aainstitutetext: Department of Physics, College of Sciences, Northeastern
University
No. 3-11, Wenhua Road, Shenyang, 110819, China

# Hawking–Page phase transitions of the black holes in a cavity

Wen-Bo Zhao, a Guo-Rong Liu a,1 and Nan Li 111Corresponding author.
<EMAIL_ADDRESS>
###### Abstract

The Hawking–Page phase transitions of the Schwarzschild and charged black
holes are investigated in an extended phase space, in which the black holes
are enclosed in a spherical cavity of radius $r_{B}$ in asymptotically flat
space. An effective thermodynamic volume $V=4\pi r_{B}^{3}/3$ is introduced
for the black hole, and an effective pressure $p$ is defined as the conjugate
variable of $V$. The phase transition temperature $T_{\rm HP}$ and the Gibbs
free energy $G$ are systematically studied in a grand canonical ensemble with
fixed electric potential $\Phi$, and $T_{\rm HP}$ is found to increase with
$p$ and decrease with $\Phi$. If the phase transition occurs, $\Phi$ must have
an upper bound, such that the cavity radius is always larger than the black
hole horizon radius. These phase transition behaviors are further compared to
those in the anti-de Sitter space, and the remarkable similarities and notable
differences are also discussed in depth. Our work reveals the relationship of
the thermodynamic properties of black holes and their specific boundary
conditions in different extended phase spaces.

##  1 Introduction

During the last half century, black hole thermodynamics has been one of the
most important and fruitful research areas in theoretical physics, and has
been continuously offering profound and fundamental insights to our
understanding of quantum gravity. The establishment of the four laws of black
hole thermodynamics law  explicitly indicates that a black hole is not simply
a mathematical singularity, but should rather be considered as a complicated
thermodynamic system with temperature and entropy Bekenstein .

However, despite the striking resemblance between black hole thermodynamics
and traditional thermodynamics, there remain two obvious discrepancies between
them. One is that there is no $p$–$V$ term in the first law of black hole
thermodynamics, and the other is that a black hole in asymptotically flat
space has negative heat capacity and is thus thermodynamically unstable.
Therefore, when a black hole radiates via the Hawking mechanism rad , it can
no longer maintain thermal equilibrium with the environment and will evaporate
eventually. These two aspects make a black hole still somehow different from a
usual thermodynamic system (e.g., a $p$–$V$–$T$ system). In fact, both these
two aspects can be tackled to be consistent with traditional thermodynamics,
and the idea is just to impose appropriate boundary conditions and to
introduce an effective pressure $p$ and an effective volume $V$ to the black
hole system. In this way, two new dimensions are added into the black hole
phase space, and such theories are thus named as black hole thermodynamics in
the extended phase space.

Generally speaking, there are two natural choices of the boundary conditions
and the ways to restore the $p$–$V$ term. The first is to confine a black hole
in the anti-de Sitter (AdS) space, and an effective pressure can be introduced
as Kastor ; Dolan

$\displaystyle p=-\frac{\Lambda}{8\pi},$

where $\Lambda$ is the cosmological constant. Furthermore, if $\Lambda$ is
allowed to vary, $p$ will behave as a thermodynamic variable rather than a
fixed background, and an effective volume of a black hole can be defined as
the conjugate quantity of $p$ as $V=({\partial M}/{\partial p})_{S}$, with $M$
being the black hole mass KM . By this means, the first law of thermodynamics
is consistent with the Smarr relation, and the analogy between black hole
thermodynamics in the extended phase space and traditional thermodynamics
becomes more complete. Moreover, it is found that there are two branches of
black hole solutions in the extended phase space, a stable large black hole
with larger event horizon radius and positive heat capacity and an unstable
small black hole with smaller event horizon radius and negative heat capacity,
respectively. The former possesses an equation of state very similar to that
of a non-ideal fluid and exhibits rich thermodynamic behaviors, such as
various critical phenomena. All these relevant topics have been extensively
studied in the literature in recent years (see Ref. rev  for a review and the
references therein).

The essential reason that a black hole in the AdS space is thermodynamically
stable is based upon the fact that the gravitational potential in the AdS
space increases at large distances, acting as a box of finite volume.
Therefore, the second way to restore the $p$–$V$ term and to stabilize the
black hole is just to enclose the black hole in asymptotically flat space in a
cavity, on the wall of which the metric is fixed (i.e., with the Dirichlet
boundary condition). Thus, the wall of cavity, as a reflecting boundary, will
equivalently play a role of the AdS space. In a recent work Wang , the authors
considered a new extended phase space for the black hole in a cavity. In a
spherical cavity of radius $r_{B}$, an effective volume of the black hole is
first introduced as

$\displaystyle V=\frac{4\pi}{3}r^{3}_{B},$

and $r_{B}$ is regarded as a thermodynamic variable. Next, an effective
pressure is defined as the conjugate quantity of $V$ as $p=-({\partial
E}/{\partial V})_{S}$, with $E$ being the thermal energy of the black hole. It
is interesting to see that the order of introduction of the $p$–$V$ term in a
cavity is opposite to that in the AdS space, but the thermodynamic properties
of black holes are strikingly analogous in these two different extended phase
spaces. For example, there are also two black hole solutions, and the large
black hole is thermodynamically stable and has an equation of state like a
non-ideal fluid Wang .

The explorations of the black holes in a cavity had a long history. In Ref.
York , via the approach of the Euclidean action, York studied the phase
structures and transitions of the Schwarzschild black hole in a cavity and in
the AdS space, and found remarkable similarities in between. This observation
was later confirmed for the charged black hole in a canonical ensemble Carlip
; Lundgren  and in a grand canonical ensemble Braden . Later on, similar
investigations were extended to other black holes Parentani:1994wr ;
Akbar:2004ke ; 29 ; 30 ; Dias11 ; Dias12 ; Wang:2019urm ; 35 ; 36 ; 37 ;
Tzikas , the black branes Emparan:2012be ; 21 ; 22 ; 23 ; 24 ; 25 ; 26 , the
boson stars 31 ; 32 , and the stabilities of stars, solitons, and black holes
in a cavity Gregory:2001bd ; Dolansss ; 27 ; Ponglertsakul1 ; 28 ;
Ponglertsakul2 ; Sanchis-Gualhhh . Simultaneously, some dissimilarities in the
phase structures were discovered between the Born–Infeld black hole in a
cavity and their AdS counterparts Wang:2019kxp ; Liang:2019dni . Similarly,
the thermodynamic geometry and the weak cosmic censorship conjecture were also
found to be different between the cavity and the AdS cases Wang:2019cax ;
Wang:2020osg . Altogether, the similarities in the relevant works further
supported the analogy between the cavity and the AdS cases; meanwhile, the
discrepancies revealed the potential relations between the thermodynamic
properties of black holes and the specific boundary conditions.

Actually, both in the AdS space and in a cavity, the precondition of the
existence of black hole solution is that the black hole temperature must be
higher than some value. Below this temperature, there will be no black hole
solution any longer, and the black hole will turn into the thermal gas. In the
AdS space, this process is the famous Hawking–Page (HP) phase transition. It
was first investigated for the Schwarzschild black hole in Ref. HP  and then
for the charged black hole in Refs. Chamblin1 ; Chamblin2 ; Peca . The HP
phase transitions have been broadly studied in the literature Witten ; Cald ;
Birmingham:2002ph ; Herzog:2006ra ; Cai:2007wz ; Nicolini:2011dp ; Eune:2013qs
; Adams:2014vza ; Banados:2016hze ; Czinner:2017tjq ; Aharony:2019vgs ;
Mejrhit:2019oyi ; Lala:2020lge , especially in the extended phase space in the
AdS space Spallucci:2013jja ; Altamirano:2014tva ; Xu:2015rfa ; Maity:2015ida
; Liu:2016uyd ; Hansen:2016ayo ; Sahay:2017hlq ; Mbarek:2018bau ; sanren ;
HPwo ; Astefanesei:2019ehu ; Wu:2020tmz ; DeBiasio:2020xkv ; Wei:2020kra ;
Li:2020khm ; Wang:2020pmb ; Belhaj:2020mdr ; ydw . Consequently, it is quite
natural to reconsider it in the new extended phase space in a cavity. This
issue was briefly mentioned in Ref. Wang , with some graphical illustration
but no analytical result. Therefore, the purpose of our present work is to
provide a thorough exploration of the HP phase transitions of the black holes
in a cavity. The HP temperature, the minimum temperature, and the Gibbs free
energy of the black holes will be systematically studied. Besides numerical
methods, all analytical results will be given. The comparison with the AdS
case will also be shown when necessary, so that we can have a global
understanding of the HP phase transitions in different extended phase spaces.

This paper is organized as follows. In Sect. 2, we list the thermodynamic
properties of the black holes in a cavity and discuss the HP phase transition
in detail. In Sect. 3, we explore the HP phase transitions of the neutral and
charged black holes. The similarities and differences with the results in the
AdS space are also presented. We conclude in Sect. 4. In this paper, we work
in the natural system of units and set $c=G_{\rm N}=\hbar=k_{\rm B}=1$.

##  2 Black hole thermodynamics in a cavity

In this section, we first outline black hole thermodynamics in the extended
phase space in a cavity and then show the criterion of the HP phase
transition.

###  2.1 Extended phase space in a cavity

We start from the action of the four-dimensional Einstein–Maxwell gravity,

$\displaystyle I=\frac{1}{16\pi}\int{\rm
d}^{4}x\,\sqrt{-g}(R-F_{\mu\nu}F^{\mu\nu}),$

where $R$ is Ricci scalar and $F_{\mu\nu}$ is the electromagnetic field
tensor. Then, the metric of the charged black hole (i.e., the
Reissner–Nordström black hole) reads

$\displaystyle{\rm d}s^{2}=-f(r)\,{\rm d}t^{2}+\frac{{\rm
d}r^{2}}{f(r)}+r^{2}\,{\rm d}\theta^{2}+r^{2}\sin^{2}\theta\,{\rm d}\phi^{2},$

where $f(r)=1-{2M}/{r}+{Q^{2}}/{r^{2}}$, and $M$ and $Q$ are the mass and
electric charge of the black hole, with the electric potential being
$A=A_{t}(r)\,{\rm d}t=-Q\,{\rm d}t/r$. The event horizon radius $r_{+}$ can be
fixed as the largest root of $f(r_{+})=0$, so $r_{+}=M+\sqrt{M^{2}-Q^{2}}$. By
this means, $f(r)$ can be reexpressed as

$\displaystyle
f(r)=\left(1-\frac{r_{+}}{r}\right)\left(1-\frac{Q^{2}}{r_{+}r}\right).$

To avoid naked singularity, $r_{+}$ must be positive. This condition leads to
a constraint $M>Q$, so $r_{+}>Q$, with $Q$ being the event horizon radius of
the extremal black hole. Moreover, the Bekenstein–Hawking entropy is one
quarter of the event horizon area,

$\displaystyle S=\pi r_{+}^{2},$

and the Hawking temperature can be calculated as

$\displaystyle T_{\rm H}=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{1}{4\pi
r_{+}}\left(1-\frac{Q^{2}}{r_{+}^{2}}\right).$

Now, we consider black hole thermodynamics in a cavity in asymptotically flat
space. In the following discussion on the HP phase transition, we will work in
a grand canonical ensemble, in which the cavity plays a role of reservoir,
with fixed temperature $T$ and electric potential $\Phi$ on its wall located
at radius $r_{B}$ Braden . Naturally, $r_{B}$ should be larger than the event
horizon radius $r_{+}$, so we have the following constraint,

$\displaystyle Q<r_{+}<r_{B}.$

When the charged black hole is enclosed in such a cavity, its Hawking
temperature $T_{\rm H}$ is related to $T$ as

$\displaystyle T=\frac{T_{\rm
H}}{\sqrt{f(r_{B})}}=\frac{1-\frac{Q^{2}}{r_{+}^{2}}}{4\pi
r_{+}\sqrt{\left(1-\frac{r_{+}}{r_{B}}\right)\left(1-\frac{Q^{2}}{r_{+}r_{B}}\right)}},$
(1)

meaning that the temperature $T$ measured at $r_{B}$ is blue-shifted from the
temperature $T_{\rm H}$ measured at infinity by a factor $1/\sqrt{f(r_{B})}$.
Also, the electric potential $\Phi$ is

$\displaystyle\Phi=\frac{A_{t}(r_{B})-A_{t}(r_{+})}{\sqrt{f(r_{B})}}=\frac{Q\left(1-\frac{r_{+}}{r_{B}}\right)}{r_{+}\sqrt{\left(1-\frac{r_{+}}{r_{B}}\right)\left(1-\frac{Q^{2}}{r_{+}r_{B}}\right)}}.$
(2)

Furthermore, the thermal energy $E$ of the charged black hole in a cavity can
be obtained via the Euclidean action method as Braden

$\displaystyle E$
$\displaystyle=r_{B}\left[1-\sqrt{f(r_{B})}\right]=r_{B}\left[1-\sqrt{\left(1-\frac{r_{+}}{r_{B}}\right)\left(1-\frac{Q^{2}}{r_{+}r_{B}}\right)}\right].$
(3)

In previous works, the cavity radius $r_{B}$ was always considered as a
constant, so the cavity acts merely as a fixed background, and the $p$–$V$
term is still absent in black hole thermodynamics. Recently, in Ref. Wang ,
the authors introduced an effective volume $V$ for the black hole in a cavity,

$\displaystyle V=\frac{4\pi}{3}r^{3}_{B},$ (4)

and allowed $r_{B}$ to vary. Thus, $V$ becomes a thermodynamic variable, and
an effective pressure $p$ can be further defined as its conjugate quantity,

$\displaystyle p=-\left(\frac{\partial E}{\partial V}\right)_{S}=\frac{1}{4\pi
r_{B}^{2}}\left[\frac{2-\frac{r_{+}}{r_{B}}-\frac{Q^{2}}{r_{+}r_{B}}}{2\sqrt{\left(1-\frac{r_{+}}{r_{B}}\right)\left(1-\frac{Q^{2}}{r_{+}r_{B}}\right)}}-1\right].$
(5)

In this way, the missing $p$–$V$ term is restored in a new extended phase
space, complementary to that in the AdS space. In this new extended phase
space, a direct differentiation of Eq. (3) yields the first law of black hole
thermodynamics Wang ,

$\displaystyle{\rm d}E=T\,{\rm d}S-p\,{\rm d}V+\Phi\,{\rm d}Q.$

Moreover, a scaling argument directly leads to the corresponding Smarr
relation as $E=2TS-3pV+\Phi Q$ Wang . Therefore, the analogy between
traditional thermodynamics and black hole thermodynamics in the new extended
phase space in a cavity is as complete as that in the AdS space.

Equation (5) can also be viewed as the equation of state of the black hole in
a cavity. In principle, from Eqs. (1) and (4), solving $r_{+}$ and $r_{B}$ in
terms of $T$ and $V$, we may arrive at its usual form $p=p(V,T)$.
Unfortunately, Eq. (1) is an algebraic equation about $r_{+}$ of higher degree
and is analytically unsolvable, so it is impossible to show the explicit form
of $p(V,T)$. However, we may investigate two limiting cases: the Schwarzschild
black hole with $Q\to 0$ and the extremal black hole with $Q\to M$. Under
these circumstances, the analytical forms of $p(V,T)$ do exist, and these
results can be used for further reference.

First, for the Schwarzschild black hole in a cavity with $Q\to 0$, we obtain
two equations of state (i.e., the large or small black hole branch with larger
or smaller $r_{+}$, respectively),

$\displaystyle p$
$\displaystyle=\frac{1}{4\pi}\left(\frac{4\pi}{3V}\right)^{2/3}\left[\frac{5-2A}{2\sqrt{6(1-A)}}-1\right],$
(6) $\displaystyle p$
$\displaystyle=\frac{1}{4\pi}\left(\frac{4\pi}{3V}\right)^{2/3}\left[\frac{5+A-\sqrt{3(1-A^{2})}}{2\sqrt{6+3A-3\sqrt{3(1-A^{2})}}}-1\right].$
(7)

where $4A^{3}-3A=1-27a/2$ and $a=1/(4\pi
T\sqrt[3]{3V/4\pi})^{2}=(r_{+}/r_{B})^{2}(1-r_{+}/r_{B})$. In the limit
$r_{+}/r_{B}\to 1$, we have $a\to 0$ and $A\to 1-3a/2$, and Eq. (6) reduces to

$\displaystyle p=\frac{T}{2(3V/4\pi)^{1/3}}=\frac{T}{v},$

where $v=2r_{B}=2(3V/4\pi)^{1/3}$ is the specific volume of the black hole.
This result indicates that the large Schwarzschild black hole in a cavity
behaves quite like an ideal gas in the extended phase space, exactly the same
as the case in the AdS space. Similarly, in the limit $r_{+}/r_{B}\to 0$, Eq.
(7) reduces to

$\displaystyle
p=\frac{1}{512\pi^{3}T^{2}(3V/4\pi)^{4/3}}=\frac{1}{32\pi^{3}T^{2}v^{4}}.$

It is easy to find that the small black hole has negative heat capacity, so it
cannot establish thermal equilibrium with the environment and is thus
unstable.

Second, for the extremal black hole in a cavity with $Q\to M$ and $r_{+}\to
Q$, we obtain its equation of state as

$\displaystyle p=\frac{\pi Q^{4}T^{2}}{2(3V/4\pi)^{4/3}}=\frac{8\pi
Q^{4}T^{2}}{v^{4}}.$

This result also resembles that of a non-ideal fluid.

Last, the critical point of the charged black hole in a cavity is determined
by

$\displaystyle\left(\frac{\partial p}{\partial
v}\right)_{T}=\left(\frac{\partial^{2}p}{\partial v^{2}}\right)_{T}=0.$

From Eqs. (1), (4), and (5), the critical pressure, special volume, and
temperature can be obtained numerically as

$\displaystyle p_{\rm c}=\frac{0.0115}{Q^{2}},\quad v_{\rm c}=4.14Q,\quad
T_{\rm c}=\frac{0.117}{Q},$

so the critical ratio of $p_{\rm c}v_{\rm c}/T_{\rm c}$ is 0.405 Wang . These
values only slightly deviate from those in the AdS space: $p_{\rm c}=1/(96\pi
Q^{2})$, $v_{\rm c}=2\sqrt{6}Q$, $T_{\rm c}=1/(3\sqrt{6}\pi Q)$, and $p_{\rm
c}v_{\rm c}/T_{\rm c}=3/8$ KM .

###  2.2 HP phase transition

Due to the Hawking mechanism rad , a black hole may not only absorb but also
radiate energy to the environment. This process will establish thermal
equilibrium between the stable large black hole and the surrounding thermal
gas, and the phase transition in this black hole–thermal gas system is the HP
phase transition. Before detailed discussion, an important point should be
made clear. Because of the conservation of charge, a black hole with fixed
charge cannot undergo the HP phase transition to the neutral thermal gas. As a
result, the HP phase transition should be considered in a grand canonical
ensemble, in which the electric potential $\Phi$ is fixed on the wall of
cavity and the electric charge $Q$ is allowed to vary, so the thermodynamic
potential of interest is the Gibbs free energy $G$.

On the one hand, the Gibbs free energy of the charged black hole is
constructed as

$\displaystyle G=E+pV-TS-\Phi Q.$

Substituting all the relevant expressions in Sect. 2.1 into $G$, we obtain

$\displaystyle
G=r_{B}\left[\frac{7\frac{r_{+}}{r_{B}}+\frac{Q^{2}}{r_{+}r_{B}}-8}{12\sqrt{\left(1-\frac{r_{+}}{r_{B}}\right)\left(1-\frac{Q^{2}}{r_{+}r_{B}}\right)}}+\frac{2}{3}\right].$
(8)

(Here, $G$ is still expressed in terms of $Q$, and its explicit form as a
function of $\Phi$ will be shown later in Sect. 3.2.) On the other hand,
because the total number of particles of the thermal gas is not conserved but
varies with temperature, its Gibbs free energy is always vanishing. In Sect.
3, it will be shown that the Gibbs free energy of the black hole monotonically
decreases with temperature. Therefore, at low temperatures, the thermal gas
phase with vanishing $G$ is more stable; at high temperatures, the black hole
phase with negative $G$ is globally preferred, and the thermal gas will
collapse into it. Consequently, the criterion of the HP phase transition in
the black hole–thermal gas system is

$\displaystyle G=0,$ (9)

and the HP temperature $T_{\rm HP}$ can be fixed accordingly.

##  3 HP phase transitions of the black holes in a cavity

Below, we study the HP transitions of the Schwarzschild and charged black
holes in the extended phase space in a cavity. We focus on three aspects: the
HP temperature $T_{\rm HP}$, the minimum black hole temperature $T_{0}$, and
the Gibbs free energy $G$. All analytical results will be shown if possible,
and the similarities and dissimilarities between the cavity and AdS cases will
also be discussed when needed. For simplicity, we only constrain our attention
to the situation $T>T_{\rm c}$; that is to say, we only consider the HP phase
transitions, not the van der Waals-like phase transitions below the critical
temperature $T_{\rm c}$.

###  3.1 HP phase transition of the Schwarzschild black hole in a cavity

From Eqs. (1), (5), and (8), the temperature, pressure, and Gibbs free energy
of the Schwarzschild black hole in a cavity reduce to

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi
r_{+}\sqrt{1-\frac{r_{+}}{r_{B}}}},$ (10) $\displaystyle p$
$\displaystyle=\frac{1}{4\pi
r_{B}^{2}}\left(\frac{2-\frac{r_{+}}{r_{B}}}{2\sqrt{1-\frac{r_{+}}{r_{B}}}}-1\right),$
(11) $\displaystyle G$
$\displaystyle=r_{B}\left(\frac{7\frac{r_{+}}{r_{B}}-8}{12\sqrt{1-\frac{r_{+}}{r_{B}}}}+\frac{2}{3}\right).$
(12)

In principle, from Eqs. (10)–(12), it is straightforward to obtain the
explicit form of $G(T,p)$, but the full expression is unnecessarily tedious,
and there is no need to show it. Therefore, we will face a series of
parametric equations in terms of $r_{+}$ and $r_{B}$.

First, at the HP phase transition point, solving the criterion in Eq. (9)
yields

$\displaystyle r_{B}=\frac{49}{48}r_{+}.$ (13)

Substituting Eq. (13) into Eqs. (10) and (11), we have

$\displaystyle T=\frac{7}{4\pi r_{+}},\quad p=\frac{10368}{16807\pi
r_{+}^{2}},$

so the HP temperature $T_{\rm HP}$ as a function of pressure $p$ (i.e., the
equation of coexistence line) is obtained as

$\displaystyle T_{\rm
HP}=\frac{343}{288}\sqrt{\frac{7p}{2\pi}}=1.257\sqrt{p}.$ (14)

This result was also shown numerically in Ref. Wang . Moreover, it is
analogous to that in the AdS space HPwo ,

$\displaystyle T_{\rm HP}=\sqrt{\frac{8p}{3\pi}}=0.921\sqrt{p}.$

These results confirm the similarity between black hole thermodynamics in a
cavity and in the AdS space. Both coexistence lines in these two extended
phase spaces are shown in Fig. 1. From Eq. (14), there is no terminal point in
the $T_{\rm HP}$–$p$ curve, so the HP phase transition can occur at all
pressures, with no critical point. Hence, it is more like a solid–liquid phase
transition, rather than a liquid–gas phase transition, and the thermal gas
phase even plays a role of solid 1404.2126 , as it is thermodynamically
preferred at low temperatures and lies below the coexistence line in Fig. 1.

Figure 1:  The coexistence lines for the Schwarzschild black hole in a cavity
and in the AdS space. There is no terminal point in the $T_{\rm HP}$–$p$
curve, and the HP phase transition can occur at all pressures. The thermal gas
phase lies below the coexistence line, behaving like a solid. The HP
temperature $T_{\rm HP}$ increases with $p$ and is a little higher in a cavity
than in the AdS space.

Next, we study the phase structure of the Schwarzschild black hole in the HP
phase transition. Here, we should first point out a minimum temperature
$T_{0}$, above which there are large and small black hole solutions but below
which there is none. To calculate $T_{0}$, we solve $r_{B}$ in terms of
$r_{+}$ from Eq. (10),

$\displaystyle r_{B}=\frac{u^{2}}{(u+1)(u-1)}r_{+},$

where $u=4\pi r_{+}T$. Substituting $r_{B}$ into Eq. (11), we have

$\displaystyle\frac{p}{2\pi T^{2}}=\frac{(u+1)^{2}(u-1)^{4}}{u^{7}}=f(u).$

It is direct to find the maximum point of $f(u)$ is at $u=1+2\sqrt{2}$, and
$f(u)\leq\frac{256(1+\sqrt{2})^{2}}{(1+2\sqrt{2})^{7}}$. Therefore, the black
hole temperature has a lower bound $T_{0}$ as

$\displaystyle T\geq
T_{0}=\frac{(1+2\sqrt{2})^{7/2}}{16(1+\sqrt{2})}\sqrt{\frac{p}{2\pi}}=1.134\sqrt{p}.$

This result is also consistent with the numerical value in Ref. Wang .
Meanwhile, the minimum temperature of the Schwarzschild black hole in the AdS
space is HPwo

$\displaystyle T\geq T_{0}=\sqrt{\frac{2p}{\pi}}=0.798\sqrt{p}.$

We observe that both $T_{\rm HP}$ and $T_{0}$ of the Schwarzschild black hole
in a cavity are higher than the AdS counterparts.

With the above preparations, we are able to investigate the Gibbs free energy
of the black hole–thermal gas system. As already mentioned, we utilize the
simple parametric equations for $T$ and $G$ instead of the lengthy explicit
expression. For this purpose, we first solve $r_{+}$ from Eq. (11),

$\displaystyle r_{+}=xr_{B},$ (15)

where

$\displaystyle x=4\sqrt{2\pi pr_{B}^{2}(2\pi pr_{B}^{2}+1)}\Big{(}\sqrt{2\pi
pr_{B}^{2}+1}-\sqrt{2\pi pr_{B}^{2}}\Big{)}^{2}.$

It is easy to find $x$ is less than 1, so $r_{+}<r_{B}$ is always ensured for
the Schwarzschild black hole in a cavity (Attention, this is not the case for
the charged black hole in a cavity, see Sect. 3.2). Substituting Eq. (15) into
Eqs. (10) and (12), we can reexpress $T$ and $G$ in terms of $r_{B}$,

$\displaystyle T$ $\displaystyle=\frac{1}{4\pi r_{B}x\sqrt{1-x}}=T(p,r_{B}),$
(16) $\displaystyle G$
$\displaystyle=r_{B}\left(\frac{7x-8}{12\sqrt{1-x}}+\frac{2}{3}\right)=G(p,r_{B}).$
(17)

By means of these parametric equations, we plot the $G$–$T$ curves of the
Schwarzschild black holes in a cavity in Fig. 2, with different pressures
[these curves can also be plotted via Eqs. (6) and (7)]. We observe two
branches of the $G$–$T$ curves, corresponding to the large and small black
holes, and these two curves meet with a cusp at the minimum black hole
temperature $T_{0}$, below which no black hole solution exists any longer.
Both Gibbs free energies of the large and small black holes decrease with
temperature. However, from the heat capacity at constant pressure,
$C_{p}=T(\partial S/\partial T)_{p}=-T(\partial^{2}G/\partial T^{2})_{p}$, the
small black holes have negative $C_{p}$ (their $G$–$T$ curves are always
concave), so they are unstable and cannot maintain thermal equilibrium with
the environment. Moreover, their $G$–$T$ curves only tend to but never reach
the $T$-axis, so the HP phase transition never occurs. On the contrary, the
large black holes are stable, and their $G$–$T$ curves are convex and cross
the $T$-axis at the HP temperature $T_{\rm HP}$. With $p$ increasing, $T_{\rm
HP}$ moves rightward, meaning that $T_{\rm HP}$ increases at high pressures,
consistent with the coexistence line in Fig. 1. Furthermore, below $T_{\rm
HP}$, the thermal gas phase with vanishing Gibbs free energy is more stable,
and above $T_{\rm HP}$, the large black hole phase with negative Gibbs free
energy is thermodynamically preferred. Altogether, there is a discontinuity in
the slopes of the $G$–$T$ curves of the black hole–thermal gas system,
corresponding to the first-order HP phase transition at $T_{\rm HP}$.

Figure 2:  The Gibbs free energy of the black hole–thermal gas system in a
cavity, with different values of pressure $p$. The $G$–$T$ curves of the
thermal gas and the stable large black holes intersect at the HP temperature
$T_{\rm HP}$, which increases at large pressures. Below or above $T_{\rm HP}$,
the thermal gas phase or the large black hole phase is globally preferred. The
$G$–$T$ curves of the unstable small black holes are concave and always above
the $T$-axis, so the HP phase transition never occurs. (LBH and SBH stand for
large and small black hole, respectively.)

For comparison, we also plot the $G$–$T$ curves of the Schwarzschild black
holes in a cavity and in the AdS space together in Fig 3. The shapes of these
curves are similar, only with the cavity case having higher $T_{\rm HP}$ and
$T_{0}$ than the AdS case at the same pressure. This observation further
verifies the analogy between these two extended phase spaces.

Figure 3:  The Gibbs free energy of the Schwarzschild black holes in a cavity
and in the AdS space. Both the HP temperature $T_{\rm HP}$ and the minimum
black hole temperature $T_{0}$ in the cavity case are higher than those in the
AdS case at the same pressure.

###  3.2 HP phase transition of the charged black hole in a cavity

Now, we continue to discuss the HP phase transition of the charged black hole
in a cavity and compare the relevant results with those in the AdS
counterparts. Generally speaking, the following procedure is in parallel to
that in Sect. 3.1, but as it will be shown, there are several intrinsic
differences in between. First, we will work in the grand canonical ensemble
with fixed electric potential $\Phi$. Second, if the HP phase transition
occurs, $\Phi$ must have an upper bound, and the charged black hole with
$\Phi$ larger than this bound cannot have the HP phase transition. Third, due
to the constraint $r_{+}<r_{B}$, the large black hole temperature also has an
upper bound (i.e., its $G$–$T$ curve has a terminal point). All these issues
will be explained in depth below.

We start our discussion by reexpressing everything in terms of the electric
potential $\Phi$ instead of the charge $Q$. From Eq. (2), we have

$\displaystyle Q=\frac{\Phi r_{+}}{\sqrt{1-(1-\Phi^{2})\frac{r_{+}}{r_{B}}}},$

and it is not hard to see the upper bound of $\Phi$,

$\displaystyle\Phi<\frac{Q}{r_{+}}<1.$ (18)

Substituting $Q$ into Eqs. (1), (5), and (8), we obtain the temperature,
pressure, and Gibbs free energy of the charged black hole in a cavity in terms
of $\Phi$,

$\displaystyle T$ $\displaystyle=\frac{1-\Phi^{2}}{4\pi
r_{+}\sqrt{1-(1-\Phi^{2})\frac{r_{+}}{r_{B}}}},$ (19) $\displaystyle p$
$\displaystyle=\frac{1}{4\pi
r_{B}^{2}}\left[\frac{2-(1-\Phi^{2})\frac{r_{+}}{r_{B}}}{2\sqrt{1-(1-\Phi^{2})\frac{r_{+}}{r_{B}}}}-1\right],$
(20) $\displaystyle G$
$\displaystyle=r_{B}\left[\frac{7(1-\Phi^{2})\frac{r_{+}}{r_{B}}-8}{12\sqrt{1-(1-\Phi^{2})\frac{r_{+}}{r_{B}}}}+\frac{2}{3}\right].$
(21)

Naturally, these results reduce to those in Eqs. (10)–(12) when $\Phi$
vanishes.

Again, at the HP phase transition point, from the criterion in Eq. (9), we
have

$\displaystyle r_{B}=(1-\Phi^{2})\frac{49}{48}r_{+}.$ (22)

Taking into account the constraint $r_{+}<r_{B}$, we find that, if the HP
phase transition occurs, $\Phi$ must have a much tighter upper bound than that
in Eq. (18),

$\displaystyle\Phi<\frac{1}{7}.$

In other words, when $1/7<\Phi<1$, even if there is the charged black hole
solution in a cavity, the HP phase transition cannot occur. This is quite
different from the case of the Schwarzschild black hole in Sect. 3.1, where
the HP phase transition always occurs.

Substituting Eq. (22) into Eqs. (19) and (20), we obtain the equation of
coexistence line for the charged black hole in a cavity,

$\displaystyle T_{\rm
HP}=(1-\Phi^{2})^{2}\frac{343}{288}\sqrt{\frac{7p}{2\pi}}.$ (23)

Compared to the result in the AdS space HPwo ,

$\displaystyle T_{\rm HP}=\sqrt{\frac{(1-\Phi^{2})8p}{3\pi}},$

the HP temperature $T_{\rm HP}$ decreases with $\Phi$ even faster. The
coexistence lines for the charged black hole in a cavity are shown in Fig. 4,
with different electric potentials.

Figure 4:  The coexistence lines for the charged black hole in a cavity, with
different values of electric potential $\Phi$. The HP phase transition can
occur at all pressures, and $T_{\rm HP}$ decreases with $\Phi$ at a fixed
pressure. $\Phi$ has an upper bound of $1/7$, above which there is no HP phase
transition.

Next, the minimum black hole temperature $T_{0}$ can also be obtained by the
same method in Sect. 3.1,

$\displaystyle T\geq
T_{0}=(1-\Phi^{2})^{2}\frac{(1+2\sqrt{2})^{7/2}}{16(1+\sqrt{2})}\sqrt{\frac{p}{2\pi}},$
(24)

and $T_{0}$ is modified with the same factor $(1-\Phi^{2})^{2}$ as that in
$T_{\rm HP}$ in Eq. (23).

Last, we explore the Gibbs free energy for the charged black hole–thermal gas
system. Again, we first solve $r_{+}$ from Eq. (20),

$\displaystyle r_{+}=\frac{x}{1-\Phi^{2}}r_{B}.$ (25)

However, before substituting $r_{+}$ into $T$ and $G$, we should emphasize an
important point in advance. Different from Eq. (15), where $r_{+}=xr_{B}$ is
always less than $r_{B}$, now due to the factor $1-\Phi^{2}$ in the
denominator, we must set a new constraint as

$\displaystyle x$ $\displaystyle=4\sqrt{2\pi pr_{B}^{2}(2\pi
pr_{B}^{2}+1)}\Big{(}\sqrt{2\pi pr_{B}^{2}+1}-\sqrt{2\pi
pr_{B}^{2}}\Big{)}^{2}<1-\Phi^{2},$ (26)

such that $r_{+}<r_{B}$. Since $x$ is a monotonically increasing function of
$r_{B}$, solving the above inequality, we find that, in the grand canonical
ensemble with fixed electric potential, the cavity radius $r_{B}$ has an upper
bound as

$\displaystyle r_{B}<\frac{1-\Phi}{2\sqrt{2\pi p\Phi}}.$ (27)

This result is significantly different from the Schwarzschild case, where
$r_{B}$ can take any value. Of course, in the limit $\Phi\to 0$, the upper
bound of $r_{B}$ goes to infinity.

Substituting Eq. (25) into Eqs. (19) and (21), we can reexpress $T$ and $G$ in
terms of $r_{B}$,

$\displaystyle T$ $\displaystyle=\frac{(1-\Phi^{2})^{2}}{4\pi
r_{B}x\sqrt{1-x}}=T(p,r_{B}),$ (28) $\displaystyle G$
$\displaystyle=r_{B}\left(\frac{7x-8}{12\sqrt{1-x}}+\frac{2}{3}\right)=G(p,r_{B}).$
(29)

From Eq. (28), we find that $T$ is modified by the same factor
$(1-\Phi^{2})^{2}$ as $T_{\rm HP}$ and $T_{0}$. Meanwhile, from Eq. (29), $G$
remains the same expression as that in the Schwarzschild case in Eq. (17).
With these parametric equations, we plot the $G$–$T$ curves of the charged
black holes in a cavity in Fig. 5, with different pressures and electric
potentials. We observe that $T_{\rm HP}$ increases with $p$ and decreases with
$\Phi$, consistent with Fig. 4.

Here, we point out a characteristic feature in Fig. 5. Because of the
constraints in Eqs. (26) and (27), the $G$–$T$ curves of the large charged
black holes in a cavity have the terminal points at

$\displaystyle(T,G)=\left((1+\Phi)\sqrt{\frac{p}{2\pi\Phi}},-\frac{(1-\Phi)^{2}(1-7\Phi)}{24\sqrt{2\pi
p\Phi^{3}}}\right).$

This means that the temperature of the large charged black hole has an upper
bound. In particular, in the right panel of Fig. 5, as $\Phi$ increases, the
terminal points move toward the upper left corner of the $G$–$T$ plane, and
finally when $\Phi\to 1/7$, the terminal point is located exactly on the
$T$-axis at the point $(T_{\rm HP},0)=(\sqrt{32p/7\pi},0)$, which is just the
result extracted from Eq. (23). Also, when $\Phi\to 0$, there is no terminal
point in the $G$–$T$ curve, as already shown for the Schwarzschild black hole
in a cavity in Fig. 2.

Figure 5:  The Gibbs free energy of the charged black hole–thermal gas system
in a cavity, with different values of pressure $p$ and electric potential
$\Phi$. The HP temperature $T_{\rm HP}$ increases with $p$ and decreases with
$\Phi$. The small charged black holes have no terminal point in their $G$–$T$
curves and also no HP phase transition. The large charged black holes have the
HP phase transitions, but also have the terminal points in their $G$–$T$
curves. In the right panel, when $\Phi=0$, the $G$–$T$ curve extends to
infinity; when $\Phi$ increases, the terminal points move up left; when
$\Phi\to 1/7$, the terminal point is located just on the $T$-axis at $(T_{\rm
HP},0)$.

We need not repeat the comparison with the AdS case, and the situation is
rather similar as that in Fig. 3. The unique difference is that there is no
terminal point in the $G$–$T$ curves of the large charged black holes in the
AdS case, as the event horizon radius $r_{+}$ can take any value, without
constraint like $r_{+}<r_{B}$ in a cavity.

##  4 Conclusion

From the thermodynamic perspective, a black hole resembles a complicated
system with temperature and entropy. However, a black hole in asymptotically
flat space has negative heat capacity and is thermodynamically unstable. This
instability can be avoided by imposing appropriate boundary conditions. By
doing so, the effective pressure and volume can also be introduced
simultaneously, and such theories are thus named as black hole thermodynamics
in the extended phase space. There can be two complementary approaches to
extend the black hole phase space. One is to enclose the black hole in the AdS
space and to introduce the effective pressure via the varying cosmological
constant $\Lambda$ KM , and the other is to confine the black hole in a cavity
in asymptotically flat space and to introduce the effective volume via the
varying cavity radius $r_{B}$ Wang . The former has received great interest in
recent years, but the relevant studies on the latter are just starting.

In our present work, we investigate one of the most important issues in black
hole thermodynamics: the HP phase transitions between the thermal gas and the
black holes in a cavity. Our basic conclusions can be drawn as: (1) there can
be large and small black hole solutions in a cavity, when the black hole
temperature is above a minimum temperature $T_{0}$. (2) The Gibbs free
energies of the small black holes are always positive, so there is no HP phase
transition totally. The Gibbs free energies of the large black holes decrease
with temperature, and there are HP phase transitions at the HP temperature
$T_{\rm HP}$. (3) For the Schwarzschild black hole in a cavity, the HP phase
transition occurs at all pressures, and $T_{\rm HP}$ increases with pressure
$p$. (4) For the charged black hole in a cavity, in the grand canonical
ensemble with fixed electric potential $\Phi$, $T_{\rm HP}$ increases with $p$
and decreases with $\Phi$. (5) If the HP phase transition occurs, $\Phi$ has
an upper bound of $1/7$, and above this value, no HP phase transition exists,
although there are still black hole solutions.

Furthermore, we compare these results to our previous work in the AdS space in
Ref. HPwo  and find remarkable similarities in $T_{\rm HP}$–$p$ and $G$–$T$
curves, both for the Schwarzschild and charged black holes. At the same time,
there is a notable difference in between. The $G$–$T$ curves for the charged
black holes in the grand canonical ensemble in the AdS space can extend to
infinity, but have terminal points in the cavity case, and when $\Phi\to 1/7$,
the terminal point lies exactly on the $T$-axis at $(T_{\rm HP},0)$. This
particular discrepancy stems from the fact that, when a black hole is in a
cavity, its event horizon radius $r_{+}$ must be smaller than the cavity
radius $r_{B}$. However, in the AdS space, there is no such constraint.
Altogether, we intend to provide a whole picture of the HP phase transitions
in the two extended phase spaces.

The above observations naturally motivate further investigations on these two
extended phase spaces. Of course, they are equivalent in the limits
$\Lambda\to 0$ and $r_{B}\to\infty$. However, at finite values of $\Lambda$
and $r_{B}$, there still seem to be some issues sensitive to the boundary
conditions. Therefore, it would be highly necessary and interesting to explore
beyond the present level. For instance, going to higher dimensions is a next
step naturally, in which some typical phenomena are expected (e.g., the
reentrant phase transitions). Besides, the inclusion of angular momentum of
the black hole is also a straightforward generalization. However, from our
experience in Ref. HPwo , it is not very hopeful to gain much new physical
insight due to rotation, except some mathematical complexity. In fact, all the
relevant results obtained in the AdS space in the previous literature can be
reevaluated in the new extended phase space in a cavity. In this way, we wish
to achieve a deeper understanding of black hole thermodynamics in different
extended phase spaces with various boundary conditions.

###### Acknowledgements.

We are very grateful to Bing-Yu Su and Peng Wang for fruitful discussions.
This work is supported by the Fundamental Research Funds for the Central
Universities of China (No. N182410001).

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