# Thermodynamic Relations for Kiselev and Dilaton Black Hole

Authors: Bushra Majeed; Mubasher Jamil; Parthapratim Pradhan
Journal: Advances in High Energy Physics
 (2015)

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## 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.


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## Body

## 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 (T) and entropy (S) with surface gravity (κ) and area (A) 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 (H-) and an event horizon(H+), 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 ω 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 Section2, 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 H± 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 G=ħ=c=1.


## 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 as(1)ds2=-frdt2+1frdr2+r2dθ2+sin2⁡θdϕ2,where(2)fr=1-2Mr+Q2r2-σr3ω+1;here M is mass of the black hole, Q is the electric charge, σ is normalization parameter, and ω is the state parameter of the matter around black hole. We consider the cases when RN black hole is surrounded by radiation (ω=1/3) and dust (ω=0). For ω=1/3 two horizons of the black hole are obtained from(3)1-2Mr+Q2r2-σrr2=0;that is,(4)r±=M±M2-Q2+σr.Here σr denotes the normalization parameter for radiation case, with dimensions, [σr]=L2, where L denotes length, r+ is the outer horizon named as event horizon H+, and r- is the inner horizon known as Cauchy horizon H-. 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,(5)r+r-=Q2-σr,is independent of the mass of the black hole but depends on electric charge and σr. Areas of two horizons of the black hole are(6)A±=∫02π∫0πgθθgϕϕdθdϕ=4πr±2=4π2Mr±-Q2+σr.The corresponding semiclassical Bekenstein-Hawking entropy at the horizons H± is [24](7)S±=A±4=π2Mr±-Q2+σr.Hawking temperature of the horizons is determined by using the formula(8)T±=14πdfdrr=r±=14πr±2-Q2+σrr±3=r±-M2π2Mr±-Q2+σr,where we have used Q2=2Mr±+σr-r±2.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](9)κ±=12dfdrr=r±=2πT±,κ±=r±-M2Mr±-Q2+σr. The Komar energy of the black hole given by [26](10)E±=2S±T±=r±-M.Products of surface gravities and surface temperatures at H± are(11)κ+κ-=4π2T+T-=Q2-σr-M2Q2-σr2.Product of the Komar energies at H± is(12)E+E-=Q2-M2-σr.It is clear that all these products (except product of horizons r+r-) are depending on mass of the black hole, so these quantities are not universal. We also calculate the products of areas and entropies at H±, which turn out to(13)A+A-=16S+S-=16π2Q2-σr2.Note that both the products are independent of mass, so these are universal quantities.


### 2.1. Smarr Formula for Cauchy Horizon(H-)

The expression for area of the black hole can be rewritten using the idea proposed by Smarr [27, 28] as(14)A=4π2M2-Q2+σr+2MM2-Q2+σr.The area of both horizons must be constant given by(15)A±=4π2M2-Q2+σr±2MM2-Q2+σr.Using (15) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as follows:(16)M2=A±16π+Q4πA±+Q22-2πσrQ2A±-σr2+πσr2A±=A±16π+πQ4A±+Q2A±-4πσr2A±-σrA±-2πσr2A±.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:(17)dM=T±dA±+Φ±dQ,where T± and A± are the physical invariants of both horizons, defined as (18)T±=effective  surface  tension  at  horizons=1M132π-Q4π2A±2+Q2σrπA±2-πσr22A±Φ±=electromagnetic  potentials  at  horizons=1M2πQ3A±-2πQσrA±+Q2.We can rewrite effective surface tension as(19)T±=132πM1-16π2Q4-2Q2σr+σr2A±2=116πM1-2M2-Q2+σrr±2=18πr±-M2Mr±-Q2+σr=κ±8π.That 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 Mirr of a black hole. In fact, most processes result in an increase in Mirr and during reversible process this quantity also does not change. Also, the surface area of a black hole has behavior [29]:(20)dA±≥0, so there exists a relation between area and irreducible mass. Mirr 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]:(21)A±16π=Mirr±=r±24. The irreducible mass defined on inner and outer horizons is Mirr- and Mirr+, respectively. The product of the irreducible mass at the horizons H± is(22)Mirr+Mirr-=A+A-16π2=Q2-σr4.This 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](23)M2=Mirr±+Q24Mirr±2+J24Mirr±2.For RN black hole surrounded by radiation expression of mass in terms of irreducible mass becomes(24)M2=ρ±4+Q44ρ±2+π2σr2-2σrQ2+2Mirr+Mirr-, where ρ±=2Mirr±.


### 2.3. Heat CapacityC± on H±

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(25)C±=∂M∂T±, where mass M in terms of r± is(26)M=r±2+Q2-σr2r±. The partial derivatives of mass M and temperature T± with respect to r+ are(27)∂M∂r±=r±2-Q2+σr2r±2, and from (8) we have(28)∂T∂r±=12π3Q2-r±2-3σrr±4. The expression for heat capacity C±=∂M/∂r±∂r±/∂T± for RN black hole surrounded by radiation at horizons becomes(29)C±=2πr±2r±2-Q2+σr3Q2-3σr-r±2.Note that there are two possible cases for heat capacity to be positive.Case 1.
When bothr±2-Q2+σr and 3Q2-3σr-r±2 are positive.Case 2.
When bothr±2-Q2+σr and 3Q2-3σr-r±2 are negative.Since we are interested in positiver only, so Case 1 implies that heat capacity is positive if(30)Q2-σr<r<3Q2-σr; from Case 2 we get(31)3Q2-σr<r2<Q2-σ,which is not possible, so we exclude this case.Heat capacity is negative if the following cases hold.Case a. r±2-Q2+σr>0 and 3Q2-3σr-r±2<0.Case b. r±2-Q2+σr<0 and 3Q2-3σr-r±2>0.Casea implies that for(32)r>3Q2-3σr, heat capacity is negative. From Case b we get negative capacity for(33)-Q2-σ<r<Q2-σ; we are interested in positive r 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 0<r<0.4898 and r>0.8485, while positive in 0.4898<r<0.8485. Interestingly, the product of heat capacity on H± becomes(34)C+C-=4π2Q2-σr2Q2-σr-M24Q2-σr-3M2;the product depends on mass parameter and charge parameter. Thus the product of specific heats is not universal.Figure 1
Heat capacity undergoes phase transition from instability to stability, diverges atr=3(Q2-σr), and again goes to instable region; we chose σr=0.01 and Q=0.5.


## 2.1. Smarr Formula for Cauchy Horizon(H-)

The expression for area of the black hole can be rewritten using the idea proposed by Smarr [27, 28] as(14)A=4π2M2-Q2+σr+2MM2-Q2+σr.The area of both horizons must be constant given by(15)A±=4π2M2-Q2+σr±2MM2-Q2+σr.Using (15) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as follows:(16)M2=A±16π+Q4πA±+Q22-2πσrQ2A±-σr2+πσr2A±=A±16π+πQ4A±+Q2A±-4πσr2A±-σrA±-2πσr2A±.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:(17)dM=T±dA±+Φ±dQ,where T± and A± are the physical invariants of both horizons, defined as (18)T±=effective  surface  tension  at  horizons=1M132π-Q4π2A±2+Q2σrπA±2-πσr22A±Φ±=electromagnetic  potentials  at  horizons=1M2πQ3A±-2πQσrA±+Q2.We can rewrite effective surface tension as(19)T±=132πM1-16π2Q4-2Q2σr+σr2A±2=116πM1-2M2-Q2+σrr±2=18πr±-M2Mr±-Q2+σr=κ±8π.That 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 Mirr of a black hole. In fact, most processes result in an increase in Mirr and during reversible process this quantity also does not change. Also, the surface area of a black hole has behavior [29]:(20)dA±≥0, so there exists a relation between area and irreducible mass. Mirr 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]:(21)A±16π=Mirr±=r±24. The irreducible mass defined on inner and outer horizons is Mirr- and Mirr+, respectively. The product of the irreducible mass at the horizons H± is(22)Mirr+Mirr-=A+A-16π2=Q2-σr4.This 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](23)M2=Mirr±+Q24Mirr±2+J24Mirr±2.For RN black hole surrounded by radiation expression of mass in terms of irreducible mass becomes(24)M2=ρ±4+Q44ρ±2+π2σr2-2σrQ2+2Mirr+Mirr-, where ρ±=2Mirr±.


## 2.3. Heat CapacityC± on H±

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(25)C±=∂M∂T±, where mass M in terms of r± is(26)M=r±2+Q2-σr2r±. The partial derivatives of mass M and temperature T± with respect to r+ are(27)∂M∂r±=r±2-Q2+σr2r±2, and from (8) we have(28)∂T∂r±=12π3Q2-r±2-3σrr±4. The expression for heat capacity C±=∂M/∂r±∂r±/∂T± for RN black hole surrounded by radiation at horizons becomes(29)C±=2πr±2r±2-Q2+σr3Q2-3σr-r±2.Note that there are two possible cases for heat capacity to be positive.Case 1.
When bothr±2-Q2+σr and 3Q2-3σr-r±2 are positive.Case 2.
When bothr±2-Q2+σr and 3Q2-3σr-r±2 are negative.Since we are interested in positiver only, so Case 1 implies that heat capacity is positive if(30)Q2-σr<r<3Q2-σr; from Case 2 we get(31)3Q2-σr<r2<Q2-σ,which is not possible, so we exclude this case.Heat capacity is negative if the following cases hold.Case a. r±2-Q2+σr>0 and 3Q2-3σr-r±2<0.Case b. r±2-Q2+σr<0 and 3Q2-3σr-r±2>0.Casea implies that for(32)r>3Q2-3σr, heat capacity is negative. From Case b we get negative capacity for(33)-Q2-σ<r<Q2-σ; we are interested in positive r 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 0<r<0.4898 and r>0.8485, while positive in 0.4898<r<0.8485. Interestingly, the product of heat capacity on H± becomes(34)C+C-=4π2Q2-σr2Q2-σr-M24Q2-σr-3M2;the product depends on mass parameter and charge parameter. Thus the product of specific heats is not universal.Figure 1
Heat capacity undergoes phase transition from instability to stability, diverges atr=3(Q2-σr), and again goes to instable region; we chose σr=0.01 and Q=0.5.


## 3. RN Black Hole Surrounded by Dust

Metric of RN black hole surrounded by dust is the same as in (1), f(r) is defined in (2) with ω=0, and σ=σd becomes(35)fr=1-2Mr+Q2r2-σdr, where [σd]=L. The horizons are(36)r±=2M+σd±2M+σd2-4Q22.Area of the horizons H± is(37)A±=π22M+σd2-4Q2±22M+σd2M+σd2-4Q2=4π2M+σdr±-Q2.Entropy of the horizons is(38)S±=π2M+σdr±-Q2.Surface gravity and Hawking temperature of horizons are, respectively,(39)κ±=2r±-2M+σd22M+σdr±-Q2,(40)T±=2r±-2M+σd4π2M+σdr±-Q2=14πr±2-Q2r+3,where we have used r±2=(2M+σd)r±-Q2. The Komar energy becomes(41)E±=2r±-2M+σd2.Product of surface gravities and temperatures at the horizons is(42)κ+κ-=4π2T+T-=-2M+σd2-4Q24Q4.Product of Komar energies at the horizons is(43)E+E-=4Q2-2M+σd24.Note that all products are mass dependent, so these quantities are not universal. Products of areas and entropies at both horizons H± are(44)A+A-=16S+S-=16π2Q4.It is clear that area product and entropy product are universal entities.


### 3.1. Smarr Formula for Cauchy Horizon (H-)

Area of both horizons must be constant given by(45)A±=π22M+σd2-4Q2±22M+σd2M+σd2-4Q2.Using (45) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as(46)M2+Mσd=A±16π+πQ4A±-σd2-2Q24.Differential of mass could be expressed in terms of physical invariants of the horizons:(47)dM=T±dA±+Φ±dQ,where(48)T±=12M+σd116π-πQ4A±2,Φ±=12M+σdQ+4πQ3A±.We can rewrite effective surface tension as(49)T±=116π2M+σd1-16π2Q4A±2=18π2M+σd1-4M2+Mσd+σd2-2Q22r±2or(50)T=±2M+σd2-4Q216π2M+σdr±-Q2=κ±8π. So the first law of black hole thermodynamics is verified, for RN black hole surrounded by dust, using the Smarr formula approach.


### 3.2. Christodoulou-Ruffini Mass Formula for RN Black Hole Surrounded by Dust

The expression for irreducible mass for RN black hole surrounded by dust is(51)Mirr±=2M+σd±2M+σd2-4Q24. Here Mirr- and Mirr+ are irreducible masses defined on inner and outer horizons, respectively. Area of H±, in terms of Mirr±, is(52)A±=16πMirr±2.Product of the irreducible mass at the horizons H± is(53)Mirr+Mirr-=A+A-16π2=Q24.This product is independent of mass of the black hole. Mass of the black hole expressed in terms of its irreducible mass and charge is(54)M2+Mσd=ρ±4+Q44ρ±2-σd+2Mirr+Mirr-.


### 3.3. Heat CapacityC± on H±

Mass of RN black hole surrounded by dust in terms ofr± is(55)M=r±2+Q2-σdr±2r±.Partial derivatives of mass M and temperature T± with respect to r± are(56)∂M∂r±=r±2-Q2+σdr±2r±2,∂T±∂r±=14π3Q2-r±2r±4,where T is given in (40). The heat capacity C=∂M/∂r±∂r±/∂T at the horizons is(57)C±=2πr±2r±2-Q2+σdr±r±2-Q2.In this case the product formula for heat capacity is found to be(58)C+C-=4π2Q44Q4-Q22M+σd2+σd2Q24Q4-Q22M+σd2.It is clear that the product formula does depend on mass parameter, so it is not universal in nature. Note that there are two possible cases for heat capacity, C, to be positive.Case 1.
When bothr±2-Q2+σdr± and r2-Q2 are positive.Case 2.
When bothr±2-Q2+σdr± and r2-Q2 are negative.Consideringσdr± as a positive quantity (for physically accepted region, r), from Case 1, we can say that C is positive for only r±2-Q2>0; that is,(59)Q2<r<-Q2, while from Case 2 we can say that C is negative for only r±2-Q2+σdr±<0; that is,(60)r<-σd2+124Q2+σd2. Heat capacity is negative if the following cases hold.Case a. r±2-Q2+σdr±>0 and r±2-Q2<0.Case b. r±2-Q2+σdr±<0 and r±2-Q2>0.Caseb is not possible mathematically since σdr±>0, while in Case a heat capacity is negative in the region, where r satisfies both of the following conditions:(61)-σd2+124Q2+σd2<r<-σd2-124Q2+σd2,-Q2<r<Q2, simultaneously. We consider that both σ and Q are positive in all the calculations. The behavior of the heat capacity given in (57) is shown in Figure 2.Figure 2
Heat capacity undergoes phase transition from stability to instability region; we choseσd=0.01, Q=0.5, and M=1.


## 3.1. Smarr Formula for Cauchy Horizon (H-)

Area of both horizons must be constant given by(45)A±=π22M+σd2-4Q2±22M+σd2M+σd2-4Q2.Using (45) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as(46)M2+Mσd=A±16π+πQ4A±-σd2-2Q24.Differential of mass could be expressed in terms of physical invariants of the horizons:(47)dM=T±dA±+Φ±dQ,where(48)T±=12M+σd116π-πQ4A±2,Φ±=12M+σdQ+4πQ3A±.We can rewrite effective surface tension as(49)T±=116π2M+σd1-16π2Q4A±2=18π2M+σd1-4M2+Mσd+σd2-2Q22r±2or(50)T=±2M+σd2-4Q216π2M+σdr±-Q2=κ±8π. So the first law of black hole thermodynamics is verified, for RN black hole surrounded by dust, using the Smarr formula approach.


## 3.2. Christodoulou-Ruffini Mass Formula for RN Black Hole Surrounded by Dust

The expression for irreducible mass for RN black hole surrounded by dust is(51)Mirr±=2M+σd±2M+σd2-4Q24. Here Mirr- and Mirr+ are irreducible masses defined on inner and outer horizons, respectively. Area of H±, in terms of Mirr±, is(52)A±=16πMirr±2.Product of the irreducible mass at the horizons H± is(53)Mirr+Mirr-=A+A-16π2=Q24.This product is independent of mass of the black hole. Mass of the black hole expressed in terms of its irreducible mass and charge is(54)M2+Mσd=ρ±4+Q44ρ±2-σd+2Mirr+Mirr-.


## 3.3. Heat CapacityC± on H±

Mass of RN black hole surrounded by dust in terms ofr± is(55)M=r±2+Q2-σdr±2r±.Partial derivatives of mass M and temperature T± with respect to r± are(56)∂M∂r±=r±2-Q2+σdr±2r±2,∂T±∂r±=14π3Q2-r±2r±4,where T is given in (40). The heat capacity C=∂M/∂r±∂r±/∂T at the horizons is(57)C±=2πr±2r±2-Q2+σdr±r±2-Q2.In this case the product formula for heat capacity is found to be(58)C+C-=4π2Q44Q4-Q22M+σd2+σd2Q24Q4-Q22M+σd2.It is clear that the product formula does depend on mass parameter, so it is not universal in nature. Note that there are two possible cases for heat capacity, C, to be positive.Case 1.
When bothr±2-Q2+σdr± and r2-Q2 are positive.Case 2.
When bothr±2-Q2+σdr± and r2-Q2 are negative.Consideringσdr± as a positive quantity (for physically accepted region, r), from Case 1, we can say that C is positive for only r±2-Q2>0; that is,(59)Q2<r<-Q2, while from Case 2 we can say that C is negative for only r±2-Q2+σdr±<0; that is,(60)r<-σd2+124Q2+σd2. Heat capacity is negative if the following cases hold.Case a. r±2-Q2+σdr±>0 and r±2-Q2<0.Case b. r±2-Q2+σdr±<0 and r±2-Q2>0.Caseb is not possible mathematically since σdr±>0, while in Case a heat capacity is negative in the region, where r satisfies both of the following conditions:(61)-σd2+124Q2+σd2<r<-σd2-124Q2+σd2,-Q2<r<Q2, simultaneously. We consider that both σ and Q are positive in all the calculations. The behavior of the heat capacity given in (57) is shown in Figure 2.Figure 2
Heat capacity undergoes phase transition from stability to instability region; we choseσd=0.01, Q=0.5, and M=1.


## 4. Schwarzschild Black Hole Surrounded by Quintessence

Metric for Schwarzschild black hole surrounded by quintessence is the same as that defined in (1) and f(r) defined in (2) with ω=-2/3, σ=σq, and Q=0 becomes(62)fr=1-2Mr-σqr,where dimensions of σq are that of L-1. The horizons, r±, of the black hole are(63)r±=1±1-8Mσq2σq.Product of the two horizons yields(64)r+r-=2Mσq,and it is depending on mass of the black hole and σq. Areas of the horizons are(65)A±=4πr±-2Mσq.Entropy at the horizons H± is(66)S±=πσqr±-2M.Hawking temperature of the horizons is(67)T±=14π1-2σqr±r±,and the surface gravity on the black hole horizons H± is given by(68)κ±=121-2σqr±r±.The Komar energy is given by(69)E±=2M1+2σqr±-r±2σqr±.Product of surface gravities and temperatures of H± is(70)κ+κ-=4π2T+T-=8Mσq-1σq8M.Product of Komar energies of the horizons is(71)E+E-=M8Mσq-12σq,respectively. It is clear that all these products are depending on the mass of the black hole, so these quantities are not universal. The products of areas and entropies at H± are(72)A+A-=16S+S-=8πMσq2;again both products are not universal quantities.


### 4.1. Smarr Formula for Cauchy Horizon(H-)

Write area of both horizons of the black hole as(73)A±=πσq22-8Mσq±21-8Mσq.Using (73) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as(74)4M2+MAσqπ=116π4A±-A±2σq2π.Differential of mass, expressed in terms of physical invariants of the horizons, is(75)dM=T±dA±,where(76)T±=18πM+σqA±4π-16πσqM-2A±σq216π=116πA±8πM-A±σq=κ±8π,where we have used M=(r±-σqr±2)/2 and κ is defined in (68). Hence the first law of thermodynamics is satisfied by Schwarzschild black hole surrounded by quintessence.


### 4.2. Christodoulou-Ruffini Mass Formula for Schwarzschild Black Hole Surrounded by Quintessence

The irreducible mass of Schwarzschild black hole surrounded by quintessence is(77)Mirr±=1±1-8Mσq4σq. Here Mirr- and Mirr+ are irreducible masses defined on inner and outer horizons, respectively. Area of H±, in terms of Mirr±, is(78)A±=16πMirr±2.Product of the irreducible mass at the horizons H± is(79)Mirr+Mirr-=A+A-16π2=M2σ.This product is depending on mass of the black hole. Expression of mass given in (74), in terms of irreducible mass, becomes(80)8σ2Mirr+2Mirr-2+4πρ±2Mirr+Mirr-=4ρ±21-4σ2.


### 4.3. Heat CapacityC± on H±

The mass of Schwarzschild black hole surrounded by quintessence in terms ofr± is(81)M=r±-σqr±22. The partial derivative of mass M with respect to r± is(82)∂M∂r±=1-2σqr±2, and using (67) we get(83)∂T±∂r±=-14πσqr±2-2M. The expression for heat capacity C=∂M/∂r±∂r±/∂T at the horizon becomes(84)C±=-2π1-2σqr±r±-2Mσq.Heat capacity would be positive if the following cases hold.Case 1.
1
-
2
σ
q
r
±
<
0 and r±-2M>0.Case 2.
1
-
2
σ
q
r
±
>
0 and r±-2M<0.Heat capacity is negative if the following cases hold.Case a. 1-2σqr±>0 and r±-2M>0.Case b. 1-2σqr±<0 and r±-2M<0.The behavior of the heat capacity given in (84) is shown in Figure 3 for M=1 and σd=0.01; heat capacity is negative for 2<r<50 and positive for 0<r<2 and r>50; it is zero at r=2 and r=50. A comparison of all the parameters calculated for Kiselev solutions is shown in Table 1.Table 1
A comparison of thermodynamical parameters for RN-radiation, RN-dust, and Schwarzschild-quintessence black holes.
Parameter
RN-radiation
RN-dust
Schwarzschild-quintessence
r
±
M
±
M
-
Q
2
+
σ
r
2
M
+
σ
d
±
2
M
+
σ
d
2
-
4
Q
2
2
1
±
1
-
8
M
σ
q
2
σ
q
A
±
4
π
2
M
r
±
-
Q
2
+
σ
r
4
π
2
M
+
σ
d
r
±
-
Q
2
4
π
r
±
-
2
M
σ
q
S
±
π
2
M
r
±
-
Q
2
+
σ
r
π
2
M
+
σ
d
r
±
-
Q
2
π
σ
q
r
±
-
2
M
T
±
r
±
-
M
2
π
2
M
r
±
-
Q
2
+
σ
r
2
r
±
-
2
M
+
σ
d
4
π
2
M
+
σ
d
r
±
-
Q
2
1
4
π
1
-
2
σ
q
r
±
r
±
κ
±
r
±
-
M
2
M
r
±
-
Q
2
+
σ
r
2
r
±
-
2
M
+
σ
d
2
2
M
+
σ
d
r
±
-
Q
2
1
2
1
-
2
σ
q
r
±
r
±
E
±
r
±
-
M
2
r
±
-
2
M
+
σ
d
2
2
M
1
+
2
σ
q
r
±
-
r
±
2
σ
q
r
±
κ
+
κ
-
Q
2
-
σ
r
-
M
2
Q
2
-
σ
r
2
-
2
M
+
σ
d
2
-
4
Q
2
4
Q
4
8
M
σ
q
-
1
σ
q
8
M
T
+
T
-
Q
2
-
M
2
-
σ
r
4
π
2
Q
2
-
σ
r
2
4
Q
2
-
2
M
+
σ
d
2
16
π
2
Q
4
8
M
σ
q
-
1
σ
q
32
π
2
M
E
+
E
-
Q
2
-
M
2
-
σ
r
4
Q
2
-
2
M
+
σ
d
2
4
M
8
M
σ
q
-
1
2
σ
q
A
+
A
-
16
π
2
Q
2
-
σ
r
2
16
π
2
Q
4
8
π
M
σ
q
2
S
+
S
-
π
2
Q
2
-
σ
r
2
π
2
Q
4
2
π
M
σ
q
2
M
i
r
r
+
M
i
r
r
-
Q
2
-
σ
r
4
Q
2
4
M
2
σ
C
±
2
π
r
±
2
r
±
2
-
Q
2
+
σ
r
3
Q
2
-
3
σ
r
-
r
±
2
2
π
r
±
2
r
±
2
-
Q
2
+
σ
d
r
±
r
±
2
-
Q
2
-
2
π
1
-
2
σ
q
r
±
r
±
-
2
M
σ
q
C
+
C
-
4
π
2
Q
2
-
σ
r
2
Q
2
-
σ
r
-
M
2
4
Q
2
-
σ
r
-
3
M
2
4
π
2
Q
4
4
Q
4
-
Q
2
2
M
+
σ
d
2
+
σ
d
2
Q
2
4
Q
4
-
Q
2
2
M
+
σ
d
2
16
π
2
M
2
8
M
σ
-
1
σ
2Figure 3
Heat capacity phase transition of Schwarzschild black hole surrounded by quintessence. We choseσq=0.01 and M=1. Heat capacity is positive for 0<r<2 and r>50 and negative for 2<r<50 and is zero at r=2 and r=50.


## 4.1. Smarr Formula for Cauchy Horizon(H-)

Write area of both horizons of the black hole as(73)A±=πσq22-8Mσq±21-8Mσq.Using (73) mass of the black hole or ADM mass is expressed in terms of the areas of horizons as(74)4M2+MAσqπ=116π4A±-A±2σq2π.Differential of mass, expressed in terms of physical invariants of the horizons, is(75)dM=T±dA±,where(76)T±=18πM+σqA±4π-16πσqM-2A±σq216π=116πA±8πM-A±σq=κ±8π,where we have used M=(r±-σqr±2)/2 and κ is defined in (68). Hence the first law of thermodynamics is satisfied by Schwarzschild black hole surrounded by quintessence.


## 4.2. Christodoulou-Ruffini Mass Formula for Schwarzschild Black Hole Surrounded by Quintessence

The irreducible mass of Schwarzschild black hole surrounded by quintessence is(77)Mirr±=1±1-8Mσq4σq. Here Mirr- and Mirr+ are irreducible masses defined on inner and outer horizons, respectively. Area of H±, in terms of Mirr±, is(78)A±=16πMirr±2.Product of the irreducible mass at the horizons H± is(79)Mirr+Mirr-=A+A-16π2=M2σ.This product is depending on mass of the black hole. Expression of mass given in (74), in terms of irreducible mass, becomes(80)8σ2Mirr+2Mirr-2+4πρ±2Mirr+Mirr-=4ρ±21-4σ2.


## 4.3. Heat CapacityC± on H±

The mass of Schwarzschild black hole surrounded by quintessence in terms ofr± is(81)M=r±-σqr±22. The partial derivative of mass M with respect to r± is(82)∂M∂r±=1-2σqr±2, and using (67) we get(83)∂T±∂r±=-14πσqr±2-2M. The expression for heat capacity C=∂M/∂r±∂r±/∂T at the horizon becomes(84)C±=-2π1-2σqr±r±-2Mσq.Heat capacity would be positive if the following cases hold.Case 1.
1
-
2
σ
q
r
±
<
0 and r±-2M>0.Case 2.
1
-
2
σ
q
r
±
>
0 and r±-2M<0.Heat capacity is negative if the following cases hold.Case a. 1-2σqr±>0 and r±-2M>0.Case b. 1-2σqr±<0 and r±-2M<0.The behavior of the heat capacity given in (84) is shown in Figure 3 for M=1 and σd=0.01; heat capacity is negative for 2<r<50 and positive for 0<r<2 and r>50; it is zero at r=2 and r=50. A comparison of all the parameters calculated for Kiselev solutions is shown in Table 1.Table 1
A comparison of thermodynamical parameters for RN-radiation, RN-dust, and Schwarzschild-quintessence black holes.
Parameter
RN-radiation
RN-dust
Schwarzschild-quintessence
r
±
M
±
M
-
Q
2
+
σ
r
2
M
+
σ
d
±
2
M
+
σ
d
2
-
4
Q
2
2
1
±
1
-
8
M
σ
q
2
σ
q
A
±
4
π
2
M
r
±
-
Q
2
+
σ
r
4
π
2
M
+
σ
d
r
±
-
Q
2
4
π
r
±
-
2
M
σ
q
S
±
π
2
M
r
±
-
Q
2
+
σ
r
π
2
M
+
σ
d
r
±
-
Q
2
π
σ
q
r
±
-
2
M
T
±
r
±
-
M
2
π
2
M
r
±
-
Q
2
+
σ
r
2
r
±
-
2
M
+
σ
d
4
π
2
M
+
σ
d
r
±
-
Q
2
1
4
π
1
-
2
σ
q
r
±
r
±
κ
±
r
±
-
M
2
M
r
±
-
Q
2
+
σ
r
2
r
±
-
2
M
+
σ
d
2
2
M
+
σ
d
r
±
-
Q
2
1
2
1
-
2
σ
q
r
±
r
±
E
±
r
±
-
M
2
r
±
-
2
M
+
σ
d
2
2
M
1
+
2
σ
q
r
±
-
r
±
2
σ
q
r
±
κ
+
κ
-
Q
2
-
σ
r
-
M
2
Q
2
-
σ
r
2
-
2
M
+
σ
d
2
-
4
Q
2
4
Q
4
8
M
σ
q
-
1
σ
q
8
M
T
+
T
-
Q
2
-
M
2
-
σ
r
4
π
2
Q
2
-
σ
r
2
4
Q
2
-
2
M
+
σ
d
2
16
π
2
Q
4
8
M
σ
q
-
1
σ
q
32
π
2
M
E
+
E
-
Q
2
-
M
2
-
σ
r
4
Q
2
-
2
M
+
σ
d
2
4
M
8
M
σ
q
-
1
2
σ
q
A
+
A
-
16
π
2
Q
2
-
σ
r
2
16
π
2
Q
4
8
π
M
σ
q
2
S
+
S
-
π
2
Q
2
-
σ
r
2
π
2
Q
4
2
π
M
σ
q
2
M
i
r
r
+
M
i
r
r
-
Q
2
-
σ
r
4
Q
2
4
M
2
σ
C
±
2
π
r
±
2
r
±
2
-
Q
2
+
σ
r
3
Q
2
-
3
σ
r
-
r
±
2
2
π
r
±
2
r
±
2
-
Q
2
+
σ
d
r
±
r
±
2
-
Q
2
-
2
π
1
-
2
σ
q
r
±
r
±
-
2
M
σ
q
C
+
C
-
4
π
2
Q
2
-
σ
r
2
Q
2
-
σ
r
-
M
2
4
Q
2
-
σ
r
-
3
M
2
4
π
2
Q
4
4
Q
4
-
Q
2
2
M
+
σ
d
2
+
σ
d
2
Q
2
4
Q
4
-
Q
2
2
M
+
σ
d
2
16
π
2
M
2
8
M
σ
-
1
σ
2Figure 3
Heat capacity phase transition of Schwarzschild black hole surrounded by quintessence. We choseσq=0.01 and M=1. Heat capacity is positive for 0<r<2 and r>50 and negative for 2<r<50 and is zero at r=2 and r=50.


## 5. Charged Dilaton Black Hole

The action for charged black hole in string theory is [31](85)S=∫d4x-g-R+2∇φ2+e-2aφF2,where F is the Maxwell field, φ is the scalar field, and a is an arbitrary parameter specifying the strength of dilaton and the Maxwell field’s coupling. We are going to derive the area product formula and entropy product formula for a spherically symmetric dilaton black hole [31] whose metric can be written in Schwarzschild-like coordinates as(86)ds2=-Nrdt2+dr2Nr+Rr2dθ2+sin2θdϕ2,where the function N(r) is defined by(87)Nr=1-r+r1-r-r1-a2/1+a2,R2r=r21-r-r2a2/1+a2.In these equations, r+ and r- are constants, which are related to mass and charge of the black hole as(88)M=r+2+1-a21+a2r-2,Q=r+r-1+a2, where as usual M is mass of the black hole and Q is electric charge of the black hole. It may be noted that Q and a are positive. The horizons of the black hole are determined by the function N(r)=0 which yields(89)r+=M+M2-2n1+nQ2,r-=1nM+M2-2n1+nQ2,and n is defined by(90)n=1-a21+a2. Here r+ and r- are called event horizon (H+) or outer horizon and Cauchy horizon (H-) or inner horizon, respectively, and r+=r- or M2=1+n/2Q2 corresponding to the extreme charged dilaton black hole.Case I. When a=0 or n=1, the metric corresponds to RN black hole.Case II. When a=1 or n=0, the metric corresponds to Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) black hole. The expressions for surface gravity of dilaton black hole at both horizons (H±) are(91)κ+=12r+r+-r-r+n,κ-=0.The black hole temperature or Hawking temperature at H± is(92)T+=14πr+r+-r-r+n,T-=0.Areas of the horizons (H±) are(93)A+=4πr+2r+-r-r+1-n,A-=0.Interestingly, the area of both horizons goes to zero at the extremal limit (r+=r-) which is quite different from the well-known RN and Schwarzschild black hole. The other characteristic of this spacetime is that there is a curvature singularity at r=r-.Now the entropies of both horizons (H±) are(94)S+=πr+2r+-r-r+1-n,S-=0.Finally, the Komar energy is given by(95)E+=r+-r-2,E-=0.Now we compute products of all the parameters given above:(96)A+A-=0,S+S-=0,κ+κ-=0,T+T-=0,E+E-=0.Interestingly their products go to zero value and independent of mass; thus they are universal quantities. All of the above thermodynamical quantities must satisfy the first law of thermodynamics:(97)dM=T±dA±+Φ±dQ,where(98)Φ±=2n1+nQr±.The irreducible mass at H± for this black hole is(99)Mirr+=r+2r+-r-r+1-n/2,Mirr-=0.Their product yields(100)Mirr+Mirr-=0.The heat capacity for this dilaton black hole is calculated to be(101)C+=-2πr+21-2n/1+nQ2/r+1-2/1+nQ2/r+1-21+2n/1+nQ2/r+1-2/1+nQ2/r+n.Due to curvature singularity at r=r- the heat capacity at the Cauchy horizon diverges. Thus the product of heat capacity diverges. Note that for odd n, heat capacity would be positive if (102)r<21+2nQ21+nr>2nQ21+n,or(103)r>21+2nQ21+n,r<2nQ21+n, and heat capacity is negative for(104)r>21+2nQ21+nr>2nQ21+n, or(105)r<21+2nQ21+n,r<2nQ21+n. The behavior of the heat capacity is shown in Figure 4. For Q=0.5 and n=3, the given expression of heat capacity (101) diverges at r=0.125 and r=0.875; it is positive for 0.375<r<0.875 and negative for 0<r<0.125, 0.125<r<0.375, and 0.875<r<1.5 (for r∈(0,1.5)).Figure 4
Heat capacity undergoes phase transition from instability to stability region and again to instability region, with divergence atr=2Q2/(1+n) and 2(1+n)Q2/(1+n); we chose Q=0.5, n=3, and M=1.


## 6. Conclusion

We have studied the thermodynamical properties on the inner and outer horizons of Kiselev solutions (RN black hole surrounded by energy-matter (radiation and dust) and Schwarzschild black hole surrounded by quintessence) and charged dilaton black hole. We have studied some important parameters of black hole thermodynamics with reference to their event and Cauchy horizons. We derive the expressions for temperatures and heat capacities of all the black holes mentioned above. It is observed that the product of surface gravities, surface temperature product, and product of Komar energies at the horizons are not universal quantities for the Kiselev’s solutions while products of areas and entropies at both horizons are independent of mass of black hole (except for Schwarzschild black hole surrounded by quintessence). For dilaton black hole these products are universal except the products of specific heat which has shown divergent properties due to the curvature singularity at the Cauchy horizon. Thus the implication of these thermodynamical products may somehow give us further understanding of the microscopic nature of black hole entropy (both exterior and interior) in the black hole physics.Using the heat capacity expressions, stability regions of the black holes are also observed graphically. Figures1–4 show that the above-mentioned black holes undergo a phase transition under certain conditions on r. For RN black hole surrounded by radiation and dust and Schwarzschild black hole surrounded by quintessence, we derived the first law of thermodynamics using the Smarr formula approach. It is observed that the third law of thermodynamics which states that “surface gravity, κ, of a black hole cannot be reduced to zero in a finite sequence of processes” holds for all the above-mentioned black holes. The derived expressions of κ show that it is zero for extreme black holes only.



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*Source: 124910-2015-09-08.xml*