Thermodynamics and Phase Transition of a Nonlinear Electrodynamics Black Hole in a Cavity

TL;DR

This paper explores the thermodynamics and phase transitions of nonlinear electrodynamics black holes in a cavity, revealing diverse phase behaviors and transitions depending on the nonlinearity strength, with implications differing from AdS boundary conditions.

Contribution

It provides a detailed analysis of the thermodynamics and phase structure of Born-Infeld black holes in a cavity, highlighting differences from AdS boundary conditions and identifying phase transition types.

Findings

Hawking-Page-like and van der Waals-like phase transitions occur in weak nonlinearity regimes.

Only Hawking-Page-like phase transitions occur in strong nonlinearity regimes.

A tricritical point appears in the phase diagram for weak nonlinearities.

Abstract

We discuss the thermodynamics of a general nonlinear electrodynamics (NLED) asymptotically flat black hole enclosed in a finite spherical cavity. A canonical ensemble is considered, which means that the temperature and the charge on the wall of the cavity are fixed. After the free energy is obtained by computing the Euclidean action, it shows that the first law of thermodynamics is satisfied at the locally stationary points of the free energy. Focusing on a Born-Infeld (BI) black hole in a cavity, the phase structure and transition in various regions of the parameter space are investigated. In the region where the BI electrodynamics has weak nonlinearities, Hawking-Page-like and van der Waals-like phase transitions occur, and a tricritical point appears. In the region where the BI electrodynamics has strong enough nonlinearities, only Hawking-Page-like phase transitions occur. The phase diagram of the BI black hole in a cavity can have dissimilarity from that of a BI black hole using asymptotically anti-de Sitter boundary conditions. The dissimilarity may stem from a lack of an appropriate reference state with the same charge and temperature for the BI-AdS black hole.

Equations

$$\mathcal{S}={\int }_{\mathcal{M}}{d}^{4}x\sqrt{-g}\left[R+\mathcal{L}\left(s,p\right)\right]+{\mathcal{S}}_{\text{surf}},$$

$$s=-\frac{1}{4}{F}^{\mu \nu }{F}_{\mu \nu }\text{ and }p=-\frac{1}{8}{ϵ}^{\mu \nu \rho \sigma }{F}_{\mu \nu }{F}_{\rho \sigma }\text{,}$$

$$G^{\mu \nu }=-\frac{\partial \mathcal{L}\left(s,p\right)}{\partial F_{\mu \nu }}={\mathcal{L}}^{\left(1,0\right)}\left(s,p\right)F^{\mu \nu }+\frac{1}{2}{\mathcal{L}}^{\left(0,1\right)}\left(s,p\right){ϵ}^{\mu \nu \rho \sigma }{F}_{\rho \sigma }.$$

$${\mathcal{S}}_{\text{surf}}=-2{\int }_{\partial \mathcal{M}}{d}^{3}x\sqrt{-\gamma }\left(K-K_0\right)-{\int }_{\partial \mathcal{M}}{d}^{3}x\sqrt{-\gamma }{n}_{\nu }{G}^{\mu \nu }{A}_{\mu }.$$

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }$$

$${\nabla }_{\mu }{G}^{\mu \nu }$$

$$T_{\mu \nu }=g_{\mu \nu }\left[\mathcal{L}\left(s,p\right)-p{\mathcal{L}}^{\left(0,1\right)}\left(s,p\right)\right]+{\mathcal{L}}^{\left(1,0\right)}\left(s,p\right)F_{\mu }^{\rho }{F}_{\nu \rho }\text{.}$$

$$d{s}^2$$

$$A$$

$$-1+f\left(r\right)+r{f}^{\prime }\left(r\right)$$

$$2f^{\prime }\left(r\right)+r{f}^{\prime \prime }\left(r\right)$$

$${\left[r^2{G}^{rt}\right]}^{\prime }$$

$$s=\frac{A_t^{\prime 2}\left(r\right)}{2}\text{ and }{G}^{rt}=-{\mathcal{L}}^{\left(1,0\right)}\left(s,0\right)A_t^{\prime }\left(r\right)\text{.}$$

$$G^{tr}=\frac{q}{r^2}\text{,}$$

$$Q=-\frac{1}{4\pi }{\int }_{S^2}{d}^{2}x\sqrt{\sigma }{n}_{\mu }{l}_{\nu }{G}^{\mu \nu },$$

$$Q=\frac{1}{4\pi }\int d\theta d\varphi r_B^2\sin \theta \frac{q}{r_B^2}=q.$$

$${\mathcal{L}}^{\left(1,0\right)}\left(\frac{A_t^{\prime 2}\left(r\right)}{2},0\right)A_t^{\prime }\left(r\right)=\frac{Q}{r^2}.$$

$$\Phi =4\pi {\int }_{r_{+}}^{r_B}{A}_t^{\prime }\left(r\right)=\frac{4\pi A_t\left(r_B\right)}{\sqrt{f\left(r_B\right)}},$$

$$f\left(r\right)=1-\frac{M}{8\pi r}-\frac{1}{2r}{\int }_r^{\infty }dr{r}^2\left[\mathcal{L}\left(\frac{A_t^{\prime 2}\left(r\right)}{2},0\right)-A_t^{\prime }\left(r\right)\frac{Q}{r^2}\right],$$

$$f\left(r\right)=1-\frac{r_{+}}{r}+\frac{1}{2r}{\int }_{r_{+}}^{r}dr{r}^2\mathcal{L}\left(\frac{A_t^{\prime 2}\left(r\right)}{2},0\right)-\frac{Q}{2r}{A}_t\left(r\right).$$

$$Z\simeq e^{-{\mathcal{S}}^E},$$

$$A_{\tau }=-i{A}_t\text{,}$$

$$G_{r\tau }=-i\frac{Q}{r^2}.$$

$$\Phi =\frac{4\pi i{A}_{\tau }\left(r_B\right)}{\sqrt{f\left(r_B\right)}}.$$

$$\int \sqrt{f\left(r_B\right)}d\tau =T^{-1}.$$

$${\mathcal{S}}^E$$

$$-16{\pi }^2{r}_{+}^2-\frac{4\pi }{T\sqrt{f\left(r_B\right)}}{\int }_{r_{+}}^{r_B}dr{r}^2\mathcal{L}\left(-\frac{A_{\tau }^{\prime 2}\left(r\right)}{2},0\right)+\frac{4\pi i{A}_{\tau }\left(r_B\right)Q}{T\sqrt{f\left(r_B\right)}}.$$

$${\mathcal{S}}^E=\frac{16\pi r_B}{T}\left[1-\sqrt{f\left(r_B\right)}\right]-S,$$

$$f\left(r_B\right)=1-\frac{M}{8\pi r_B}+\frac{Q^2}{4r_B^2}+\mathcal{O}\left(r_B^{-4}\right).$$

$${\mathcal{S}}^E=\frac{1}{T}\left(M-TS\right),$$

$$F=-T\ln Z=T{\mathcal{S}}^E.$$

$$F=F\left(r_{+};T,Q,r_B\right),$$

$$\frac{dF\left(r_{+};T,Q,r_B\right)}{d{r}_{+}}=0⟹-\frac{d\left[r_{B}f\left(r_B\right)\right]/d{r}_{+}}{2\sqrt{f\left(r_B\right)}}=2\pi r_{+}T⟹f^{\prime }\left(r_{+}\right)=4\pi T\sqrt{f\left(r_B\right)},$$

Full text

††preprint: CTP-SCU/2019001

1. I Introduction

2. II NLED Black Hole in a Cavity

3. II.1 Black Hole Solution

4. II.2 Euclidean Action

5. II.3 Thermodynamics

6. III Born-Infeld Black Hole in a Cavity

7. IV Discussion and Conclusion

8. A Reduced Action

I Introduction

A Schwarzschild black hole in asymptotically flat space has negative specific heat and hence radiates more when it is smaller. To make this system thermally stable, appropriate boundary conditions must be imposed. One popular choice is putting the black hole in anti-de Sitter (AdS) space, which has a negative cosmological constant. The black hole becomes thermally stable since the AdS boundary acts as a reflecting wall. The thermodynamic properties of AdS black holes were first studied by Hawking and Page IN-Hawking:1982dh , who discovered the Hawking-Page phase transition, i.e., a phase transition between the thermal AdS space and the Schwarzschild-AdS black hole. Later, with the advent of the AdS/CFT correspondence IN-Maldacena:1997re ; IN-Gubser:1998bc ; IN-Witten:1998qj , there has been much interest in studying the phase transitions of AdS black holes IN-Witten:1998zw ; IN-Chamblin:1999tk ; IN-Chamblin:1999hg ; IN-Caldarelli:1999xj ; IN-Cai:2001dz ; IN-Kubiznak:2012wp . However, it is not clear whether the duality between the black hole and a boundary field theory is independent of the details of the boundary conditions, or just the special result of the asymptotically AdS space. It is therefore interesting to investigate thermodynamics of black holes in the case of different boundary conditions.

Alternatively, one can place the black hole inside a cavity in asymptotically flat space, on the wall of which the metric is fixed. In IN-York:1986it , York showed that a Schwarzschild black hole in a cavity can be thermally stable and experiences a Hawking-Page-like transition to the thermal flat space as the temperature decreases. Later, the thermodynamics of a Reissner-Nordstrom (RN) black hole in a cavity was discussed in a grand canonical ensemble IN-Braden:1990hw and a canonical ensemble IN-Carlip:2003ne ; IN-Lundgren:2006kt . Similar to a RN-AdS black hole, it was found that a Hawking-Page-like phase transition occurs in the grand canonical ensemble, and a van der Waals-like phase transition occurs in the canonical ensemble. Note that a van der Waals-like phase transition consists of a first-order phase transition between two black hole phases of different sizes and a critical point, where the first-order phase transition ends, and a second order phase transition takes place. The phase structures of several black brane systems in a cavity were investigated in a series of paper IN-Lu:2010xt ; IN-Wu:2011yu ; IN-Lu:2012rm ; IN-Lu:2013nt ; IN-Zhou:2015yxa ; IN-Xiao:2015bha , where Hawking-Page-like or van der Waals-like phase transitions were always found except for some special cases. Including charged scalars, boson stars and hairy black holes in a cavity were considered in IN-Basu:2016srp ; IN-Peng:2017gss ; IN-Peng:2017squ ; IN-Peng:2018abh , which showed that the phase structure of the gravity system in a cavity is strikingly similar to that of holographic superconductors in the AdS gravity. The stabilities of solitons, stars and black holes in a cavity were also studied in IN-Sanchis-Gual:2015lje ; IN-Dolan:2015dha ; IN-Ponglertsakul:2016wae ; IN-Sanchis-Gual:2016tcm ; IN-Ponglertsakul:2016anb ; IN-Sanchis-Gual:2016ros ; IN-Dias:2018zjg ; IN-Dias:2018yey . It was found that the nonlinear dynamical evolution of a charged black hole in a cavity could end in a quasi-local hairy black hole. Recently, McGough, Mezei and Verlinde IN-McGough:2016lol proposed that the $T\bar{T}$ deformed CFT${}_{\text{2}}$ locates at the finite radial position of AdS${}_{\text{3}}$, which further motivates us to explore the properties of a black hole in a cavity.

Taking quantum contributions into account, nonlinear corrections are usually added to the Maxwell Lagrangian, which gives the nonlinear electrodynamics (NLED). Coupling NLED fields to gravity, various NLED charged black holes were derived and discussed in a number of papers IN-Soleng:1995kn ; IN-Maeda:2008ha ; IN-Hendi:2017mgb ; IN-Tao:2017fsy ; IN-Guo:2017bru ; IN-Mu:2017usw ; IN-Hendi:2012um ; IN-Mo:2016jqd ; IN-Nam:2018tpf ; IN-Dehghani:2018eps . It is interesting to note that some NLED black holes can be regular black hole models IN-AyonBeato:1998ub ; IN-AyonBeato:1999rg . As pointed out in IN-Bronnikov:2000vy , a globally regular NLED black hole requires vanishing electric charge and a finite NLED Lagrangian (or in the FP dual theory). Born-Infeld (BI) electrodynamics was first introduced to incorporates maximal electric fields and smooths divergences of the electrostatic self-energy of point charges IN-Born:1934gh . Later, it is realized that BI electrodynamics can come from the low energy limit of string theory and encodes the low-energy dynamics of D-branes. The BI black hole solution was obtained in IN-Dey:2004yt ; IN-Cai:2004eh . The thermodynamic behavior and phase transitions of BI black holes in various gravities were investigated in IN-Fernando:2003tz ; IN-Fernando:2006gh ; IN-Banerjee:2010da ; IN-Banerjee:2011cz ; IN-Lala:2011np ; IN-Banerjee:2012zm ; IN-Azreg-Ainou:2014twa ; IN-Hendi:2015hoa ; IN-Zangeneh:2016fhy ; IN-Zeng:2016sei ; IN-Li:2016nll ; IN-Zou:2013owa ; IN-Hendi:2017oka . Especially, the thermodynamics of a 4D BI-AdS black hole was studied in IN-Gunasekaran:2012dq ; IN-Dehyadegari:2017hvd ; IN-Wang:2018xdz , where a reentrant phase transition was always observed in a certain region of the parameter space.

In this paper, we first investigate the thermodynamic behavior of a 4D general NLED asymptotically flat black hole enclosed in a cavity. Then, we turn to study the phase structure and transition of a BI black hole in a cavity. We find that Hawking-Page-like and van der Waals-like phase transitions can occur while there is no reentrant phase transition. The rest of this paper is organized as follows. In section II, we compute the Euclidean action for the general NLED black hole in a cavity and discuss the thermodynamic properties of the system in the canonical ensemble. In section III, we focus on the BI black hole case to discuss the phase structure and transition. The phase diagrams of the BI black hole in a cavity is given in FIG. 1, from which one can read the black hole’s phase structure and transition. In the appendix, we present an alternative derivation of the Euclidean action for a general NLED black hole in a cavity using the reduced action method proposed in IN-Braden:1990hw .

II NLED Black Hole in a Cavity

In this section, we consider a NLED charged black hole inside a cavity, on the boundary of which the temperature and charge are fixed. That said, the thermodynamics of the black hole is discussed in a canonical ensemble.

II.1 Black Hole Solution

First, we will consider the black hole solution in a $\left(3+1\right)$ dimensional model of gravity coupled to a nonlinear electromagnetic field $A_{\mu}$. On a spacetime manifold $\mathcal{M}$ with a time-like boundary $\partial\mathcal{M}$, the action is given by

[TABLE]

where we take $16\pi G=1$ for simplicity, $\mathcal{L}\left(s,p\right)$ is a general NLED Lagrangian, and $S_{\text{surf}}$ are the surface terms on $\partial\mathcal{M}$. Here, $s$ and $p$ are two independent nontrivial scalars built from the field strength tensor $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ and none of its derivatives:

[TABLE]

where $\epsilon^{\mu\nu\rho\sigma}\equiv-\left[\mu\text{ }\nu\text{ }\rho\text{ }\sigma\right]/\sqrt{-g}$ is a totally antisymmetric Lorentz tensor, and $\left[\mu\text{ }\nu\text{ }\rho\text{ }\sigma\right]$ is the permutation symbol. For later use, we define

[TABLE]

where we denote $\mathcal{L}^{\left(1,0\right)}\left(s,p\right)\equiv\frac{\partial\mathcal{L}\left(s,p\right)}{\partial s}$ and $\mathcal{L}^{\left(0,1\right)}\left(s,p\right)\equiv\frac{\partial\mathcal{L}\left(s,p\right)}{\partial p}$, respectively. Note that the general NLED theories with the Lagrangian $\mathcal{L}\left(s,p\right)$ were first considered in NBHC-Pellicer:1969cf ; NBHC-Pleb:1969 . The surface terms of the action $\left(\ref{eq:Action}\right)$ are

[TABLE]

The first term above is the Gibbons-Hawking-York surface term, where $K$ is the extrinsic curvature, $\gamma$ is the metric on the boundary, and $K_{0}$ is a subtraction term to make the Gibbons-Hawking-York term vanish in flat spacetime. When the metric on $\partial\mathcal{M}$ is fixed, the Gibbons-Hawking-York term is crucial to obtain the correct the equations of motion from performing the variation. The second term, where $n^{\mu}$ is the unit outward-pointing normal vector of $\partial\mathcal{M}$, is included to keep the charge fixed on $\partial\mathcal{M}$, instead of the potential, when one varies the action to have the correct equations of motion IN-Wang:2018xdz . Varying the action $\left(\ref{eq:Action}\right)$ in terms of $g_{\mu\nu}$ and $A_{\mu}$ with the metric and the charge fixed on $\partial\mathcal{M}$, we find that the equations of motion are

[TABLE]

where $T_{\mu\nu}$ is the energy-momentum tensor for the NLED field:

[TABLE]

We consider a static spherically symmetric black hole solution with the metric and the NLED field of the form

[TABLE]

Moreover, we assume that the black hole lives in a spherical cavity, which has a boundary $\partial\mathcal{M}$ at $r=r_{B}$. The spacelike slices with constant $t$ of $\partial\mathcal{M}$ are $2$-spheres $S^{2}$ whose radii are $r_{B}$. The equations of motion then reduce to

[TABLE]

where

[TABLE]

It can show that eqn. $\left(\ref{eq:thetathetaEOM}\right)$ can be derived from eqns. $\left(\ref{eq:ttEOM}\right)$ and $\left(\ref{eq:NLEDEOM}\right)$.

Solving eqn. $\left(\ref{eq:NLEDEOM}\right)$, we find that

[TABLE]

where $q$ is a constant. The charge of the system inside the cavity is defined as IN-Wang:2018xdz .

[TABLE]

where $l^{\mu}$ is the unit normal vector of the constant $t$ hypersurface, and $\sigma$ is the induced metric on $S^{2}$. Using eqn. $\left(\ref{eq:Grt}\right)$, one finds that the charge inside the cavity becomes

[TABLE]

From eqns. $\left(\ref{eq:SGrt}\right)$ and $\left(\ref{eq:Grt}\right)$, $A_{t}^{\prime}\left(r\right)$ is determined by

[TABLE]

The gauge potential measured on $\partial\mathcal{M}$ with respect to the horizon is

[TABLE]

where the blueshift factor $1/\sqrt{f\left(r_{B}\right)}$ relates $A_{t}$ to the proper orthonormal frame component of the potential one-form $A$ IN-Braden:1990hw , and we fix the gauge field $A_{t}\left(r\right)$ at the horizon to be zero, i.e., $A_{t}\left(r_{+}\right)=0$.

By integrating eqn. $\left(\ref{eq:ttEOM}\right)$, we have

[TABLE]

where $M$ is the mass of the black hole IN-Wang:2018xdz . Suppose that $r_{+}$ is the outer event horizon radius of the black hole. Since $f\left(r_{+}\right)=0$, we can express $f\left(r\right)$ in terms of $r_{+}$:

[TABLE]

II.2 Euclidean Action

In the semiclassical approximation, one can relate the on-shell Euclidean action to the thermal partition function:

[TABLE]

where $\mathcal{S}^{E}$ is the Euclidean continuation of the action $\mathcal{S}$: $\mathcal{S}^{E}=i\mathcal{S}$. The Euclidean time $\tau$ is obtained from Lorentzian time $t$ by the analytic continuation $t=i\tau$. From $A_{\tau}d\tau=A_{t}dt$, it follows that

[TABLE]

which gives $G_{r\tau}=-iG_{rt}$. So eqn. $\left(\ref{eq:Grt}\right)$ becomes

[TABLE]

Moreover, the gauge potential on $\partial\mathcal{M}$ is

[TABLE]

Since the temperature $T$ is fixed on the boundary of the cavity, we can impose the boundary condition at $r=r_{B}$ in terms of the reciprocal temperature:

[TABLE]

which identifies the Euclidean time $\tau$ as $\tau\sim\tau+\frac{1}{T\sqrt{f\left(r_{B}\right)}}$, and hence the period of $\tau$ is $\frac{1}{T\sqrt{f\left(r_{B}\right)}}$.

For the black hole solution $\left(\ref{eq:ansatz}\right)$, one can evaluate the Euclidean action by integrating over angles and performing the integration by parts:

[TABLE]

After eqn. $\left(\ref{eq:f(r)}\right)$ is plugged into eqn. $\left(\ref{eq:EActionf}\right)$, a straightforward calculation gives

[TABLE]

where $S=16\pi^{2}r_{+}^{2}$ is the entropy of the black hole.

For large values of $r_{B}$, one finds that

[TABLE]

In the limit of $r_{B}\rightarrow\infty$, the Euclidean action then reduces to

[TABLE]

as expected.

II.3 Thermodynamics

Various thermodynamic quantities can be derived from the Euclidean action $\left(\ref{eq:EAction}\right)$, which is related to the free energy $F$ in the semiclassical approximation by

[TABLE]

From eqns. $\left(\ref{eq:f(r)}\right)$ and $\left(\ref{eq:EAction}\right)$, one finds that the free energy $F$ is a function of the temperature $T$, the charge $Q$, the cavity radius $r_{B}$ and the horizon radius $r_{+}$:

[TABLE]

where $T$, $Q$ and $r_{B}$ are parameters of the canonical ensemble. The only variable $r_{+}$ can be determined by extremizing the free energy $F\left(r_{+};T,Q,r_{B}\right)$ with respect to $r_{+}$:

[TABLE]

where we use $d\left[r_{B}f\left(r_{B}\right)\right]/dr_{+}=-r_{+}f^{\prime}\left(r_{+}\right)$. That said, the solution $r_{+}=r_{+}\left(T,Q,r_{B}\right)$ of eqn. $\left(\ref{eq:frpus}\right)$ corresponds to a locally stationary point of $F\left(r_{+};T,Q,r_{B}\right)$. It is interesting to note that eqn. $\left(\ref{eq:frpus}\right)$ can be written as

[TABLE]

where

[TABLE]

is the Hawking temperature of the black hole. The temperature $T$ on $\partial\mathcal{M}$ is thus blueshifted from $T_{h}$, which is measured at infinity.

After obtained $r_{+}=r_{+}\left(T,Q,r_{B}\right)$, we can evaluate $F\left(r_{+};T,Q,r_{B}\right)$ at the locally stationary point $r_{+}=r_{+}\left(T,Q,r_{B}\right)$:

[TABLE]

For later convenience, we shall suppress $T,Q$ and $r_{B}$ in $F\left(r_{+};T,Q,r_{B}\right)$ and $F\left(T,Q,r_{B}\right)$ and denote $F\left(r_{+};T,Q,r_{B}\right)$ and $F\left(T,Q,r_{B}\right)$ as $F\left(r_{+}\right)$ and $F$, respectively. The thermal energy of the black hole in the cavity is

[TABLE]

Using eqn. $\left(\ref{eq:f(r)inf}\right)$, we can express the ADM mass of the black hole $M$ in terms of $E$ and $Q$:

[TABLE]

where the second and third terms on left-hand side can be interpreted as the gravitational and electrostatic binding energies, respectively. Using eqn. $\left(\ref{eq:f(r)}\right)$, we can express the thermal energy $E$ in terms of the entropy $S$, the charge $Q$ and the cavity radius $r_{B}$. Differentiating $E$ with respect to $S$ and $Q$, respectively, gives

[TABLE]

From the energy $E$, we can define a thermodynamic surface pressure by

[TABLE]

From eqns. $\left(\ref{eq:ESQ}\right)$ and $\left(\ref{eq:lamda}\right)$, the first law of thermodynamics can be established:

[TABLE]

where $A\equiv 4\pi r_{B}^{2}$ is the surface area of the cavity.

To obtain the proper Smarr relation for the black hole, we need to consider the dimensionful couplings $a_{i}$ in the NLED Lagrangian $\mathcal{L}\left(s,p\right)$, which have $\left[a_{i}\right]=L^{c_{i}}$. Since the dimensional analysis give that $\left[Q\right]=\left[M\right]=L$, $\left[T\right]=\left[\lambda\right]=L^{-1}$and $\left[A\right]=\left[S\right]=L^{2}$, the Euler scaling argument leads to the Smarr relation

[TABLE]

where we introduce the conjugates $\mathcal{A}_{i}$ associated with $a_{i}$:

[TABLE]

We now discuss the thermodynamic stability of the black hole in the cavity against thermal fluctuations. In the canonical ensemble, one considers the specific heat at constant electric charge:

[TABLE]

When $C_{Q}>0$, the system is thermally stable. Thus, a thermally stable black hole phase has $\frac{\partial r_{+}\left(T,Q,r_{B}\right)}{\partial T}>0$. Since $\partial^{2}F/\partial^{2}T=-C_{Q}$, the thermally stable/unstable phases have concave downward/upward $F$-$T$ curves. On the other hand, it can show that, at $r_{+}=r_{+}\left(T,Q,r_{B}\right)$,

[TABLE]

which means that the black hole phase is thermally stable/unstable if $r_{+}\left(T,Q,r_{B}\right)$ is a local minimum/maximum of $F\left(r_{+}\right)$.

To find the global minimum of $F\left(r_{+}\right)$ over the space of the variable $r_{+}$ with fixed values of $T,Q$ and $r_{B}$, we also need to consider the values of $F\left(r_{+}\right)$ at the edges of the space of $r_{+}$. In fact, the physical space of $r_{+}$ is constrained by

[TABLE]

where $r_{e}$ is the horizon radius of the extremal black hole with the charge being $Q$. If there exists no extremal black hole solution for $Q$, one can simply set $r_{e}=0$. For simplicity, the global minimum of $F\left(r_{+}\right)$ at the edges is dubbed ”edge state (ES)” in our paper.

III Born-Infeld Black Hole in a Cavity

BI electrodynamics is described by the Lagrangian density

[TABLE]

where the coupling parameter $a$ is related to the string tension $\alpha^{\prime}$ as $a=\left(2\pi\alpha^{\prime}\right)^{2}>0$. For $a=0$, the BI Lagrangian would reduce to the Maxwell Lagrangian. Solving eqn. $\left(\ref{eq:QAt}\right)$ for $A_{t}^{\prime}\left(r\right)$ gives

[TABLE]

where $Q$ is the charge of the BI black hole. From eqn. $\left(\ref{eq:f(r)}\right)$, one can express $f\left(r\right)$ in terms of the horizon radius $r_{+}$:

[TABLE]

where ${}_{2}F_{1}\left(a,b,c;x\right)$ is the hypergeometric function.

It is convenient to express quantities in units of $r_{B}$:

[TABLE]

where $r_{+}$ is the horizon radius. We then use eqns. $\left(\ref{eq:EAction}\right)$ and $\left(\ref{eq:BIBHf(r)}\right)$ to find the free energy as a function of $x$:

[TABLE]

where

[TABLE]

The Hawking temperature of the BI black hole can be calculated from eqn. $\left(\ref{eq:HT}\right)$:

[TABLE]

The locally stationary points of $\tilde{F}\left(x\right)$ are determined by $d\tilde{F}\left(x\right)/dx=0$, which becomes

[TABLE]

As shown in IN-Tao:2017fsy , there are two types of BI black holes depending on the minimum value of $\tilde{T}_{h}$:

• •

RN type: $4\tilde{a}\leq\tilde{Q}^{2}\leq 4\left(1+\tilde{a}\right)$. This type of BI black holes can have extremal black hole solutions like RN black holes. In fact, the Hawking temperature $\tilde{T}_{h}=0$ has one single solution $x=x_{e}\equiv\frac{1}{2}\sqrt{\tilde{Q}^{2}-4\tilde{a}}$, where $x_{e}r_{B}$ is the horizon radius of the extremal BI black hole with $Q$ and $a$. In this case, we must have $x_{e}\leq x\leq 1$. Note that requiring $x_{e}\leq 1$ puts an upper bound on $\tilde{Q}^{2}$: $\tilde{Q}^{2}\leq 4\left(1+\tilde{a}\right)$. Another way to understand this upper bound is that, when $\tilde{Q}^{2}>4\left(1+\tilde{a}\right)$, $f\left(x\right)$ is always negative and hence $\tilde{F}\left(x\right)$ is not a real-valued function for any $x$. When $x$ $=x_{e}$ we have an extremal BI black hole, and when $x=1$ the horizon merges with the boundary.

• •

Schwarzschild-like type: $\tilde{Q}^{2}<4\tilde{a}$. This type of BI black holes have only one horizon like Schwarzschild black holes. The Hawking temperature $\tilde{T}_{h}$ has a positive minimum value and goes to $+\infty$ as $x\rightarrow 0$. In this case, we can have $0\leq x\leq 1$, over which $\tilde{F}\left(x\right)$ is well-defined. It can show that $\tilde{T}_{h}/\sqrt{f\left(x\right)}$ has a minimum value of $\tilde{T}_{\min}>0$ over $0\leq x\leq 1$. Eqn. $\left(\ref{eq:TBIBH}\right)$ implies that the locally stationary points of $\tilde{F}\left(x\right)$, corresponding to BI black hole solutions, only exist for $\tilde{T}\geq\tilde{T}_{\min}$. When $x$ $=0$, one finds that eqn. $\left(\ref{eq:BIBHf(r)}\right)$ becomes

[TABLE]

If $Q=0$, $f\left(r\right)=1$, and hence the edge state at $x=0$ is just the thermal flat space. For $Q>0$, we have

[TABLE]

where $R$ is the Ricci scalar. So the metric has a physical singularity at $r=0$ although $f\left(0\right)$ is finite. It can show that $f\left(r\right)>0$, and hence there exists no horizon. The edge state with $Q>0$ at $x=0$ is thus a naked singularity.

To find the phase structure and transition of a BI black hole in a cavity, we need to analyze the locally stationary points of $\tilde{F}\left(x\right)$ and find the global minimum value of $\tilde{F}\left(x\right)$. In fact, with fixed value of $\tilde{a}$, the locally stationary points of $\tilde{F}\left(x\right)$ can be determined by solving eqns. $\left(\ref{eq:TBlue}\right)$ and $\left(\ref{eq:HTBIBH}\right)$ for $x$ in terms of $\tilde{T}$ and $\tilde{Q}$. The solution $x(\tilde{T},\tilde{Q})$ is often a multivalued function, each branch of which corresponds to a family of BI black hole solutions. When $x(\tilde{T},\tilde{Q})$ is multivalued, there is more than one family of BI black holes of different sizes with fixed values of $\tilde{T}$ and $\tilde{Q}$. We find that there are five regions in the $\tilde{a}$-$\tilde{Q}$ phase space of the BI black hole, in each of which the BI black hole has distinct behavior of the branches of $x(\tilde{T},\tilde{Q})$ and phase structure. The five regions of the $\tilde{a}$-$\tilde{Q}$ phase space are mapped in FIG. 1. We plot $x(\tilde{T},\tilde{Q})$ against $\tilde{T}$ for various values of $\tilde{Q}$ with $\tilde{a}=0.05$ in FIG. 2, which shows the general behavior of $x(\tilde{T},\tilde{Q})$ in these five regions. In what follows, we discuss the phase structure and transition of the BI black hole in the five regions:

• •

Region I: As shown in FIG. 2, there is only one branch for $x(\tilde{T},\tilde{Q})$ with fixed values of $\tilde{a}$ and $\tilde{Q}$, on which the BI black hole is thermally stable. Since BI black holes in this region satisfy $\tilde{Q}\geq 4\tilde{a}$, they are RN type, and hence $\tilde{T}$ can go to zero. For the BI black hole with $\tilde{a}=0.01$ and $\tilde{Q}=0.6$ in this region, we plot the free energy $\tilde{F}\left(x\right)$ in FIG. 3(a), which shows that the endpoints always have higher free energy, and the local minimum of $\tilde{F}\left(x\right)$ is also the global minimum. There is only one phase in this region.

• •

Region II: BI black holes in this region are RN type. As shown in FIG. 2, there are three branches for $x(\tilde{T},\tilde{Q})$ with fixed values of $\tilde{a}$ and $\tilde{Q}$, which are denoted by Small BH (green), Large BH (blue) and Intermediate BH (brown). Both the Small BH and Large BH branches are thermally stable. For the BI black hole with $\tilde{a}=0.01$ and $\tilde{Q}=0.3$ in this region, we plot the free energy $\tilde{F}\left(x\right)$ in FIG. 3(b), which shows that the endpoints always have higher free energy than the global minimum. $\tilde{F}\left(x\right)$ has the global minimum at Small BH for small enough $\tilde{T}$ and Large BH for large enough $\tilde{T}$, respectively. The free energies of the three branches are plotted in the right panel of FIG. 3(b), which shows that there is a first-order phase transition between Small BH and Large BH.

• •

Region III: BI black holes in this region are Schwarzschild-like type. For $\tilde{T}<\tilde{T}_{\min}$, $\tilde{F}\left(x\right)$ is a strictly increasing function (see $\tilde{T}=0.10$ in the left panel of FIG. 4), and hence $\tilde{F}\left(x\right)$ has the global minimum at $x=0$, dubbed the edge state. For $\tilde{T}\geq\tilde{T}_{\min}$, as shown in FIG. 2, there are four branches for $x(\tilde{T},\tilde{Q})$ with fixed values of $\tilde{a}$ and $\tilde{Q}$, which are denoted by Small BH (green), Large BH (blue) and Intermediate BH (brown and red). Both the Small BH and Large BH branches are thermally stable. The free energies of the four branches and the edge state are plotted in the right panel of FIG. 4. As $\tilde{T}$ increases from $\tilde{T}_{\min}$, the free energy of Small BH decrease while $\tilde{F}\left(0\right)$ is constant. They cross each other at some point, where a first-order transition occurs, and Small BH becomes globally stable. Further increasing $\tilde{T}$, Large BH appears, and its free energy decrease more rapidly than that of Small BH. So they cross each other at some point, where another first-order transition occurs, and Large BH then becomes the globally stable one.

• •

Region IV: BI black holes in this region are also Schwarzschild-like type. As in Region III, the edge state at $x=0$ is globally stable for $\tilde{T}<\tilde{T}_{\min}$. For $\tilde{T}\geq\tilde{T}_{\min}$, there are also four branches for $x(\tilde{T},\tilde{Q})$ with fixed values of $\tilde{a}$ and $\tilde{Q}$, which are denoted by Small BH (green), Large BH (blue) and Intermediate BH (brown and red). The free energies of the four branches and the edge state are plotted in FIG. 5(a). Unlike Region III, the temperature at which Small BH appears is too high, such that $\tilde{F}\left(0\right)$ does not cross the free energy of Small BH or $\tilde{F}\left(0\right)$ crosses the free energy of Large BH before it crosses that of Small BH. Hence as $\tilde{T}$ increases, there is only one first-order transition from the edge state to Large BH at some temperature, where the free energy of Large BH and $\tilde{F}\left(0\right)$ cross each other.

• •

Region V: BI black holes in this region are also Schwarzschild-like type. As in Regions III and IV, the edge state at $x=0$ is globally stable for $\tilde{T}<\tilde{T}_{\min}$. However for $\tilde{T}\geq\tilde{T}_{\min}$, there are two branches for $x(\tilde{T},\tilde{Q})$ with fixed values of $\tilde{a}$ and $\tilde{Q}$, which are denoted by BH (green) and Intermediate BH (red). The BH branch is thermally stable. The free energies of the two branches and the edge state are plotted in FIG. 5(b), which shows that there is a first-order transition from the edge state to Large BH as $\tilde{T}$ increases.

In FIG. 1, the boundary between the region in which $x(\tilde{T},\tilde{Q})$ has $n$ branches and that in which $x(\tilde{T},\tilde{Q})$ has $n+2$ branches is the critical line. There are 3 such boundaries in FIG. 1, i.e., $\tilde{Q}_{12}\left(\tilde{a}\right)$, $\tilde{Q}_{35}\left(\tilde{a}\right)$ and $\tilde{Q}_{45}\left(\tilde{a}\right)$, where $\tilde{Q}_{ij}\left(\tilde{a}\right)$ is the boundary between Region $i$ and Region $j$. However, FIG. 5 shows that $\tilde{Q}_{45}\left(\tilde{a}\right)$ is not physical since it does not globally minimize the free energy. Thus physical critical line only consists of $\tilde{Q}_{12}\left(\tilde{a}\right)$ and $\tilde{Q}_{35}\left(\tilde{a}\right)$, which terminates at $\left\{\tilde{a}_{c},\tilde{Q}_{c}\right\}\simeq\left\{0.092,0.524\right\}$. The line $\tilde{Q}_{12}\left(\tilde{a}\right)$ is reminiscent of RN black holes.

The phase diagram with $\tilde{a}=0.05$ in the $\tilde{Q}$-$\tilde{T}$ phase space is displayed in the left panel of FIG. 6. There is a LBH/SBH first-order phase transition for some range of $\tilde{Q}$, a Hawking-Page-like ES/SBH first-order phase transition for some smaller range of $\tilde{Q}$ and another Hawking-Page-like ES/BH first-order phase transition for some larger range of $\tilde{Q}$. The LBH/SBH phase transition line is a van der Waals-like phase transition and hence terminates at the critical point, represented by the red dot. These three first-order phase transition lines merge together at the tricritical point, marked by the green dot. The phase diagram with $\tilde{a}=0.15$ in the $\tilde{Q}$-$\tilde{T}$ phase space is plotted in the right panel of FIG. 6, which is simpler than that with $\tilde{a}=0.05$. Since $\tilde{a}=0.15>\tilde{a}_{c}$, there is no critical point or tricritical point in this phase diagram. There are two phases, namely the edge state and BH, which are separated by a Hawking-Page-like first-order phase transition line.

IV Discussion and Conclusion

In the first part of this paper, we calculated the Euclidean action of a general NLED black hole in a finite spherical cavity and investigated the corresponding thermodynamic behavior in a canonical ensemble. Specifically, the Euclidean action was given by eqn. $\left(\ref{eq:EAction}\right)$, which could be interpreted as the free energy of the black hole. It was then demonstrated that the first law of thermodynamics and the Smarr relation were satisfied at the locally stationary points of the free energy. It also showed that the local minimum of the free energy corresponds to the locally stable phase of the system. To determine the globally stable phase, the edge states are needed to be considered as well.

In the second part, we examined the phase structure and transition of a BI black hole in a cavity. In FIG. 1, we mapped five regions in the $\tilde{a}$-$\tilde{Q}$ phase space, each of which has different phase behavior. Regions I and II are reminiscent of RN black holes, where there exist extremal black hole solutions. In these two regions, the global minimum of the free energy is always at one of the locally stationary points. There is only one branch of black hole solutions in Region I, while in Region II, there is a band of temperatures where three branches coexist, and a first-order LBH/SBH phase transition occurs. In Regions III, IV and V, for low temperatures, the global minimum of the free energy is at $x=0$, which describes a naked singularity. For high enough temperatures, the global minimum is at one of the locally stationary points, and hence there is a first-order phase transition from the edge state at $x=0$ to the black hole phase as the temperature increases. In Region III, further increasing the temperature will lead to another first-order phase transition from Small BH to Large BH. The phase diagrams with $\tilde{a}=0.05$ and $0.15$ in the $\tilde{Q}$-$\tilde{T}$ phase space were plotted in FIG. 6, respectively. For $\tilde{a}=0.05$, there are three first-order phase transition lines merging together at a tricritical point. At the critical temperature, there is a critical point, corresponding to a second-order phase transition, beyond which there is only one phase. For $\tilde{a}=0.15$, there is only one phase transition line, which separates the edge state from the black hole phase. Note that we only focus on spherical topology in our paper, so it is possible that there are some other states of lower free energy in a different topological sector with the same charge and temperature. If this happens, the stable phases discussed above are only metastable.

Using asymptotically AdS boundary conditions, the thermodynamics of BI black holes was considered in IN-Gunasekaran:2012dq ; IN-Dehyadegari:2017hvd ; IN-Wang:2018xdz . For the RN type in Regions I and II, the phase structure of a BI black hole in a cavity is analogous to that of the corresponding BI-AdS black hole. However for the Schwarzschild-like type in Regions III, IV and V, we find that there are some differences between the thermodynamics of the BI black holes under these two boundary conditions. For example, a LBH/SBH/LBH reentrant phase transition, which consists of a LBH/SBH first-order phase transition and a LBH/SBH zeroth-order phase transition, could occur for the BI-AdS black holes in Region V of IN-Wang:2018xdz . On the other hand, such reentrant phase transition is not observed for any BI black hole in a cavity. Nevertheless, it is naive to claim that the phase structure of BI black holes depends on details of the boundary conditions. A crucial observation is that, if there were no edge states, the phase structure of the BI black hole in a cavity would be quite similar to that of the BI-AdS black hole. In fact, if the edge state is ignored, the inset of the bottom left panel in FIG. 5(a) shows that, as the temperature increases, there is a finite jump in free energy leading to a zeroth-order phase transition from Large BH to Small BH, which is followed by a first-order phase transition returning to Large BH. This LBH/SBH/LBH transition is just the reentrant phase transition.

In asymptotically AdS spaces, the Euclidean action needs to be regulated to cancel the divergences coming from the asymptotic region. One can use the background-substraction method to regularize the Euclidean action by subtracting a contribution from a reference background. The reference background and the edge state play a similar role in determining the global minimum of the free energy. For a Schwarzschild-AdS black hole, the reference background is just the thermal flat space, which is the same as the edge state at $x=0$ for a Schwarzschild black hole in a cavity. As the temperature decreases, both the Schwarzschild black hole in a cavity and the Schwarzschild-AdS black hole thus experience the Hawking-Page transition to the thermal flat space IN-York:1986it . Although there is ambiguity about the reference background of charged black holes, one can circumvent this by using the counterterm subtraction method CON-Balasubramanian:1999re ; CON-Emparan:1999pm , in which the Euclidean action is regularized in a background-independent fashion by adding a series of boundary counterterms to the action. In IN-Gunasekaran:2012dq ; IN-Wang:2018xdz , the counterterm subtraction method was used to compute the Euclidean action for a BI-AdS black hole. For a RN black hole in a cavity, the global minimum of the free energy is never at the endpoints, which explains that the phase structures of the RN black hole in a cavity and the RN-AdS black hole have extensive similarities IN-Lundgren:2006kt . However for a BI black hole in a cavity, there are some regions in the $\tilde{Q}$, $\tilde{a}$ and $\tilde{T}$ parameter space, where the global minimum of the free energy is at $x=0$. Different phase structure from that of a BI-AdS black hole appears there. Our results imply that, in some region of the parameter space of the BI-AdS black hole, there might be other states of lower free energy with the same charge and temperature. Inspired by Schwarzschild-AdS black holes, one natural candidate is the thermal AdS space filled with charged particles. However, the backreaction of the charged particles on the AdS geometry should be considered, which could lead to formation of a naked singularity. It is inspiring to explore the possible equilibrium phases of lower free energy for charged AdS black holes.

Acknowledgements.

We are grateful to Zheng Sun and Zhipeng Zhang for useful discussions and valuable comments. This work is supported in part by NSFC (Grant No. 11005016, 11175039 and 11375121).

Appendix A Reduced Action

In this appendix, we use the reduced action method proposed in IN-Braden:1990hw to evaluate the Euclidean action of a NLED black hole in a cavity. Instead of employing ansatz functions to solve the equations of motion of the NLED-gravity system, we here start with the usual form of the static spherically symmetric metric and consider the reduced action for this specific metric first. In fact, the Euclidean continuation of the static spherically symmetric metric takes the form

[TABLE]

where $b$ and $\alpha$ are free functions of $r$. We suppose that the Euclidean time is periodic with period $2\pi$. In a canonical ensemble, the temperature $T$ on the boundary of the cavity at $r=r_{B}$ is fixed, which impose the boundary condition in terms of the reciprocal temperature:

[TABLE]

At the event horizon at $r=r_{+}$, one has $b\left(r_{+}\right)=0$, and hence the $\tau-r$ part of the metric $\left(\ref{eq:metric}\right)$ looks like a disc. To avoid a conical singularity at $r=r_{+}$, we require that

[TABLE]

For the metric $\left(\ref{eq:metric}\right)$, the Euclidean continuation of the action $\left(\ref{eq:Action}\right)$ becomes

[TABLE]

Varying the above action with respect to $\alpha\left(r\right)$ and $b\left(r\right)$ gives that

[TABLE]

where $T_{\mu\nu}$, the energy-momentum tensor for the NLED field, is given by $\left(\ref{eq:stNLED}\right)$, and $E_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$ is the Einstein tensor. Note that $E_{\tau}^{\tau}$ and $E_{r}^{r}$ are

[TABLE]

For the NLED field, we take the static spherical symmetry and gauge symmetry into account and assume that

[TABLE]

For this ansatz, the equations of motion $\left(\ref{eq:REOM}\right)$ becomes

[TABLE]

where $G_{r\tau}=\mathcal{L}^{\left(1,0\right)}\left(-\frac{\alpha^{-2}b^{-2}}{2}A_{\tau}^{\prime 2},0\right)A_{\tau}^{\prime}$. Moreover, varying the reduced action $\left(\ref{eq:Raction}\right)$ with respect to $A_{\tau}$ leads to

[TABLE]

Substracting eqns. $\left(\ref{eq:Rtt}\right)$ from $\left(\ref{eq:Rrr}\right)$, one finds that

[TABLE]

where $C$ is a constant. Actually, $C$ can be determined by eqn. $\left(\ref{eq:ab}\right)$:

[TABLE]

We can rescale $\tau\rightarrow C\tau$, and hence we have $b\rightarrow C^{-1}b=\alpha^{-1}$. So after this rescaling, the metric $\left(\ref{eq:metric}\right)$ becomes

[TABLE]

where we define $f\left(r\right)=\alpha^{-2}\left(r\right)$, and the period of $\tau$ is $2\pi C$.

Since $A_{\tau}$ and $G^{r\tau}$ is rescaled by $A_{\tau}\rightarrow C^{-1}A_{\tau}$ and $G^{r\tau}\rightarrow CG^{r\tau}$, respectively, eqns. $\left(\ref{eq:Rtt}\right)$ and $\left(\ref{eq:RNLED}\right)$ becomes

[TABLE]

which are just the Euclidean version of eqns. $\left(\ref{eq:ttEOM}\right)$ and $\left(\ref{eq:NLEDEOM}\right)$. Therefore, the solutions to the above equations are

[TABLE]

where $Q$ is the charge of the black hole as discussed before. Plugging the solutions $\left(\ref{eq:ESolution}\right)$ into the the Euclidean action $\left(\ref{eq:Raction}\right)$, we find that

[TABLE]

where $S=16\pi^{2}r_{+}^{2}$ is the entropy of the black hole.