Advances in High Energy Physics

Volume 2015 (2015), Article ID 124910, 11 pages

http://dx.doi.org/10.1155/2015/124910

## Thermodynamic Relations for Kiselev and Dilaton Black Hole

^{1}School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), H-12, Islamabad 44000, Pakistan^{2}Department of Physics, Vivekananda Satavarshiki Mahavidyalaya, Vidyasagar University, Manikpara, Jhargram, West Midnapur, West Bengal 721513, India

Received 23 May 2015; Revised 12 August 2015; Accepted 19 August 2015

Academic Editor: George Siopsis

Copyright © 2015 Bushra Majeed 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

We investigate the thermodynamics and phase transition for Kiselev black hole and dilaton black hole. Specifically we consider Reissner-Nordström black hole surrounded by radiation and dust and Schwarzschild black hole surrounded by quintessence, as special cases of Kiselev solution. We have calculated the products relating the surface gravities, surface temperatures, Komar energies, areas, entropies, horizon radii, and the irreducible masses at the Cauchy and the event horizons. It is observed that the product of surface gravities, product of surface temperature, and product of Komar energies at the horizons are not universal quantities for the Kiselev solutions while products of areas and entropies at both the horizons are independent of mass of the above-mentioned black holes (except for Schwarzschild black hole surrounded by quintessence). For charged dilaton black hole, all the products vanish. The first law of thermodynamics is also verified for Kiselev solutions. Heat capacities are calculated and phase transitions are observed, under certain conditions.

#### 1. Introduction

Black holes are the most exotic objects in physics and their connection with thermodynamics is even more surprising. Discovery of Hawking radiations leads to the identification of black holes as thermodynamic objects with physical temperature and entropy. Once black holes are identified as thermodynamical objects, it is quite natural to find whether they also behave as familiar thermodynamic systems. The analogy between the black hole thermodynamics and four laws of thermodynamics was first proposed in the 1970s [1–6]. The important results of the black hole thermodynamics are the association of temperature () and entropy with surface gravity and area of the black hole event horizon, respectively. Laws of black hole thermodynamics are studied in the literature [7–10]. In [11] universal properties of black holes and the first law of black hole inner mechanics are discussed. In [12] horizon entropy sums in A(dS) spacetimes are studied. In [13] the authors have discussed the spin entropy of a rotating black hole. The study of phase transition in black holes is a fascinating topic. The phenomenon of phase transition in black hole thermodynamics was first observed long ago [14–16]. If a black hole has a Cauchy horizon () and an event horizon(), then it is quite interesting to study different quantities like the product of areas of a black hole on these horizons.

In [17–19] the thermal products for rotating black holes are studied. In [20] area products for stationary black hole horizons are calculated. The calculations show that sometimes these products depend not only on the ADM (Arnowitt-Deser-Misner) mass parameter but also on the charge and angular momentum. The relations that are independent of the black hole mass are of particular interest because these may turn out to be universal and hold for more general solutions with nontrivial surroundings too.

Kiselev [21] considered Einstein’s field equation surrounded by quintessential matter and proposed new solutions, dependent on state parameter of the matter surrounding black hole. Recently, some dynamical aspects, that is, collision between particles and their escape energies after collision around Kiselev black hole [22], have been studied. In this work we consider the solution of Reissner-Nordström (RN) black hole surrounded by energy-matter, derived by Kiselev, and study the important thermodynamic features of black hole at the horizons and generalize some already existing results for Cauchy horizon. We also consider the solution of Schwarzschild black hole surrounded by energy-matter and analyzed its different thermodynamic products. Furthermore, we have considered the charged dilaton black hole and computed its various thermodynamic products.

The plan of the work is as follows. In Section 2, we discuss the basic aspects of RN black hole surrounded by radiation. We have shown that products of surface gravity, temperature, and Komar energy calculated on the inner and outer horizons are not universal due to dependance on mass of the black hole, while products of horizons and products of area and entropy calculated at are independent of mass of black hole. In Section 2.1, the Smarr formula for the black hole is derived and, using the obtained expression of mass, calculations for the first law of thermodynamics are given there. In Section 2.2, we have discussed the irreducible mass and the rest mass is written in terms of irreducible mass. In Section 2.3, heat capacity of black hole is calculated, and phase transition is discussed. In Section 3, the metric of RN black hole surrounded by dust is discussed and some basic aspects of black hole thermodynamics are discussed. In Sections 3.1, 3.2, and 3.3, we study the irreducible mass, Smarr formula, and heat capacity of the black hole, respectively. In Section 4, the metric of Schwarzschild black hole surrounded by quintessence is discussed. In Section 4.1, the Smarr formula and first law of thermodynamics are discussed, irreducible mass is studied in Section 4.2, and analysis of heat capacity of the black hole is completed in Section 4.3. In Section 5, we compute thermodynamic product relations for dilaton black hole. In the last section we concluded the work. We use units in which .

#### 2. RN Black Hole Surrounded by Radiation

The spherically symmetric and static solutions for Einstein’s field equations, surrounded by energy-matter, as investigated by Kiselev [21] can be written aswherehere is mass of the black hole, is the electric charge, is normalization parameter, and is the state parameter of the matter around black hole. We consider the cases when RN black hole is surrounded by radiation () and dust (). For two horizons of the black hole are obtained fromthat is,Here denotes the normalization parameter for radiation case, with dimensions, , where denotes length, is the outer horizon named as event horizon , and is the inner horizon known as Cauchy horizon . Cauchy horizon is a null surface of infinite blue-shift, while the event horizon is an infinite red-shift surface [23]. The product of the two horizons,is independent of the mass of the black hole but depends on electric charge and . Areas of two horizons of the black hole areThe corresponding semiclassical Bekenstein-Hawking entropy at the horizons is [24]Hawking temperature of the horizons is determined by using the formulawhere we have used .

Surface gravity is the acceleration due to gravity at the horizon of a black hole. It is defined as the force required to an observer at infinity, for holding a particle (of unit mass) in place at the event horizon, given as [25] The Komar energy of the black hole given by [26]Products of surface gravities and surface temperatures at areProduct of the Komar energies at isIt is clear that all these products (except product of horizons ) are depending on mass of the black hole, so these quantities are not universal. We also calculate the products of areas and entropies at , which turn out toNote that both the products are independent of mass, so these are universal quantities.

##### 2.1. Smarr Formula for Cauchy Horizon

The expression for area of the black hole can be rewritten using the idea proposed by Smarr [27, 28] asThe area of both horizons must be constant given byUsing (15) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as follows:Since the first law of thermodynamics states that change in mass of a black hole is related to change in its area and electric charge and also the effective surface tensions at the horizons are proportional to the temperatures of the black hole horizons, we can write:where and are the physical invariants of both horizons, defined as We can rewrite effective surface tension asThat is, RN black hole surrounded by radiation satisfies the first law of thermodynamics.

##### 2.2. Christodoulou-Ruffini Mass Formula for RN Black Hole Surrounded by Radiation

Christodoulou and Christodoulou and Ruffini [1, 2] had shown that the mass of a black hole could be increased or decreased, but there is no way to decrease the irreducible mass of a black hole. In fact, most processes result in an increase in and during reversible process this quantity also does not change. Also, the surface area of a black hole has behavior [29]: so there exists a relation between area and irreducible mass. is proportional to the square root of the black hole’s area. Since the RN-radiation space-time has regular event horizon and Cauchy horizons, the irreducible mass of a black hole is proportional to the square root of its surface area [2]: The irreducible mass defined on inner and outer horizons is and , respectively. The product of the irreducible mass at the horizons isThis product is universal because it does not depend on the mass of the black hole. The expression for the rest mass of the rotating charged black hole given by Christodoulou and Ruffini in terms of its irreducible mass, angular momentum, and charge is [2]For RN black hole surrounded by radiation expression of mass in terms of irreducible mass becomes where .

##### 2.3. Heat Capacity on

Another important measure to study the thermodynamic properties of a black hole is the heat capacity of black hole. The nature (positivity or negativity) of heat capacity reflects the change in the stability properties of the thermal system (black hole). A black hole with negative heat capacity is in unstable equilibrium state; that is, by emitting Hawking radiations, it may decay to a hot flat space or by absorbing a radiation it may grow without limit [30]. Heat capacity of a black hole is given by where mass in terms of is The partial derivatives of mass and temperature with respect to are and from (8) we have The expression for heat capacity for RN black hole surrounded by radiation at horizons becomesNote that there are two possible cases for heat capacity to be positive.

*Case 1. *When both and are positive.

*Case 2. *When both and are negative.

Since we are interested in positive only, so Case implies that heat capacity is positive if from Case we getwhich is not possible, so we exclude this case.

Heat capacity is negative if the following cases hold.

*Case a*. and .

*Case b*. and .

Case implies that for heat capacity is negative. From Case we get negative capacity for we are interested in positive only. The region where heat capacity is negative corresponds to an instable region around black hole, whereas a region in which the heat capacity is positive represents a stability region. Behavior of heat capacity given in (29) is shown in Figure 1. Heat capacity is negative in the regions and , while positive in . Interestingly, the product of heat capacity on becomesthe product depends on mass parameter and charge parameter. Thus the product of specific heats is not universal.