Advances in High Energy Physics

Volume 2018, Article ID 4350287, 10 pages

https://doi.org/10.1155/2018/4350287

## Phase Transition of RN-AdS Black Hole with Fixed Electric Charge and Topological Charge

Department of Physics, Lingnan Normal University, Zhanjiang, 524048, Guangdong, China

Correspondence should be addressed to Shan-Quan Lan; moc.621@nalnauqnahs

Received 5 September 2018; Revised 28 October 2018; Accepted 25 November 2018; Published 11 December 2018

Academic Editor: Elias C. Vagenas

Copyright © 2018 Shan-Quan Lan. 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

Phase transition of RN-AdS black hole is investigated from a new perspective. Not only is the cosmological constant treated as pressure but also the spatial curvature of black hole is treated as topological charge . We obtain the extended thermodynamic first law from which the mass is naturally viewed as enthalpy rather than internal energy. In canonical ensemble with fixed topological charge and electric charge , interesting van der Waals like oscillatory behavior in and graphs and swallow tail behavior in and graphs is observed. By applying the Maxwell equal area law and analysing the Gibbs free energy, we obtain analytical phase transition coexistence curves which are consistent with each other. The phase diagram is four dimensional with .

#### 1. Introduction

Black hole is a simple object which can be described by only a few physical quantities, such as mass, charge, and angular momentum, while it is also a complicate thermodynamic system. Since the discovery of black hole’s entropy [1], the four-thermodynamic law [2], and the Hawking radiation [3] in 1970s, thermodynamic of black hole has become an interesting and challenging topic. Especially, in the anti-de Sitter (AdS) space, there exists Hawking-Page phase transition between stable large black hole and thermal gas [4]. Due to the AdS/CFT correspondence [5–7], the Hawking-Page phase transition is explained as the confinement/deconfinement phase transition of a gauge field [8].

When the AdS black hole is electrically charged, its thermodynamic properties become more rich. In the canonical ensemble with fixed electric charge, there is a first-order phase transition between small and large black holes [9–12]. Increasing the temperature, the phase transition coexistence curve ends at the critical point, where the first-order phase transition becomes a second-order one. In the grand canonical ensemble with fixed temperature, there is also a critical temperature. Below the critical temperature, is a single-valued function, where is electric charge and is the conjugate potential. Above the critical temperature, is a multivalued function with phase transitions [12]. The phase transition behavior of AdS black hole is reminiscent to the liquid-gas phase transition in a van der Waals system.

Viewing the cosmological constant as a dynamical pressure and the black hole volume as its conjugate quantity [13], the analogy of charged AdS black hole as a van der Waals system has been further enhanced in Ref. [14]. Both the systems share the same oscillatory behavior in pressure-volume graph and swallow tail behavior in Gibbs free energy- temperature (pressure) graph. What’s more, they have very similar phase diagrams and have exactly the same critical exponents. The phase transition property is also investigated in temperature-entropy graph [15]. Later, this analogy has been generalized to different AdS black holes, such as rotating black holes, higher dimensional black holes, Gauss-Bonnet black holes, f(R) black holes, black holes with scalar hair, etc [15–57], where more interesting phenomena are found.

Recently, the spatial curvature of electrically charged AdS black hole is viewed as variable and treated as topological charge [58, 59] in Einstein-Maxwell’s gravity and Lovelock-Maxwell theory. The authors found that the topological charge naturally arose in holography. What is more, together with all other known charges (electric charge, mass, and entropy), they satisfy an extended first law and the Gibbs-Duhem-like relation as a completeness. In our last paper [60], when the cosmological constant is not viewed as variable, we find a van der Waals type but new phase transition relating to the topological charge, while in this paper, we will treat both the cosmological constant and the spatial curvature as variables, then following one of their methods to derive the extended first law, from which one can see the cosmological constant is naturally viewed as pressure and the mass is viewed as enthalpy. Based on the extended first law, the black hole’s phase transition property will be investigated in canonical ensemble with fixed electric charge and topological charge.

This paper is organized as follows. In Section 2, following the method in Ref. [59], we will derive the extended first law in d dimensional space-time. In Section 3, by analysing the specific heat, the phase transition of AdS black hole in 4-dimensional space-time is studied and the critical point is determined. In Section 4, the van der Waals like oscillatory behavior is observed in both and graphs. Then we use the Maxwell equal area law to obtain the phase transition coexistence curve. In Section 5, the van der Waals like swallow tail behavior is observed in and graphs, then we will obtain the phase transition coexistence curve by analysing the gibbs free energy. Finally, we summarize and discuss our results in Section 6.

#### 2. The Extended Thermodynamic First Law

The d dimensional space-time AdS black hole solutions with maximal symmetry in the Einstein-Maxwell theory arewhere are related to the ADM mass , electric charge , and cosmological constant byand is the volume of the “unit" sphere, plane, or hyperbola, and stands for the spatial curvature of the black hole. Under suitable compactifications for , we assume that the volume of the unit space is a constant hereafter [58, 59].

Following [59], the first law of thermodynamics can be derived. As the first law of thermodynamics is about the differential relation of every physical quantity, one can first find an equation containing these physical quantities and then differentiate it to obtain the first law of thermodynamics. Considering an equipotential surface with fixed (here set ), we variate both sides of the equation and obtainwhere is the radius of event horizon. Notingwe obtainMultiplying both sides with an constant factor , the above equation becomeswhere is the temperature, is the entropy, and is the electric potential. If we introduce a new “charge" as in [58, 59]then its conjugate potential is obtained as . If we define the black hole volume as , then its conjugate pressure is naturally arisen as , and the black hole mass is naturally viewed as enthalpy instead of energy. Finally, the extended first law is obtained as

#### 3. The Specific Heat and Phase Transition

Hereafter, the investigation will be limited in dimensional space-time and in canonical ensemble with fixed electric charge and topological charge, leaving other situations for further study. First of all, we would like to analyse the behavior of the specific heat and the related possible phase transition phenomena. The first law can be rewritten in terms of energy ,So the isobaric specific heat can be written asSince we are in canonical ensemble, can be abbreviated as . From the denominator, we can conclude the following:

When , has two diverge points atwhich signals a phase transition.

When , the two diverge points of merge into one atwhich is the phase transition critical point. The critical temperature .

When , is always larger than zero, so there is no phase transition.

Comparing with the van der Waals equation, the specific volume is defined as [14]At the critical point, we obtain an interesting relationwhich is exactly the same as for the van der Waals fluid and RN-AdS black holes. Note that this number which seems to be universal does not depend on the topological charge or electric charge.

All the physical quantities can be rescaled by those at the critical point. Definingthe isobaric specific heat becomes

The behaviors of the rescaled specific heat for the cases , , and are shown in Figure 1. The curve of specific heat for has two divergent points which divide the region into three parts. Both the large radius region and the small radius region are thermodynamically stable with positive specific heat, while the medium radius region is unstable with negative specific heat. So there is a phase transition which takes place between small black hole and large black hole. The curve of specific heat for has only one divergent point and always positive denoting that is exactly the critical point. The curve of specific heat for has no divergent point and is always positive, implying that the black holes are stable and no phase transition will take place. This behavior of specific heat is very similar to that of the liquid-gas var der Waals system.