# Quantum statistical mechanics near a black hole horizon

Eirini<EMAIL_ADDRESS>and Charis
<EMAIL_ADDRESS>
Department of Physics, University of Patras, 26504 Patras, Greece

###### Abstract

We undertake a first-principles analysis of the thermodynamics of a small body
near a black hole horizon. In particular, we study the paradigmatic system of
a quantum ideal gas in a small box hovering over the Schwarzschild horizon. We
describe the gas in terms of free quantum fields, bosonic and fermionic,
massive and massless. We identify thermodynamic properties through the
microcanonical distribution. We first analyse the more general case of a box
in Rindler spacetime, and then specialize to the black hole case. The physics
depends strongly on the distance of the box from the horizon, which we treat
as a macroscopic thermodynamic variable. We find that the effective dimension
of the system transitions from three-dimensional to two-dimensional as we
approach the horizon, that Bekenstein’s bound fails when the box is
adiabatically lowered towards the black hole, and that the pressure is highly
anisotropic. The pressure difference between the upper and lower wall leads to
an effective force that must be added to the gravitational acceleration. We
also show that the approximation of quantum fields propagating on a fixed
background for matter breaks down when the system is brought to microscopic
distances from the horizon, in which case backreaction effects must be
included.

##  1 Introduction

###  1.1 Motivation

Bekenstein’s proposal of black hole entropy [1] and Hawking’s derivation of
black hole radiation [2] signal our understanding of black holes as
thermodynamic objects. Furthermore, the generalized second law of
thermodynamics (GSL)[1, 3] asserts that black hole entropy adds up with matter
entropy, and that their sum never decreases with time.

The GSL originates from the thermodynamic analysis of small bodies falling
into a black hole, where by small we mean that their self-gravity is
negligible. This analysis assumes that the entropy of those small bodies is
the same that they would have in Minkowski spacetime. This assumption appears
reasonable if the dimensions of the body are much smaller than the curvature
radius of the spacetime. Nonetheless, it is problematic; black holes affect
the spacetime causal structure, and hence, the Hamiltonian of any small body
in its vicinity. Consequently, the presence of the horizon affects the density
of states that defines the entropy in the microcanonical distribution.

In this paper, we undertake a first-principles analysis of the thermodynamics
of a small body near a black hole horizon. The body under consideration is an
ideal quantum gas that is contained in a box hovering above the horizon. The
box is assumed to be sufficiently small, so that both self-gravity and the
backreaction to the black hole is negligible. Then, the relevant theory for
the particles in the box is Quantum Field Theory in curved spacetime (QFTCS).
For non-interacting particles, the evolution equation of the fields are
linear; hence, all properties of the QFTCS are determined by the structure of
the field modes. This allows us to evaluate the eigenvalues of the quantum
field Hamiltonian, and from them to identify all thermodynamic properties of
the system through the microcanonical distribution. We find a strong
dependence of thermodynamic variables on the distance of the box from the
horizon.

This work is firstly motivated by the need to further understand the GSL, by
constructing more accurate models. Bekenstein has argued that the GSL is
guaranteed if all thermodynamic systems satisfy an entropy bound (EB) [4],
namely, the ratio of the entropy $S$ to the energy $U$ obeys

$\displaystyle\frac{S}{U}\leq 2\pi D.$ (1)

Here, $D$ is a characteristic length scale, usually taken equal to the radius
of a sphere that encloses the system. There is independent evidence that this
bound is satisfied in flat-space systems [5], in weakly gravitating systems
[6, 7], and in strongly gravitating, static systems [8]. However, there is as
yet no analysis for matter in the vicinity of a black hole horizon, which was
the original set-up for Bekenstein’s EB. We undertake such an analysis in this
work.

Our second aim is to understand how black hole horizons affect the statistical
properties of matter. Quantum effects near the horizon may be manifested at a
macroscopic scale, for example, as the quantum buoyancy force of Unruh and
Wald [9, 10] that originates from Unruh radiation. This topic is significantly
under-represented in the literature. We believe that it is important as a
stepping stone towards a more nuanced description of the horizon that will
also involve the effects of quantum backreaction.

Our third motivation comes from the foundations of statistical mechanics. It
is far from obvious how to extend the usual recipes of statistical mechanics
to general relativity, where there is no preferred notion of time translation
[11]. The fact that gravity is a long-range force complicates the
thermodynamic description [12], for example, the canonical and the micro-
canonical ensembles may be inequivalent. In stationary spacetimes, a preferred
notion of time translation exists, so it is possible to define a time-
independent Hamiltonian, and hence, to analyze thermodynamic properties using
the microcanonical distribution. Whether the standard procedures of
statistical mechanics remain physically meaningful in setups so much removed
from the theory’s ordinary applications is an open issue, that can be explored
by the systems that we analyse in this paper.

###  1.2 Relation to past work

Padmanabhan et al analyzed the thermodynamical properties of a classical gas
in a box near the horizon [13, 14]. They used both a hydrodynamics description
and a description in terms of classical statistical mechanics. These works
look for an interplay between thermodynamic entropy of matter and the entropy
of the horizon. For this reason they focus on a box that lies at a proper
distance of the order of the Planck length from the horizon. Our analysis here
considers quantum gases and covers a broader range of distances. We also show
that backreaction cannot be ignored at small distances from the horizon, so
that the idea of a test body at a Planck-scale proper distance from the
horizon is not consistent. In a related context, Ref. [15] studied the
thermodynamics of a classical box of gas in Rindler spacetime, but did not
consider the black hole regime.

In Ref. [16], ’t Hooft analyzed the statistical mechanics of a gas of massive
particles inside a large spherical shell that surrounds a black hole. The
internal radius of the shell is barely larger than the Schwarzschild radius,
while the external radius is very large. As in our analysis, matter inside the
shell is treated at the level of QFTCS, i.e., backreaction is ignored. ’t
Hooft identifies a divergent entropy term $S_{div}$ as the internal radius of
the shell approaches the Schwarzschild radius, and he eventually identifies
$S_{div}$ with the Bekenstein-Hawking entropy of the black hole. The small box
analysed here is very different from the shell in ’t Hooft’s model, in
particular, it can be treated as a test body, for which backreaction is
negligible. Still, the spherical symmetry of the system allows us to reproduce
$S_{div}$ from the entropy of the small box.

Our analysis is also influenced by models of equilibrium black holes, i.e.,
models of black holes inside a box that coexist with their Hawking radiation
[18, 19, 20, 21, 17]. Ref. [17] incorporated backreaction in this setting, and
showed that a thermodynamically consistent description of black holes in a box
implies the existence of a thin but macroscopic shell around the horizon,
where Einstein’s equations fail. The shell likely consists of radiation with
exotic thermodynamics—see, also Ref. [22] for a similar conclusion in a
different context. However, [17] works only at the level of thermodynamics,
and a deeper analysis, at the level of quantum statistical mechanics is
required.

###  1.3 Our results

We study the thermodynamics of a quantum gas in a box hovering above a
Schwarzschild black hole horizon at a proper distance significantly smaller
than the Schwarzschild radius. There are different levels of description for
this system, in increasing degree of generality: (i) a purely hydrodynamic /
macroscopic analysis, (ii) a microscopic analysis using classical physics,
(iii) a microscopic analysis with quantum physics and (iv) a microscopic
quantum description that incorporates backreaction and self-gravity effects.
In this paper, we work at level (iii), moving beyond past analyses that worked
at levels (i) and (ii). To this end, it is necessary to describe matter with
QFTCS, which provides the only consistent quantum description of matter in a
background curved spacetime. For free fields in a static spacetime, the QFTCS
description is constructed solely from the properties of the single-particle
Hamiltonian. We identify a general exact form for the density of states of
this Hamiltonian, and then, we evaluate the relevant coefficients using a
semi-classical approximation. We derive all thermodynamic quantities using
standard methods from statistical mechanics.

The spacetime metric near the Schwarzschild horizon is approximately Rindler.
For this reason, we start our analysis from the thermodynamics of a box with
matter in Rindler spacetime. This allows us to incorporate in our study the
effects of acceleration on thermodynamics in settings other than black holes,
for example, finding correction to the equations of state in presence of weak
gravity.

Our most important findings are the following.

  1. 1.

We construct the thermodynamic potentials for quantum fields in an accelerated
cavity, modeled by imposing Dirichlet boundary conditions on free quantum
fields in Rindler spacetime. Thermodynamic quantities like entropy and
internal energy are well defined.

  2. 2.

We apply the results above to a small box near a Schwarzschild horizon. Then,
we check for the validity of the Bekenstein bound. In the simplest
interpretation, where $D$ in Eq. (1) is identified with the proper radius of
the box, we identify a regime of physically admissible parameters in which the
bound is violated. In fact, a box that is lowered adiabatically towards the
black hole will at some point violate Bekenstein’s bound. We discuss possible
ways that the bound may be restored.

  3. 3.

For radiation, the scaling of entropy transitions from three-dimensional to
effectively two-dimensional as we approach the horizon. For particles of mass
$m$, and temperatures $T<<m$, the system always scales as two-dimensional near
the horizon.

  4. 4.

In an accelerated box, the pressure at the bottom wall is always larger than
the pressure at the top wall. The pressure differential leads to an effective
acceleration on the box that is induced by the horizon. This is a novel
phenomenon, we can only compare it with the Unruh-Wald buoyancy, even if the
physical origins are very different. However, the acceleration we derive here
points towards the horizon, and it adds to the gravitational acceleration.

  5. 5.

Our approximations fail when we bring the box to microscopic distances from
the horizon. The whole system of the box and of the mechanism that keeps it
static cannot be viewed as a test system, and its backreaction to the black
hole must be taken into account. This means that the limit of taking the box
to proper distance of order of the Planck length from the horizon is not well
defined.

The structure of this paper is the following. In Sec. 2, we analyse the
quantum theory of a scalar field in a box in Rindler spacetime. Then, we
consider the case of a Dirac field. In Sec. 3, we analyse the thermodynamic
properties of an accelerated box. In Sec. 4, we focus on gases of massless
bosons in the vicinity of the black hole horizon. In Sec. 5, we study massive
bosons at low temperatures near the horizon. In Sec. 6, we summarize our
results and discuss their implications. In the Appendix, we extend our
analysis to fermions.

##  2 The quantum theory of a gas in a box in Rindler spacetime

###  2.1 Setup

In this section, we construct the density of states of a bosonic field
enclosed in a rectilinear box at rest in Rindler spacetime with acceleration
$a$

$\displaystyle ds^{2}\ =\ -a^{2}x^{2}\ dt^{2}\ +\ dx^{2}\ +\ dy^{2}\ +\
dz^{2},$ (2)

expressed in terms of the standard coordinates $(t,x,y,z)$. The lowest wall of
the box lies at $x_{1}$, and the upper wall at $x_{2}$. The proper length in
the $x$ direction is $H_{x}=x_{2}-x_{1}$. We denote the length of the box in
the $y$ coordinate by $L_{y}$ and in the $z$ coordinate by $L_{z}$.

We analyse a free scalar field inside the box, subject to the Klein-Gordon
equation,

$\displaystyle\ddot{\Phi}-ax\
\partial_{x}(ax)\partial_{x}\Phi-(ax)^{2}\big{(}\partial_{x}^{2}+\partial_{y}^{2}+\partial_{z}^{2}-m^{2}\big{)}\Phi\
=\ 0\ .$ (3)

We solve the Klein-Gordon equation subject to Dirichlet boundary conditions on
the box

$\displaystyle\Phi(x_{1},z,t)=\Phi(x_{2},y,z,t)\ =\ 0\ ,$
$\displaystyle\Phi(x,0,z,t)=\Phi(x,L_{y},z,t)\ =\ \Phi(x,y,0,t)\ =\
\Phi(x,y,L_{z},t)\ =\ 0\ .$ (4)

The Dirichlet boundary conditions mimic walls that reflect particles
elastically.

We look for mode solutions to Eq. (3) of the form $\exp[-i(\omega
t-k_{y}y-k_{z}z)]f(x)$. Substituting into Eq. (3), we obtain

$\displaystyle x^{2}f^{{}^{\prime\prime}}\ +\ xf^{{}^{\prime}}\ +\
\bigg{[}\dfrac{\omega^{2}}{a^{2}}-\kappa^{2}\ x^{2}\bigg{]}f\ =\ 0\ ,$ (5)

where $\kappa=\sqrt{k_{y}^{2}+k_{z}^{2}+m^{2}}$.

The general solution of Eq. (5) is a linear combination of a modified Bessel
function of the first type $I_{i\omega/a}(\kappa x)$ and of a modified Bessel
function of the second type $K_{i\omega/a}(\kappa x)$ [23]. The implementation
of the Dirichlet boundary conditions with the modified Bessel functions leads
to an algebraic equation from which we can determine the energy eigenvalues
only numerically. In this paper, we will obtain analytic expressions through
the use of a semi-classical approximation.

###  2.2 The general form of the density of states

The thermodynamic analysis of a free-particle system does not require an
explicit form of the energy eigenvalues, rather it suffices to identify the
single-particle density of states at high energies. We will show that the
general form of this quantity can be determined without any approximation. We
will only employ a semiclassical approximation in order to determine specific
coefficients that appear in the density of states. The semi-classical
evaluation is accurate at high energies, hence gas temperatures. In Sec. 3, we
will see that this approximation is reliable for temperatures significantly
larger than the black hole’s Hawking temperature.

We go back to Eq. (5), and we change variables to $\xi=\ln(ax)$. Eq. (5)
becomes a Schrödinger-type equation

$\displaystyle f^{\prime\prime}(\xi)+[E-U(\xi)]f(\xi)=0\ ,$ (6)

with ‘energy’ $E=\dfrac{\omega^{2}}{a^{2}}$ and ‘potential’
$U(\xi)=\frac{\kappa^{2}}{a^{2}}e^{2\xi}$.

Let us denote by $V(\omega,\kappa)$ the number of eigenstates of Eq. (6) with
energy less than $\dfrac{\omega^{2}}{a^{2}}$. For large energies, where $E$
becomes approximately continuous, $V(\omega,\kappa)$ coincides with the area
(actually length) of the classical energy surface

$\displaystyle\left(a\frac{k_{\xi}}{\omega}\right)^{2}+\frac{\kappa^{2}}{\omega^{2}}e^{2\xi}=1$

on the $\xi-k_{\xi}$ phase space. The symmetry of the energy surface under
rescaling $k_{\xi}\rightarrow\lambda k_{\xi}$,
$\kappa\rightarrow\lambda\kappa$, and $\omega\rightarrow\lambda\omega$,
implies that the function $V(\omega,\kappa)$ is homogeneous of first order
with respect to its arguments. Hence, we can express $V(\omega,\kappa)$ in the
convenient form $V(\omega,\kappa)=\omega\zeta(\rho)$ for some function $\zeta$
of $\rho:=(\kappa/\omega)^{2}$.

Then, the number $\Omega(\omega)$ of mode solutions to Eq. (3) with energy
less that $\omega$ is obtained by integrating $V(\omega,\kappa)$ with respect
to $k_{y}$ and $k_{z}$,

$\displaystyle\Omega(\omega)\ =\ \frac{L_{y}L_{z}}{\pi^{2}}\int
dk_{y}dk_{z}V(\omega,\kappa)\ =\ \frac{L_{y}L_{z}}{\pi}\
\omega^{3}\int_{\rho_{min}}^{\rho_{max}}d\rho\zeta(\rho)\ \ .$ (7)

The maximal value $\rho_{max}$ is fixed by the boundary conditions. The
minimal value $\rho_{min}$ is $m^{2}/\omega^{2}$ and it is obtained for
$k_{y}=k_{z}=0$. Hence,

$\displaystyle\Omega(\omega)\ =\ \frac{L_{y}L_{z}}{\pi}\ \omega^{3}\
[F(\rho_{max})-F(m^{2}/\omega^{2})]\ ,$ (8) $\displaystyle\text{where}\hskip
142.26378ptF(\rho):=\int_{0}^{\rho}d\rho^{\prime}\zeta(\rho^{\prime}).\hskip
170.71652pt$ (9)

The density of states $g(\omega)$ is the derivative of $\Omega(\omega)$.

Eq. (8) applies to the regime where $\omega$ can be treated as continuous, and
it does not involve any approximation. In particular, it implies that for
$m=0$, $\Omega(\omega)$ is always proportional to $\omega^{3}$. In this case,
it is convenient to define the effective length $\bar{L}_{x}$ in the $x$
direction as

$\displaystyle\bar{L}_{x}:=\frac{3\pi}{2}F(\rho_{max})\ ,$ (10)

so that

$\displaystyle\Omega(\omega)=\frac{2\bar{L}_{x}L_{y}L_{z}}{3\pi^{2}}\omega^{3}.$
(11)

Then, the number-of-states function is formally identical with that of a gas
of massless particles in flat spacetime: the Rindler acceleration has been
fully incorporated in the definition of the effective length. The factor
$\frac{2}{3}$ is due to the Dirichlet boundary conditions.

For Neumann or Robin boundary conditions on the box, the density of states
$g(\omega)$ remains proportional to $\omega^{2}$ and only a proportionality
constant changes. Thermodynamic variables are not significantly affected.

###  2.3 Density of states in a semi-classical approximation

Next, we evaluate the function $F(\rho)$ of Eq. (9) in a semi-classical
approximation. The Wigner-Kramers-Brillouin (WKB) solutions to Eq. (6) are of
the form

$\displaystyle f(\xi)\ =\ \left\\{\begin{array}[]{cc}\dfrac{1}{\sqrt{k(\xi)}}\
\big{(}Ae^{iS_{1}(\xi_{1},\xi)}+Be^{-iS_{1}(\xi_{1},\xi)}\big{)}\
,&\xi<\xi_{c}\\\ \dfrac{1}{\sqrt{\lambda(\xi)}}\
\Big{(}Ce^{-S_{2}(\xi_{c},\xi)}+De^{S_{2}(\xi_{c},\xi)}\Big{)}\
,&\xi>\xi_{c}\end{array}\right.$ (14)

where

$\displaystyle k(s)$ $\displaystyle=$ $\displaystyle\sqrt{E-U(s)}\ ,\hskip
17.07182pt\lambda(s)=\sqrt{U(s)-E},$ (15) $\displaystyle
S_{1}(\xi,\xi^{\prime})$ $\displaystyle=$
$\displaystyle\int_{\xi}^{\xi^{\prime}}k(s)ds,\hskip
11.38092ptS_{2}(\xi,\xi^{\prime})=\int_{\xi}^{\xi^{\prime}}\lambda(s)ds,$ (16)

and $\xi_{c}=-\ln\sqrt{\rho}$ is the turning point. The form of the solution
depends on the relation of $\xi_{c}$ to $\xi_{1}$ and $\xi_{2}$. In
particular, the box interior

  * •

is a classically forbidden region, if $\xi_{c}<\xi_{1}$;

  * •

is a classically allowed region, if $\xi_{c}>\xi_{2}$;

  * •

contains a classically allowed and a classically forbidden region, if
$\xi_{1}<\xi_{c}<\xi_{2}$.

Since we focus on the density of states $g(\omega)$ at the continuum limit,
and not on a precise evaluation of all eigenvalues, we can ignore the
contribution from the classically forbidden regions. This approximation is
often referred to as the geometric optics approximation to Schrödinger’s
equation. Hence, we consider only the oscillating solutions in Eq. (14)
subject to the boundary conditions

  * •

$f(\xi_{1})=0$

  * •

$f(\xi_{2})=0$ if $\xi_{c}>\xi_{2}$; $f(\xi_{c})=0$, if $\xi_{c}<\xi_{2}$, .

It follows that $\sin S_{1}(\xi_{1},\xi_{2})=0$ for $\xi_{c}>\xi_{2}$, and
$\sin S_{1}(\xi_{1},\xi_{c})=0$ for $\xi_{c}<\xi_{2}$. Equivalently,
$S_{1}(\xi_{1},\xi_{2})=\ell\pi$ for $\xi_{c}>\xi_{2}$ and
$S_{1}(\xi_{1},\xi_{c})=\ell\pi$ for $\xi_{c}<\xi_{2}$, $\ell=1,2,\ldots$.

Changing the integration variable in $S_{1}$ to $q:=\rho e^{2\xi}$, we obtain

$\displaystyle\frac{2\ell\pi a}{\omega}=\int_{\rho s_{1}}^{c(\rho
s_{2})}dq\frac{\sqrt{1-q}}{q},$ (17)

where we defined $s_{1}=(ax_{1})^{2}$, $s_{2}=(ax_{2})^{2}$, and

$\displaystyle c(x)=\left\\{\begin{array}[]{cc}x\ ,&x<1\\\ 1\ ,&\ \ \ x>1\ \
.\end{array}\right.$ (20)

Given the eigenvalues (17), we straightforwardly evaluate the number-of-states
function $V(\kappa,\omega)=\omega\zeta(\rho)$, where

$\displaystyle\zeta(\rho)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
a}\int_{\rho s_{1}}^{c(\rho s_{2})}dq\frac{\sqrt{1-q}}{q}.\ \ $ (21)

The maximum value of $\rho$ is $\rho_{max}=\frac{1}{s_{1}}$. Then, the
function $F(\rho)$ of Eq. (8) is

$\displaystyle F(\rho)\ =\ \frac{1}{2\pi
a}\times\left\\{\begin{array}[]{cc}\frac{u(\rho s_{2})}{s_{2}}-\frac{u(\rho
s_{1})}{s_{1}},&\rho<\frac{1}{s_{2}}\\\ \frac{u(1)}{s_{2}}-\frac{u(\rho
s_{1})}{s_{1}},&\rho>\frac{1}{s_{2}}\end{array}\right.$ (24)

where

$\displaystyle u(x)\ =\
\frac{2}{3}\sqrt{1-x}(1+2x)-\frac{2}{3}-2x\tanh^{-1}\sqrt{1-x}\ .$ (25)

The function $F$ is increasing by construction; it is plotted in Fig. 1.

Figure 1: The function $F$ multiplied with $2\pi as_{2}$ as a function of
$\rho s_{2}$ for of $s_{2}/s_{1}=10$.

Eq. (24) implies that

$\displaystyle F(\rho_{max})=F(s_{1}^{-1})=\ \dfrac{1}{2\pi a}\ \frac{2}{3}\
\Big{(}s_{1}^{-1}-s_{2}^{-1}\Big{)}\ .$ (26)

It follows that

$\displaystyle\bar{L}_{x}=\frac{1}{2a^{3}}(x_{1}^{-2}-x_{2}^{-2}).$ (27)

##  3 Thermodynamic properties of the accelerated box

In this section, we analyse the thermodynamical properties of the accelerated
box. We consider massless and massive bosons, and we also examine the weak-
acceleration limit.

###  3.1 Entropy

By Eqs. (8) and (26), the single-particle density of states for $m=0$ is

$\displaystyle
g(\omega):=\frac{d\Omega(\omega)}{d\omega}=\frac{2\bar{L}_{x}L_{y}L_{z}}{\pi^{2}}\omega^{2}=\frac{2\bar{V}}{\pi^{2}}\omega^{2},$
(28)

where we wrote $\bar{V}=\bar{L}_{x}L_{y}L_{z}$.

The density of states is given by the standard expression for photons in flat
spacetime, modulo the substitution of $V$ with $\bar{V}$. The partition
function is standardly evaluated [24],

$\displaystyle\log Z(\beta,\bar{V})=-\int_{0}^{\infty}d\omega
g(\omega)\log(1-e^{-\beta\omega})=\beta\int_{0}^{\infty}d\omega\frac{\Omega(\omega)}{e^{\beta\omega}-1}=\frac{2\pi^{2}}{45}\bar{V}\beta^{-3}.$
(29)

In the canonical distribution, we identify $-\beta^{-1}\log Z$ with the
Helmholtz free energy, and thereby obtain the thermodynamic description of the
system. However, the appropriate ensemble for an isolated box with adiabatic
walls is the microcanonical one. Since acceleration renders the system non-
extensive, the canonical and the microcanonical ensemble are not guaranteed to
be equivalent [12].

The partition function $Z(\beta)$ is the Laplace transform of the density of
states function $\Gamma(E)$, which defines the microcanonical entropy
$S(E)=\log\Gamma(E)$. We evaluate the inverse Laplace transform of $Z(\beta)$
in the first-order saddle-point approximation [24], to obtain

$\displaystyle\Gamma(E)=\left[8\pi(2\pi^{2}/15)^{-1/4}E^{5/4}\bar{V}^{-1/4}\right]^{-1/2}\exp\left(\frac{4}{3}(2\pi^{2}/15)^{1/4}E^{3/4}\bar{V}^{1/4}\right).$
(30)

The standard expression for the photon entropy follows, modulo terms of order
$\ln E/E$ and $\ln\bar{V}/\bar{V}$,

$\displaystyle
S(E,\bar{V})=\frac{4}{3}\left(\frac{2\pi^{2}}{15}\right)^{1/4}E^{3/4}\bar{V}^{1/4}.$
(31)

Identifying $E$ with the internal energy $U$, we obtain results fully
consistent with the canonical distribution. The difference is that the
temperature is now defined as $T:=(\partial U/\partial S)_{V}$, and we need
make no reference to a heat bath. Since energy is defined with respect to
asymptotic time translations, and entropy is a scalar, the temperature $T$
refers to observers at infinity. Assuming Tolman’s law, we can define a local
temperature $T_{loc}(x)=T\sqrt{-g_{00}(x)}=T/(ax)$ associated to planes of
constant Rindler coordinate $x$ inside the box.

The equations for entropy and internal energy are

$\displaystyle S=\frac{8\pi^{2}}{45}\bar{V}T^{3},\hskip
28.45274ptU=\frac{2\pi^{2}}{15}\bar{V}T^{4}.$ (32)

What changes from flat space is the proportionality constant $\sigma$ (the
Stefan-Boltzmann constant) in the relation $U=\sigma VT^{4}$,

$\displaystyle\sigma=\frac{2\pi^{2}}{15}\frac{\bar{L}_{x}}{L_{x}}.$ (33)

Weak acceleration limit. The usual spacetime metric for a weak homogeneous
gravitational field

$\displaystyle
ds^{2}=-(1+a\bar{x})^{2}dt^{2}+\frac{d\bar{x}^{2}}{(1+a\bar{x})^{2}}+dy^{2}+dz^{2}$
(34)

can be brought into the Rindler form (2) with a change of coordinates
$x=\frac{1}{a}+\bar{x}+O(a)$. Since $\bar{x}$ is a Cartesian coordinate, the
proper length $H_{x}$ is identified with the coordinate length $L_{x}$. We
shift the coordinate $\bar{x}$, so that $x_{1}$ corresponds to $\bar{x}=0$.
Then, $x_{1}=\frac{1}{a}$, $x_{2}=\frac{1}{a}+L_{x}$, and

$\displaystyle\bar{L}_{x}=L_{x}+\frac{3}{2}L_{x}a+O[\left(L_{x}a\right)^{2}].$
(35)

Hence, black-body energy to leading order in $L_{x}a$ is
$U=U_{0}(1+\frac{3}{2}L_{x}a)$, where $U_{0}$ is the energy in absence of
gravity. The relative size of the correction term for a box of length
$L_{x}\sim 100$m in the gravitational field of the Earth is of the order of
$10^{-14}$.

###  3.2 Pressure

The entropy (31) is a function on the thermodynamic state space $\Gamma$,
i.e., a manifold spanned by the variables $U,L_{y},L_{z},x_{1}$ and $x_{2}$.
The ‘lengths’ $L_{x},L_{y},L_{z},$ and $\epsilon$ are coordinates of a
submanifold $\Gamma_{m}$ of $\Gamma$ for the mechanical degrees of freedom.
Any variation of the internal energy with fixed entropy defines an one-form on
$\Gamma_{m}$

$\displaystyle(dU)_{\Gamma_{m}}=-\left(\frac{\partial U}{\partial
y^{a}}\right)_{S}dy^{a},$ (36)

where we used $y^{a}$ as a shorthand for the coordinates
$(L_{x},L_{y},L_{z},\epsilon)$.

All variables of $\Gamma_{m}$ are lengths that corresponds to changes in the
location of a wall of the box. The derivatives in Eq. (36) lead to the
definition of pressures. For this reason, we will refer to $(dU)_{\Gamma_{m}}$
as the pressure one-form. Since the system is not homogeneous, it cannot be
described by a single scalar for pressure. Each vector field $X^{a}$ on
$\Gamma_{m}$ defines a different change of the walls, hence, each contraction
$(dU)_{\Gamma_{m}}(X)$ corresponds to a different type of pressure.

We proceed to define pressures with respect to the Rindler coordinate system.
For the horizontal pressure $P_{h}$, we choose either of the vector fields
$X_{1}=\dfrac{1}{H_{x}L_{z}}\dfrac{\partial}{\partial L_{y}}$ or
$X_{2}=\dfrac{1}{H_{x}L_{y}}\dfrac{\partial}{\partial L_{z}}$. Then,

$\displaystyle P_{h}:=-\frac{1}{H_{x}L_{z}}\left(\frac{\partial U}{\partial
L_{y}}\right)_{S,L_{z},x_{1},x_{2}}=\frac{1}{3}\rho,$ (37)

where

$\displaystyle\rho=\frac{U}{H_{x}L_{y}L_{z}}=\frac{2\pi^{2}}{15}T^{4}\frac{\bar{L}_{x}}{H_{x}}$
(38)

is the average energy density in the box.

The pressure $P_{b}$ at the bottom of the box corresponds to a vector field
$X_{b}=\dfrac{1}{L_{y}L_{z}}\dfrac{\partial}{\partial x_{1}}$. Then,

$\displaystyle
P_{b}:=\frac{2\pi^{2}T^{4}}{15a^{3}x_{1}^{3}}=P_{h}\frac{x_{2}^{2}}{\frac{1}{2}(x_{1}+x_{2})x_{1}}.$
(39)

Similarly, we define the pressure at the top of the box in terms of the vector
field $X_{t}=-\dfrac{1}{L_{y}L_{z}}\dfrac{\partial}{\partial x_{2}}$,

$\displaystyle
P_{t}:=\frac{2\pi^{2}T^{4}}{15a^{3}x_{2}^{3}}=P_{h}\frac{x_{1}^{2}}{\frac{1}{2}(x_{1}+x_{2})x_{2}}.$
(40)

The three pressures always satisfy

$\displaystyle P_{b}>P_{h}>P_{t}.$ (41)

We rescale coordinates so that $ax_{1}=1$. Then, the quantities
$\Pi_{h,b,t}=\frac{45a^{3}x_{2}^{3}}{2\pi^{2}T^{4}}P_{h,b.t}$ are functions
solely of $aH_{x}$,

$\displaystyle\Pi_{h}=\frac{1+\frac{1}{2}H_{x}a}{(1+H_{x}a)^{2}},\;\;\;\Pi_{b}=1,\;\;\;\Pi_{t}=\frac{1}{(1+H_{x}a)^{3}}.$
(42)

These quantities are plotted in Fig. 2.

Figure 2: The quantities $\Pi_{h},\Pi_{b},\Pi_{t}$ expressed as functions of
$H_{x}a$. The inequality (41) is evident.

###  3.3 Gas of massive bosons

For $m\neq 0$, the number-of-states function has a more complex dependence on
the energy $\omega$, and there are several different thermodynamic regimes
depending on the relative values of the various parameters that characterize
the system. Here, we will consider the ‘non-relativistic’ regime $T<<m$.

For $m\neq 0$, Eq. (8) implies that there is minimum value of $\omega$,

$\displaystyle\omega_{min}(\epsilon)=m/\sqrt{\rho_{max}}=m\sqrt{s_{1}},$ (43)

which functions as an effective rest mass of the bosons.

For $T<<m$, the values of $\omega$ around $\omega_{min}$ dominate. We evaluate
the number-of-states function (8), by expanding $F(\rho)$ around $\rho_{max}$.
To this end, we define $\xi:=1-\rho/\rho_{max}\in[0,1)$, and we expand

$\displaystyle u(1-\xi)\simeq-\dfrac{2}{3}\ +\ \dfrac{4}{15}\ \xi^{5/2}\ +\
\dfrac{4}{35}\ \xi^{7/2}+\mathcal{O}(\xi^{9/2}).{}$

Then, to leading order in $\xi$, we obtain

$\displaystyle F(s_{1}^{-1})-F(\xi)=\left\\{\begin{array}[]{cc}\frac{2}{15\pi
as_{1}}\left[\xi^{5/2}-\left(\frac{s_{2}}{s_{1}}\right)^{3/2}(\xi-1+\frac{s_{1}}{s_{2}})^{5/2}\right],&\xi>1-\frac{s_{1}}{s_{2}}\\\
\frac{2}{15\pi as_{1}}\xi^{5/2},&\xi\leq
1-\frac{s_{1}}{s_{2}}\end{array}\right.$ (46)

It is convenient to express the number-of-states function $\Omega$ in terms of
$q:=\omega-\omega_{min}$, the effective kinetic energy of a particle. By
definition, $\dfrac{m^{2}}{\omega^{2}}\ =\ \rho_{max}(1-\xi)$, hence,
$\omega/\omega_{min}=(1-\xi)^{-1/2}=1+q/\omega_{min}$. Hence, to leading-order
in $\xi$, $\xi\simeq 2q/(m\sqrt{s_{1}})$ and we write

$\displaystyle\Omega(\xi)=\frac{2L_{y}L_{z}m^{3}}{15\pi^{2}as_{1}}\left\\{\begin{array}[]{cc}\xi^{5/2}-\left(\frac{s_{2}}{s_{1}}\right)^{3/2}\left(\xi-1+\frac{s_{1}}{s_{2}}\right)^{5/2}&\xi>1-\frac{s_{1}}{s_{2}}\\\
\xi^{5/2}&\xi<1-\frac{s_{1}}{s_{2}}\end{array}\right..$ (49)

The number-of-states function is very sensitive on the ratio $s_{1}/s_{2}$.
The regime where $s_{1}/s_{2}<<1$ is appropriate to a box near a black hole
horizon, and it is explored in detail in Sec. 5. Here, we consider the
‘classical’ regime that corresponds to a dilute gas, in which case the
partition function can be evaluated using the Boltzmann distribution. For a
gas of $N$ particles, $Z=Z_{1}^{N}$, where
$Z_{1}=\int_{0}^{\infty}dqg(q)e^{-\beta(m\sqrt{s_{1}}+q)}=e^{-\beta
m\sqrt{s_{1}}}\beta\int_{0}^{\infty}dq\Omega(q)e^{-\beta q}$. We calculate,

$\displaystyle Z_{1}=\sqrt{\frac{2}{\pi
m}}\frac{L_{y}L_{z}}{\pi^{2}\beta^{5/2}a(ax_{1})^{3/4}}e^{-\beta
max_{1}}\left[1-\left(\frac{x_{2}}{x_{1}}\right)^{3}e^{-\frac{1}{2}\beta
max_{1}(1-x_{1}^{2}/x_{2}^{2})}\right].$ (50)

The partition function leads to analytic expressions for all thermodynamic
observables. In particular, the internal energy per particle is

$\displaystyle
U/N=max_{1}+\frac{5}{2\beta}-\frac{\frac{1}{2}max_{1}(1-x_{1}^{2}/x_{2}^{2})}{e^{\frac{1}{2}\beta
max_{1}(1-x_{1}^{2}/x_{2}^{2})}-1}.$ (51)

This quantity is an increasing function of $x_{2}/x_{1}$, and it converges to
$max_{1}+\frac{5}{2\beta}$ as $x_{2}/x_{1}\rightarrow\infty$. In the weak
gravity limit ($ax_{1}=1$, $x_{2}=x_{1}+L_{x}$),
$U/N=m+\frac{3}{2}T+\frac{1}{2}maL_{x}$, i.e., the leading order correction is
the dynamical energy of the center of the mass.

We straightforwardly evaluate the pressures,

$\displaystyle P_{h}$ $\displaystyle=$ $\displaystyle\frac{N}{\beta
H_{x}L_{y}L_{z}},$ (52) $\displaystyle P_{b}$ $\displaystyle=$ $\displaystyle
P_{h}\,\beta
maH_{x}\left(1+\frac{\frac{3x_{1}^{2}}{2x_{2}^{2}}-\frac{1}{2}}{\left(\frac{x_{1}}{x_{2}}\right)^{3}e^{\frac{1}{2}\beta
max_{1}(1-x_{1}^{2}/x_{2}^{2})}-1}\right),$ (53) $\displaystyle P_{t}$
$\displaystyle=$ $\displaystyle P_{h}\,\beta
maH_{x}\dfrac{\frac{x_{1}^{3}}{x_{2}^{3}}}{\left(\frac{x_{1}}{x_{2}}\right)^{3}e^{\frac{1}{2}\beta
max_{1}(1-x_{1}^{2}/x_{2}^{2})}-1}$ (54)

In Fig. 3, we plot the ratios $P_{t}/P_{h}$ and $P_{b}/P_{h}$. The pressures
satisfy again inequality (41).

Figure 3: The logarithms of the ratios $P_{t}/P_{h}$ and $P_{b}/P_{h}$ as a
function of $H_{x}a$ for $ax_{1}=1$.

##  4 Massless particles near a Schwarzschild horizon

In this section, we apply the analysis of the previous section to a box at
rest near a black hole horizon, i.e., we assume that the box is accelerated so
that the acceleration cancels free fall. We will work with the massless case
here (radiation), and take up the massive case in the following section.

###  4.1 Setup

We consider a Schwarzschild spacetime of mass $M$,

$\displaystyle ds^{2}\ =\ -\Bigg{(}1-\dfrac{2GM}{r}\Bigg{)}dt^{2}\ +\
\Bigg{(}1-\dfrac{2GM}{r}\Bigg{)}^{-1}dr^{2}\ +\
r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),\;\;\;\;\;\;\;$ (55)

in terms of the standard coordinates $(t,r,\theta,\phi)$.

We assume that the box is much smaller than the Schwarzschild radius
$r_{*}=2GM$. We take the lower wall of the box to lie at radial coordinate
$r=2GM+\epsilon$, and the upper wall to lie at coordinate
$r=2GM+\epsilon+L_{x}$.

We assume that $\epsilon\ll 2GM$ and that $L_{x}\ll 2GM$, but we make no
assumption about the relation between $\epsilon$ and $L_{x}$. Then, the
geometry near the horizon is approximately Rindler, as in Eq. (2). In the
present context, $a=(4GM)^{-1}$ is the surface gravity and
$x=\sqrt{8GM(r-2GM)}$ is the radial-proper length coordinate; $y$ and $z$ are
local Cartesian coordinates on spheres of constant area. Hence, the proper
length of the box is

$\displaystyle H_{x}=\sqrt{8GM(\epsilon+L_{x})}-\sqrt{8GM\epsilon}.$ (56)

The description of the local geometry near the horizon with the Rindler metric
also applies to spherical symmetric dirty black holes [25], i.e., black holes
that coexist with a spherically symmetric matter distribution. Hence, it also
works for models of black holes in a box and in thermal equilibrium with their
Hawking radiation as in Ref. [17].

The idealization of matter in a box does not make sense for black holes of
stellar mass or smaller, because any physical box would be crashed by the
strong tidal forces. However, such forces are much weaker in super-massive
black holes, and in this case the idealization of a gas in a bounding box is
physically meaningful. For a black hole with $M=10^{9}M_{\odot}$, the
Schwarzschild radius is of the order of $10^{9}$km. Hence, the geometry in a
box of proper length $1$m at proper distance $1$km from the horizon is well
described by the metric (2); and so is the geometry of a box of proper length
$1$km at proper distance $1$m from the horizon.

We also note that the location of a bounding box in thermodynamics can only be
viewed as a macroscopic variable (or a macroscopic constraint [26]). Since a
physical box is made of atoms, its location can only be specified with an
error larger than the atomic scale. Any suggestion that the error can be made
as small as Planck’s length has no physical justification. This is an
important point because the models of Padmanabhan [13] and ’t Hooft [16]
mainly concentrate on a box and a shell at Planck proper distance from the
horizon. This is only possible if one invokes new physics of Planck-scale
origin—’t Hooft explicitly makes such an assumption [27].

In this section, we specialize to the case of massless particles and we assume
that the particle number is not preserved. Then, our system is
thermodynamically identical with the photon gas, if we ignore the effects of
polarization (scalar photons).

We will work in the thermodynamic state space $\Gamma$ that is spanned by the
variables $U,L_{y},L_{z},L_{x}$ and $\epsilon$. To this end, we write the
effective length, Eq. (10), as a function of $\epsilon$ and $L_{x}$,

$\displaystyle\bar{L}_{x}\ =\ (2GM)^{2}\ \Bigg{(}\dfrac{1}{\epsilon}\ -\
\dfrac{1}{\epsilon+L_{x}}\Bigg{)}\ .$ (57)

Substituting $\bar{L}_{x}$ into Eq. (31), we obtain the entropy as a function
on $\Gamma$.

Note the importance of deriving Eq. (31) in the microcanonical distribution.
The canonical distribution presupposes that the walls of the box are
diathermal, and that the box is surrounded by a thermal bath at temperature
$\beta^{-1}$. The presence of such a bath would greatly complicate the
analysis in a gravitational system, because it would gravitate and thus,
change the spacetime geometry. A box in thermal contact with a gravitating
thermal bath would not have the same thermodynamic properties with an isolated
box in Schwarzschild spacetime [28]. We also note that no thermal fluid can
coexist with a black hole horizon in a spherically symmetric system [29].
While there exist static, spherically symmetric solutions with Yang-Mills-
Higgs and other types of fields outside the horizon [30, 31], to the best of
our knowledge, no such solution has an interpretation as a thermal bath.

When quantum effects are included, the natural QFT vacuum for a black hole in
a box is the Hartle-Hawking vacuum. In this vacuum, Hawking radiation acts as
a thermal bath of temperature $T_{H}=(8\pi GM)^{-1}$. However, the Hawking
temperature is too low. We will show that the consistency of the statistical-
mechanics analysis requires that $T>>T_{H}$. Note also that the Hawking
temperature is fixed uniquely by the black hole mass, so in the canonical
distribution we would be restricted to boxes of temperature $T_{H}$ in a given
black hole. We conclude that the usual conditions for the validity of the
canonical distributions are not available, and neither is the interpretation
of $\beta^{-1}$ as a temperature of a surrounding heat bath.

It is convenient to express the pressures in terms of the Schwarzschild
coordinates $\epsilon$ and $L_{x}$. Denoting the pressures in this frame by an
overbar, we find that

$\displaystyle\bar{P}_{h}=\frac{H_{x}}{L_{x}}P_{h},\;\;\;\;\bar{P}_{b}=\frac{x_{1}}{2\epsilon}P_{b},\;\;\;\;\bar{P}_{t}=\frac{x_{2}}{2(L_{x}+\epsilon)}P_{t}.$
(58)

###  4.2 The small-box regime

The thermodynamics of the box with radiation depends crucially on the ratio
$\epsilon/L_{x}$. First, we explore the regime where $\epsilon>>L_{x}$, i.e.,
we take the dimensions of the box to be much smaller than its radial distance
from the horizon. Then,
$\bar{L}_{x}\simeq\frac{8}{3}\left(\frac{GM}{\epsilon}\right)^{2}L_{x}$, and
$\bar{V}=\frac{8}{3}\left(\frac{GM}{\epsilon}\right)^{2}V$. Entropy and
internal energy are proportional to the coordinate volume $V$. Hence, in this
regime, energy and entropy behave like extensive quantities, and we can define
an energy density $\rho=U/V$ and an entropy density $s=S/V$. Furthermore,
$P_{b}\simeq P_{h}\simeq P_{t}$, i.e., pressure is almost isotropic, and the
radiation satisfies the equation of state $P=\frac{1}{3}\rho$.

We cannot take the box to be arbitrarily small. In Minkowski spacetime, the
length $L$ of a box bounding radiation must be much larger than the typical
wave-length of the photons, which is of order $T^{-1}$. This is also necessary
in order to describe the single-photon density of states $g(\omega)$ as a
continuous function. It follows that $TL>>1$.

In the present context, the conditions $TL_{y}>>1$ and $TL_{z}>>1$ follow as
in flat space. Regarding the $x$ direction, we note that the continuum
approximation to the eigenvalues of Eq. (6) follows from the requirement that
$(\omega/a)(\xi_{2}-\xi_{1})>>1$. The relevant scale for $\omega$ is given by
$T$, hence,

$\displaystyle T>>\frac{1}{2GM\log(1+L_{x}/\epsilon)}.$ (59)

For $L_{x}/\epsilon<<1$, Eq. (59) implies that

$\displaystyle L_{x}T>>\frac{\epsilon}{2GM}.$ (60)

Eq. (60) can also be written as $L_{x}>>4\pi\epsilon(T_{H}/T)$, where $T_{H}$
is the Hawking temperature. We conclude that a thermodynamic description for
$L_{x}<<\epsilon$ is meaningful only for temperatures much larger than the
black hole’s Hawking temperature. The typical wavelength of Hawking radiation
is of the order of the Schwarzschild radius, hence, it cannot be localized in
a box of much smaller dimensions.

###  4.3 The near-horizon regime

If $\epsilon<<L_{x}$, then $\bar{L}_{x}\sim\epsilon^{-1}$. Internal energy and
entropy diverge as $\epsilon\rightarrow 0$,

$\displaystyle S=\frac{8\pi S_{BH}}{45}\frac{AT^{3}}{\epsilon},\hskip
28.45274ptU=\frac{2\pi S_{BH}}{15}\frac{AT^{4}}{\epsilon},$ (61)

where $S_{BH}=4\pi G^{2}M^{2}$ is the Bekenstein-Hawking entropy of the black
hole. However, their ratio $S/U$ remains constant.

For fixed $\epsilon$, both entropy and energy are proportional to the area
$A=L_{y}L_{z}$. There is a transition from a three-dimensional to a two-
dimensional system as we approach the horizon. We see this in Fig. 4, where we
plot the effective scaling dimension
$d_{eff}:=\log_{2}\left[S(T,\epsilon,2L_{x},2L_{y},2L_{z})/S(T,\epsilon,L_{x},L_{y},L_{z})\right]$
as a function of $L_{x}/\epsilon$.

Figure 4: The effective scaling dimension of a box of radiation as a function
of the ratio $L_{x}/\epsilon$.

However, we should refrain from interpreting the photon gas in this regime as
a two-dimensional fluid, because the pressure in the horizontal direction
$P_{h}$ is much smaller than the bottom pressure $P_{b}$. The system is highly
non-extensive: we would not be able to construct hydrodynamic variables (e.g.,
an effective stress tensor), even if we restrict to a two dimensional brane
around the horizon.

The divergence of the internal energy as $\epsilon\rightarrow 0$ implies that
the approximation of treating the field as propagating on a background
Schwarzschild spacetime breaks down near the horizon. Backreaction has to be
taken into account. Keeping the box static near the horizon requires an
external force. As the box approaches the horizon, the force becomes
increasingly large and it must be incorporated into a stress-energy tensor
which will deform the horizon.

We can ignore backreaction as long as the box can be treated as a test body in
Schwarzschild spacetime, i.e., as long as $U<<M$. By Eq. (32), we obtain a
constraint for the proper distance $x_{1}$ from the horizon,

$\displaystyle\frac{x_{1}}{l_{Pl}}>>(T\sqrt{A})\left(\frac{M}{m_{Pl}}\right)\left(\frac{T}{T_{Pl}}\right),$
(62)

where $l_{Pl},m_{Pl}$ and $T_{Pl}$, are the Planck length, Planck mass and
Planck temperature respectively.

For $T\sim 10^{3}K$, $M$ corresponding to a super-massive black hole and
$L_{y},L_{z}$ of the order of one centimeter, Eq. (62) implies that
$\frac{x_{1}}{l_{Pl}}>>10^{20}$, or, $x_{1}>>10^{-15}m$. From an operational
point of view, the proper distance only takes macroscopic values, certainly it
must be larger than the atomic scale, so the condition (62) poses no
constraint. However, Eq. (62) demonstrates that we cannot take $x_{1}$ to be
the Planck scale while ignoring backreaction. We emphasize that inequality
(62) refers solely to the validity of a QFTCS description without the
inclusion of backreaction. The lower bound to $x_{1}$ does not reflect any
intrinsic uncertainty in the location of the box, as in the corresponding
analyses of Refs. [13] and [16].

Finally, we note that the consistency condition (59) becomes in this regime

$\displaystyle T>>\frac{4\pi T_{H}}{\log(L_{x}/\epsilon)}=\frac{2\pi
T_{H}}{\log(H_{x}/x_{1})}.$ (63)

The requirement that $T>>T_{H}$ suffices to guarantee the validity of Eq.
(63). However, for sufficiently large values of $H_{x}/x_{1}$, a temperature
$T\sim T_{H}$ might be acceptable; the logarithmic dependence on $H_{x}/x_{1}$
makes it difficult to establish a sharp bound in an order-of-magnitude
estimation.

The ratio $S/A$ given by Eq. (61) coincides with the expression obtained by ’t
Hooft [16] as the divergent term in the entropy of a large shell around the
horizon for $\epsilon\rightarrow 0$. This coincidence is a consequence of
spherical symmetry, and it occurs despite the fact that the two expressions
are obtained for systems of different size and through calculations that
involve sums over different modes.

Note that the inner surface of the shell in Ref. [16] is taken at proper
distance of the order of the Planck length from the horizon, while we found
that in this regime the QFTCS description is inadequate and backreaction must
be taken into account. Backreaction is a more severe problem in t’ Hooft’s
model, as a shell can hardly be considered as a test body like the one
considered here. The restriction that $U<<M$ leads to a more stringent
restriction for $x_{1}$ than Eq. (63) in ’t Hooft’s model for $T>>T_{H}$.
However, it has been argued [32] that for $T=T_{H}$, the inclusion of a
negative contribution from the Boulware vacuum renders the total energy
sufficiently small, so that the QFTCS description suffices.

###  4.4 Entropy bounds

The Bekenstein bound usually applies to isolated, weakly-gravitating systems,
because of possible ambiguities in the definition of energy and of the
characteristic length scale $D$ in other contexts. The system studied here is
certainly not isolated, so the Bekenstein bound here, strictly speaking, does
not apply here. Nonetheless, some entropy bound should exist also in the
present context. After all, entropy bounds originate from the requirement that
the GSL applies also to setups that involve test bodies falling into the black
hole.

Here, we proceed with the simple assumption that the length scale $D$ in Eq.
(1) is the proper length of the smallest sphere that encloses the box, that
is, $D=\frac{1}{2}\sqrt{H_{x}^{2}+L_{y}^{2}+L_{z}^{2}}$. By Eq. (32),
$\frac{S}{U}=\frac{4}{3T}$, hence, Eq. (1) becomes

$\displaystyle\frac{3\pi}{4}T\sqrt{H_{x}^{2}+L_{y}^{2}+L_{z}^{2}}\geq 1.$ (64)

Clearly if $L_{y},L_{z}$ are of order $H_{x}$ or larger, Eq. (64) holds easily
because $TL_{y}>>1$ and $TL_{z}>>1$. This is only possible if
$L_{x}>>L_{y},L_{z}$ and the box is not very close to the horizon.

This means that the contribution from the horizontal directions to any
possible violation of the EB is negligible, so we can ignore it. Hence, we
look for possible violations of the inequality

$\displaystyle\frac{3\pi}{4}TH_{x}\geq 1.$ (65)

In the small box regime, Eq. (65) becomes

$\displaystyle L_{x}T>\frac{4}{3\pi}\sqrt{\frac{\epsilon}{2GM}}.$ (66)

Eq. (66) is compatible with the condition (60) for the thermodynamic limit,
but it is not guaranteed by it. Hence, there is a physically allowed regime of
parameters, for which the EB in the form (64) fails.

In the near-horizon regime, Eq. (65) becomes
$\frac{9\pi}{16}(L_{x}T)(T/T_{H})>1$. The EB holds, if we can keep the
temperature of the gas sufficiently large close to the horizon. This may not
be always possible.

###  4.5 Adiabatic lowering of the box

Suppose we prepare the box with the lower wall at some distance from the
horizon. If the walls are adiabatic, then a quasi-static lowering of the box
preserves the entropy $S$. By Eq. (31), the internal energy $U$ in an
adiabatic process is proportional to $\bar{V}^{-1/3}$. This means that
$U\sim\bar{L}_{x}^{-1/3}$. As the box approaches the horizon,
$U\sim\epsilon^{1/3}$, i.e., the energy remains finite, and the approximation
of the box as a test body remains accurate. This does not contradict Eq. (62),
because the temperature $T$ also decreases when we lower the box
adiabatically.

The adiabatic lowering of the box leads to a decrease of the internal energy
while entropy, and the proper length $H_{x}$ remain constant. Hence, if
adiabatic lowering continues up to small distances from the horizon,
Bekenstein’s bound will be violated. It is straightforward to show that in the
near-horizon regime

$\displaystyle\frac{S}{U}=2GM\left(\frac{2^{11}\pi^{2}A}{3^{5}\ 5\
Sx_{1}^{2}}\right)^{1/3}.$ (67)

In this regime, $D_{x}\simeq\frac{1}{2}\sqrt{8GML_{x}+L_{y}^{2}+L_{z}^{2}}$
does not depend on $x_{1}$. Hence, the Bekenstein bound is violated for

$\displaystyle x_{1}<1.3\sqrt{\frac{A(GM)^{3}}{SH_{x}^{3}}},$ (68)

which corresponds to macroscopic values of $x_{1}$. In fact, the Bekenstein
bound fails well before we reach the near-horizon regime.

There are two possible explanations for this failure. The first explanation is
that the condition (64) is not an accurate representation of the entropy bound
because the box is not isolated. We would have to go back to first principles,
and find a new expression for the entropy bound that is valid for small
material bodies near a black hole horizon. The other explanation is that the
energy of the system must also include the energy of the walls enclosing it
[33]. As we will see, the latter diverges as the system approaches the
horizon, hence, its contribution would keep the ratio $S/U$ small.

The average energy density $\rho$ is proportional to $\bar{L}_{x}^{-1/3}$. As
the box approaches the horizon, the horizontal pressure vanishes,
$P_{h}\sim\epsilon^{1/3}$; the pressure at the top wall also vanishes,
$P_{t}\sim\epsilon^{4/3}$; however, the pressure at the bottom diverges,
$P_{b}\sim\epsilon^{-2/3}$. Hence, the energy of the walls that keep the
system confined increases as we approach the horizon. The Bekenstein bound
(64) may persist if the energy of the walls is included in the total
energy[33]. To test this possibility, we need an explicit model for the
confining energy of the system—see, for example, [34]. However, we must point
out that the divergence of the pressure and hence of the confinement energy
occurs in the near-horizon regime, while the breakdown of Eq. (64) occurs well
before the system enters this regime.

In general, the fact that $P_{b}\neq P_{t}$ implies that there is a net total
force acting on the box. We can evaluate this force by specifying how it is
transmitted from the bottom to the top. Each mode of transmission corresponds
to a vector field on $\Gamma_{m}$ that is contracted with the pressure form
(36). The simplest assumption is that the force is transmitted through rigid
box walls, that is, the proper length $H_{x}$ remains constant in any
displacement of the box.

Hence, for an adiabatic lowering of the box, the acceleration due to the
pressure differential is

$\displaystyle a_{S}:=-\frac{1}{U}\left(\frac{\partial
U}{\partial\epsilon}\right)_{S,H_{x},L_{y},L_{z}}=-\frac{1}{3L_{x}}\left(1+\frac{L_{x}}{\epsilon}-\frac{1}{\sqrt{1+\frac{L_{x}}{\epsilon}}}\right)<0.$
(69)

The negative sign of $a_{S}$ means that the acceleration is directed towards
the black hole.

Note that if the box is lowered isothermally, the acceleration is

$\displaystyle a_{T}:=-\frac{1}{U}\left(\frac{\partial
U}{\partial\epsilon}\right)_{T,H_{x},L_{y},L_{z}}=-3a_{S},$ (70)

i.e., it acts as buoyancy, analogous to the one proposed by Unruh and Wald
[9]. However, as we mentioned, we are not aware of any static solution to
Einstein’s equation with a heat bath at temperature $T$ coexisting with a
black hole horizon. Hawking radiation could provide such a bath, but, as shown
in Sec. 3.3., its temperature is so low that the usual thermodynamic
approximations for the radiation in the box do not apply.

In the small-box regime, $a_{S}=-(6\epsilon)^{-1}$. This acceleration is
comparable with the radial acceleration $a_{r}=-\epsilon^{-1}$ of a static
point-particle at radial distance $\epsilon$ from the horizon. Near the
horizon, $a_{S}=-(3\epsilon)^{-1}$, and, expectedly, it diverges as
$\epsilon\rightarrow 0$.

It is remarkable that, in the small-box regime, the acceleration $a_{S}$ does
not depend on temperature or on properties of the box. It appears to depend on
the properties of the particles— for massive particles at low temperature,
$a_{S}=-\epsilon^{-1}$—see below in Sec. 4.3. In this paper, we have seen this
force only for static (or quasi-static) bodies. However, the mechanism that
generates the force—the pressure difference in the radial direction—appears to
be generic, and it is a consequence of the near-horizon geometry. Therefore,
we conjecture that this acceleration could appear in other systems, for
example, macroscopic bodies freely falling into the black hole. Since this
acceleration depends on the properties of the gas, it is not equivalent to a
change of geodesics due a quantum deformation of the metric, like, for
example, that of Ref. [36]. The derivation any observational consequences of
this acceleration in astrophysical bodies would require a first-principles
analysis, and an extension of our results to distances from the horizon larger
than the ones considered in this paper.

##  5 Massive particles near a Schwarzschild horizon

###  5.1 The number-of-states function

We analyse the thermodynamics of a box of massive particles (bosons and
fermions) near the Schwarzschild horizon. We will mainly work in the regime
where the temperature $T$ is much smaller than the mass $m$. This condition is
also satisfied for Bose-Einstein condensates—see Ref. [35] for a study of
Bose-Einstein condensates in the brick wall model.

Note that the opposite regime, of very high temperatures, $T>>m$, can be
studied by expanding $F(m^{2}/\omega^{2})$ in Eq. (8) as a series in
$(m/\omega)^{2}<<1$. This leads to

$\displaystyle\Omega(\omega)=\frac{V}{\pi^{2}}\omega^{3}+\frac{2GMAm^{2}}{\pi^{2}}\log(1+L_{x}/\epsilon)\omega+\frac{4Vm^{4}}{3\pi^{2}}\omega^{-1}+\ldots,$
(71)

where $A=L_{y}L_{z}$.

It is straightforward to show that the $m$-dependent terms only cause a small
correction to the expressions for the thermodynamic variables in the massless
case. There is no qualitative change, and for this reason, we will not
elaborate on this regime.

To analyse the regime of $T<<m$, we first note that the minimum value of
$\omega$ for a box in the vicinity of the Schwarzschild horizon is

$\displaystyle\omega_{min}(\epsilon)=m\sqrt{\frac{\epsilon}{2GM}},$ (72)

i.e., $\omega_{min}$ vanishes for $\epsilon\rightarrow 0$.

By Eqs. (8) and (46), the number $\Omega(q)$ of states with energy less than
$\omega_{min}\ +\ q$ for a box close to the horizon is

$\displaystyle\Omega(q)=\eta(\epsilon)Aq^{5/2}.$ (73)

where

$\displaystyle\eta(\epsilon)=\frac{64\sqrt{2}G^{2}M^{2}}{15\pi^{2}\epsilon}\sqrt{\omega_{min}(\epsilon)}=\frac{16\sqrt{2m}}{15\pi^{2}}\dfrac{(2GM)^{7/4}}{\epsilon^{3/4}}.$
(74)

This expansion of $\Omega$ applies as long as $q<<\omega_{min}$. Since the
average value of $q$ is of the order of the temperature $T$, hence, the
expansion (73) is accurate for

$\displaystyle\frac{T}{m}<<\sqrt{\frac{\epsilon}{2GM}}.$ (75)

This condition is much stronger than the usual condition $T<<m$ for the non-
relativistic regime. It implies that we cannot bring the box arbitrarily close
to the horizon and still employ the expansion (73). The proper distance
$x_{1}$ from the horizon satisfies $x_{1}/2r_{S}>>T/m$, where $r_{S}$ is the
Schwarzschild radius. For $T\sim 300{}^{o}K$, particle mass $m\sim 100$ amu
and $M\sim 10^{9}M_{\odot}$, $x_{1}>>1$m.

It is straightforward to show that as the system approaches the horizon
($s_{1}\rightarrow 0$), $F(m^{2}/\omega^{2})$ approaches a constant $-(3\pi
s_{2})^{-1}$ that is mass independent and much smaller than $F(\rho_{max})$;
$F(\rho_{max})$ diverges on the horizon. This means that the dominant
contribution to the density of states near the horizon is of the form (28)
that corresponds to massless particles. Hence, the near-horizon behavior is
mass independent and coincides with that of Sec. 3.4.

###  5.2 Thermodynamic functions

We proceed to the evaluation of thermodynamic observables using the grand-
canonical distribution. First, we write the number of particles

$\displaystyle N=\
\int_{0}^{\infty}\frac{dqg(q)}{z^{-1}e^{\beta(\omega_{min}+q)}-1}\
+N_{0}=\dfrac{A}{\lambda_{2D}^{2}}\mbox{Li}_{5/2}(ze^{-\beta\omega_{min}(\epsilon)})\
+\ N_{0}\ ,$ (76)

where $\mbox{Li}_{a}(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{n^{a}}$ is the
polylogarithm function, $N_{0}=(z^{-1}e^{\beta\omega_{min}}-1)^{-1}$ is the
average occupation number for the single-particle level with q = 0333This term
originates from the divergence of the Bose-Einstein distribution at
$ze^{-\beta\omega_{min}}=1$, and it is evaluated prior to the continuous
limit. Here, $\omega_{min}$ is defined as the lowest energy state after the
continuous limit has been imposed. The actual minimum energy state $E_{0}$
differs from $\omega_{min}$ by a term of order unity. However, the explicit
value of $E_{0}$ affects only the vacuum energy and pressure and not the other
thermodynamic quantities of the system. , and

$\displaystyle\lambda_{2D}(\beta,\epsilon)=\sqrt{\frac{8\beta^{5/2}}{15\sqrt{\pi}\eta(\epsilon)}}\
$ (77)

is the effective de Broglie wave-length of the system. The label 2D emphasizes
that in this regime the system becomes effectively two-dimensional, as the
particle number scales with the area $A$ and not with the volume
$V=L_{x}L_{y}L_{z}$, or with an effective volume like $\bar{V}$ that was
defined for photons. Note that $\lambda_{2D}\sim\epsilon^{3/8}$, and it
becomes increasingly smaller as we approach the horizon.

The grand potential

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\beta^{-1}\int_{0}^{\infty}dqg(q)\log(1-ze^{-\beta(\omega_{min}+q)}\
)\ +\beta^{-1}\log(1+N_{0})$ (78) $\displaystyle=$
$\displaystyle-\dfrac{A\beta^{-1}}{\lambda_{2D}^{2}}\mbox{Li}_{7/2}(ze^{-\beta\omega_{min}(\epsilon)})+\beta^{-1}\
\log(1+N_{0}),$

is also proportional to the area $A$, modulo the ground state contribution
$\log(1+N_{0})$.

The lowest energy for a single particle is $\omega_{min}(\epsilon)$, so the
internal energy has a vacuum contribution $N\omega_{min}(\epsilon)$,

$\displaystyle U=N\omega_{min}(\epsilon)-\ \frac{5}{2\beta}\Big{(}\Phi\ -\
\log(1+N_{0})\Big{)}.\;\;\;\;\;\;$ (79)

The internal energy splits as $U=U_{0}+U_{th}$, where
$U_{0}=N\omega_{min}(\epsilon)$ is the ground-state energy and
$U_{th}=-\dfrac{5}{2\beta}\Big{(}\Phi\ -\ \log(1+N_{0})\Big{)}$ is the thermal
energy; we call it thermal because it corresponds to the kinetic energy of the
particles.

We also evaluate the entropy

$\displaystyle S$ $\displaystyle=$
$\displaystyle\dfrac{A}{(\lambda_{2D})^{2}}\left[(\beta\omega_{min}-\log
z)\mbox{Li}_{5/2}(ze^{-\beta\omega_{min}})+\frac{7}{2}\mbox{Li}_{7/2}(ze^{-\beta\omega_{min}})\right]$
(80) $\displaystyle+\left[(N_{0}+1)\log(N_{0}+1)-N_{0}\log N_{0}\right].$

The height of the box $L_{x}$ does not appear explicitly in the equation for
entropy. It drops out completely in this regime. This is the reason why the
system scales with area and behaves essentially as two dimensional. When the
temperature drops to the regime $T<<m$, all gas molecules are constrained to
move along the bottom of the box.

By Eq. (76), if $N\lambda_{2D}^{2}/A\ >\ \zeta(\frac{5}{2})$, then $N_{0}>0$,
i.e., there is a non-zero number of particles in the Bose-Einstein condensate
phase. This condition can be written equivalently as $T<T_{c}$, where

$\displaystyle
T_{c}=\frac{\pi^{3/5}}{2[2\zeta(\frac{5}{2})]^{2/5}}\left(\frac{N}{A}\right)^{\frac{2}{5}}\left(\frac{2\epsilon^{3}}{(GM)^{7}m^{2}}\right)^{\frac{1}{10}}.$
(81)

is the critical temperature for Bose-Einstein condensation. It follows that

$\displaystyle\dfrac{N_{0}}{N}=\begin{cases}{}\ \ \ 0\ ,\ \ \ \ \ \ \ T\geq
T_{c}\\\ \\\\[2.0pt] {}\ \ \ 1\ -\ \Big{(}\dfrac{T}{T_{c}}\Big{)}^{5/2}\ ,\
T\leq T_{c}\ \ .\end{cases}$ (82)

###  5.3 Specific regimes

Dilute gas.  The dilute gas regime corresponds to
$ze^{-\beta\omega_{min}(\epsilon)}\ll 1$, which implies that
$\mbox{Li}_{\alpha}(z^{\prime})\simeq z^{\prime}$. There is no Bose-Einstein
condensate. We obtain

$\displaystyle N\simeq\dfrac{A}{\lambda_{2D}^{2}}\
ze^{-\beta\omega_{min}(\epsilon)}\hskip 11.38092pt,\hskip 17.07182ptU_{th}\
\simeq\dfrac{5}{2}NT\ .$ (83)

The entropy is

$\displaystyle S=\
\dfrac{7}{2}N+N\log\Bigg{(}\dfrac{A}{N\lambda_{2D}^{2}}\Bigg{)}=\dfrac{7}{2}N\
+\
N\log\Bigg{[}\Bigg{(}\dfrac{15\sqrt{\pi}A\eta(\epsilon)}{8N}\Bigg{)}\Bigg{(}\dfrac{2}{5}\
\dfrac{U_{th}}{N}\Bigg{)}^{5/2}\Bigg{]}.$ (84)

Since $L_{x}$ does not appear explicitly in the fundamental equation (84),
there is no pressure at the top of the box. It is straightforward to calculate
the horizontal pressure

$\displaystyle\bar{P}_{h}=\frac{2U_{th}}{5V}=\frac{NT}{V}\ ,$ (85)

and the pressure at the bottom

$\displaystyle\bar{P}_{b}=\dfrac{N\omega_{min}}{2A\epsilon}+\dfrac{3U_{th}}{10A\epsilon}=\dfrac{N}{2A\epsilon}\left(\omega_{min}+\frac{3}{2}NT\right)\
.$ (86)

We also evaluate the acceleration on a box that is lowered adiabatically

$\displaystyle
a_{S}=-\frac{L_{x}}{2\epsilon(L_{x}+\epsilon)\left(1-\frac{1}{\sqrt{1+\frac{L_{x}}{\epsilon}}}\right)}+O(T/m).$
(87)

This expression differs from that for massless or ultra-relativistic
particles. In the small-box regime, $a_{S}=-\epsilon^{-1}$ and in the near-
horizon regime, $a_{S}=-(2\epsilon)^{-1}$.

Dense gas. Bose-Einstein condensates correspond to $z=1$. In this case,
entropy and thermal energy scale with the number of particles in the gas
phase,

$\displaystyle N-N_{0}=\zeta(5/2)\dfrac{A}{\lambda_{2D}^{2}},\;\;\;U_{th}=\
\dfrac{5\zeta(7/2)}{2\zeta(5/2)}(N-N_{0})T,\;\;\;S=\frac{7U_{th}}{5T}.$ (88)

The equations for pressure and acceleration are the same as in the dilute gas
case.

The Bose-Einstein condensate phase exists for $T<T_{c}$. A thermodynamic
description is possible only if $T_{c}>>T_{H}$, which implies that

$\displaystyle\epsilon>>\frac{[\zeta(5/2)^{4/3}]}{2^{17/3}\pi^{2}}\frac{m^{2/3}}{GM(N/A)^{4/3}}\simeq
0.003\frac{m^{2/3}}{GM(N/A)^{4/3}},$ (89)

or equivalently, $x_{1}>>0.5\frac{m^{1/3}}{(N/A)^{2/3}}$.

For $m=100$amu and $\sqrt{A/N}\sim 3$ nm (air in room temperature), we find
that $x_{1}>>10^{-34}$m, i.e., no significant constraint to $x_{1}$. Note that
by Eq. (75), the Bose-Einstein condensate phase cannot persist in the near-
horizon regime.

Entropy bounds. In the regime where $L_{x}>>\epsilon$, the gas of massive
particles and the gas of massless particles have the same thermodynamic
behavior. Hence, as shown in Sec. 3.5, the EB is violated.

For a dilute gas and for $L_{x}>>\epsilon$, Eqs. (83) and (84) apply. We note
that entropy $S$ varies as $\log\epsilon$ while $U\simeq N\omega_{min}$
decreases with $\sqrt{\epsilon}$. Hence, we expect that there is a range of
parameters such that the EB is violated when we bring the box sufficiently
close to the horizon, while preserving the condition $L_{x}>>\epsilon$. The
reader can straightforwardly verify a violation of the EB for $T=1K$,
$H_{x}=L_{y}=L_{z}=1$cm and $x_{1}=1$m. Again, the EB is eventually violated
when we lower a box adiabatically, where $U$ decreases with $\sqrt{\epsilon}$
while $S$ remains constant.

In contrast, the entropy content of a Bose-Einstein condensate is negligible,
hence, there is no violation of the EB, as long as $L_{x}>>\epsilon$.

##  6 Conclusions

We presented our motivation and our results in the Introduction. Here, we
comment on the implications of our work.

First, the combination of QFTCS for the ideal gas dynamics and of the
microcanonical distribution leads to a self-consistent description of the
systems that were studied here. We expect that the results can be generalized
to larger class of small systems near the black hole horizon, including self-
gravitating systems. We saw that there is a large and yet unexplored
thermodynamic phenomenology of such systems, including the presence of
acceleration due to anisotropic pressure. Its detailed study will enable,
among others, a deeper understanding of the GSL, especially its grounding on
the quantum-field theoretic description of thermodynamic systems.

Second, our results highlight the importance of backreaction for any
discussion of horizon thermodynamics, as QFTCS fails for macroscopic distances
beyond the horizon, well before we approach the Planck length. We expect that
a combination of the methods of Ref. [17] and the methods developed here could
prove fruitful for implementing backreaction.

Finally, we note that the formulation of entropy bounds for small systems near
a black hole need to be reappraised. We saw that the simplest implementation
of the Bekenstein bound fails, possibly because this bound applies primarily
to isolated systems. It is important to understand how or whether this failure
can be remedied, before proceeding to the formulation of a general entropy
bound applicable to this class of systems.

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##  Appendix A The case of fermions

In this Appendix, we analyze the case of a box with a fermion gas near a black
hole horizon. The overall behavior is very similar with the bosonic case
analyzed in the main text, the differences are primarily technical.

###  A.1 Density of states for fermions

The Dirac equation in a curved spacetime with metric $g_{\mu\nu}$ reads

$\displaystyle[i\gamma^{\mu}(\partial_{\mu}-\Gamma_{\mu})-m]\hat{\psi}=0,$
(90)

where the curved-spacetime Dirac matrices $\gamma^{\mu}$ are defined in terms
of the Minkowski Dirac matrices $\gamma_{a}$ by
$\gamma^{\mu}=E^{\mu}_{a}\gamma^{a}$. Here, $E^{\mu}_{a}$ the inverse of the
tetrad field associated to $g_{\mu\nu}$, and
$\Gamma_{\mu}:=\frac{1}{8}E^{\nu}_{a}E_{b\nu;\mu}[\gamma^{a},\gamma^{b}]$ is
the spin connection.

For the Rindler metric, the Dirac equation becomes

$\displaystyle a^{-1}\frac{\partial}{\partial
t}\hat{\psi}=(-ix\;{\bf\alpha}\cdot{\bf\nabla}-\frac{i}{2}a^{1}\psi+mx\beta)\hat{\psi},$
(91)

where $\alpha^{i}=\gamma^{0}\gamma^{i}$ and $\beta=\gamma^{0}$. We consider
modes of the form $e^{-i\omega t+ik_{y}y+ik_{z}z}f(x)$, where $f(x)$ is a
Dirac spinor. One can show [37, 38] that there are two linearly independent
positive-frequency solutions for $f$, and these are obtained from the solution
of the scalar equation

$\displaystyle x^{2}f^{{}^{\prime\prime}}\ +\ xf^{{}^{\prime}}\ +\
\bigg{[}\left(\dfrac{\omega}{a}\pm\frac{i}{2}\right)^{2}-\kappa^{2}\
x^{2}\bigg{]}f\ =\ 0\ ,$ (92)

The modes $f$ still satisfy Schrödinger’s equation (6), albeit with complex
energy
$E=\left(\dfrac{\omega}{a}\pm\frac{i}{2}\right)^{2}=\frac{\omega^{2}}{\alpha^{2}}-\frac{1}{4}\pm
i\frac{\omega}{a}$. For a complex energy eigenvalue $E_{R}+iE_{I}$ the WKB
solutions are of the form $Re^{\pm iS}$, where now

$\displaystyle R=\frac{1}{4}\int
dx^{\prime}\frac{-V^{\prime}(x^{\prime})+E_{I}}{E_{R}-V(x^{\prime})},$ (93)

but the Hamilton-Jacobi action $S$ is unaffected. Therefore, the eigenvalue
equation (17) remains unchanged modulo the change of $\omega$ with
$\sqrt{\omega^{2}-\frac{1}{4}a^{2}}$, together with the restriction that
$\omega>\frac{1}{2}a$.

This change in $\omega$ is negligible for any temperature $T$ that are much
larger than the Unruh temperature $T_{U}=\frac{a}{2\pi}$ of the accelerated
box. As shown in Sec. 4.2, the condition $T>>T_{U}$ is also necessary for the
validity of the semi-classical approximation and of the thermodynamic
description. Hence, the density of states is simply twice that of scalar
particles, corresponding to the two spin directions of the Dirac particle.

###  A.2 Density of states

We will first analyse the case of ultra-relativistic fermions ($m=0$). The
density of states is $g(\omega)=\frac{4\bar{V}}{\pi^{2}}\omega^{2}$ (twice
that of the scalar case). We will use the grand-canonical distribution. For
ideal gases, and in the limit of a large number of particles, the grand
canonical distribution gives the same entropy with the microcanonical
distribution [24]. However, in the present context, the parameters $\beta$ and
$z$ that appear in the grand-canonical distribution (see below) are not to be
viewed as the inverse temperature and the fugacity of an external heat bath,
but as arbitrary constants (Lagrange multipliers).

The grand potential $\Phi$ and the particle number $N$ for one particle
species are defined as

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle-\beta^{-1}\int d\omega
g(\omega)\ln(1+ze^{-\beta\omega})=-\int
d\omega\frac{\Omega(\omega)}{z^{-1}e^{\beta\omega}+1}$ (94) $\displaystyle N$
$\displaystyle=$ $\displaystyle\int
d\omega\frac{g(\omega)}{z^{-1}e^{\beta\omega}+1},$ (95)

where $z=e^{\beta\mu}$ with $\mu$ the chemical potential.

For ultra-relativistic fermions, the particle number is not preserved in
equilibrium. What is preserved is the lepton number $N_{l}=N-\bar{N}$, where
$N$ is the number of particles and $\bar{N}$ is the number of anti-particles.
The grand potential is a sum of the grand potential for particles and for
antiparticles, while the chemical potential of anti-particles is the opposite
of the chemical potential for particles. Then, we calculate

$\displaystyle\Phi=-\frac{8\bar{V}}{\pi^{2}\beta^{4}}[f_{4}(z)+f_{4}(z^{-1})],$
(96) $\displaystyle
N_{l}=\frac{8\bar{V}}{\pi^{2}\beta^{3}}[f_{3}(z)-f_{3}(z^{-1})],$ (97)

We identify the three pressures
$P_{h}:=-\frac{1}{H_{x}L_{z}}\left(\frac{\partial\Phi}{\partial
L_{y}}\right)_{z,\beta,L_{z},x_{1},x_{2}}$,
$P_{b}:=-\frac{1}{L_{y}L_{z}}\left(\frac{\partial\Phi}{\partial
x_{1}}\right)_{z,\beta,L_{y},L_{z},x_{2}}$ and
$P_{t}:=-\frac{1}{L_{y}L_{z}}\left(\frac{\partial\Phi}{\partial
x_{2}}\right)_{z,\beta,L_{y},L_{z},x_{1}}$. Since $\Phi$ depends on the length
parameters only through $\bar{V}$, Eqs. (37,39, 40) still apply. The only
difference from the bosonic case is the explicit expression for the energy
density $\rho=U/(H_{x}L_{y}L_{z})$, where $U=-3\Phi$,

$\displaystyle\rho=\frac{24T^{4}}{\pi^{2}}\frac{\bar{L}_{x}}{H_{x}}[f_{4}(z)+f_{4}(z^{-1})],$
(98)

where $z$ is expressed in terms of $N_{l}$ by Eq. (97).

We also consider massive fermions at zero temperature. To be precise, we
consider finite temperature $T>>T_{U}$, but small enough so that the entropy
of the gas is negligible. In this regime, there is no particle-antiparticle
pair creation, hence, the number of particles $N$ is conserved. The chemical
potential coincides with the Fermi energy $\epsilon_{F}$, defined by
$\Omega(\epsilon_{F})=N$. For $\Omega$ given by twice the expression (49), we
obtain

$\displaystyle\Phi=-\frac{4L_{y}L_{z}m^{3}}{105\pi^{2}a^{2}x_{1}}\xi_{F}^{7/2}\times\left\\{\begin{array}[]{cc}1-\left(\frac{x_{2}}{x_{1}}\right)^{3}\left[1-\frac{1-(x_{1}/x_{2})^{2}}{\xi_{F}}\right]^{7/2},&\xi_{F}\geq
1-\frac{x_{1}^{2}}{x_{2}^{2}}\\\
1&\xi_{F}<1-\frac{x_{1}^{2}}{x_{2}^{2}}\end{array}\right.,$ (101)

where $\xi_{F}$ is defined by the condition $\Omega(\xi_{F})=N$. Given
Eq.(101), it is straightforward to compute all thermodynamic quantities,
including the pressures. The latter simplify significantly in the regime where
$\xi_{F}<1-\frac{x_{1}^{2}}{x_{2}^{2}}$ that is, as we will see, relevant to
the black hole case. We find that

$\displaystyle
P_{h}=-\frac{\Phi}{H_{x}L_{y}L_{z}},\;\;\,P_{b}=-\frac{\Phi}{x_{1}L_{y}L_{z}},\;\;\;P_{t}=0.$
(102)

###  A.3 Gas of massless fermions

There are few differences in the behavior of a fermionic gas of massless
particles near the horizon. As shown in Sec. 3.4, the pressures behave exactly
as in the bosonic case. The only difference is the dependence of entropy and
total energy on the total lepton number $N_{l}$. In the grand ensemble,
$S=(U-\Phi-\mu N_{l})/T$. Since $U=-3\Phi$, Eqs. (96,97) yield

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{24\bar{V}T^{4}}{\pi^{2}}[f_{4}(z)+f_{4}(z^{-1})]$ (103)
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{8\bar{V}T^{3}}{\pi^{2}}\left[4[f_{4}(z)+f_{4}(z^{-1})]-\ln
z[f_{3}(z)-f_{3}(-z)]\right].$ (104)

To express $U$ and $S$ as a function of $N_{l}$, we must solve Eq. (97) for
$z$ and substitute in the equations above.

At high-temperatures, $z\simeq 1+\nu$, where $\nu<<1$, and
$f_{a}(z)=f_{a}(1)+\nu f^{\prime}_{a_{1}}(1)$. Then, the dominant contribution
to $U$ and $S$ are $N_{l}$-independent

$\displaystyle U=\frac{7\pi^{2}\bar{V}T^{4}}{15},\hskip
28.45274ptS=\frac{28\pi^{2}\bar{V}T^{4}}{45}$ (105)

Hence, the analysis of the breakdown of the Bekenstein bound and of the QFTCS
approximation remain unchanged modulo numerical constants of order unity.

In the low temperature regime, either $z\rightarrow\infty$ or $z\rightarrow
0$. We will consider the case $z\rightarrow\infty$, which corresponds to a
surplus of particles. The extension to $z\rightarrow 0$ (surplus of
antiparticles) is straightforward. In this regime, $f_{n}(z)\simeq(\ln
z)^{n}/n!+\frac{\pi^{2}}{6(n-2)!}(\ln z)^{n-2}$ and $f_{n}(z^{-1})\simeq
z^{-1}$. We straightforwardly solve Eq. (97) for $z$,

$\displaystyle\ln
z=\left(\frac{3\pi^{2}N}{4\bar{V}T^{3}}\right)^{1/3}-\frac{\pi^{2}}{3}\left(\frac{3\pi^{2}N}{4\bar{V}T^{3}}\right)^{-1/3}.$
(106)

Then, we find

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{3}{4}(3\pi^{2}/4)^{1/3}\frac{N_{l}^{4/3}}{\bar{V}^{1/3}}+\frac{1}{6}(3\pi^{2}/4)^{2/3}T^{2}N_{l}^{2/3}\bar{V}^{1/3},$
(107) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{2}{3}(3\pi^{2}/4)^{2/3}TN_{l}^{2/3}\bar{V}^{1/3}.$ (108)

In the near horizon regime,

$\displaystyle U$ $\displaystyle=$
$\displaystyle\frac{3^{5/3}\pi}{8}\frac{N_{l}^{4/3}}{A^{1/3}S_{BH}^{1/3}}\epsilon^{1/3}+\frac{3^{1/3}\pi}{12}T^{2}N_{l}^{2/3}S_{BH}^{1/3}A^{1/3}\epsilon^{-1/3},$
(109) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{3^{1/3}\pi}{3}TN_{l}^{2/3}S_{BH}^{1/3}A^{1/3}\epsilon^{-1/3}.$
(110)

The dominant contribution to the internal energy vanishes as
$\epsilon\rightarrow 0$, but in this limit the correction term diverges.
Hence, the ratio $S/U$ remains always finite. Of course, the low temperature
expansion applies only if the correction term is significantly smaller than
the dominant term in $U$. Still, it is straightforward to show that in the low
temperature regime, $S/U<4/T$.

When the box is adiabatically lowered, $T\sim\epsilon$, and the correction
term in $U$ also decreases with $\epsilon^{1/3}$. Hence, for a sufficiently
dense gas ($N_{l}\rightarrow\infty$) the first term always dominates. Hence,
the ratio $S/U$ diverges as $\epsilon\rightarrow 0$, and we find that the
Bekenstein bound is violated for

$\displaystyle
x_{1}<0.02\frac{A^{1/2}S^{3/2}}{H_{x}^{3/2}N_{l}^{2}}(GM)^{1/2}.$ (111)

In principle, the violation can take place at macroscopic distance from the
horizon if $M$ is taken sufficiently large, but, expectedly, it is much weaker
than the corresponding violations at high temperature.