Research of Canada and by Deutsche
Forschungsgemeinschaft (V.F.).

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