# Gauge theories with nontrivial boundary conditions:
Black holes

Peng Cheng<EMAIL_ADDRESS>Center for Joint Quantum Studies and
Department of Physics, School of Science, Tianjin University, 135 Yaguan Road,
Tianjin 300350, China  Institute for Theoretical Physics, University of
Amsterdam, 1090 GL Amsterdam, Netherlands

###### Abstract

We study the partition function and entropy of U(1) gauge theories with
multiple boundaries on the black holes background. The nontrivial boundary
conditions allow residual zero longitudinal momentum modes and Wilson lines
stretched between boundaries. Topological modes of the Wilson lines and other
modes are also analyzed in this paper. We study the behavior of the partition
function of the theory in different temperature limits, and find the
transitions of dominances of different modes as we vary the temperature.
Moreover, we find two different area contributions plus logarithm corrections
in the entropy. One being part of the bulk fluctuation modes can be seen for
finite-temperature black holes, and the other coming from vacuum degeneracy
can only be seen in the superlow temperature limit. We have confirmed the
mechanism and entropy found in the superlow temperature limit also persist for
extremal black holes. The gauge fluctuation on the black hole background might
help us understand some fundamental aspects of quantum gravity related to
gauge symmetries.

##  I Introduction

It is well known that black holes are thermodynamic systems as seen by outside
observers Hawking1971 ; Christodoulou1970 ; Christodoulou1971 ; HAWKING1974 ;
Bekenstein1973 ; Bardeen1973 ; Bekenstein1972 ; CARTER1972 , which might
reflect the properties of the underlying microscopic structures. The so-called
central dogma Almheiri2020  claims that the number of the microscopic degrees
of freedom of a black hole should be the Bekenstein-Hawking entropy

$S=\frac{\text{Area}}{4\hbar G_{N}}\propto\frac{\text{Area}}{l_{p}^{2}}\,,$
(1)

and the whole system unitarily evolves under time evolution. $l_{p}$ is the
Planck length.

So, one of the most important questions in black hole physics is “what is the
theory that describes the microscopic structure of a black hole”. Many
different research programs are trying to give a microscopic explanation and
reproduce the Bekenstein-Hawking entropy by counting the microstates. Those
research programs can be more or less divided into two categories.

  * •

The first train of thought is to add or find microscopic states near the
horizon. Usually, one expects the quantum fields living close to the horizon
or entanglement pairs across the horizon would have an entropy contribution
proportional to the area tHooft:1984kcu ; Srednicki1993 ; Susskind1994 ;
Solodukhin2011 ; Blommaert2018a ; Donnelly:2014fua ; Donnelly2015a ; Wei2015am
; Haco2018 . There are more proposals, like loop quantum gravity Krasnov1996 ;
Rovelli1996 ; Ashtekar1997 ; Domagala2004 ; Meissner2004 ; Ghosh2006  that can
also be categorized here.

  * •

The second set of ideas is to explain the Bekenstein-Hawking entropy by
finding hidden symmetries, which are mainly used to understand the entropy of
(near-)extremal black holes. To reproduce the universal coefficient $1/4$ in
the Bekentein-Hawking entropy, one always needs to use a powerful tool:
symmetry. If there are hidden conformal symmetries near the horizon, the
density of states of two-dimensional CFT is controlled by the Cardy formula
Cardy1986 ; Bloete1986 . The universality of the Cardy formula can be used to
reproduce the Bekenstein-Hawking entropy with the coefficient $1/4$
Strominger1998 ; Birmingham1998 ; Guica2009 ; Carlip2011 . Hidden symmetries
(can also be Bondi-Metzner-Sachs, Carrollian, and others) and their relation
with black hole entropy are extensively studied in the literature Brown1986 ;
Solodukhin1999 ; Carlip1999a ; Carlip1999 ; Silva2002 ; Carlip2007 ;
Castro2010 ; Wang2010 ; Chen2010 ; Carlip2011a ; Compere2017 ; Song2012 ;
Carlip2018 ; Carlip2019 . The holographic or stringy methods Strominger1996 ;
Callan1996 ; Peet2000 ; Das2001 ; Maldacena1996 ; Emparan2006  also heavily
rely on the symmetries of spacetime. However, those methods related to
symmetries usually work only in (near-)extremal black hole cases, and are
difficult to be generalized to finite-temperature black holes.

Despite completely different starting points, lots of theories in both
categories successfully reproduced the Bekenstein-Hawking entropy. Is it a
problem to have so many different explanations for black hole microstates?
There are two obvious responses to the above question. i) There are common
structures that lie behind those theories, and we may find a unified
description that explains why those different theories are able to count the
microscopic structures of black holes. ii) Some of the theories describe the
wrong physics, and the corresponding entropy was reproduced by counting fake
microscopic structures of the black hole. This paper aims to address this
problem using a toy model of gravitational fluctuations on a black hole
background. Our final conclusion tends to say that the two categories of
theories are both counting the right microscopic structures of the black hole
although they are counting different things. Moreover, the microstates that
are used to explain the entropy of finite temperature black holes are not the
same ones for extremal black holes.

Gauge theories on a fixed background can be regarded as good toy models of
gravitational fluctuations around saddle points of gravity theory. The one-
loop correction around a fixed saddle is captured by the linearized Einstein-
Hilbert action, which is a massless quadratic Fierz-Pauli action. The massless
Fierz-Pauli theory is a gauge theory with two physical polarizations; thus, we
can use gauge theories on classical solutions of Einstein’s theory as toy
models of metric perturbations. The U(1) gauge theory has a simpler structure
to deal with and can properly capture the gauge subtleties of the gravity
theory, which is the theory we will focus on in this paper.

As a brief summary of the main results, we study the Euclidean path integral
of U(1) gauge theory living between two parallel boundaries. The system is put
on a black hole background with the boundaries perpendicular to the radius
direction of the black hole $r$. We allow residual degrees of freedom of
$A_{r}$ on the boundaries. In the first paper of the series Cheng2022 , we
studied the flat case with two boundaries and found nonlocal effects due to
the interplay between the boundaries. Here, the analysis is generalized to the
curved spacetime, and we carefully study different behavior of the gauge
theory on the black hole background. The physical variables of the theory are
bulk fluctuation modes, zero longitudinal momentum mods of $A_{r}$, Wilson
lines stretched between the two boundaries, and other modes.111Like the
topological modes of the Wilson lines.

The bulk fluctuation modes whose entropy scales as the volume of the region
are always dominant at high temperatures. The bulk part also contains a
contribution proportional to the area of the horizon plus logarithm
corrections due to the highly curved spacetime near the horizon. As the
temperature cools down, we would mainly see the fluctuation modes of edge
residual degrees of freedom and the Wilson lines. The entropy of those modes
scales as the area of the boundaries multiplied by the temperature squared.
For superlow temperatures, there is a localization of modes on
$\partial_{\tau}\phi=\partial_{\tau}W=0$. Performing the path integral, the
corresponding entropy of those modes scales as the horizon area divided by the
Planck length squared. The constant modes and topological modes of the
boundary-stretched Wilson lines contribute logarithm corrections to the
entropy at superlow temperatures. So, we also get area contribution plus
logarithm corrections at superlow temperatures.

Now, we have seen that the Bekenstein-Hawking-like entropies and logarithm
corrections come from two different places. The first contribution comes from
the bulk fluctuation modes, which agrees with the entropy found in the “brick
wall” model tHooft:1984kcu . Those degrees of freedom correspond to the
degrees of freedom living very close to the horizon. The second contribution
is because the zero longitudinal momentum modes of $A_{r}$ and the boundary-
stretched Wilson lines $W$ are localized in the phase space at
$\partial_{\tau}\phi=\partial_{\tau}W=0$, whose entropy is coming from the
zero-point energy. Those modes correspond to the vacuum degeneracy near the
horizon and can be explained by symmetry-breaking patterns. Note that the
fluctuation modes dying off at superlow temperature was well-studied
Amsel:2009ev ; Dias:2009ex ; Johnstone:2013ioa ; Hajian:2013lna ;
Hajian:2014twa , so the extremal black hole entropy coming from a different
mechanism, like the zero-point energy discussed in the current paper, is a
reasonable way out.

It is interesting to notice that the first contribution only appears in
finite-temperature black holes, and the second one only appears in the
superlow temperature limit. We thus infer that the two categories of theories
might correspond to different modes found in this calculation. The attempts
that are trying to add or find quantum structure near the horizon are counting
the same thing as the contributions contained in the bulk fluctuation modes.
Furthermore, the attempts that are trying to find hidden symmetries for
extremal black holes and explaining the black hole entropy from a symmetry-
breaking viewpoint are counting the same thing as the modes dominating at the
superlow temperatures. The black hole entropy for finite-temperature black
holes and extremal black holes might be counting completely different things.

In this paper, we study the partition function of U(1) gauge theories with
multiple boundaries on a black hole background. The paper is organized as
follows. In Sec. II, we briefly review the main results from the first paper
of this series, where we studied the flat case. The flat case can be regarded
as a guide for our black hole calculation. And then, we analyze the allowed
boundary conditions and derive effective actions for different modes in Sec.
III. We perform the Euclidean path integral in Sec. IV, and discuss different
behavior of the fields at different temperatures. The same mechanism is
confirmed in the extremal black hole case in Sec. V. Sec. VI is a summary of
the whole paper and we also provide some further physical discussion there.
Appendix A is devoted to studying a different but still interesting set of
boundary conditions. More details of deriving the effective action are exiled
to the Appendixes B and C. The path integral of fluctuation modes in curved
spacetime is demonstrated in Appendix D.

##  II A brief review of the flat case

Figure 1: Sketch of the entropy of gauge fields with different temperatures in
the flat case. The red line demonstrates the dominant contribution, and the
overall entropy is the sum of different contributions.

In the first paper of the series Cheng2022 , we studied gauge theories between
two parallel boundaries with nontrivial boundary conditions on a flat
background. We allowed residual degrees of freedom of $A_{r}$ to survive on
the boundary, with the direction perpendicular to the boundary denoted as $r$.
Besides the residual modes due to the boundary conditions, we also found
Wilson lines stretched between different boundaries because the interplay
between boundaries is an interesting mode of the system. The symplectic form
and the canonical commutation relations were carefully studied, which helped
us to confirm the dynamical variables and measures in the phase space. It was
shown that there are bulk fluctuation modes, boundary edge modes, Wilson
lines, and other interesting modes in the phase space. With the canonical
analysis, we derived the partition function and entropy via the Euclidean path
integral method, and studied the different behavior of those modes by varying
the temperature of the system.

The main results are shown in Fig. 1. At high temperatures, the partition
function and entropy are the standard ones for blackbody radiation at
temperature $T$. The entropy contains two copies of physical polarizations and
is proportional to the volume between the two boundaries multiplied by
$T^{3}$. As the temperature becomes cooler than before, the surviving zero
longitudinal momentum modes $\phi$ start to be the most important
contribution. Also, boundary-stretched Wilson lines $W$ have similar behavior.
$\phi$ and $W$ just behave like two lower-dimensional massless scalar fields
living on the boundary. At relatively high temperatures, the fluctuation modes
of $\phi$ and $W$ are dominant and their entropies scale as the area of the
boundary multiplied by $T^{2}$. At superlower temperatures, all the
fluctuation contributions die off; and we are left with the constant and
winding modes of $W$, whose entropies more or less scale as the logarithm of
the length scales of the theory, as shown in Fig. 1. Note that the winding
modes of $W$ arise due to the map between the Euclidean time circle and U(1)
symmetry.

There are two transitions of dominance as can be seen from Fig. 1. The bulk
fluctuation modes always dominate in the high-temperature limit, and the
entropy scales as the volume multiplied by temperature cubed. As lower
temperature comes, the area contribution starts to dominate. At superlow
temperatures, the fluctuation contribution is not important anymore, and the
only contribution is from the constant modes and winding modes of the field
$W$.

The flat case was regarded as a toy model of the more general situation, for
example, curved spacetime as the background. With the canonical analysis and
partition function calculation of the flat case, we expect to see a similar
set of dynamical variables in the phase space. Moreover, we can follow a
similar logic in decomposing the variables and calculating the partition
function and entropy.

##  III Gauge fields on Schwarzschild black hole

Black holes are systems associated with temperatures and entropies, and the
Euclidean method for finite temperature systems also works for black holes.
However, the black hole system is special compared to the flat case discussed,
because the Bekenstein-Hawking entropy is always proportional to the area of
the horizon in units of $G_{N}$. As has been reviewed in the previous section,
there are interesting transitions of dominances of different modes arising due
to different reasons, one might speculate that a similar phenomenon would also
happen in black hole systems, which might explain something deep in the black
hole mechanism.

The background we are interested in is a Euclidean Schwarzschild black hole
with metric

$ds^{2}=(1-\frac{r_{s}}{r})d\tau^{2}+(1-\frac{r_{s}}{r})^{-1}dr^{2}+r^{2}d\Omega^{2}\,,$
(2)

where the Schwarzschild radius is $r_{s}=2G_{N}M$. Note that one important
difference between the flat case and black hole case is that we are using
spherical coordinates rather than Cartesian here. The inverse temperature of
the system (2) is identified with the periodicity of the Euclidean time
$\beta$, to avoid conical singularity at the horizon. The geometry can be
illustrated as the so-called “cigar” geometry, as shown in Fig. 2. Following
’t Hooft tHooft:1984kcu , we put two boundaries on this background and study
different temperature limits of the system, as shown in Fig. 3. The first
boundary is located at a very small distance $\varepsilon$ away from the
horizon, known as the “stretched horizon” Susskind:1993if . The other boundary
is at the distance $L$ away from the horizon, which is the surface of the box
similar to the flat case.

We will mainly discuss two different limits of the black hole system, as shown
in Fig. 3. In the high-temperature limit $L\gg r_{s}$, shown in the first
panel, the situation is supposed to be similar to the flat case because the
existence of the small black hole merely gives a periodicity to the Euclidean
time, i.e a temperature $T=1/\beta$, to the system. On the other hand, in the
low temperature limit $L\ll r_{s}$, we expect the results to be qualitatively
different from the high-temperature limit and share some similarities with the
extremal black hole case, the behavior of which was also well studied in
literature Strominger1996 ; Callan1996 ; Maldacena:1996ds ; Maldacena:1997ih ;
Peet2000 ; Banerjee:2010qc ; Sen:2012cj ; Iliesiu:2020qvm . This means that
there will be transitions of different behaviors, which might be explained by
the symmetry-breaking pattern.

We can always redefine $\rho=r-r_{s}$ and write the metric as

$ds^{2}=\frac{\rho}{\rho+r_{s}}d\tau^{2}+\frac{\rho+r_{s}}{\rho}d\rho^{2}+(\rho+r_{s})^{2}d\Omega^{2}\,.$
(3)

Now the two boundaries are at $\rho=\varepsilon$ and $\rho=L$ separately. In
order to derive the lower-dimensional effective action by dimensional
reduction, it is convenient to define the proper distance $y$ along the radius
direction

$dy=\sqrt{\frac{\rho+r_{s}}{\rho}}d\rho\,.$ (4)

The proper distance $y$ can be integrated out and read as

$y=\sqrt{\rho~{}(\rho+r_{s})}+r_{s}\text{~{}arcsinh}\sqrt{\frac{\rho}{r_{s}}}\,.$
(5)

Inverting the above equation, one can express $\rho$ as a function of $y$ and
rewrite the black hole metric as

$ds^{2}=\frac{\rho(y)}{\rho(y)+r_{s}}d\tau^{2}+dy^{2}+[\rho(y)+r_{s}]^{2}d\Omega^{2}\,,$
(6)

which will be the metric we mainly work with. Note that $y$ takes value from
$y_{1}$ to $y_{2}$, with

$y_{1}=\sqrt{\varepsilon(\varepsilon+r_{s})}+r_{s}\text{~{}arcsinh}\sqrt{\frac{\varepsilon}{r_{s}}}\,,~{}~{}~{}~{}~{}~{}y_{2}=\sqrt{L(L+r_{s})}+r_{s}\text{~{}arcsinh}\sqrt{\frac{L}{r_{s}}}\,.$
(7)

Figure 2: Geometry of a Euclidean black hole, where every point on the cigar
is an ${S}^{2}$. We put two boundaries on this background, at the stretched
horizon ($r=r_{s}+\varepsilon$) and distance $L$ away from the horizon
($r=r_{s}+L$) separately.

Figure 3: Different temperature limits of black holes in a box. Left: high
temperature limit $L\gg r_{s}$, i.e. the small black hole case. The
temperature of the system is $T\propto 1/r_{s}$. The mere role of the black
hole is to give a very high temperature $T$ to the system. We expect the final
results to be similar to the high-temperature limit of the flat case. Right:
low temperature limit of the system $L\ll r_{s}$, i.e. large black hole case.
We expect the fluctuation modes should be less important, but we still see
Bekenstein-Hawking-like entropy.

Before moving to the gauge theory and boundary conditions, let us briefly
discuss issues related to the stretched horizon. Here the stretched horizon
can be regarded as an auxiliary timelike surface, where we can put boundary
conditions on. There are different opinions towards the stretched horizon
222Mathematically, one can put boundaries at any location. The original
physics can be recovered by gluing the two sides along the edge, which means
one needs to integrate over the boundary conditions along the boundary. So it
is not strange to put a boundary near the horizon. However, whether is it
reasonable to put a physical boundary near the horizon is unknown because any
object near the horizon would fall into the black hole. Although we largely
mimic the flat case calculation, one may need to keep in mind that the black
hole situation might not be the same as the case of the parallel plates.; for
example, it is believed that only the physics outside of the stretched horizon
is well described by local QFT. In this case, the stretched horizon is a
physical surface Susskind2004 , and $\varepsilon$ can be regarded as the UV
regulator of the QFT outside of the horizon and can be taken to but not exceed
the Planck scale $l_{p}$. Whether the stretched horizon is a physical surface
or an auxiliary surface does not make much difference for our calculations
here, and $\varepsilon$ will be made very small either way. However, one does
need to pay attention to the order of taking different limits of length
scales. As we will see, in the low-temperature limit $\beta\gg L$, taking the
$\varepsilon\to 0$ limit would introduce localizations on the spaces of some
specific modes, which is essential to get the area entropy contribution.
However, one may not see this phenomenon if the order of taking the above
limits is messed up.

###  III.1 Boundary conditions

Now, let us put U(1) gauge theories between the boundaries, with Euclidean
action

$S_{E}=\frac{1}{4e^{2}}\int_{\mathcal{M}}d^{4}x\sqrt{g}~{}F^{\mu\nu}F_{\mu\nu}\,.$
(8)

As analyzed in Cheng2022 , to have a well-defined variation principle, there
are two sets of interesting boundary conditions.

  * •

The most interesting boundary condition for this paper is the situation where
we allow residual $A_{r}$ modes on boundaries

$\delta A_{a}\Big{|}_{\partial\mathcal{M}}=0,~{}~{}~{}~{}~{}~{}\delta
A_{r}\Big{|}_{\partial\mathcal{M}}=f(x^{a})\,,$ (9)

where $f(x^{a})$ can have local dependence of $x^{a}=(\tau,\theta,\phi)$ and
$A_{a}\big{|}_{\partial\mathcal{M}}$ are fixed configurations.

  * •

The other interesting boundary condition is Neumann-like:

$F^{\mu\nu}\Big{|}_{\partial\mathcal{M}}=0\,.$ (10)

There can be interesting physics as well, for example, the would-be pure gauge
modes on the boundary are not fixed by the boundary condition. Although this
boundary condition is not the main focus of this paper, we will briefly
discuss the physics related to (10) in Appendix A.

We are mainly going to focus on the boundary condition (9) on the black hole
background. To be precise, the boundary condition $A_{a}$ components are fixed
to

$\displaystyle A_{a}\Big{|}_{y=y_{\alpha}}$ $\displaystyle=$ $\displaystyle
0\,,$ (11)

where $\alpha$ labels different boundaries. As for the $A_{y}$ component, we
can separate it as

$A_{y}(x^{\mu})=\hat{A}_{y}(x^{\mu})+\frac{\phi(x^{a})}{|y|}\,,$ (12)

with $\hat{A}_{y}(x^{\mu})\big{|}_{y=y_{1}}=0$ and $|y|=y_{2}-y_{1}$. So,
$\phi(x^{a})$ more or less captures the degrees of freedom at the left
boundary $y=y_{1}$. Besides $\phi(x^{a})$, there are residual modes at the
boundary $y=y_{2}$. Following a similar logic from the flat case, the
difference between the two boundaries can be captured by Wilson lines
stretched between the two boundaries

$W(x^{a})=i\int_{y_{1}}^{y_{2}}dy~{}\hat{A}_{y}.$ (13)

The boundary-stretched Wilson lines capture the physical non-local effects and
will play an important role in our main calculation. There is no constraint on
the boundary configurations of $A_{y}$, so we need to integrate all the
boundary configurations of $A_{y}$ into the path integral.

One can work out the Hamiltonian formulation and symplectic form to study the
phase space of the theory with the indicated boundary condition. The dynamical
variables in the phase space would be clear from the canonical analysis, which
tells us which degrees of freedom should be included in the path integral
formulation. We have done an analog analysis for the flat case in paper
Cheng2022 , while we will not go through all the canonical analysis here.
Similar to the flat case, we would also have bulk modes
$\hat{A}_{\mu}(x^{\mu})$, zero momentum modes along the $y$ direction
$\phi(x^{a})$, and Wilson lines stretched between the two boundaries
$W(x^{a})$ on a black hole background. We will justify that they are physical
by working out their effective actions.

###  III.2 The effective action

The partition function can be calculated by the Euclidean functional integral
over all the dynamical variables in the theory. For the theory we are
studying, we have bulk fluctuation modes $\hat{A}_{\mu}$ and the modes arise
because of the nontrivial boundary condition: $\phi$ and $W$. So the partition
function can be written as

$Z=\int\mathcal{D}\hat{A}_{\mu}~{}\mathcal{D}\phi~{}\mathcal{D}W~{}\exp\left\\{-S_{E}[\hat{A}_{\mu},\phi,W]\right\\}\,.$
(14)

The effective action $S_{E}$ is worked out in Appendix B, which reads as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}+\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]\,$ (15)
$\displaystyle-\frac{1}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}i\partial_{a}W\partial_{b}\phi\,.$

The first part can be directly put into the path integral, so we can write the
partition function for the bulk fluctuation modes as

$Z_{\hat{A}}=\int\mathcal{D}\hat{A}_{\mu}\exp\left[-\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}\right]\,.$ (16)

We will discuss bulk gauge fixing condition and evaluate the above partition
function in the next subsection. The rest of the action is

$\displaystyle S[\phi,W]=\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{1}{e^{2}|y|}\int
d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}i\partial_{a}W\partial_{b}\phi\,.$
(17)

Note that $S[\phi,W]$ contains three-dimensional and four-dimensional parts,
while the fields $\phi(x^{a})$ and $W(x^{a})$ only depend on the transverse
coordinates $x^{a}$. In order to perform the path integral, we are going to
rewrite $S[\phi,W]$ into a three-dimensional action

$\displaystyle S^{(3)}[\phi,W]$ $\displaystyle=$
$\displaystyle\frac{1}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}\left[h^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{i}{2e^{\prime
2}}\int d\tau d^{2}x\sqrt{h}$ (18)
$\displaystyle\times\left[\gamma_{1}~{}h^{\tau\tau}\partial_{\tau}W\partial_{\tau}\phi+\gamma_{2}~{}h^{\theta\theta}\partial_{\theta}W\partial_{\theta}\phi+\gamma_{3}~{}h^{\varphi\varphi}\partial_{\varphi}W\partial_{\varphi}\phi\right]\,.$

$e^{\prime}$, $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ are undetermined
parameters, which will be worked out soon. This is more or less like the
Kaluza-Klein reduction from a higher-dimensional action to a lower-dimensional
one.

Now we are going to work out all the parameters in action (18). More details
of the calculations can be found in Appendix C. Before actually doing that,
let us first assume that the three-dimensional theory uses the same time
coordinate $\tau$ as the original one, and the three-dimensional metric takes
the following form

$h_{ab}=\text{diag}(h_{\tau\tau},~{}R^{2},~{}R^{2}\sin^{2}\theta)\,.$ (19)

$R^{2}$ is a parameter in the metric that we are going to fix by dimensional
reduction. With this assumption, we have
$\sqrt{h}=\sqrt{h_{\tau\tau}}~{}R^{2}\sin\theta$. To match the first parts in
actions (18) and (17), we need to work out the following problem

$\displaystyle\frac{1}{2e^{2}|y|^{2}}\int d\tau
d^{2}x~{}(\int_{y_{1}}^{y_{2}}dy\sqrt{g}g^{ab})~{}\partial_{a}\phi\partial_{b}\phi$
$\displaystyle=$ $\displaystyle\frac{1}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}\left[h^{ab}\partial_{a}\phi\partial_{b}\phi\right]\,$ (20)

The solution can be easily obtained and can be written as

$\displaystyle
h_{\tau\tau}=\frac{LR^{2}}{F}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}\frac{1}{e^{\prime
2}}=\frac{L}{e^{2}|y|^{2}}\sqrt{h^{\tau\tau}}=\frac{1}{e^{2}|y|^{2}}\frac{\sqrt{L~{}F}}{R}\,,$
(21)

with

$F\approx~{}3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}\,.$
(22)

More details can be found in Appendix C. Now we have obtained the three-
dimensional metric $h_{ab}$ and the effective coupling constant $e^{\prime
2}$. The next step is to calculate the parameters $\gamma_{1}$, $\gamma_{2}$,
and $\gamma_{3}$ in the action (18). We need to match the rest of the action
in (17) and (18), i.e.

$\displaystyle\frac{1}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{\tau\tau}\right]_{y=y_{1}}^{y=y_{2}}\partial_{\tau}W\partial_{\tau}\phi$
$\displaystyle=$ $\displaystyle\frac{\gamma_{1}}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}~{}\left[h^{\tau\tau}\partial_{\tau}W\partial_{\tau}\phi\right]\,,$
$\displaystyle\frac{1}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{\theta\theta}\right]_{y=y_{1}}^{y=y_{2}}\partial_{\theta}W\partial_{\theta}\phi$
$\displaystyle=$ $\displaystyle\frac{\gamma_{2}}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}~{}\left[h^{\theta\theta}\partial_{\theta}W\partial_{\theta}\phi\right]\,,$
(23) $\displaystyle\frac{1}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{\varphi\varphi}\right]_{y=y_{1}}^{y=y_{2}}\partial_{\varphi}W\partial_{\varphi}\phi$
$\displaystyle=$ $\displaystyle\frac{\gamma_{3}}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}~{}\left[h^{\varphi\varphi}\partial_{\varphi}W\partial_{\varphi}\phi\right]\,.$

The solution of the equations (23) can be obtained as

$\displaystyle\gamma_{1}=-2|y|\sqrt{\frac{r_{s}}{\varepsilon}}\frac{r_{s}^{2}}{F}\,,~{}~{}~{}~{}~{}~{}~{}~{}\gamma_{2}=\gamma_{3}=\frac{2|y|}{L}\sqrt{\frac{L}{L+r_{s}}}\,.$
(24)

Now, all the undetermined parameters in the effective action (18) are worked
out.

With the effective action (18) at hand, we can perform the path integral to
calculate the partition function. The Gaussian path integral for field $\phi$
can be easily worked out, and we get an effective action for field $W$ in the
meantime. The path integral over $\phi$ gives $\det(\partial^{2})^{-1/2}$.
Then, the corresponding action for $W$ can be expressed as

$S_{W}=-\frac{1}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}~{}(\gamma^{2}_{1}~{}h^{\tau\tau}\partial_{\tau}W\partial_{\tau}W+\gamma^{2}_{2}~{}h^{\theta\theta}\partial_{\theta}W\partial_{\theta}W+\gamma^{2}_{3}~{}h^{\varphi\varphi}\partial_{\varphi}W\partial_{\varphi}W)\,.$
(25)

One can always rewrite $\det(\partial^{2})^{-1/2}$ as a path integral over
field $\phi$, and the overall partition function can be expressed as

$\displaystyle Z_{\phi,W}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\phi~{}\mathcal{D}W~{}\exp\left[-S_{\phi,W}\right]\,.$
(26)

with the effective action

$\displaystyle S_{\phi,W}$ $\displaystyle=$ $\displaystyle\frac{1}{2e^{\prime
2}}\int d\tau d^{2}x\sqrt{h}$ (27) $\displaystyle\times$
$\displaystyle\left(h^{ab}\partial_{a}\phi\partial_{b}\phi+\gamma^{2}_{1}~{}h^{\tau\tau}(\partial_{\tau}W)^{2}+\gamma^{2}_{2}~{}h^{\theta\theta}(\partial_{\theta}W)^{2}+\gamma^{2}_{3}~{}h^{\varphi\varphi}(\partial_{\varphi}W)^{2}\right)\,.$

A different way of saying rewriting $\det(\partial^{2})^{-1/2}$ as a path
integral over fields $\phi$ is that the path integrals with effective actions
(18) or (27) are the same. The effective action (27) can be explicitly
expressed as

$S_{\phi,W}=S_{\phi}+S_{W}\,$ (28)

with

$\displaystyle S_{\phi}=\frac{L}{2e^{2}|y|^{2}}\int d\tau
d^{2}x~{}R^{2}\sin\theta~{}\Big{(}\frac{F}{LR^{2}}\partial_{\tau}\phi\partial_{\tau}\phi+\frac{1}{R^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+\frac{1}{R^{2}\sin^{2}\theta}\partial_{\varphi}\phi\partial_{\varphi}\phi\Big{)}\,,$
$\displaystyle S_{W}=\frac{2}{e^{2}(L+r_{s})}\int d\tau
d^{2}x~{}r_{s}^{2}\sin\theta~{}\Big{(}\frac{r_{s}}{\varepsilon}\frac{Lr_{s}^{2}+r_{s}^{3}}{F}\partial_{\tau}W\partial_{\tau}W+\frac{1}{r_{s}^{2}}~{}\partial_{\theta}W\partial_{\theta}W+\frac{1}{r_{s}^{2}\sin^{2}\theta}\partial_{\varphi}W\partial_{\varphi}W\Big{)}\,.$

The two symbols $F$ and $|y|$ in the above actions are

$\displaystyle F$ $\displaystyle=$ $\displaystyle
3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}\,,$
(29) $\displaystyle|y|$ $\displaystyle=$
$\displaystyle\sqrt{L(L+r_{s})}+r_{s}\text{~{}arcsinh}\sqrt{\frac{L}{r_{s}}}\,.$
(30)

Equations (26) and (28) are the partition function and the effective action
that we will mainly focus on. With that at hand, we can evaluate the partition
functions of fields $\hat{A}_{\mu}$, $\phi$, and $W$ by the path integral.

Note that it is important to keep the original Euclidean time coordinate
$\tau$ as the time coordinate for the three-dimensional theory because the
periodicity of coordinate time $\tau$ is the physical inverse temperature for
the observer who uses the Schwarzschild metric. If one uses different time
coordinates, the physics would be difficult to discuss. For example, the low-
temperature limit for the coordinate observer can be high temperatures for an
observer using a different coordinate system. So we would always keep $\tau$
as our time coordinate.

##  IV Behavior of different modes

In this section, we are going to evaluate the partition function of the theory
in different temperature limits. The different temperature limits are
illustrated in Fig. 3. The partition function for $W$ and $\phi$ will be
treated separately, and we will see different behavior in different
temperature limits. The overall partition function can be written as three
parts: bulk contribution $Z_{\hat{A}}$ multiplied by $Z_{\phi}$ and $Z_{W}$

$\displaystyle Z$ $\displaystyle:=$ $\displaystyle Z_{\hat{A}}\times
Z_{\phi}\times Z_{W}$ $\displaystyle=$
$\displaystyle\int\mathcal{D}\hat{A}_{\mu}~{}e^{-\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}}\times\int\mathcal{D}\phi~{}e^{-S_{\phi}}\times\int\mathcal{D}W~{}e^{-S_{W}}\,.$

We are going to calculate the partition function for the bulk fluctuation
modes $Z_{\hat{A}}$ in the next subsection. After that, we will evaluate
$Z_{\phi}$ and $Z_{W}$ in the high- and low-temperature limits in Secs. IV.2
and IV.3.

###  IV.1 Bulk fluctuation modes

Now, let us evaluate the entropy of bulk fluctuation modes. The details of the
calculation are exiled to Appendix D, to avoid being distracted from the main
text. This is the standard blackbody calculation on curved spacetime. For
electromagnetism, we have two physical polarizations in the bulk, which are
both bosonic and massless. The bulk partition function reads as

$\ln
Z_{\hat{A}}=-\frac{4\pi^{3}}{45}\frac{1}{\beta^{3}}\frac{r_{s}^{4}}{\varepsilon}-\frac{16\pi^{3}}{45}\frac{r_{s}^{3}}{\beta^{3}}\ln\frac{L}{\varepsilon}-\frac{4\pi^{3}}{45}\frac{1}{\beta^{3}}\left(-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\right)\,.$
(32)

and the corresponding entropy can be written as

$\displaystyle\mathcal{S}_{\hat{A}}$ $\displaystyle=$
$\displaystyle\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\frac{r_{s}^{4}}{\varepsilon}+\frac{64\pi^{3}}{45}\frac{r_{s}^{3}}{\beta^{3}}\ln\frac{L}{\varepsilon}+\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\left(-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\right)\,.$
(33)

Let us look at the first two terms

$\mathcal{S}_{0}=\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\frac{r_{s}^{4}}{\varepsilon}+\frac{64\pi^{3}}{45}\frac{r_{s}^{3}}{\beta^{3}}\ln\frac{L}{\varepsilon}\,.$
(34)

First of all, those contributions cannot be seen in the extremal black hole
(or superlow temperature) case where we have $\beta\to\infty$ while keeping
the radius of the black hole to be finite. For a finite temperature black
hole, defining the proper distance from the real horizon to the stretched
horizon as $\delta$, we have

$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\int_{r_{s}}^{r_{s}+\varepsilon}\sqrt{g_{rr}}~{}dr\approx
2\sqrt{\varepsilon~{}r_{s}}\,.$ (35)

Thus, we have $\delta^{2}\approx 4\varepsilon~{}r_{s}$. For the finite
temperature black hole, with the inverse temperature

$\beta=8\pi G_{N}M\,,$ (36)

the entropy (34) can be written as

$\mathcal{S}_{0}\propto\frac{r_{s}^{2}}{\delta^{2}}+\ln\frac{Lr_{s}}{\delta^{2}}\,.$
(37)

Those contributions arise because of the redshift between the horizon and the
coordinate observer sitting at infinity. Any finite frequency modes near the
horizon would have zero frequency as seen by a coordinate observer. We have an
infinite number of states with zero energy, and summing over all of those
states with a UV cutoff would give us the above result. Those very dense
ground states mainly come from the near-horizon region; thus we have an area
contribution and corrections Susskind2004 . This result was used to understand
the Bekenstein-Hawking entropy by some authors, for example tHooft:1984kcu ,
and it was also interpreted as the entanglement entropy across the stretched
horizon Srednicki:1993im . 333However there are different opinions. For
instance, Susskind showed that the contribution should be absorbed by the
renormalization of Newton’s constant $G_{N}$ due to the loop contribution
Susskind:1994sm .

The other terms

$\displaystyle\mathcal{S}_{\hat{A}}=\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\left(-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\right)\propto\text{Volume}\times
T^{3}\,$ (38)

are entropy of the thermal fluctuation modes of gauge fields living in the
bulk, which can be regarded as the curved spacetime analog of the blackbody
result. $S_{\hat{A}}$ is finite and more or less proportional to the volume
between the two boundaries multiplied by $T^{3}$.

###  IV.2 Zero longitudinal momentum modes of $A_{r}$

In this section, we evaluate the partition function for $\phi$, which is zero
longitudinal momentum modes of $A_{r}$. The partition function $Z_{\phi}$ can
always be written as

$Z_{\phi}=\int\mathcal{D}\phi~{}e^{-S_{\phi}}\,,$ (39)

with the action

$S_{\phi}=\frac{L}{2e^{2}|y|^{2}}\int d\tau
d^{2}x~{}R^{2}\sin\theta~{}\Big{(}\frac{F}{LR^{2}}\partial_{\tau}\phi\partial_{\tau}\phi+\frac{1}{R^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+\frac{1}{R^{2}\sin^{2}\theta}\partial_{\varphi}\phi\partial_{\varphi}\phi\Big{)}\,,$
(40)

with

$\displaystyle
F=3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}\,,~{}~{}~{}|y|=\sqrt{L(L+r_{s})}+r_{s}\text{~{}arcsinh}\sqrt{\frac{L}{r_{s}}}\,.$
(41)

The action (40) has different behavior in different temperature limits, and we
will see how the entropy contribution from $\phi$ changes as we vary the
temperature of the system. One obvious thing to notice is that there is a
transition of dominance in function $F$ shown in Eq. (41) at different
temperatures. The $L^{3}$ term is the dominant contribution for high
temperatures $L\gg r_{s}$, and the UV cutoff $\varepsilon$ is absent in
$S_{\phi}$. While at low temperatures $L\ll r_{s}$, the most important term is
$r_{s}^{3}\ln L/\varepsilon$, and we will see different behavior of the
corresponding entropy. We always assume that the short distance cutoff
$\varepsilon$ is always much smaller than $L$ and $r_{s}$.

####  IV.2.1 High-temperature limit

Let us first discuss the high-temperature limit $r_{s}\ll L$, as shown in the
first panel of Fig. 3. If

$r_{s}^{3}\ln\frac{L}{\varepsilon}\ll L^{3}\,$ (42)

is satisfied, then $L^{3}$ is the most important term in $F$. In this case,
$F$ can be approximated as

$F\approx\frac{L^{3}}{3}\,.$ (43)

Function arcsinh$(x)$ is always much smaller than $x$ for large value of $x$,
so $|y|$ can be written as $|y|\approx L$. Then, the effective action in the
high-temperature limit can be expressed as

$\displaystyle S_{\phi}$ $\displaystyle=$ $\displaystyle\frac{1}{2e^{2}L}\int
d\tau
d^{2}x~{}L^{2}\sin\theta~{}\Big{(}\frac{1}{3}\partial_{\tau}\phi\partial_{\tau}\phi+\frac{1}{L^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+\frac{1}{L^{2}\sin^{2}\theta}\partial_{\varphi}\phi\partial_{\varphi}\phi\Big{)}\,.$
(44)

The effective action for $\phi$ is just a scalar field living on a
${S}^{1}\times{S}^{2}$, where the length scale of ${S}^{1}$ is $\beta$, and
the length scale of ${S}^{2}$ is $L$.

Equipped with the effective action (44), we can directly calculate the
partition function $Z_{\phi}$ shown in (39). We can absorb the finite constant
$1/3$ in front of $\partial_{\tau}\phi\partial_{\tau}\phi$ term in the action
into the redefinition of $\tau$ to $\tau^{\prime}$. Let us suppose the
fluctuation modes of $\phi$ are

$\displaystyle\phi(x^{a})$ $\displaystyle=$
$\displaystyle\mathcal{N}_{\phi}\cdot\sum_{\omega}\sum_{l,m}e^{i\omega\tau^{\prime}}Y_{lm}(\theta,\varphi)\tilde{\phi}(\omega,l,m)\,,$
(45)

where $\mathcal{N}_{\phi}$ is a normalization constant. The partition function
for $\phi$ in the canonical ensemble can be written as

$\ln Z_{\phi}=-\sum_{\omega}\ln(1-e^{-\beta^{\prime}\omega})\,.$ (46)

The calculation is similar to the bulk fluctuation modes calculation
demonstrated in Appendix D. We can change the summation of $\omega$ to
integration by introducing density of state $g(\omega)$, which gives out

$\displaystyle\ln Z_{\phi}$ $\displaystyle=$
$\displaystyle-\int_{0}^{\infty}g(\omega)\ln(1-e^{-\beta^{\prime}\omega})d\omega=\beta^{\prime}\int_{0}^{\infty}\frac{\Gamma(\omega)}{e^{\beta^{\prime}\omega}-1}d\omega\,.$
(47)

$\Gamma(\omega)$ is defined by $d\Gamma=g(\omega)d\omega$. According to the
dispersion relation in this background, $\Gamma(\omega)$ can be counted as

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\sum_{l,m}\frac{\beta^{\prime}}{2\pi}\sqrt{\frac{l(l+1)}{L^{2}}}=\frac{\beta^{\prime}}{2\pi}\sum_{l}(2l+1)\sqrt{\frac{l(l+1)}{L^{2}}}\,.$
(48)

The summation is from $l=0$ to the level with energy $\omega$. Putting
everything back into Eq. (47) and changing the summation of $l$ into the
integral, we can write the partition function as

$\displaystyle\ln Z_{\phi}$ $\displaystyle=$ $\displaystyle\frac{\beta^{\prime
2}}{2\pi}L^{2}\int_{0}^{\infty}\frac{d\omega}{e^{\beta^{\prime}\omega}-1}\int_{l}d\left(\frac{l(l+1)}{L^{2}}\right)~{}\sqrt{\frac{l(l+1)}{L^{2}}}\,.$
(49)

$\sqrt{l(l+1)/L^{2}}$ is exactly the energy carried by a particle with angular
momentum $l(l+1)$. Thus we can change a variable and get

$\ln Z_{\phi}=\frac{\beta^{\prime
2}L^{2}}{2\pi}\int_{0}^{\infty}\frac{d\omega}{e^{\beta^{\prime}\omega}-1}\int_{0}^{\omega^{2}}dx~{}\sqrt{x}=\frac{\beta^{\prime
2}L^{2}}{3\pi}\int_{0}^{\infty}\frac{\omega^{3}d\omega}{e^{\beta^{\prime}\omega}-1}\,.$
(50)

Then the above expression can be worked out, and the logarithm of the
partition function can be expressed as

$\ln Z_{\phi}=\frac{\pi^{3}}{45}\frac{L^{2}}{\beta^{\prime
2}}=\frac{\pi^{3}}{135}\frac{L^{2}}{\beta^{2}}\,.$ (51)

Then, the corresponding entropy can be written as

$\mathcal{S}_{\phi}=\frac{\pi^{3}}{45}\frac{L^{2}}{\beta^{2}}\,.$ (52)

The above result can be compared with the entropy of $\phi$ in the flat case
in Cheng2022 . Thus, we can conclude that the zero modes $\phi$ share similar
properties with the flat case in the high-temperature limit $r_{s}\ll L$.

So the high-temperature behavior of field $\phi$ is just like a three-
dimensional scalar field with inverse temperature $\beta$, whose entropy is
standard, as shown in Eq. (52). There is not much difference with the flat
case.

####  IV.2.2 Low-temperature limit

As the temperature becomes superlow, $r_{s}$ can be larger than $L$, and we
arrive at the low-temperature limit of the system with $r_{s}\gg L$, as shown
in the second panel of Fig. 3. As we can see, there is a transition of
dominance between different terms in $F$, and in the low-temperature limit $F$
can be approximated as

$F\approx r_{s}^{3}\ln\frac{L}{\varepsilon}\,.$ (53)

We also have $|y|^{2}\approx L(L+r_{s})$. Then, the effective action for
$\phi$ in the low-temperature limit can be written as

$\displaystyle S_{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2e^{2}(L+r_{s})}\int d\tau
d^{2}x~{}r_{s}^{2}\sin\theta\Big{(}\frac{r_{s}}{L}\ln\frac{L}{\varepsilon}\partial_{\tau}\phi\partial_{\tau}\phi+\frac{1}{r_{s}^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+\frac{1}{r_{s}^{2}\sin^{2}\theta}\partial_{\varphi}\phi\partial_{\varphi}\phi\Big{)}\,.$
(54)

As discussed before, the time of the three-dimensional action should always be
chosen as the coordinate time $\tau$. Doing so, the coefficient in front of
$(\partial_{\tau}\phi)^{2}$ is fixed and always much larger than 1. Especially
when we take the real event horizon limit, i.e. the $\varepsilon\to 0$ limit,
the path integral localizes on the space of zero energy modes
$\partial_{\tau}\phi=0$. Then the partition function of the field $\phi$ only
depends on the radius of ${S}^{2}$.

Because of the localization discussed above, the logarithm of the partition
function for the zero-point energy is no longer linear in $\beta$ but only
depends on the radius of $S^{2}$ and UV cutoff, which can be expressed as

$\ln Z_{\phi}\propto r_{s}^{2}\cdot\Lambda^{2}\,.$ (55)

Let us assume the UV cutoff is the Planck scale, then the corresponding
entropy is of Bekenstein-Hawking entropy magnitude

$\mathcal{S}\propto\frac{r_{s}^{2}}{l_{p}^{2}}\,.$ (56)

So, we can say that, in the low-temperature limit, the entropy of $\phi$ can
be written as the area of the horizon divided by the Planck area $l_{p}^{2}$.

Now, let us summarize what we have got for the entropy of zero longitudinal
momentum modes $\phi$. The entropy of the whole system is illustrated in Fig.
4. At high temperatures, $L\gg r_{s}$, the entropy of $\phi$ is more or less
like what we have in the flat case, namely the area of the box times
temperature squared $L^{2}\times T^{2}$. Hence, the presence of a small black
hole merely gives a temperature to the system. This amount of entropy then has
competition with the bulk fluctuation modes. At very high temperatures, the
volume times temperature cubed wins, and the area times temperature squared
wins at relatively lower temperatures. This is more or less the same story
told in Cheng2022 , reviewed in Sec. II. The difference starts to show up at
superlow temperatures; one can compare Figs. 4 and 1 to see the difference. In
the black hole case, the entropy for $\phi$ is shown in Eq. (56) at superlow
temperature, which is reminiscent of the Bekenstein-Hawking entropy. In
comparison, this phenomenon cannot be seen in the flat case, where the entropy
has a logarithm behavior as shown in Fig. 1.

Figure 4: Sketch of the entropy of the gauge theory as we vary the temperature
of the black hole. The high-temperature behavior is qualitatively like the
flat case as shown in Fig. 1. At superlow temperatures, the entropy is
proportional to the area of the horizon, multiplied by the UV cutoff $\Lambda$
squared.

###  IV.3 Boundary-stretched Wilson lines

Now, let us also discuss the different behavior of field $W$ in different
temperature limits in this subsection. The effective action for $W$ is

$\displaystyle S_{W}$ $\displaystyle=$
$\displaystyle\frac{2}{e^{2}(L+r_{s})}\int d\tau
d^{2}x~{}r_{s}^{2}\sin\theta\Big{(}\frac{r_{s}}{\varepsilon}\frac{Lr_{s}^{2}+r_{s}^{3}}{F}\partial_{\tau}W\partial_{\tau}W+\frac{1}{r_{s}^{2}}~{}\partial_{\theta}W\partial_{\theta}W+\frac{1}{r_{s}^{2}\sin^{2}\theta}\partial_{\varphi}W\partial_{\varphi}W\Big{)}\,.$
(57)

with

$F=3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}\,.$
(58)

The infinite coefficient introduces a localization on the zero energy modes
$\partial_{\tau}W=0$, and the entropy of the Boundary-stretched Wilson lines
$W$ is more or less

$\mathcal{S}\propto\frac{r_{s}^{2}}{l_{p}^{2}}\,,$ (59)

for the same reason discussed in the previous subsection.

Only when the size of the box $L$ is extremely large compared to $r_{s}$,
namely the high-temperature limit, some power of $r_{s}/L$ may overcome
$r_{s}/\varepsilon$, and we get a finite coefficient. Then, the corresponding
entropy can be expressed as

$\mathcal{S}\propto r_{s}^{2}\times T^{2}\,,$ (60)

which is a finite constant. In this limit, the entropy from $W$ can always be
ignored, because the entropy of fluctuation modes always plays the dominant
role at high temperatures.

There can also be constant modes, and topological modes due to the map between
the Euclidean time circle and the U(1) group. The entropy of the constant
modes and winding modes of $W$ are never comparable with (59), but contribute
as logarithm corrections. We have

$Z_{0}=\sqrt{\frac{e^{2}\beta(L+r_{s})}{\text{Area}}}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}Z_{w}=\sum_{n}e^{0}=\zeta(0)\,,$
(61)

which contribute as logarithms of the temperature $T$, the coupling constant
$e$, and other length scales. Note that the constant modes always contribute a
logarithm term in entropy, but the winding modes tend to be less important as
can be seen from (61). This is because for the superlow temperatures, $\beta$
is superlarge and we can only see the mode of the zero winding number. The
winding modes contribution in relatively high temperatures is more or less the
same as the flat case Cheng2022 , which is not expatiated here because those
modes are not comparable with the fluctuation modes at high temperatures. The
overall logarithm correlation from constant and winding modes is an important
aspect of the entropy but is small compared which other contributions. So, we
can say that the entropy of $W$ follows the same qualitative pattern as the
entropy behavior shown in Fig. 4.

Let us briefly summarize this section. Basically, we have three different
phases. At high temperatures, the bulk fluctuation modes shown in Sec. IV.1
are the most important contribution. The entropy is proportional to the volume
of the box multiplied by the temperature cubed, as shown by the right part of
the curve in Fig. 4. We also have a contribution (37) scale as the area of the
horizon plus logarithm correction, contained in the bulk fluctuation
contribution. As the temperature cools down, the boundary area contribution
dominates over other contributions. The corresponding entropy is shown in Eq.
(52). At superlow temperatures, we have an entropy of Bekenstein-Hawking
magnitude

$\mathcal{S}\propto\frac{r_{s}^{2}}{l_{p}^{2}}\,,$ (62)

coming from both the zero modes $\phi$ and the Wilson lines $W$. This
contribution will not die off at zero temperature. So we expect the behavior
would persist for the extremal black holes. It is interesting to further
verify this point for the extremal black hole case.

##  V Extremal black hole

The entropy in the superlow temperature limit of finite temperature black
holes is because of the localization of modes on a specific region in the
phase space. Although we inferred that the mechanism should be responsible for
the entropy of extremal black holes, a calculation on an extremal black hole
background is vital for the story. The main task of this section is to show
the same localization mechanism also works for the extremal black hole case.
We are going to evaluate the partition function and entropy of U(1) gauge
theory with nontrivial boundary conditions on multiple boundaries on a
(near-)extremal black hole background.

To get an extremal black hole, we start with the Euclidean Reissner-Nordström
metric, which can be written as

$ds^{2}=f(r)d\tau^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2}_{2}\,,$ (63)

with

$f(r)=1-\frac{2G_{N}M}{r}+\frac{G_{N}Q^{2}}{r^{2}}\,.$ (64)

The horizon is a null surface, which can be obtained by solving $g_{rr}=0$.
The inner and outer horizons can be written as

$r_{\pm}=(G_{N}M\pm\sqrt{(G_{N}M)^{2}-G_{N}Q^{2}})\,.$ (65)

The extremal limit is the limit where we have a double zero, which means

$r_{+}=r_{-}=G_{N}M\,.$ (66)

Thus, the metric of an extremal black hole can be written as

$ds^{2}=\left(1-\frac{r_{H}}{r}\right)^{2}dt^{2}+\left(1-\frac{r_{H}}{r}\right)^{-2}dr^{2}+r^{2}d\Omega_{2}^{2}\,,$
(67)

with

$r_{H}=G_{N}M\,.$ (68)

We can give this black hole system a very tiny temperature by replacing the
double zero at the horizon with two single zeros $r_{\pm}=r_{H}\pm\varepsilon$
with the short distance cutoff $\varepsilon$. The inverse temperature of the
near extremal black hole can be expressed as

$\beta=\frac{2\pi r_{H}^{2}}{\varepsilon}\,,$ (69)

which is infinity in the limit $\varepsilon\to 0$.

Now, let us put U(1) gauge theory with boundaries at the stretched horizon and
at location $r_{H}+L$ in spacetime (67). The original Euclidean action can be
written as

$S_{E}=\frac{1}{4e^{2}}\int_{\mathcal{M}}d^{4}x\sqrt{g}~{}F^{\mu\nu}F_{\mu\nu}\,.$
(70)

We are going to focus on the first boundary condition (9), where we allow
residual $A_{r}$ modes on boundaries

$\delta A_{a}\Big{|}_{\partial\mathcal{M}}=0,~{}~{}~{}~{}~{}~{}\delta
A_{r}\Big{|}_{\partial\mathcal{M}}=f(x^{a})\,.$ (71)

$f(x^{a})$ can have local dependence of $x^{a}=(\tau,\theta,\phi)$ on
different boundaries. $A_{a}\big{|}_{\partial\mathcal{M}}$ are fixed
configurations, which can be set to zero.

Following the same logic as the finite temperature case, we shift the radius
direction by $r_{H}$

$\rho=r-r_{H}\,,$ (72)

and then define the proper distance $y$ along the radius direction as

$dy=\frac{\rho+r_{H}}{\rho}d\rho\,.$ (73)

Solving the above equation, the proper distance $y$ can be expressed as

$y=\rho+r_{H}\ln\frac{\rho}{r_{H}}\,.$ (74)

$y$ takes value from $y_{1}$ to $y_{2}$, with

$y_{1}=\varepsilon+{r_{H}}\ln\frac{\varepsilon}{r_{H}}\,,~{}~{}~{}~{}~{}y_{2}=L+{r_{H}}\ln\frac{L}{r_{H}}\,.\\\
$ (75)

We have ln$(0)=-\infty$ and thus $y_{1}\to-\infty$444This is different with
the finite temperature case. In the finite temperature case, the left boundary
is localized at $y_{1}\to 0$ because of arcsinh$(0)=0$.. So $|y|$ can be
approximated as

$|y|=y_{2}-y_{1}\approx L+r_{H}\ln\frac{L}{\varepsilon}\,.$ (76)

On the stretched horizon, the residual degrees of freedom of $A_{y}$ are set
to $\phi(x^{a})/L$, and the field $A_{y}$ can be decomposed as

$A_{y}(x^{\mu})=\hat{A}_{y}(x^{\mu})+\frac{\phi(x^{a})}{|y|}\,;$ (77)

the difference between the two boundaries is captured by the Wilson lines
stretched between the two boundaries at $y=y_{1}$ and $y=y_{2}$,

$W(x^{a})=i\int_{y_{1}}^{y_{2}}dy~{}\hat{A}_{y}.$ (78)

Thus, we have divided our gauge fields into three different parts: the bulk
fluctuation modes $\hat{A}_{y}$ that vanish on both boundaries, zero
longitudinal momentum modes $\phi(x^{a})$, and boundary-stretched Wilson lines
$W(x^{a})$. The symplectic form and commutation relations of those modes were
worked out by carefully analyzing the phase space of the theory in Cheng2022 .
So one can just put those modes in the Euclidean path integral to calculate
the corresponding partition function and entropy of them.

The partition function can be formally written as

$Z=\int\mathcal{D}\hat{A}_{\mu}~{}\mathcal{D}\phi~{}\mathcal{D}W~{}\exp\left\\{-S_{E}[\hat{A}_{\mu},\phi,W]\right\\}\,.$
(79)

The action $S_{E}$ can be separated into

$S_{E}=S_{\hat{A}}+S[\phi,W]\,,$ (80)

where

$S_{\hat{A}}=\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}\,.$ (81)

The bulk fluctuation modes reviewed in the previous section will not survive,
because of the finite $r_{H}$ and the tiny temperature
$T=1/\beta\propto\varepsilon$. So we will mainly focus on fields $\phi$ and
$W$ to see the behavior of those modes. Actually, all of the fluctuation mode
contributions whose entropy is proportional to the temperature will not play
any role in the final result, and we expect to see entropy contributions that
look like the Bekenstein-Hawking entropy for extremal black holes.

Ignoring the bulk fluctuation modes, we are going to focus on fields $\phi$
and $W$, whose action can be written as

$\displaystyle S[\phi,W]=\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{i}{e^{2}|y|}\int
d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}\partial_{a}W\partial_{b}\phi\,.$
(82)

Note that the action $S[\phi,W]$ contains four-dimensional and three-
dimensional integrations, while $\phi(x^{a})$ and $W(x^{a})$ only depend on
transverse directions along the boundaries. So we can dimension reduce the
action $S[\phi,W]$ to three-dimensional and work out an effective action for
$\phi$ and $W$. Following a similar dimensional reduction procedure in the
previous section, we obtain the three-dimensional effective action for fields
$\phi$ and $W$

$S_{\phi,W}=S_{\phi}+S_{W}\,,$ (83)

with

$\displaystyle S_{\phi}$ $\displaystyle=$
$\displaystyle\frac{L}{2e^{2}|y|^{2}}\int d\tau
d^{2}x~{}r_{H}^{2}\sin\theta\left(\frac{r_{H}^{2}}{\varepsilon
L}\partial_{\tau}\phi\partial_{\tau}\phi+\frac{1}{r_{H}^{2}}\partial_{\theta}\phi\partial_{\theta}\phi+\frac{1}{r_{H}^{2}\sin^{2}\theta}\partial_{\varphi}\phi\partial_{\varphi}\phi\right)\,,$
(84) $\displaystyle S_{W}$ $\displaystyle=$
$\displaystyle\frac{2L}{e^{2}(r_{H}+L)^{2}}\int d\tau
d^{2}x~{}r_{H}^{2}\sin\theta$ (85)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\left(\frac{(r_{H}+L)^{2}}{\varepsilon
L}\partial_{\tau}W\partial_{\tau}W+\frac{1}{r_{H}^{2}}\partial_{\theta}W\partial_{\theta}W+\frac{1}{r_{H}^{2}\sin^{2}\theta}\partial_{\varphi}W\partial_{\varphi}W\right)\,.$

Note that one of the important ingredients in the above procedure is to keep
the original coordinate time $\tau$ as the time coordinate for the three-
dimensional effective actions. This is because we always need to stick to the
observer with the inverse temperature (69), otherwise, the physics can be
completely different.

As we can see from the effective actions (84) and (85), for finite $r_{s}$ and
$L$, the coefficients in front of terms $(\partial_{\tau}\phi)^{2}$ and
$(\partial_{\tau}W)^{2}$ are proportional to $1/\varepsilon$. Taking the
$\varepsilon\to 0$ limit, the coefficients are divergent, which introduces a
localization on the space of $\partial_{\tau}\phi=\partial_{\tau}W=0$ in the
path integral (79). The localization is similar to what we discussed in the
low-temperature limit of the finite-temperature black hole case, as reviewed
in the previous section. Then the entropy of those modes can be calculated as

$S\propto\frac{r_{H}^{2}}{l_{p}^{2}}\,,$ (86)

which is the same result as the superlow temperature entropy (62) in the
finite-temperature black hole case. The partition functions of the constant
modes and winding modes of $W$ can be calculated, which is the logarithm of
the length scales in the theory. One thing worth noticing is that the entropy
of constant modes proportional to

$S_{0}\propto\ln|y|\,,$ (87)

can be a large contribution.

Note that in the (near-)extremal black hole case, the localization of
$\partial_{\tau}\phi=\partial_{\tau}W=0$ space in the path integral is very
straightforward as shown in the actions (84) and (85). This is different from
the finite-temperature black hole case where we have to take different
temperature limits and the localization only shows up in the superlow
temperature limit. So the localization mechanism and the entropy (86) and
logarithm corrections are intrinsic for the (near-)extremal black hole and
only capture the low-temperature properties of the black hole system. So the
localization in $\partial_{\tau}\phi=\partial_{\tau}W=0$ space should be used
as a general mechanism to explain the entropy of the extremal black hole in a
box. Moreover, one of the surprising properties of entropy is that the modes
are effectively living near the horizon rather than the other boundary. So the
entropy (86) only depends on $r_{H}$. There might be redshift-related
arguments, but we do not have a good explanation here.

As for where the entropy (86) comes from, the zero-point energy of boundary
modes contributes as

$\beta\text{Area}\times\Lambda^{3}\in\ln Z$ (88)

in the logarithm of the partition function. This part has no contribution to
the entropy because it is linear in $\beta$.
$\partial_{\tau}\phi=\partial_{\tau}W=0$ means there is no $\beta$ dependence
in this part and correspondingly this part contributes to the entropy as
$\text{Area}\times\Lambda^{2}$. There must be fundamental explanations of
those modes from symmetry-related arguments. And, the microscopic derivations
for the Bekenstein-Hawking entropy of extremal black holes Strominger1998 ;
Birmingham1998 ; Guica2009 ; Carlip2011 ; Brown1986 ; Solodukhin1999 ;
Carlip1999a ; Carlip1999 ; Silva2002 ; Carlip2007 ; Castro2010 ; Wang2010 ;
Chen2010 ; Carlip2011a ; Compere2017 ; Song2012 ; Carlip2018 ; Carlip2019 ;
Strominger1996 ; Callan1996 ; Peet2000 ; Das2001 ; Maldacena1996 ; Emparan2006
might give the right physical explanation of those modes.

So, we have confirmed that the localization on
$\partial_{\tau}\phi=\partial_{\tau}W=0$ space in the path integral discovered
in the superlow temperature limit discussed in the previous section persists
in the (near-)extremal black hole case. The mechanism is the main reason we
get Bekenstein-Hawking-like entropies from (near-)extremal black holes.

##  VI Conclusion and discussion

This paper evaluates the partition function and entropy of U(1) gauge theory
on black hole background with nontrivial boundary conditions, using the
Euclidean path integral method. The allowed physical degrees of freedom are
the bulk fluctuation modes, the zero longitudinal momentum modes of $A_{r}$,
and the boundary-stretched Wilson lines. The effective actions for different
modes are derived, and we can separately calculate the entropy contributions
from different modes.

The high-temperature limit of the black hole is fairly similar to the flat
case reviewed in Sec. II, where the presence of the black hole merely gives a
temperature to the system. The dominant contribution comes from the bulk
fluctuation modes, whose entropy is shown in Eq. (33). Most of the entropy of
the fluctuations is proportional to the volume of the region where the
semiclassical fields live, multiplied by the temperature cubed. There is also
a contribution from the modes living very close to the horizon, whose entropy
is proportional to the area of the horizon plus logarithm corrections.
Interesting phenomena start to happen at relatively lower temperatures. As we
gradually decrease the temperature, bulk fluctuation modes became less
important, and the entropy of the zero modes $\phi$ and Wilson lines $W$
behave as the area of the box multiplied by the temperature squared are the
most important contribution as shown in Fig. 4. For superlow temperature, the
entropy of $\phi$ and $W$ scale as the area of the horizon divided by UV
cutoff squared. This is because the coefficients in front of
$(\partial_{\tau}\phi)^{2}$ and $(\partial_{\tau}W)^{2}$ in the effective
actions diverge, introducing localizations in the space of zero-energy modes
$\partial_{\tau}\phi=\partial_{\tau}W=0$ in the Euclidean path integral. Then
entropy of those modes naturally is proportional to the horizon area divided
by the Planck area. We also have extra logarithm corrections from the constant
modes and topological modes of $W$. The localization mechanism of the U(1)
gauge theory is also confirmed for the extremal black holes. The overall
behavior of the entropy is depicted in Fig. 4, and we can see the transitions
of dominances between the low-temperature black hole and high-temperature ones
clearly.

Now, the question is how to understand the large entropy (56) we have gotten
for the low-temperature black hole. Let us take $\phi$ as an example to show
where this large entropy comes from. The finite temperature partition function
for a three-dimensional massless bosonic field can be written as

$\ln Z_{\phi}\propto\beta\text{Area}\cdot\Lambda^{3}+\text{Area}\int
d^{2}p~{}\ln(1-e^{-\beta p})\,,$ (89)

where $\Lambda$ is the UV cutoff and $p=\sqrt{\vec{p}^{~{}2}}$. The first part
proportional to the volume of the whole spacetime divided by the smallest
volume unit is the zero-point energy of the field theory, which is a constant
piece in the free energy. Constant free energy does not contribute to entropy,
and thus can be ignored. The second part is finite and gives out the
$\text{Area}\times T^{2}$ contribution in entropy. However, the finite
contribution cannot be seen at superlow temperatures. Now because of the
localization, the zero-point energy of the field is not a constant free energy
anymore. The logarithm of the partition function only depends on the area of
the boundary because $\phi$ only depends on the spatial coordinates on the
boundary. So the partition function of the zero-point energy can be written as

$\ln
Z_{\phi}\propto\text{Area}\cdot\Lambda^{2}=\frac{\text{Area}}{l_{p}^{2}}\,,$
(90)

if we suppose the UV cutoff is at the Planck scale. The corresponding entropy
is of Bekenstein-Hawking entropy magnitude. There are also constant modes and
winding mode contributions at superlow temperatures, whose entropy scales as
the logarithm of the coupling constant and other length scales. It is
interesting to notice that the area shown in Eq. (90) only contains the area
of the horizon not the area of the other boundary.

The above argument suggests that the low-temperature Bekenstein-Hawking
entropy might come from the zero modes $\phi$ and boundary-stretched Wilson
lines $W$. For the transitions of dominances for the finite temperature black
hole, especially at the low temperature, there should be some symmetry-
breaking pattern to explain them. As shown in Fig. 1, the low-temperature
entropy of the flat case scales as a logarithm because the degeneracy manifold
is a circle in the symmetry-breaking phase of U(1) global symmetry. This can
be seen from the bottom of the “Mexican hat” potential. In such a sense, the
low-temperature Bekenstein-Hawking-like entropy

$\mathcal{S}\propto\frac{\text{Area}}{l_{p}^{2}}\,,$

can also be explained from a symmetry-breaking viewpoint. We are sitting in a
global symmetry-breaking phase at low temperature with degeneracy
$\exp\mathcal{S}$. Note that those amounts of entropy come from the modes in
the limit

$\lim_{\omega\to 0}~{}\tilde{\phi}(\omega,x^{2},x^{3})e^{i\omega\tau}\,$ (91)

because of the localization on $\partial_{\tau}\phi=0$ space. This might can
be thought of as calculating the entropy of the soft hair of the black hole
system Hawking:2016sgy ; He:2014laa ; Haco:2018ske ; Hawking:2016msc ;
Strominger2018 ; Cheng:2020vzw ; Cheng:2022xyr ; Cheng:2022xgm . The exact
relation between our story and the soft hair of black holes needs further
studies.

Note that we have gotten two Bekenstein-Hawking-like entropies in this paper.
The original point of this paper is that the zero modes $\phi$ and the Wilson
lines $W$ have an entropy proportional to the horizon area divided by the
Planck area for superlow temperature black holes, which might be used to
understand the Bekenstein-Hawking entropy because of the right magnitude. Note
that there is also part of the entropy (37) in the bulk fluctuation modes for
the finite temperature black hole system, which is also proportional to the
area divided by the UV cutoff squared. This entropy was used to understand the
microscopic degrees of freedom of finite-temperature black hole systems
tHooft:1984kcu , and was interpreted as the entanglement entropy across the
stretched horizon Srednicki:1993im . If we accept the above argument, we might
tend to interpret that the finite-temperature Bekenstein-Hawking entropy comes
from some extra microscopic structure near the horizon, like entanglement
across the horizon, but the (near-)extremal black hole entropy comes from
different places as the finite-temperature black hole. The (near-)extremal
black hole entropy only appears in low temperatures and comes from the
breaking of global symmetries. Thus, we have two different types of
Bekenstein-Hawking-like entropies for finite-temperature and (near-)extremal
black holes, and they both behave like the area of the horizon divided by the
UV cutoff squared. We leave the symmetry-breaking explanation of the phase
transitions and other related topics for further study.

###### Acknowledgements.

We would like to thank Ankit Aggarwal, Jan de Boer, Diego Hofman, and Pujian
Mao for their useful discussions. This work is supported by the National
Natural Science Foundation of China (NSFC) under Grants No. 11905156 and No.
11935009.

##  Appendix A Would-be gauge degrees of freedom

In this appendix, let us briefly discuss the physics related to the Neumann-
like boundary condition (10), where the boundary would-be gauge degrees of
freedom are allowed. 555A more careful analysis of those modes can be found in
appendix A of Cheng2022

The boundary configurations that respect (10) are the flat boundary
configurations. We can find bulk modes $B_{\mu}$ that correspond to those
boundary configurations and integral over those modes in the Euclidean path
integral to evaluate the partition function. So, the main task in this
appendix is to find a solution to the bulk equation of motion on the Euclidean
Schwarzschild background. To avoid complicating the story, we can solve the
problem using the original metric (3)

$ds^{2}=\frac{\rho}{\rho+r_{s}}d\tau^{2}+\frac{\rho+r_{s}}{\rho}d\rho^{2}+(\rho+r_{s})^{2}d\Omega^{2}\,.$
(92)

After getting the solutions, we can perform a coordinate transformation from
$x^{\mu}=(\tau,\rho,\theta,\varphi)$ to
$x^{\mu^{\prime}}=(\tau,y,\theta,\varphi)$ to obtain the solutions on the
infalling coordinate system.

In coordinate system (3), the problem reads as

$\displaystyle\nabla_{\mu}F^{\mu\nu}_{(B)}$ $\displaystyle=$ $\displaystyle
0\,$ $\displaystyle B_{a}\big{|}_{\rho=\varepsilon}$ $\displaystyle=$
$\displaystyle f_{a}^{(l)}(x^{a})\,$ (93) $\displaystyle
B_{a}\big{|}_{\rho=L}$ $\displaystyle=$ $\displaystyle f_{a}^{(r)}(x^{a})\,.$

Now, we use $f_{a}^{\alpha}(x^{a})$ with $\alpha=(r)$ or $(l)$ to denote the
boundary configuration of $B_{a}$. $B_{\rho}$ should change accordingly. We
will fix the boundary configurations to be flat later. The bulk equation of
motion can be further written as

$\displaystyle\nabla_{\mu}F_{(B)}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}F_{(B)}^{\mu\nu}+\Gamma^{\mu}_{\mu\lambda}F_{(B)}^{\lambda\nu}+\Gamma^{\nu}_{\mu\lambda}F_{(B)}^{\mu\lambda}=\partial_{\mu}F_{(B)}^{\mu\nu}+\Gamma^{\mu}_{\mu\lambda}F_{(B)}^{\lambda\nu}$
(94) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}F_{(B)}^{\mu\nu})=0.$

We may find a solution that satisfies the following equations separately

$\displaystyle\frac{1}{\sqrt{g}}\partial_{\rho}(\sqrt{g}F_{(B)}^{\rho\nu})=0\,,$
(95)
$\displaystyle\frac{1}{\sqrt{g}}\partial_{a}(\sqrt{g}F_{(B)}^{a\nu})=0\,.$
(96)

First of all, let us look at the $\nu=2$ component of (95)

$\frac{1}{\sqrt{g}}\partial_{\rho}(\sqrt{g}F_{(B)}^{\rho 2})=0$ (97)

which is satisfied if

$F^{(B)}_{\rho
2}=\partial_{\rho}B_{2}-\partial_{2}B_{\rho}=\frac{D_{1}(x^{a})}{(\rho+r_{s})^{2}}\,.$
(98)

Supposing $B_{2}$ takes the form

$B_{2}=-\frac{D_{1}(x^{a})}{\rho+r_{s}}+D_{2}(x^{a})\,,$ (99)

the boundary condition fixes the coefficients $D_{1}$ and $D_{2}$

$\displaystyle
B_{2}\big{|}_{\rho=\varepsilon}=-\frac{D_{1}}{r_{s}+\varepsilon}+D_{2}=f_{2}^{(l)}(x^{a})$
$\displaystyle
B_{2}\big{|}_{\rho=L}=-\frac{D_{1}}{r_{s}+L}+D_{2}=f_{2}^{(r)}(x^{a})\,.$
(100)

We can get that

$\displaystyle
D_{1}=\frac{(r_{s}+\varepsilon)(r_{s}+L)}{L-\varepsilon}\left(f_{2}^{(r)}-f_{2}^{(l)}\right)\,,$
$\displaystyle
D_{2}=\frac{r_{s}+L}{L-\varepsilon}f_{2}^{(r)}-\frac{r_{s}+\varepsilon}{L-\varepsilon}f_{2}^{(l)}\,.$
(101)

$\varepsilon$ is much smaller than $r_{s}$ and $L$, so we can write
$r_{s}\approx r_{s}+\varepsilon$ and $L\approx L-\varepsilon$. Thus the
solution for $B_{2}$ can be written as

$\displaystyle B_{2}$ $\displaystyle=$
$\displaystyle-\frac{r_{s}(r_{s}+L)}{(\rho+r_{s})L}(f_{2}^{(r)}-f_{2}^{(l)})+\frac{r_{s}+L}{L}f_{2}^{(r)}-\frac{r_{s}}{L}f_{2}^{(l)}\,.$
(102)

For a similar reason, we can get the solution for other components of $B_{a}$,
and the solution of $B_{a}$ can be written as

$\displaystyle B_{a}$ $\displaystyle=$
$\displaystyle-\frac{r_{s}(r_{s}+L)}{(\rho+r_{s})L}(f_{a}^{(r)}-f_{a}^{(l)})+\frac{r_{s}+L}{L}f_{a}^{(r)}-\frac{r_{s}}{L}f_{a}^{(l)}\,.$
(103)

With the solution $B_{a}$, one can then further fix $B_{\rho}$ such that the
field strength satisfies the bulk equation of motion. We have

$\displaystyle\partial_{\rho}B_{a}=\frac{1}{(\rho+r_{s})^{2}}\frac{r_{s}(r_{s}+L)}{L}(f_{a}^{(r)}-f_{a}^{(l)})$
$\displaystyle\partial_{a}B_{\rho}=\frac{1}{(\rho+r_{s})^{2}}\frac{r_{s}(r_{s}+L)}{L}(f_{a}^{(r)}-f_{a}^{(l)})\,.$

Note that the field strength is set to zero here. Assuming flat boundary
configurations

$f_{a}^{(r)}=C_{a}^{(r)}-\partial_{a}\lambda^{(r)}\,,~{}~{}~{}~{}f_{a}^{(l)}=C_{a}^{(l)}-\partial_{a}\lambda^{(l)}\,.$
(104)

We can get the solution for $B_{\rho}$ by integrating $\partial_{a}B_{\rho}$
over $x^{a}$

$\displaystyle
B_{\rho}=\frac{1}{(\rho+r_{s})^{2}}\frac{r_{s}(r_{s}+L)}{L}\times[x^{a}\cdot(C_{a}^{(r)}-C_{a}^{(l)})+(\lambda^{(r)}-\lambda^{(l)})]\,,$
(105)

As a double check, one can put the solutions (102), (105), and (103) into Eq.
(A) to check if it is satisfied or not.

Now, we can perform the coordinate transformation and transfer everything into
coordinate (6). We have

$\frac{d\rho}{dx}=\sqrt{\frac{\rho}{\rho+r_{s}}}\,,$ (106)

thus $B_{y}$ can be written as

$\displaystyle
B_{y}=\frac{\sqrt{\rho}}{(\rho+r_{s})^{5/2}}\frac{r_{s}(r_{s}+L)}{L}\times[x^{a}\cdot(C_{a}^{(r)}-C_{a}^{(l)})+(\lambda^{(r)}-\lambda^{(l)})]\,.$
(107)

So, the solutions can be summarized as

$\displaystyle
B_{a}=-\frac{r_{s}(r_{s}+L)}{(\rho+r_{s})L}[C_{a}^{(r)}-C_{a}^{(l)}+\partial_{a}(\lambda^{(r)}-\lambda^{(l)})]$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}+\frac{r_{s}+L}{L}(C_{a}^{(r)}+\partial_{a}\lambda^{(r)})-\frac{r_{s}}{L}(C_{a}^{(l)}+\partial_{a}\lambda^{(l)})\,,$
$\displaystyle
B_{y}=\frac{\sqrt{\rho}}{(\rho+r_{s})^{5/2}}\frac{r_{s}(r_{s}+L)}{L}\times[x^{a}\cdot(C_{a}^{(r)}-C_{a}^{(l)})+(\lambda^{(r)}-\lambda^{(l)})]\,$

Then, we need to add those modes in the Euclidean path integral of the U(1)
gauge theory. Correspondingly, there would be new contributions to the
partition function and the thermal entropy. Those bulk modes corresponding to
the bulk would-be pure gauge configurations. It would be interesting to see
the relationship among those modes, the soft hair of black holes
Hawking:2016sgy ; He:2014laa ; Haco:2018ske ; Hawking:2016msc ; Strominger2018
; Cheng:2020vzw ; Cheng:2022xyr ; Cheng:2022xgm , Barnich’s nonproper gauge
degrees of freedom Barnich:2018zdg ; Barnich:2019qex ; Alessio:2020lpk ;
Aggarwal:2022rrp , and other boundary modes Park:2022nkb .

##  Appendix B Effective action for fields $\phi$ and $W$

In this appendix, we derive the effective action (15) from the original action
of the U(1) gauge theory. The Euclidean action for Maxwell’s theory on a
curved Euclidean background can be written as

$S_{E}=\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}F^{\mu\nu}F_{\mu\nu}$ (110)

Now we are going to work out the above Euclidean action in terms of fields
$\hat{A}_{\mu}$, $\phi$, and $W$. Working on the Euclidean Schwarzschild black
hole background (3), the action can be separated into two parts with regard to
(3+$y$) decomposition as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}F^{ab}F_{ab}+\frac{1}{2e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}F^{ya}F_{ya}\,.$ (111)

Putting the field decomposition (77) into the action, the above effective
action can be further written as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}+\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]\,$ (112)
$\displaystyle-\frac{1}{e^{2}|y|}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}(\partial_{y}\hat{A}_{a}-\partial_{a}\hat{A}_{y})\partial_{b}\phi\right]\,.$

Denoting the first part in the above action as $\hat{S}_{0}$, the above
effective action can be further written as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\hat{S}_{0}+\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{1}{e^{2}|y|}\int_{\mathcal{M}}d\tau
d^{2}xdy~{}\sqrt{g}\left[g^{ab}(\partial_{y}\hat{A}_{a}-\partial_{a}\hat{A}_{y})\partial_{b}\phi\right]\,,$

which can be further simplified as

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\hat{S}_{0}+\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{i}{e^{2}|y|}\int
d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}\partial_{a}W\partial_{b}\phi$
$\displaystyle+\frac{1}{e^{2}|y|}\int_{\mathcal{M}}d\tau
d^{3}x~{}\partial_{y}(\sqrt{g}g^{ab})\left[\hat{A}_{a}-\partial_{a}(\int
dy\hat{A}_{y})\right]\partial_{b}\phi\,.$

Note that the last term in the above expression can be set to zero by gauge
choice. So we are going to ignore the last term and only focus on the
following Euclidean action as shown in Eq. (15)

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\frac{1}{4e^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}~{}\hat{F}^{\mu\nu}\hat{F}_{\mu\nu}+\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]$ (114)
$\displaystyle-\frac{i}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}\partial_{a}W\partial_{b}\phi\,.$

##  Appendix C Dimensional reduction

This appendix performs the dimensional reduction in Sec. III.2, which is to
reduce the higher dimensional action $S[\phi,W]$ to three-dimensional action
$S_{\phi,W}$ shown below. We have

$\displaystyle S[\phi,W]$ $\displaystyle=$
$\displaystyle\frac{1}{2e^{2}|y|^{2}}\int_{\mathcal{M}}d\tau
d^{3}x\sqrt{g}\left[g^{ab}\partial_{a}\phi\partial_{b}\phi\right]-\frac{i}{e^{2}|y|}\int
d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}\partial_{a}W\partial_{b}\phi\,$
(115)

and

$\displaystyle S_{\phi,W}$ $\displaystyle=$ $\displaystyle\frac{1}{2e^{\prime
2}}\int d\tau
d^{2}x\sqrt{h}\left[h^{ab}\partial_{a}\phi\partial_{b}\phi\right]$ (116)
$\displaystyle-\frac{i}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}\left[\gamma_{1}~{}h^{\tau\tau}\partial_{\tau}W\partial_{\tau}\phi+\gamma_{2}~{}h^{\theta\theta}\partial_{\theta}W\partial_{\theta}\phi+\gamma_{3}~{}h^{\varphi\varphi}\partial_{\varphi}W\partial_{\varphi}\phi\right]\,.$

We are going to divide the problem into two steps. The first is to solve the
low dimensional metric $h_{ab}$ and coupling constant $e^{\prime}$. Then we
are able to solve $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$ with the
results from the first step.

###  C.1 Metric and coupling constant $e^{\prime}$

As discussed in the main context, we assume the three-dimensional metric takes
the following form

$h_{ab}=\text{diag}(h_{\tau\tau},~{}R^{2},~{}R^{2}\sin^{2}\theta)\,,$ (117)

with topology ${S}^{1}\times{S}^{2}$. The radius of ${S}^{1}$ is $\beta$ and
radius of ${S}^{2}$ is $R$. We just need to solve the following equations

$\displaystyle\frac{1}{2e^{2}|y|^{2}}\int d\tau
d^{2}x~{}(\int_{y_{1}}^{y_{2}}dy\sqrt{g}g^{ab})~{}\partial_{a}\phi\partial_{b}\phi=\frac{1}{2e^{\prime
2}}\int d\tau
d^{2}x\sqrt{h}\left[h^{ab}\partial_{a}\phi\partial_{b}\phi\right]\,.$

The above equations can be simplified as

$\frac{1}{2e^{2}|y|^{2}}\int_{\varepsilon}^{L}d\rho~{}\frac{\partial
y}{\partial\rho}\sqrt{g}g^{ab}=\frac{1}{2e^{\prime 2}}\sqrt{h}h^{ab}\,.$ (118)

We are left with two independent components

$\displaystyle\frac{1}{e^{2}|y|^{2}}(3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln\frac{L}{\varepsilon})=\frac{1}{e^{\prime
2}}\sqrt{h^{\tau\tau}}R^{2}\,,$ (119)
$\displaystyle\frac{L}{e^{2}|y|^{2}}=\frac{1}{e^{\prime
2}}\sqrt{h_{\tau\tau}}\,.$ (120)

There are three unknown variables and only two independent equations. So we
are going to write $h_{\tau\tau}$ and coupling constant $e^{\prime 2}$ as a
function of radius $R$. The solution can be written as

$\displaystyle h_{\tau\tau}$ $\displaystyle=$
$\displaystyle\frac{LR^{2}}{3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}}\,,$
(121) $\displaystyle\frac{1}{e^{\prime 2}}$ $\displaystyle=$
$\displaystyle\frac{L}{e^{2}|y|^{2}}\sqrt{h^{\tau\tau}}=\frac{\sqrt{L}}{e^{2}|y|^{2}}\frac{\sqrt{3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}}}{R}\,.$

###  C.2 $\gamma$ couplings

The next step is to calculate $\gamma_{1}$, $\gamma_{2}$, and $\gamma_{3}$. To
do that we just need to match the rest of the actions (115) and (116), which
read as

$\displaystyle\frac{1}{e^{2}|y|}\int d\tau
d^{2}x\left[\sqrt{g}~{}g^{ab}\right]_{y=y_{1}}^{y=y_{2}}\partial_{a}W\partial_{b}\phi=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\frac{1}{2e^{\prime 2}}\int d\tau
d^{2}x\sqrt{h}\left[\gamma_{1}~{}h^{\tau\tau}\partial_{\tau}W\partial_{\tau}\phi+\gamma_{2}~{}h^{\theta\theta}\partial_{\theta}W\partial_{\theta}\phi+\gamma_{3}~{}h^{\varphi\varphi}\partial_{\varphi}W\partial_{\varphi}\phi\right]\,.$

The useful information from the above equation is

$\displaystyle\frac{1}{e^{2}|y|}\left[\sqrt{g}~{}g^{\tau\tau}\right]_{y=y_{1}}^{y=y_{2}}$
$\displaystyle=$ $\displaystyle\frac{\gamma_{1}}{2e^{\prime
2}}\sqrt{h}~{}h^{\tau\tau}\,,$ (124)
$\displaystyle\frac{1}{e^{2}|y|}\left[\sqrt{g}~{}g^{\theta\theta}\right]_{y=y_{1}}^{y=y_{2}}$
$\displaystyle=$ $\displaystyle\frac{\gamma_{2}}{2e^{\prime
2}}\sqrt{h}~{}h^{\theta\theta}\,,$ (125) $\displaystyle\gamma_{2}$
$\displaystyle=$ $\displaystyle\gamma_{3}\,.$ (126)

Writing everything explicitly, we have

$\displaystyle\frac{1}{e^{2}|y|}\left[\sqrt{\frac{\rho(y)+r_{s}}{\rho}}(\rho(y)+r_{s})^{2}\right]_{y=y_{1}}^{y=y_{2}}$
$\displaystyle=$
$\displaystyle\gamma_{1}~{}\frac{L}{2e^{2}|y|^{2}}h^{\tau\tau}R^{2}\,,$ (127)
$\displaystyle\frac{1}{e^{2}|y|}\left[\sqrt{\frac{\rho(y)}{\rho(y)+r_{s}}}\right]_{y=y_{1}}^{y=y_{2}}$
$\displaystyle=$ $\displaystyle\gamma_{2}~{}\frac{L}{2e^{2}|y|^{2}}\,,$ (128)
$\displaystyle\gamma_{2}$ $\displaystyle=$ $\displaystyle\gamma_{3}\,.$ (129)

The above equations can be solved as

$\displaystyle\gamma_{1}$ $\displaystyle=$
$\displaystyle-2|y|\sqrt{\frac{r_{s}}{\varepsilon}}\frac{r_{s}^{2}}{3r_{s}^{2}L+\frac{3}{2}r_{s}L^{2}+\frac{L^{3}}{3}+r_{s}^{3}\ln{L}/{\varepsilon}}\,$
$\displaystyle\gamma_{2}$ $\displaystyle=$
$\displaystyle\gamma_{3}=\frac{2|y|}{L}\sqrt{\frac{L}{L+r_{s}}}\,.$ (130)

##  Appendix D Bulk partition function

In this appendix, we give a detailed analysis of the partition function of the
bulk fluctuation modes. The gauge fixing condition can be imposed by inserting
the following identity in the path integral.

$1=\int\mathcal{D}\lambda\det\left(\frac{\partial
G}{\partial\lambda}\right)\delta(G-0)\,,$ (131)

with gauge fixing condition $G$. After gauge fixing, there are only two
polarization degrees of freedom for Maxwell’s theory in the bulk, which can be
simulated by two massless scalar fields. One can repeat the calculation using
the standard Faddeev-Popov method. Here we would just assume that on black
hole background, the gauge fields $A_{\mu}$ are also left with two massless
bosonic components after gauge fixing.

The metric of Schwarzschild is shown in (2). For one free massless particle
living on this background, the motion for geodesics can be expressed as

$g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0\,,$ (132)

with $\lambda$ the parameter along the trajectory. For a Schwarzschild black
hole (See Sec. 5.4 in carrollSpacetimeGeometry2019  for more details), the
Killing vector associated with energy can be written as

$K^{\mu}=(\partial_{\tau})^{\mu}=(1,0,0,0)\,~{}~{}~{}~{}\Bigg{(}\text{or
}K_{\mu}=\Big{(}1-\frac{2G_{N}M}{r},0,0,0\Big{)}\Bigg{)}\,.$ (133)

And the Killing vector associated with angular momentum is

$R^{\mu}=(\partial_{\varphi})^{\mu}=(0,0,0,1)\,~{}~{}~{}~{}(\text{or
}R_{\mu}=(0,0,0,r^{2}\sin^{2}\theta))\,.$ (134)

There are two conserved charges energy and angular momentum on the equatorial
plane because of the Killing vectors. The conserved quantities can be
expressed as

E $\displaystyle=$ $\displaystyle-
K_{\mu}\frac{dx^{\mu}}{d\lambda}=-\left(1-\frac{2G_{N}M}{r}\right)\frac{d\tau}{d\lambda}$
L $\displaystyle=$ $\displaystyle
R_{\mu}\frac{dx^{\mu}}{d\lambda}=r^{2}\frac{d\varphi}{d\lambda}\,.$ (135)

On the other hand, we have

$p_{\mu}=g_{\mu\nu}\frac{dx^{\nu}}{d\lambda}\,.$ (136)

Then expression (132) can be expressed by the conserved energy and angular
momentum as

$-\left(1-\frac{2G_{N}M}{r}\right)^{-1}\textbf{E}^{2}+\left(1-\frac{2G_{N}M}{r}\right)p_{r}^{2}+\frac{\textbf{L}^{2}}{r^{2}}=0\,,$
(137)

which can be rewritten as

$p_{r}^{2}=\left(1-\frac{2G_{N}M}{r}\right)^{-1}\left[\left(1-\frac{2G_{N}M}{r}\right)^{-1}\textbf{E}^{2}+\frac{\textbf{L}^{2}}{r^{2}}\right]\,.$
(138)

For a massless scalar field with mode expansion

$\bar{A}(\tau,r,\theta,\varphi)=\sum_{\omega}\sum_{l,m}e^{-i\omega\tau}Y_{lm}(\theta,\phi)\tilde{A}(\omega,l,m,r)\,,$
(139)

with

$\partial_{r}\tilde{A}=ip_{r}\tilde{A}\,,$ (140)

the above expression (138) can be written as the dispersion relation for $A$

$p_{r}^{2}=\left(1-\frac{2G_{N}M}{r}\right)^{-1}\left[\left(1-\frac{2G_{N}M}{r}\right)^{-1}\omega^{2}+\frac{l(l+1)}{r^{2}}\right]\,.$
(141)

This expression can also be obtained from different methods, for example from
the equation of motion of massless field on curved space time tHooft:1984kcu .

Let us study the statistical properties of those bulk fluctuation modes.
Because of the boundary conditions on $r=r_{s}+\varepsilon$ and $r=r_{s}+L$,
we have a standing-wave condition along the radius direction, which can be
written as

$n\pi=\int_{r_{s}+\varepsilon}^{r_{s}+L}p_{r}(r,\omega,l)~{}dr\,,$ (142)

with $n\in{Z}_{n}$. The partition function of bulk fluctuation modes can
always be written as

$\ln Z_{\bar{A}}=-2\sum_{\omega}\ln(1-e^{-\beta\omega})\,,$ (143)

where the factor 2 means we have two polarizations, and we ignored the zero-
point energy in the above expression. Now we can change the summation of
$\omega$ into integration by introducing density of state $g(\omega)$ and
regarding the spectrum to be continuous. We obtain

$\displaystyle\ln Z_{F}$ $\displaystyle=$
$\displaystyle-2\int_{0}^{\infty}g(\omega)\ln(1-e^{-\beta\omega})d\omega=-2\int_{0}^{\infty}\ln(1-e^{-\beta\omega})d\Gamma(\omega)$
(144) $\displaystyle=$
$\displaystyle-2\ln(1-e^{-\beta\omega})\Gamma(\omega)\big{|}_{0}^{\infty}+2\int_{0}^{\infty}\frac{\Gamma(\omega)e^{-\beta\omega}}{1-e^{-\beta\omega}}\beta
d\omega\,,$

where $\Gamma(\omega)$ defined by $d\Gamma=g(\omega)d\omega$ is the number of
state not exceeding $\omega$. The first part is zero when $\omega\to 0$ and
$\omega\to\infty$, so the final result for our partition function can be
written as

$\ln
Z_{F}=2\beta\int_{0}^{\infty}\frac{\Gamma(\omega)}{e^{\beta\omega}-1}d\omega\,.$
(145)

$\Gamma(\omega)$ is the number of states that have energy lower than $\omega$,
and we have

$\displaystyle\Gamma(\omega)=\frac{1}{\pi}\sum_{l}(2l+1)\int_{r_{s}+\varepsilon}^{r_{s}+L}dr\sqrt{\left(1-\frac{2G_{N}M}{r}\right)^{-1}\left[\left(1-\frac{2G_{N}M}{r}\right)^{-1}\omega^{2}+\frac{l(l+1)}{r^{2}}\right]}$
$\displaystyle\approx\frac{1}{\pi}\int_{l}(2l+1)dl\int_{r_{s}+\varepsilon}^{r_{s}+L}\frac{dr}{1-\frac{2G_{N}M}{r}}\left[\omega^{2}+\left(1-\frac{2G_{N}M}{r}\right)\frac{l(l+1)}{r^{2}}\right]^{1/2}\,.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(146)

Note that we have changed the summation of $l$ into an integral assuming the
area of the boundary is big enough. The summation or integration is from $l=0$
to the state with energy $\omega$. Now we can put (146) into Eq. (145), and
the logarithm of the partition function can be written as

$\displaystyle\ln
Z_{F}=\frac{2\beta}{\pi}\int_{0}^{\infty}\frac{d\omega}{e^{\beta\omega}-1}\int
d[l(l+1)]\int~{}\frac{dr}{1-\frac{2G_{N}M}{r}}\left[\omega^{2}+\left(1-\frac{2G_{N}M}{r}\right)\frac{l(l+1)}{r^{2}}\right]^{1/2}\,.$

Now, let us redefine $x=\left(1-\frac{2G_{N}M}{r}\right)\frac{l(l+1)}{r^{2}}$.
We obtain

$d[l(l+1)]=\frac{r^{2}}{1-\frac{2G_{N}M}{r}}dx\,,$ (147)

thus we can rewrite the integral as

$\displaystyle\ln Z_{F}$ $\displaystyle=$
$\displaystyle\frac{2\beta}{\pi}\int_{0}^{\infty}\frac{d\omega}{e^{\beta\omega}-1}\int~{}\frac{r^{2}dr}{(1-\frac{2G_{N}M}{r})^{2}}\int_{0}^{\omega^{2}}dx\left[\omega^{2}+x\right]^{1/2}$
(148) $\displaystyle=$
$\displaystyle-\frac{4\beta}{3\pi}\int_{0}^{\infty}\frac{\omega^{3}d\omega}{e^{\beta\omega}-1}\int~{}\frac{r^{4}dr}{(r-2G_{N}M)^{2}}\,.$

Those integrals are straightforward to work out, we have

$\int_{0}^{\infty}\frac{\omega^{3}d\omega}{e^{\beta\omega}-1}=\frac{\pi^{4}}{15}\frac{1}{\beta^{4}}$
(149)

and

$\displaystyle\int_{r_{s}+\varepsilon}^{r_{s}+l}~{}\frac{r^{4}dr}{(r-2G_{N}M)^{2}}\approx\frac{r_{s}^{4}}{\varepsilon}+4~{}r_{s}^{3}\ln\frac{L}{\varepsilon}-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\,.$

All in all, the logarithm of the partition function can be written as

$\displaystyle\ln Z_{F}$ $\displaystyle=$
$\displaystyle-\frac{4\pi^{3}}{45}\frac{1}{\beta^{3}}\frac{r_{s}^{4}}{\varepsilon}-\frac{16\pi^{3}}{45}\frac{r_{s}^{3}}{\beta^{3}}\ln\frac{L}{\varepsilon}-\frac{4\pi^{3}}{45}\frac{1}{\beta^{3}}\left(-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\right)\,.$
(151)

The corresponding entropy $\mathcal{S}_{F}=(1-\beta\partial_{\beta})$ can be
calculated as

$\displaystyle\mathcal{S}_{F}=\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\frac{r_{s}^{4}}{\varepsilon}+\frac{64\pi^{3}}{45}\frac{r_{s}^{3}}{\beta^{3}}\ln\frac{L}{\varepsilon}+\frac{16\pi^{3}}{45}\frac{1}{\beta^{3}}\left(-\frac{r_{s}^{4}}{L}+6~{}r_{s}^{2}L+2~{}r_{s}L^{2}+\frac{L^{3}}{3}\right)\,.$
(152)

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