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

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.


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 antide 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 [1], 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 [2][3][4], there has been much interest in studying the phase transitions of AdS black holes [5][6][7][8][9][10]. 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 [11], 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 [12] and a canonical ensemble [13,14]. 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 [15][16][17][18][19][20], 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 [21][22][23][24], 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 [25][26][27][28][29][30][31][32]. 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 [33] proposed that the TT deformed CFT 2 locates at the finite radial position of AdS 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 [34][35][36][37][38][39][40][41][42][43]. It is interesting to note that some NLED black holes can be regular black hole models [44,45]. As pointed out in [46], 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 [47]. 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 [48,49].
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 [12].

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.

A. Black Hole Solution
First, we will consider the black hole solution in a (3 + 1) dimensional model of gravity coupled to a nonlinear electromagnetic field A µ . On a spacetime manifold M with a time-like boundary ∂M, the action is given by where we take 16πG = 1 for simplicity, L (s, p) is a general NLED Lagrangian, and S surf are the surface terms on ∂M. Here, s and p are two independent nontrivial scalars built from the field strength tensor F µν = ∂ µ A ν − ∂ ν A µ and none of its derivatives: where ǫ µνρσ ≡ − [µ ν ρ σ] / √ −g is a totally antisymmetric Lorentz tensor, and [µ ν ρ σ] is the permutation symbol. For later use, we define where we denote L (1,0) (s, p) ≡ ∂L(s,p) ∂s and L (0,1) (s, p) ≡ ∂L(s,p) ∂p , respectively. Note that the general NLED theories with the Lagrangian L (s, p) were first considered in [66,67]. The surface terms of the action (1) are The first term above is the Gibbons-Hawking-York surface term, where K is the extrinsic curvature, γ 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 ∂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 µ is the unit outward-pointing normal vector of ∂M, is included to keep the charge fixed on ∂M, instead of the potential, when one varies the action to have the correct equations of motion [65]. Varying the action (1) in terms of g µν and A µ with the metric and the charge fixed on ∂M, we find that the equations of motion are where T µν is the energy-momentum tensor for the NLED field: We consider a static spherically symmetric black hole solution with the metric and the NLED field of the form Moreover, we assume that the black hole lives in a spherical cavity, which has a boundary ∂M at r = r B . The spacelike slices with constant t of ∂M are 2-spheres S 2 whose radii are r B . The equations of motion then reduce to 2f ′ (r) + rf ′′ (r) = rL (s, 0) , where It can show that eqn. (9) can be derived from eqns. (8) and (10).
Solving eqn. (10), we find that where q is a constant. The charge of the system inside the cavity is defined as [65].
where l µ is the unit normal vector of the constant t hypersurface, and σ is the induced metric on S 2 . Using eqn. (12), one finds that the charge inside the cavity becomes From eqns. (11) and (12), A ′ t (r) is determined by The gauge potential measured on ∂M with respect to the horizon is where the blueshift factor 1/ f (r B ) relates A t to the proper orthonormal frame component of the potential one-form A [12], and we fix the gauge field A t (r) at the horizon to be zero, i.e., A t (r + ) = 0.
By integrating eqn. (8), we have where M is the mass of the black hole [65]. Suppose that r + is the outer event horizon radius of the black hole. Since f (r + ) = 0, we can express f (r) in terms of r + :

B. Euclidean Action
In the semiclassical approximation, one can relate the on-shell Euclidean action to the thermal partition function: where S E is the Euclidean continuation of the action S: S E = iS. The Euclidean time τ is obtained from Lorentzian time t by the analytic continuation t = iτ . From A τ dτ = A t dt, it follows that which gives G rτ = −iG rt . So eqn. (12) becomes Moreover, the gauge potential on ∂M is 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: which identifies the Euclidean time τ as τ ∼ τ + , and hence the period of τ is . For the black hole solution (7), one can evaluate the Euclidean action by integrating over angles and performing the integration by parts: After eqn. (17) is plugged into eqn. (23), a straightforward calculation gives where S = 16π 2 r 2 + is the entropy of the black hole.
For large values of r B , one finds that In the limit of r B → ∞, the Euclidean action then reduces to as expected.

C. Thermodynamics
Various thermodynamic quantities can be derived from the Euclidean action (24), which is related to the free energy F in the semiclassical approximation by From eqns. (17) and (24), 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 + : 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 (r + ; T, Q, r B ) with respect to r + : . That said, the solution r + = r + (T, Q, r B ) of eqn. (29) corresponds to a locally stationary point of F (r + ; T, Q, r B ). It is interesting to note that eqn. (29) can be written as where is the Hawking temperature of the black hole. The temperature T on ∂M is thus blueshifted from T h , which is measured at infinity.
After obtained r + = r + (T, Q, r B ), we can evaluate F (r + ; T, Q, r B ) at the locally stationary point r + = r + (T, Q, r B ): For later convenience, we shall suppress T, Q and r B in F (r + ; T, Q, r B ) and F (T, Q, r B ) and denote F (r + ; T, Q, r B ) and F (T, Q, r B ) as F (r + ) and F , respectively. The thermal energy of the black hole in the cavity is Using eqn. (16), we can express the ADM mass of the black hole M in terms of E and Q: where the second and third terms on left-hand side can be interpreted as the gravitational and electrostatic binding energies, respectively. Using eqn. (17), 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 From the energy E, we can define a thermodynamic surface pressure by From eqns. (35) and (36), the first law of thermodynamics can be established: where A ≡ 4πr 2 B is the surface area of the cavity. To obtain the proper Smarr relation for the black hole, we need to consider the dimen- where we introduce the conjugates A i associated with a i : 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: When C Q > 0, the system is thermally stable. Thus, a thermally stable black hole phase has which means that the black hole phase is thermally stable To find the global minimum of F (r + ) 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 (r + ) at the edges of the space of r + .
In fact, the physical space of r + is constrained by 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 (r + ) 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 where the coupling parameter a is related to the string tension α ′ as a = (2πα ′ ) 2 > 0. For a = 0, the BI Lagrangian would reduce to the Maxwell Lagrangian. Solving eqn. (14) for where Q is the charge of the BI black hole. From eqn. (17), one can express f (r) in terms of the horizon radius r + : It is convenient to express quantities in units of r B : where r + is the horizon radius. We then use eqns. (24) and (45) to find the free energy as a function of x:F where The Hawking temperature of the BI black hole can be calculated from eqn. (31): The locally stationary points ofF (x) are determined by dF (x) /dx = 0, which becomes As shown in [37], there are two types of BI black holes depending on the minimum value ofT h : (45) becomes If Q = 0, f (r) = 1, and hence the edge state at x = 0 is just the thermal flat space.
For Q > 0, we have where R is the Ricci scalar. So the metric has a physical singularity at r = 0 although f (0) is finite. It can show that f (r) > 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 ofF (x) and find the global minimum value ofF (x). In fact, with fixed value ofã, the locally stationary points ofF (x) can be determined by solving eqns. (30) and (49)  Large BH branches are thermally stable. For the BI black hole withã = 0.01 and Q = 0.3 in this region, we plot the free energyF (x) in FIG. 3(b), which shows that the endpoints always have higher free energy than the global minimum.F (x) has the global minimum at Small BH for small enoughT and Large BH for large enoughT , 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.

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. (24), 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,  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 [63][64][65]. 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 [65]. 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 zerothorder 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 [11]. Although there is ambiguity about the reference background of charged black holes, one can circumvent this by using the counterterm subtraction method [68,69], in which the Euclidean action is regularized in a background-independent fashion by adding a series of boundary counterterms to the action.
In [63,65], 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 [14]. However for a BI black hole in a cavity, there are some regions in theQ,ã andT 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. r = r B is fixed, which impose the boundary condition in terms of the reciprocal temperature: b (r B ) dτ = 2πb (r B ) = T −1 . (A2) At the event horizon at r = r + , one has b (r + ) = 0, and hence the τ − r part of the metric (A1) looks like a disc. To avoid a conical singularity at r = r + , we require that For the metric (A1), the Euclidean continuation of the action (1) becomes Varying the above action with respect to α (r) and b (r) gives that where T µν , the energy-momentum tensor for the NLED field, is given by (6), and E µν = R µν − 1 2 Rg µν is the Einstein tensor. Note that E τ τ and E r r are E τ τ = 1 α 2 (r) r 2 − 1 r 2 − 2α ′ (r) α 3 (r) r and E r r = 1 α 2 (r) r 2 − 1 r 2 + 2b ′ (r) rb (r) α 2 (r) .
For the NLED field, we take the static spherical symmetry and gauge symmetry into account and assume that For this ansatz, the equations of motion (A5) becomes where G rτ = L (1,0) − α −2 b −2 2 A ′2 τ , 0 A ′ τ . Moreover, varying the reduced action (A4) with respect to A τ leads to ∂ r r 2 G rτ = 0.
Since A τ and G rτ is rescaled by A τ → C −1 A τ and G rτ → CG rτ , respectively, eqns. (A8) and (A10) becomes ∂ r r 2 G rτ = 0, which are just the Euclidean version of eqns. (8) and (10). Therefore, the solutions to the above equations are f (r) = 1 − r + r + 1 2r where Q is the charge of the black hole as discussed before. Plugging the solutions (A16) into the the Euclidean action (A4), we find that where S = 16π 2 r 2 + is the entropy of the black hole.