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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 types of 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 types of space-time is beneficial to the mutual promotion of different time-space black hole research, which is helpful to understand the thermodynamics and quantum properties of black hole in space-time.

The AdS black hole solution in four-dimensional space-time is an accurate black hole solution of the Einstein equation with negative cosmological constant in asymptotic AdS space-time [

Therefore, the thermodynamics of AdS charged black holes, in particular its phase transition, in

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 (1986) [

The local thermodynamic stability of self-gravitational systems can be analyzed by considering the extreme value of the Helmholtz 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 [

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 [

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.

To begin with, we review the thermodynamic properties of RN black holes. The metric of a charged RN black hole is given by

The critical charge and critical radius of a black hole can be determined by the following conditions:

The characteristic behavior of

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

Moreover, the state parameters of a certain system should satisfy the first law of thermodynamics

As can be seen from Figures

It is generally believed that if the corresponding state parameters in two thermodynamic systems 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 types of 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 types of 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 types of 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 types of space-time are both

To start with, when the black hole horizon

The

Secondly, when

The

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 demonstrate that the change of the two systems is the same near the critical point, as long as the variables

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 nonasymptotically 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 M-theory and some duality relations (especially AdS/CFT correspondence) and realized the holographic properties of gravitational systems [

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 different types of space-time, we have discussed the relationship between the radius of the cavity

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant nos. 11605107 and 11503001), in part by the National Natural Science Foundation of China (Grant no. 11475108), by Program for the Innovative Talents of Higher Learning Institutions of Shanxi, the Natural Science Foundation of Shanxi Province, China (Grant no. 201601D102004) and the Natural Science Foundation for Young Scientists of Shanxi Province, China (Grant no. 201601D021022), and by the Natural Science Foundation of Datong city (Grant no. 20150110).