Thermodynamic Geometry of AdS Black Holes and Black Holes in a Cavity

Abstract The thermodynamic geometry has been proved to be quite useful in understanding the microscopic structure of black holes. We investigate the phase structure, thermodynamic geometry and critical behavior of a Reissner-Nordstrom-AdS black hole and a Reissner-Nordstrom black hole in a cavity, which can reach equilibrium in a canonical ensemble. Although the phase structure and critical behavior of both cases show striking resemblance, we find that there exist significant differences between the thermodynamic geometry of these two cases. Our results imply that there may be a connection between the black hole microstates and its boundary condition.

IV. Discussion and Conclusion 20

I. INTRODUCTION
The study of black hole thermodynamics has been playing an increasingly prominent role in our understanding of the interdisciplinary area of general relativity, quantum mechanics, information theory and statistical physics. In the pioneering work [1-3], black holes were found to possess thermodynamic properties such as entropy and temperature. Later, the Hawking-Page phase transition (i.e., a phase transition between the thermal anti-de Sitter (AdS) space and a black hole) was discovered in Schwarzschild-AdS black holes [4]. Unlike Schwarzschild black holes, Schwarzschild-AdS black holes can be thermally stable since the AdS boundary acts as a reflecting wall for the Hawking radiation. With the advent of the AdS/CFT correspondence [5][6][7], there has been much interest in studying the thermodynamics and phase structure of various AdS black holes [8][9][10][11][12][13][14][15]. Specifically, it was found that Reissner-Nordstrom-AdS (RN-AdS) black holes exhibit a van der Waals-like phase transition (i.e., a phase transition consisting of a first-order phase transition terminating at a second-order critical point) in a canonical ensemble [9,10] and a Hawking-Page-like phase transition in a grand canonical ensemble [16].
Alternatively, one can make asymptotically flat black holes thermally stable by placing them inside a cavity, on the wall of which the metric is fixed. York first showed that Schwarzschild black holes in cavity can be thermally stable and have quite similar phase structure and transition to these of Schwarzschild-AdS black holes [17]. For Reissner-Nordstrom (RN) black holes in a cavity, the thermodynamics and phase structure have been studied in a grand canonical ensemble [18] and a canonical ensemble [19,20]. It also showed that the phase structure of RN black holes in a cavity and RN-AdS black holes has extensive similarities. The phase structure of various black brane systems in a cavity was investigated in [21][22][23][24][25][26], and most of the systems were found to undergo Hawking-Page-like or van der Waals-like phase transitions. Boson stars and hairy black holes in a cavity were considered in [27][28][29][30], 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. Moreover, the thermodynamic and critical behavior of de Sitter black holes in a cavity were investigated in the extended phase space [31]. Recently, we studied Born-Infeld black holes enclosed in a cavity in a canonical ensemble [32] and a grand canonical ensemble [33], respectively, and found that their phase structure has dissimilarities from that of Born-Infeld-AdS black holes.
However, although it is believed that a black hole does possess thermodynamic quantities and extremely interesting phase structure, the statistical description of the black hole microstates has not yet been fully understood. Even though a complete quantum gravity theory is still absent, there have been some attempts to understand microscopic structure of a black hole [34][35][36]. Specifically, the thermodynamic geometry method has led to many insights into microstructure of a black hole. Following the pioneering work by Weinhold [37], Ruppeiner [38] introduced a Riemannian thermodynamic entropy metric to describe the thermodynamic fluctuation theory and found a systematic way to calculate the Ricci curvature scalar R of the Ruppeiner metric. Later, R has been computed for various ordinary thermodynamic systems, such as ideal quantum gases [39], Ising models [40] and anyon gas [41]. It showed that there is a relation between the type of the interparticle interaction and the sign of R: R > 0 implies a repulsive interaction (e.g. ideal Bose gas) while R < 0 means an attractive interaction (e.g. ideal Fermi gas), and R = 0 corresponds to no interaction (e.g. ideal gas). It has also been indicated that, at a critical point, |R| diverges as the correlation volume for ordinary thermodynamic systems [38].
The Ruppeiner geometry was subsequently exploited to probe the microstructure of a black hole. Since the work of [42], thermodynamic geometry has been studied for various black holes [43][44][45][46][47][48][49][50][51][52]. The result of [53] showed that a RN black hole has a vanished R, sug-gesting it is a non-interacting system. On the other hand, a Kerr-Newmann-AdS black can reduce to a RN black hole by making certain thermodynamic variables approach zero. In these limits, a RN black hole was found to acquire a nontrivial R [54], implying that the phase space adopted in [53] may be incomplete. Recently, by choosing different thermodynamic coordinates, the authors of [55] showed that a RN black hole is an interaction system dominated by repulsive interaction. For a RN-AdS black hole, R has been calculated, and it was observed that R can be both positive and negative, and resembles the critical behavior of ordinary thermodynamic systems near a critical point [45,51,53,56,57]. The thermodynamic geometry has been investigated recently in the extended state space [58][59][60][61][62][63][64][65][66][67][68][69], in which the cosmological constant is treated as a thermodynamic variable and acts like a pressure term [70][71][72]. Inspired by the Ruppeiner geometry, a RN-AdS black hole was proposed to be built of some unknown micromolecules, interactions among which can be tested by R [60].
Recently, a new scalar curvature R was introduced for a RN-AdS black hole, and it showed that there is a large difference between the microstructure of a black hole and the Van der Waals fluid [67].
Although there have been a lot of work in progress on thermodynamic geometry for various black holes of different theories of gravity in spacetimes with differing asymptotics, little is known about thermodynamic geometry for a black hole enclosed in a cavity. Unlike RN black holes, both RN-AdS black holes and RN black holes in a cavity can be thermally stable and hence provide an appropriate scenario to explore whether or not the thermodynamic geometry is sensitive to the boundary condition of black holes. In addition, it was recently proposed that the holographic dual of TT deformed CFT 2 is a finite region of AdS 3 with the wall at finite radial distance [73,74], which further motivates us to investigate the properties of a black hole in a cavity. To this end, we undertake a study of the thermodynamic geometry for a RN black hole in a cavity. We report that, although the phase structure of a RN-AdS black hole and a RN black hole in a cavity is analogous to the van der Waals fluid, there are significant differences between the thermodynamic geometry of a RN-AdS black hole and that of a RN black hole in a cavity.
The rest of this paper is organized as follows. In section II, we first discuss the phase structure and thermodynamic geometry of a RN-AdS black hole in a canonical ensemble.
Although the thermodynamic geometry in the thermodynamic coordinates of the charge Q and potential Φ was investigated in [45,51], we carry out the analysis in a more through way with a broader survey of the parameter space and find the R > 0 region in the phase diagrams. The phase structure and thermodynamic geometry of a RN black hole in a cavity are then studied in details, starting with a discussion of its phase structure. In section III, the critical behavior of the RN-AdS black hole and the RN black hole in a cavity is obtained. We summarize our results with a brief discussion in section IV. For simplicity, we set G = = c = k B = 1 in this paper.

II. PHASE STRUCTURE AND THERMODYNAMIC GEOMETRY
In this section, we study phase structure and thermodynamic geometry of RN-AdS black holes and RN black holes in a cavity in a canonical ensemble. That said, the temperature and charge of the system are fixed. Thermodynamic geometry (Ruppeiner geometry) may provide a way to probe the microscopic structure of black holes. Adopting the Ruppeiner approach [38], one can define the Ruppeiner metric g R µν for a thermodynamic system of independent variables x µ as where S is the entropy of the system. The Ruppeiner metric can be used to measure the distance between two neighboring fluctuation states [38]. More interestingly, the Ricci scalar of the Ruppeiner metric or the Ruppeiner invariant R can shed light on some information about the microscopic behavior of the system, such as the strength and type of the dominated interaction between particles in the system.

A. RN-AdS Black Holes
The 4-dimensional static charged RN-AdS black hole solution is described by where the metric function f (r) is and l is the AdS radius. The parameters M and Q can be interpreted as the black hole mass and charge, respectively. The Hawking temperature T is given by where r + is the radius of the outer event horizon. Since f (r + ) = 0, the mass M can be expressed in terms of r + : It can show that the RN-AdS black satisfies the first law of thermodynamics where S = πr 2 + and Φ = Q/r + is the entropy and potential of the black hole, respectively. To study the phase structure of the black hole in a canonical ensemble, we need to consider the free energy. The free energy F can be obtained by computing the Euclidean action in the semiclassical approximation and is given by It was observed that the charge Q and potential Φ of a RN-AdS black hole play similar roles as the pressure P and volume V of the van der Waals-Maxwell fluid in terms of determining the phase structure [45,51]. The correspondence (Φ, Q) → (V, P ) can establish the phase structure of the RN-AdS black hole. In [45], it was also suggested that the appropriate internal energy U of the RN-AdS black hole is given by where the contribution of the static electricity to the black hole mass M is excluded. By analogy with the van der Waals fluid, we consider the parameter space coordinates x µ = (U, Φ). Therefore, the Ruppeiner metric becomes Using eqns. (4) and (5), we find that the expression of the Ruppeiner invariant R in terms of the horizon radius r + and the charge Q is has negative temperature and hence is discarded. On the red lines, one has R = −∞. The red line separating the "No BH" region and the "R < 0" region is determined by T = 0, which shows that R = −∞ for extremal RN-AdS black holes. The red line in the green region is given by C −1 Q = 0, where C Q is the heat capacity at constant Q: The divergence of C Q usually means that the black hole would undergo a phase transition.
For simplicity, we hereafter set l = 1.
To study the Ruppeiner invariant R as a function of the temperature T and the charge Q, we need to use eqn. (4) to express the horizon radius r + in terms of the temperature T : point is an inflection point and obtained by ∂T ∂r + = 0 and ∂ 2 T ∂r 2 which gives the corresponding quantities evaluated at the critical point However for Large BH, R can be negative or positive depending on the value of T . The inset shows that R > 0 at a high enough temperature, and hence the interactions between the BH molecules become repulsive. It shows that R of Large BH is negative infinity at Considering the globally stable phase, one has R = −∞ only for the extremal black hole since Small BH at T = T 2 or Large BH at T = T 1 is not globally stable. There is a crossing of R of Large BH and Small BH between T 1 and T 2 . Such R-crossing was proposed to indicate a first-order phase transition due to the equality of the correlation lengths for the phases at the phase transition [75]. It is interesting to note that

B. RN Black Holes in a Cavity
The 4-dimensional RN black hole solution is where Q b is the black hole charge, and r + is the radius of the outer event horizon. The Hawking temperature T b of the RN black hole is given by We now consider a thermodynamic system with a RN black holes enclosed in a cavity.
Suppose that the wall of the cavity enclosing the RN black hole is at r = r B , and the wall is maintained at a temperature of T and a charge of Q. For this system, the free energy F and the thermal energy E were given in [19] It also showed in [18] that the system temperature T and charge Q can be related to the black hole temperature T b and charge Q b as which means that T, measured at r = r B , is blueshifted from T b , measured at r = ∞. The potential Φ measured on the wall is [ As in the RN-AdS black hole, we define the internal energy U by excluding the contribution of the static electricity from the thermal energy E The physical space of r + is bounded by where r e = Q is the horizon radius of the extremal black hole.
In the parameter space coordinates x µ = (U, Φ), the Ruppeiner metric is and the Ruppeiner invariant R as a function of the horizon radius r + and the charge Q is We plot R against r + and Q in the right panel of FIG. 1, where R > 0 and R < 0 in the yellow and green regions, respectively, and the solution in the gray region does not satisfy the constraint (20). The red line, on which R = −∞, is also determined by C −1 Q = 0, where C Q is the heat capacity at constant Q: Unlike the RN-AdS black hole, eqn. (22) gives R = 0 for the extremal RN black hole with Q = r + . FIG. 1 shows that the behavior of R of RN-AdS black holes is quite different from that of RN black holes in a cavity. For simplicity, we hereafter set r B = 1.
It was observed that the phase structure of a RN black hole in a cavity is strikingly similar to that of a RN-AdS black hole in the Q-T space of a canonical ensemble [19,20]. Since R of Large BH above and Small BH below the transition line is negative, the type of the interaction among the microstructure of the black hole stays the same when the system undergoes the first-order transition.
We now investigate the phase structure of a RN black hole in a cavity in the Φ-T space, which has not been discussed in the literature. Similar to a RN-AdS black hole, there is phase coexistence of Large BH and Small BH when T c < T < T 0 ≈ 0.269. We plot

III. CRITICAL BEHAVIOR
In this section, we investigate the thermodynamic behavior near the critical point. The critical behavior in a RN-AdS black hole has been discussed in [45,51], which showed that the critical exponents for a RN-AdS black hole and the Van der Waals fluid are identical. However, the critical exponents for a RN black hole in cavity have not been calculated yet.
First, we define In the neighborhood of the critical point, we can expand Q in terms of t and φ Q = i,j=0 where q 00 = Q c , and eqn. (12) gives q 01 = q 02 = 0. Specifically for a RN-AdS black hole and a RN black hole in a cavity, we find respectively. Near the critical point, the Maxwell equal area rule is applicable and gives where Φ s and Φ l denote the potential of the saturated Small BH and Large BH, respectively.
We find that, at the critical point, which gives α = 0 in both cases.
Eqn. (30) gives 4.952 √ t for a RN BH in a cavity, which leads to β = 1/2 in both cases.
• Exponent γ describes the critical behavior of the isothermal compressibility: Approaching the critical point along the coexistence curve, we use eqn. (27) to obtain which gives γ = 1 in both cases.
which gives δ = 3 in both cases.
Our results show that the critical exponents of a RN black hole in cavity are also identical to these of the Van der Waals fluid predicted by the mean field theory. The critical exponents are believed to be universal since they are insensitive to the details of the physical system.
where +/− is for the saturated Large BH/Small BH, and r +l/s is the horizon radius of the saturated Large BH/Small BH. In addition, the charge Q for the coexisting line is given by Q ≃ Q c + q 03 φ 3 l/s + q 10 t + q 11 tφ l/s ≃ Q c + q 10 t.
Plugging eqns. (35) and (36) into eqns. (10) and (22), we can expand R at t = 0 and obtain where the leading order terms of the expansions are the same for the saturated Small BH and Large BH. From eqns. (31) and (37) which shows that the critical value of RC Φ t 2 may depend on the boundary conditions of the black hole. In [67], R of an RN-AdS black hole was calculated in the thermodynamic coordinates x µ = (T, V ), and it was found lim t→0 RC V t 2 = −1/8, which agrees with the numerical result of the Van der Waals fluid. However, there is a sign difference between our AdS result in eqn. (38) and the result in [67], which comes from C Φ < 0 at the critical point for a RN-AdS black hole.

IV. DISCUSSION AND CONCLUSION
In this paper, we studied the phase structure, thermodynamic geometry and critical  However, we found that the thermodynamic geometry in the AdS and cavity cases is quite different. The Ruppeiner invariant R as a function of the horizon radius r + and the charge Q was obtained in the AdS and cavity cases and plotted in FIG. 1, which showed that the R > 0 regions (yellow regions) are dissimilar in the two cases. Moreover, the extremal RN-AdS black hole has R = −∞ while R = 0 for the extremal RN black hole in a cavity.
We also discussed R as a function of T and Q and summarize the results for the AdS and cavity cases in Table I. Moreover, we found that the interactions among the microstructure of the black hole always stay attractive before and after the LBH/SBH first-order transition in the cavity case. However for a RN-AdS black hole with Q 0.097, the type of the interactions changes when the black hole undergoes the phase transition. The Ruppeiner invariant R as a function of Φ and T was also discussed, and it showed that R → −∞ as Φ → 1 with fixed T . FIGs. 5  In summary, although the phase structure of a RN-AdS black hole and a RN black hole in a cavity is strikingly similar, we found that the thermodynamic geometry in the two cases behaves rather differently. It seems that the Ruppeiner invariant R depends not only on what is inside the horizon, but also on the imposed boundary condition. Our results show that either R encodes more than the nature of black hole microscopic properties, or there may be a connection between the black hole microstates and the boundary condition. It would be interesting to study how R depends on the boundary condition in other black hole systems, which may shine further light on the connection between the internal microstructure of black holes and the boundary condition.