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Advances in High Energy Physics

Volume 2017 (2017), Article ID 3812063, 7 pages

https://doi.org/10.1155/2017/3812063

## The Thermodynamic Relationship between the RN-AdS Black Holes and the RN Black Hole in Canonical Ensemble

^{1}Department of Astronomy, Beijing Normal University, Beijing 100875, China^{2}Institute of Theoretical Physics, Shanxi Datong University, Datong 037009, China^{3}School of Physics, Shanxi Datong University, Datong 037009, China^{4}School of Physics, The University of Western Australia, Crawley, WA 6009, Australia

Correspondence should be addressed to Shuo Cao

Received 12 December 2016; Revised 21 March 2017; Accepted 9 April 2017; Published 3 May 2017

Academic Editor: Juan José Sanz-Cillero

Copyright © 2017 Yu-Bo Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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.

#### 1. Introduction

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 [1]. This solution has the same thermodynamic characteristics as the black hole solution in asymptotically flat space-time; that is, 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 -dimensional anti-de Sitter space-time was firstly discussed and extensively investigated in [2, 3], which discovered the first-order phase transition in the charged nonrotating RN-AdS black hole space-time. Recently, increasing attention has been paid to the possibility that the cosmological constant could be an independent thermodynamic parameter (pressure), and the first law of thermodynamics of AdS black hole may also be established with terms. For instance, the critical properties of AdS black hole were firstly studied in [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 (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 [5–23], 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 (1986) [24], achieving thermodynamic stability for asymptotically flat black hole system also depends on the effect of environments; that is, one needs to consider the ensemble system. Different from the general thermodynamic system, the self-gravitational 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 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 [25–31], the extreme value of the free energy of gravitational systems can be derived from the action I; that is, the partition function of the system at the zero-order approximation can be calculated by using the Gibbons-Hawking Euclidean action [32]Combining this with the Helmholtz free energy from the equation , we can obtainwhere , , and , respectively, denote the radius, temperature, and electric charge of the cavity, , and is the radius of the black hole horizon. and are the internal energy and entropy of the black hole in the cavity. Therefore, when , , and are determined, the only variable for the system is . The thermal equilibrium conditions of the black hole and environment can be determined by the following equation:The condition where 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 literatures [26–32] 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]; that is, 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 (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 nongravitational 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.

#### 2. 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 bywhereThe corresponding action is expressed as [21, 25]Here is the temperature of the cavity and . is the radius of concentricity outside a black hole horizon , which can be obtained from the following equation:The corresponding reduction quantities are defined asNote that the relation of always holds (given , ) and the reduced action takes the form asTherefore, we can obtainwhere the function of the reciprocal of temperature isThe condition for the RN black hole and cavity to reach thermal equilibrium isThus 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 (12) and (15), one can easily derive the critical charge and the critical radius of black hole as [25]The characteristic behavior curve of with respect to is shown in Figure 1.