Advances in High Energy Physics

Volume 2015 (2015), Article ID 478273, 6 pages

http://dx.doi.org/10.1155/2015/478273

## Phase Transitions of the BTZ Black Hole in New Massive Gravity

Institute of Basic Science and Department of Computer Simulation, Inje University, Gimhae 621-749, Republic of Korea

Received 29 July 2015; Accepted 8 October 2015

Academic Editor: Thomas Rössler

Copyright © 2015 Yun Soo Myung. 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

We investigate thermodynamics of the BTZ black hole in new massive gravity explicitly. For with being the mass parameter of fourth-order terms and AdS curvature radius, the Hawking-Page phase transition occurs between the BTZ black hole and AdS (thermal) soliton. For , however, this transition unlikely occurs but a phase transition between the BTZ black hole and the massless BTZ black hole is possible to occur. We may call the latter the inverse Hawking-Page phase transition and this transition is favored in the new massive gravity.

#### 1. Introduction

A black hole could be rendered thermodynamically stable by placing it in four-dimensional anti-de Sitter () spacetimes because spacetimes play the role of a confining box. Then, it is a natural question to ask how a stable black hole with positive heat capacity could emerge from thermal radiation through a phase transition. This was known to be the Hawking-Page (HP) phase transition between thermal radiation (TR) and Schwarzschild- black hole (SAdS) [1, 2]. It has shown one typical example of the first-order phase transition (TAdSsmall SAdSlarge SAdS) in the gravitational system. In the last two decades, its higher dimensional extension and its holographic duality to confinement-deconfinement transition were the hottest issues [3].

In order to study the HP phase transition in Einstein gravity, we need to know the Arnowitt-Deser-Misner (ADM) mass [4], the Hawking temperature, and the Bekenstein-Hawking (BH) entropy. These are combined to give the on-shell free energy in canonical ensemble which determines the global thermodynamic stability. The other important quantity is the heat capacity which determines the local thermodynamic stability. Employing the Euclidean action formalism, one easily finds these quantities [5]. However, a complete computation of the thermodynamic quantities was limited in fourth-order gravity because one has encountered some difficulty computing their conserved quantities in asymptotically AdS spacetimes.

In three dimensions, either third-order gravity (topologically massive gravity [6–8]) or the fourth-order gravity (new massive gravity [9]) is essential to describe spin-2 graviton because the Einstein gravity is a gauge theory without propagating degrees of freedom. Recently, there was significant progress on computation of mass and thermodynamic quantities by using the Abbott-Deser-Tekin (ADT) method [10–12]. One has to recognize that all ADT thermodynamic quantities except the Hawking temperature depend on a mass parameter . Hence, for , all thermodynamic properties are dominantly determined by Einstein gravity, while, for , all thermodynamic properties are dominantly determined by purely fourth-order curvature term. More recently, it was shown that the HP phase transition (thermal solitonBTZ black hole) occurs for in new massive gravity by computing off-shell free energies of black hole and soliton [13]. However, the role of the massless BTZ black hole was missed. The former can be completely understood if the massless BTZ black hole is introduced as a mediator. Furthermore, the present is a turnaround time to explore case of black hole thermodynamics if one wishes to study the black hole thermodynamics by employing the massive gravity theory.

On the other hand, we would like to mention that the stability condition of the BTZ black hole in the new massive gravity turned out to be regardless of the horizon size , while the instability condition is given by [14]. For , the BTZ black hole is thermodynamically unstable because of and and it is classically unstable against the metric perturbations. The latter indicates perturbative instability of the BTZ black hole that arose from the massiveness of graviton. It implies a deep connection between thermodynamic instability and classical instability for the BTZ black hole only for the new massive gravity [15]. Also, it suggests that the phase transition for is quite different from that of case. Here, we wish to explore the presumed phase transition and it will be compared with the Hawking-Page phase transition for case.

#### 2. Thermodynamics of the BTZ Black Hole

We introduce the new massive gravity (NMG) composed of the Einstein-Hilbert action with a cosmological constant and fourth-order curvature terms [9]: where is a three-dimensional Newton constant and a mass parameter with mass dimension 2. In the limit of , recovers the Einstein gravity, while reduces to purely fourth-order gravity in the limit of . The Einstein equation is given by where The BTZ black hole solution to (4) is given by [16, 17] when satisfying a condition of with being the curvature radius of spacetimes. Here, is related to the ADM mass of black hole. The horizon radius is determined by the condition of .

On the other hand, the linearized equation to (4) upon choosing the transverse-traceless gauge of and leads to the fourth-order linearized equation for the metric perturbation : which might imply the two second-order linearized equations where the mass squared of massive spin-2 graviton is given by Equation (9) describes massive graviton with 2 DOF propagating around the BTZ black hole under the gauge, while (8) indicates nonpropagating spin-2 graviton in the Einstein gravity. This explains clearly why the NMG describes massive graviton with 2 DOF. The presence of distinguishes the NMG from the Einstein gravity because it generates 2 DOF. At this stage, we briefly mention the stability of the BTZ black hole in the NMG. The stability condition of the BTZ black hole in the NMG turned out to be regardless of the horizon size , while the instability condition is given by [14]. This is valid for the NMG, not for the Einstein gravity.

Now we derive all thermodynamic quantities. The Hawking temperature is found to be which is the same for the Einstein gravity. Using the ADT method, one could derive the mass [18], heat capacity, entropy [19], and on-shell free energy: For , thermodynamic quantities in Einstein gravity are given by [20–22] which are positive regardless of the horizon size except that the free energy is negative. This means that the BTZ black hole is thermodynamically stable in Einstein gravity. Here, we check that the first law of thermodynamics is satisfied as as in Einstein gravity where “” denotes the differentiation with respect to the horizon size only. Importantly, we note that in the limit of we recover thermodynamics of the BTZ black hole in Einstein gravity, while in the limit of we recover the black hole thermodynamics in purely fourth-order gravity. The latter is similar to recovering the third-order terms of conformal Chern-Simons gravity from the topologically massive gravity [23] and conformal gravity from the Einstein-Weyl gravity [15, 24].

It is well known that the local thermodynamic stability is determined by the positive heat capacity () and the global stability is determined by the negative free energy (). Therefore, we propose that the thermodynamic stability is determined by the sign of the heat capacity while the phase transition is mainly determined by the sign of the free energy.

To investigate a phase transition, we introduce the thermal soliton (TSOL) whose thermodynamic quantities are given by [13] where with . We note that a factor of was missed in [15]. The TSOL corresponds to the spacetime picture of the NS-NS vacuum state [25].

Furthermore, we need the massless BTZ black hole (MBTZ) whose thermodynamic quantities all are zero as [21, 26] The MBTZ is called the spacetime picture of the - vacuum state.

In addition to a global mass parameter , we introduce five parameters to describe the phase transition in NMG. These are included as(i): order parameter,(ii): order parameter (on-shell temperature),(iii): control parameter (off-shell temperature),(iv): increasing (decreasing) black hole via equilibrium process,(v): increasing (decreasing) black hole via nonequilibrium process,where off-shell (on-shell) means equilibrium (nonequilibrium) configurations. In general, the equilibrium process implies a reversible process, while the nonequilibrium process implies an irreversible process. The off-shell free energy corresponds to a generalized free energy which is similar to a temperature-dependent scalar potential for a simple model of thermal phase transition where is the order parameter and is a control parameter.

#### 3. Phase Transitions

##### 3.1. HP Phase Transition

Before we proceed, we understand intuitively how the original HP transition occurs between Schwarzschild- black hole (SAdS) and thermal radiation (TR). For , the ADM mass, Hawking temperature, and the Bekenstein-Hawking entropy are given by In addition, the heat capacity and on-shell free energy are given by where blows up at (heat capacity is changed from to at ). The critical temperature is determined from the condition of for . The TR is located at in this picture.

In studying thermodynamic stability, two relevant quantities are the heat capacity which determines thermally local stability (instability) for and on-shell free energy which determines the thermally global stability (instability) for SAdS is thermodynamically stable only if and . For simplicity, we choose . We observe that the on-shell free energy (thick curve in Figure 1) is maximum at and zero at which determines the critical temperature. For , one finds negative free energy.