The thermodynamic relationship between the RN-AdS black holes and the RN black hole in canonical ensemble

In this paper, by analyzing the thermodynamic properties of charged AdS black hole and asymptotically flat space-time charged black hole in the vicinity of the critical point, we establish the correspondence between the thermodynamic parameters of asymptotically flat space-time and nonasymptotically flat space-time, based on the equality of black hole horizon area in the two different space-time. The relationship between the cavity radius (which is introduced in the study of asymptotically flat space-time charged black holes) and the cosmological constant (which is introduced in the study of nonasymptotically flat space-time) is determined. The establishment of the correspondence between the thermodynamics parameters in two different space-time is beneficial to the mutual promotion of different time-space black hole research, which is helpful to understand the thermodynamics and quantumproperties of black hole in space-time.


Introduction
The AdS black hole solution in four-dimensional spacetime is an accurate black hole solution of the Einstein equation with negative cosmological constant in asymptotic AdS space time [1]. This solution has the same thermodynamic characteristics as the black hole solution in asymptotically flat space-time, i.e., the black hole entropy is equal to a quarter of the event horizon area, while the corresponding thermodynamics quantity satisfies the law of thermodynamics of black hole. It is well known that, if taken as a thermodynamic system, the asymptotically flat black hole does not meet the requirements of thermodynamic stability due to its negative heat capacity. However, compared with the asymptotically flat space-time black hole, the AdS black hole can be in thermodynamic equilibrium and stable state, because the heat capacity of the system is positive when the system parameters take certain values.
Therefore, the thermodynamics of AdS charged black holes, in particular its phase transition in (n+1)-dimensional anti-de Sitter space-time was firstly discussed and extensively investigated in Refs. [2,3], which discovered the first order phase transition in the charged non-rotating RN-AdS black hole space-time. Recently, increasing attention has been paid to the possibility that the cosmological a e-mail: caoshuo@bnu.edu.cn constant Λ could be an independent thermodynamic parameter (pressure), and the first law of thermodynamics of AdS black hole may also be established with P − V terms. For instance, the P − V critical properties of AdS black hole was firstly studied in Ref [4], which found that the phase transition and critical behavior of RN-AdS black hole are similar to those of the general van der waals-Maxwell system. More specifically, the RN-AdS black hole exhibits the same P − V (liquid-gas phase transition) critical phase transition behavior and critical exponent as van der waals-Maxwell system. The phase transition and critical behavior of various black holes in the extended phase space of AdS have also been extensively studied in the literature [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], which showed very similar phase diagrams in different black hole systems.
The asymptotically flat black holes cannot reach thermodynamic stability, due to the inevitable so-called Hawking radiation. In order to obtain a better understanding of the thermodynamic properties and phase transition of black holes, we must ensure that the black hole can achieve stability in the sense of thermodynamics. According to the previous results obtained by York et al.(1986) [24], achieving thermodynamic stability for asymptotically flat black hole system also depends on the effect of environments, i.e., one needs to consider the ensemble system. Different from the general thermodynamic system, the selfgravitational system has inhomogeneity in space, which makes it necessary to determine the corresponding thermodynamic quantities and their characteristic values. The local thermodynamic stability of self-gravitational systems can be analyzed by considering the extreme value of the Helmholz free energy of the system. When it reaches to a minimum value, the corresponding system is at least locally stable. According to the methods extensively studied in the literature [25][26][27][28][29][30][31], the extreme value of the free energy of gravitational systems can be derived from the action I, i.e., the partition function of the system at the zero-order approximation can be calculated by using the Gibbons-Hawking Euclidean action [32] Z ≈ e −IE (1) Combining this with the Helmbolz free energy F from the equation Z = e −βF , we can obtain where r, T , Q respectively denote the radius, temperature and electric charge of the cavity, β = 1/T , and r + is the radius of the black hole horizon. E(r, Q; r + ) and S(r + ) are the internal energy and entropy of the black hole in the cavity. Therefore, when the r, T, Q is determined, the only variable for the system is r + . The thermalequilibrium conditions of the black hole and environment can be determined by the following equation The conditions that the free energy reaches to its minimum value is that the system is at local equilibrium state. In order to reach to the thermal equilibrium, it should satisfy the following criteria Utilizing the method described above, the literature [26][27][28][29][30][31][32] have studied the charged black hole and black branes, obtained the requirements to meet the thermodynamic equilibrium conditions, and discussed the phase transition and critical phenomena. More recently, Eune et al. (2015) investigated the phase transition based on the corrections of Schwarzschild black hole radiation temperature [33]. More specifically, both of the charged black hole and the radiation field outside the black hole were considered in their work, under the condition that they are both in the equilibrium state.
On the other hand, Reissner-Nordstrom (RN) black hole and RN-AdS black hole are the exact solutions of the Einstein equation. The main difference between AdS space-time and asymptotically flat space-time is the famous Hawking-Page phase transition [1], i.e., the AdS background provides a natural constraint box, which makes it possible to form a thermal equilibrium between large stable black hole and hot gas. In the recent study of the phase transition of the RN black hole, in order to meet the requirement of thermodynamic stability, one needs to artificially add a cavity concentric in the horizon of the black hole. However, the determination of the specific value of the radius of the cavity is still to be done. In the previous studies of phase transient of RN and RN-AdS black holes, it was found that both of the two types of black holes exhibit the same P − V (liquid-gas phase transition) properties as van der Waals-Maxwell system, which is also consistent with our finding through the comparison between the phase transition curves of these two kinds of black holes. Therefore, the consistency of the thermodynamic stable phase and phase transition between the cavity asymptotically flat black hole and the black hole in the AdS space seems to indicate a more profound connotation [25,26]. One of the roles played by the cavity and the AdS space is to ensure the conservation of the degree of freedom within a certain system. Naturally, the discussion of the following problems is the main motivation of our analysis: Does a duality of a gravitational theory and a non-gravitational theory exist in the cavity setting? Are these two types of black holes inherently connected? If so, can the cavity radius introduced for the RN black hole be determined by the thermodynamic properties of the RN-AdS black hole?
In this paper, by comparing the phase transition curves of the RN black hole with the RN-AdS black hole, we establish the equivalent thermodynamic relations of the two kinds of black holes. We also discuss the relationship between the radius of cavity introduced into the RN black hole, the black hole horizon radius, and the cosmological constant. Finally, we investigate the equivalent thermodynamic quantities of two kinds of black holes, which provides theoretical basis for the further exploration of their internal relation.

Thermodynamic properties of flat space-time charged black hole
To begin with, we review the thermodynamic properties of RN black holes. The metric of a charged RN black hole is given by The corresponding action expresses as [21,25] Here T B is the temperature of the cavity and β B = 1/T B . r B is the radius of concentricity outside a black hole hori-zon r + , which can be obtained from the following equation The corresponding reduction quantities are defined as Note that the relation of q < x < 1 always holds (given r + >Q , r B > r + ) and the reduced action takes the form asĪ Therefore, we can obtain where the function of the reciprocal of temperature is The condition for the RN Black hole and cavity to reach thermal equilibrium is Thus the reciprocal of the black hole radiation temperature can be written as The critical charge and critical radius of a black hole can be determined by the following conditions Combing Eqs. (12) and (15), one can easily derive the critical charge and the critical radius of black hole as [25] The characteristic behavior curve of b q (x) with respect to x is shown in Fig. 1.

P − V criticality of charged AdS black holes
To start with, we review some basic thermodynamic properties of the spherical RN-AdS black holes. In the framework of Schwarzschild-like coordinates, the metric and the U (1) field read where dΩ 2 2 stands for the standard element on S 2 and the function f is given by where l is the AdS length scale and is related to the cosmological constant as Λ = −3/l 2 . The radius of the black hole horizon, r + , satisfies the following equation and the black hole radiation temperature reads In order to explicitly illustrate the critical phenomenon of RN black hole, we present the β − r + diagram in Fig. 2, from which the phase-transition point of system could be obtained as These results are well consistent with those obtained in the previous analysis [4]. Moreover, the state parameters of a certain system should satisfy the first law of thermodynamics where the potential Φ, thermodynamic volume V and pressure P of the black hole respectively express as 4 Thermodynamic relationship of black holes in two different space-time As can be seen from Fig. 1 and 2, for the two types of black holes, the curves of the reciprocal of temperature with respect to the radius of black hole horizon are quite similar to that for the Van der Waals-Maxwell gasliquid phase transition (the corresponding critical exponents can be calculated at the critical point). Moreover, similar to the cases in the AdS space and dS space, the specific heat capacity of the black hole in the asymptotically flat space can be expressed as where T c denotes the critical temperature. These common thermodynamic properties hint us the possibility to make the thermodynamic properties of the black hole in these space-time well consistent with each other, only by adjusting some specific parameters. Therefore, a cavity radius, which may ensure the thermodynamic equilibrium stability of a system, is introduced in our discussion of the thermodynamic properties of charged black holes in the flat space-time. Meanwhile, there is no definite value for the radius in our analysis, which provides us the possibility to adjust this parameter to obtain similar thermodynamic properties from the black holes in two different space-time, and finally derive the relationship between the radius of the cavity and the radius of the black hole r + or the cosmological constant l.
It is generally believed that, if the corresponding state parameters in two thermodynamic system behave the same as the selected independent variables, these two systems will have the same thermodynamic properties. In order to obtain similar thermodynamic properties of the black holes in two different space-time, we ensure that with the change of horizon radius of black hole, the temperature and entropy of the two systems are kept to be equal. That is, the radii of the two black holes are equal to each other in two different space-time, which leads to the similar entropy in the two systems. In the vicinity of the critical point, the temperature of a black hole in the space-time varies synchronously with the radius of the horizon. Based on the assumption that the entropy and temperature of a black hole in two different space-time change the same with the horizon radius, we can obtain the relationship between the radius of the cavity introduced in the flat space-time and the radius of the black hole horizon and the cosmological constant.
Supposing the radii of the black hole horizon in the two different space-time are both r + , which can be respectively determined by Eq. (8) and Eq. (20) for RN black holes and RN-AdS black holes. The common entropy of these two systems expresses as As can be seen from the analysis in Section. 3, both of the two types of black holes show the same characteristics as Van der Waals-Maxwell system (fluid-gas phase transition), the critical points of which are respectively given by Eqs. (16) and (22). It is apparent that, Eq. (25) implies the critical point of RN black holes and RN-AdS black holes are determined when the black hole radius is the same in two different space-time, while Eq. (22) shows the critical value of the RN-AdS black hole is dependent on AdS length scale l. Therefore, when the horizon position r + and charge Q are given for such black hole, the radiation temperature depends on the cosmological constant. On the other hand, it can be seen from Eq. (13) that, when q and x are fixed, the temperature of RN black holes depends on the cavity radius r B .
To start with, when the black hole horizon r + is the same in the two different space-time, we should determine the requirement of coincidence of critical curves, which is also the requirement to make T 1 = T 2 . With the definition of y = r + /l, the combination of Eq. (13) and (21) gives to obtain equal temperature in two time-space, where Q c = l 6 , q c = √ 5 − 2. The quantitative relation between x and y is plotted in Fig. 3. Because the black hole horizon position r + is the same for both types of black holes, the x − y curve also quantifies the relation between l and r B . We can get y = y c = 0.377039 when x = x c = 5 − 2 √ 5, with which the critical curves β(x) − x of RN black hole and β(y) − x of RN-AdS black hole coincide the critical curve of Black Hole.
Secondly, when T 1 = T 2 , at the critical point y = y c , x = x c , supposing there is a deviation of the charge to the threshold, we can get ∆Q = l∆Q and ∆q from the following expression It is straightforward to obtain Substituting y c = 0.377039, x c = 5 − 2 √ 5 into the above equation, we can get the relationship between ∆Q and ∆q and the Eq. (27) can be rewritten as When ∆Q is given, one can get the value of ∆q from Eq. (28). Denoting q = q c + ∆q = √ 5 − 2 + ∆q in Eq. (29), we can obtain the y − x curve deviating from the critical charge. The results are plotted in Fig. 4, in which the blue, red and black lines respectively correspond to the three cases with the critical charge, above the critical charge and under critical charge. It is obvious that when the relation between y and x satisfies Eqs. (26) and (29), the critical curves β(x) − x of RN black holes and β(y) − y of RN-AdS black holes coincide, which implies the equal temperature of the two different space-time. Therefore, similar to the definition of entropy in Eq. (25), the heat capacity of the two systems can also be written as We remark here that in the discussion above, the cavity radius is introduced as a state parameter to test the thermodynamic properties near the critical point of the RN black hole in the asymptotically flat space-time. This procedure follows the study of the thermodynamic properties near the critical point of the RN-AdS black hole, where the cosmological constant is taken as the state parameter in the thermodynamic system. Our results demonstrates that the change of the two systems is the same near the critical point, as long as the variables y and x introduced in the two system satisfy Eqs. (26) and (29). As is shown in Figs. 1-2, the radius of the cavity is a one-to-one correspondence with the cosmological constant, and thus could be the dual of the cosmological constant in non-asymptotic flat space-time.

Conclusion
It is well known that black hole is an ideal system to understand the nature and behavior of quantum gravity. On one hand, black hole provides an ideal model to study all kinds of interesting behaviors of classical gravitation (under the sense of general relativity); on the other hand, it can be regarded as a macroscopic quantum system with unique thermodynamic properties (the entropy, temperature and holographic properties of gravitation are quantum), which provides an important window to probe the quantum gravity. More importantly, a better understanding of the black hole singularity, cosmological singularity, and cosmological inflation needs a basic theory of space-time gravitation. Up to now there is still no mature theory to precisely describe the quantum characteristics of the asymptotic flat black holes and non-asymptotically flat space-time black holes. However, string theory firstly provided the microcosmic explanations for the entropy of some AdS black holes, which predicted the existence of Mtheory and some duality relations (especially AdS / CFT correspondence) and realized the holographic properties of gravitational systems [35].
In this paper, by setting the black-hole horizon at the same value and assuming the temperature near the critical point is equal in two space-time, we have discussed the relationship between the radius of the cavity r B and the cosmological constant l. The former parameter is always introduced in studying the thermodynamics of the charged black hole in the flat space-time, while the latter is related to the non-flat space-time. Our results may provide a theoretical basis for exploring the internal relations of thermodynamics in the black space. Moreover, by establishing the correspondence between the thermodynamic parameters of black holes in the asymptotically flat space-time and non-asymptotically flat space-time, we hope to seek the correspondence between the quantum properties of the black hole in two different space-time, which is beneficial to understand the thermodynamics and quantum properties of black holes in different space-time.