1 Introduction
Abstract

In this paper, we investigate the first order correction entropy and temperature in a charged black hole with a scalar field. Here, we apply such correction for the different the case of a black hole. Also, we take advantage of such corrections and study the critically a phase transition. Also, we investigate the effect of correction on the critical point and stability of the system. We obtain modified thermodynamics and find that correction terms are important in the stability of the black hole. Finally, we compare the results of the corrected and uncorrected by thermodynamical quantities.

Keywords: Scalar field, Hairy black holes, Logarithmic corrected entropy.

The temperature and entropy corrections on the charged hairy black holes

M. Rostami 111M.rostami@iau-tnb.ac.ir, J. Sadeghi 222pouriya@ipm.ir,  S. Miraboutalebi 333S-mirabotalebi@iau-tnb.ac.ir,  B. Pourhassan 444b.pourhassan@du.ac.ir,

, , Department of Physics, Tehran Northem Branch,

Islamic Azad University, Tehran, Iran

Department of Physics, Faculty of Basic Sciences,

University of Mazandaran, P. O. Box 47416-95447, Babolsar, Iran

School of Physics, Damghan University, Damghan, 3671641167, Iran

1 Introduction

Study about the fundamental forces like gravity is one of the main subjects of the elementary particle physics, which help us to understanding of nature and hence physics laws. As we know, the gravity in (2 + 1)-dimensional space-time is a very important topic of theoretical physics which usually considered as a toy model. These studies began in the early 1980 [1, 2, 3, 4], by the discovery of BTZ [5] and MTZ [6] black holes. It become clear that the three dimension solution getting more advantage. The charged black hole with a scalar field in (2 + 1) dimensions already studied by the Ref. [7]. In that case, the scalar field couples to gravity, and it couples to itself with the self-interacting potential too. Then, the similar black hole with a rotational parameter constructed by the Ref. [8] and then developed by the Ref. [9]. In that case rotating charged hairy black hole in (2+1) dimensions considered to study Klein-Gordon equation [10]. Also, some thermodynamical studies of such kind of black hole may be found in the Refs. [11, 12]. It has been shown that the entropy of large black holes is proportional to the horizon area [13, 14]. It is important to know what happened when the black hole size reduced where thermal fluctuations of the statistical physics yield to logarithmic corrections. It is indeed the first order corrections where the canonical ensemble is stable [15]. The general form of the corrections and their dependence on the horizon area is a universal feature, which appear in almost all approaches to quantum gravity. Main differences of various approaches is in the correction coefficients. Several works, already done to find the effect of the first order correction on the small black hole thermodynamics [16]. The logarithmic corrections to black hole entropy already obtained by counting microstates in non-perturbative quantum gravity [17, 18], also by using the Cardy formula [19, 20, 21]. Moreover, there are other methods where the corrected entropy given by [22, 23, 24, 25, 26, 27],

(1)

where is Hawking temperature which will be corrected later, and is a dimensionless parameter, which introduced for the first time by the Ref. [28], also is the Bekenstein-Hawking (BH) entropy. We can trace the effect of correction using and reproduce ordinary thermodynamics when . There is also other logarithmic corrected entropy as [29],

(2)

where is ordinary specific heat which will also corrected due to the thermal fluctuations. It is interesting to check that relation (1) with and relation (2) may yield to similar results for the thermodynamics quantities. It already applied to asymptotically black holes [30]. In the Ref. [31] modified thermodynamics of a black saturn has been studied by using the corrected entropy (2). Then, the corrected thermodynamics of a charged dilatonic black saturn by using both (1) and (2) investigated by the Ref. [32] and found similar results from both relations.
There is also other logarithmic corrected entropy given by,

(3)

which will be considered in this paper. If we assume then recover the corrected entropy given by [17, 22] where rotating BTZ black hole in the Chern-Simons formulation [17] and four dimensional Schwarzschild black hole in loop quantum gravity [22] considered. In general, it is possible to state that all the different approaches to quantum gravity generate logarithmic corrections (at the first order approximation) to the area-entropy law of a black hole. It should be noted that even though the leading order corrections to this area-entropy law are logarithmic, the coefficient of such a term depends on the approach to the quantum gravity. Since the values of the coefficients depend on the chosen approach to quantum gravity, we can say that such terms are generated from quantum fluctuations of the space time geometry rather than from matter fields on that space time. Hence, we consider general form given by (3) including free parameter which is depend on the given theory. However, it is possible to consider higher order corrections on the black hole entropy [33, 34, 35, 36, 37, 38].
If the BH entropy corrected, then other thermodynamic quantities also be corrected [39]. As we know, the thermodynamic stability analyzed by the Refs. [31, 32, 39, 40] under assumption of the fixed temperature. However, it is possible to consider corrected Hawking temperature as [41, 42, 43, 44, 45, 46],

(4)

We note here the phase transition, and critical point for the black holes have been investigation by several researchers [47]. The behavior of black hole compared to van der Waals fluid are studied by the Refs. [48, 49, 50, 51, 52]. In that case, we use holographic principles, and study the charged hairy black hole via a van der Waals fluid. We use mentioned motivations to investigate logarithmic corrected entropy and temperature of a charged hairy black hole. It is important to note that such thermal fluctuations may considered as quantum gravity effect [53, 54, 55].
All above information gives us motivation to organized paper as follows. In section 2, we review the charged hairy black holes. In section 3, we consider the effects of quantum corrections on the thermodynamics of charged BTZ black holes, and study the behavior of the first-order correction on the thermodynamics of quantities and stability of black holes. In section 4, we consider an uncharged hairy AdS black hole and derive the corrected thermodynamics. We also study the global and local stability of the uncharged hairy AdS black hole. In section 5, we discuss a conformally dressed AdS black hole and derive the corrected thermodynamics due to the thermal fluctuations. We also study the global and local stability of the conformally dressed AdS black hole. Finally, we summarize our results with concluding remarks in the last section.

2 Charged hairy black holes in (2+1) dimensions

Here, we consider the solution of Einstein-Maxwell which is coupled minimally to scalar field in (2 + 1) dimensions. The hairy black hole is the same solutions, and there are lots of texts about this corresponding theory [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67]. On the other hand, the scalar field may be coupled minimally or nonminimally to gravity. Here, also the self-interacting potential play important roles as to such model. The above mentioned coupled scalar field lead us to write the corresponding action as,

(5)

where is a constant, and shown the coupling power between gravity and the scalar field. The metric function is given by following expression,

(6)

where is the electric charge, and is a relation between the black hole charge and mass. The constant related to the cosmological constant by It is negative because smooth black hole horizons can be only in the presence of a negative cosmological constant in (2+1) dimensions, and explains the radial coordinate. The relation between and scalar field is as follow,

(7)

where,

(8)

These help us to study the modified thermodynamics due to the first order correction of the black hole entropy and temperature.

3 Corrected thermodynamics for charged BTZ black hole

If we assume in the equation (6), we will have a black hole solution without a scalar field, so

(9)

where this solution corresponding to charged BTZ black hole [68, 69, 70]. Therefore, the black hole mass establish as the follow,

(10)

where is the largest root of which interpreted as the outer horizon radius. Now, by using equations (9) and (10), the Hawking temperature of corresponding to the event horizon can be calculated by,

(11)

The charged BTZ black hole entropy is given by,

(12)

Here, the negative cosmological constant could interpreted as the positive thermodynamic pressure,

(13)

Utilizing the relations (3) and (4), the first-order corrected entropy and temperature for the charged BTZ black hole is obtained by the following equations,

(14)

and,

(15)

Here, we can see quantum correction effect in Fig. 1. In that case, we draw the temperature in terms horizon radius of the charged BTZ black hole. W cane see the quantum correction was affected by the small radius of black holes. Also, we see in Fig. 1 (a) for the case of large with positive correction, we have some maximum (some critical points). But in the case of large with negative correction we have some minimum points. For a large radius of the black hole, the correction terms do not more affect the temperature. Generally one can say that for the small correction of in contrast to the ordinary case the temperature will be increased.

Figure 1: The charged BTZ black hole; (a) temperature, (b) physical mass, (c) Helmholtz free energy; in terms for all values of .

From equation (10), the corrected physical mass for the charged BTZ black hole is obtained as,

(16)

We plot in Fig. 1 (b) the behavior of physical mass in terms of horizon radius for the corresponding black hole. As we know, here we consider the as a quantum correction. Here, one can see that there exists a critical point for physical mass which increases as well as increasing the black hole charge. By increasing parameter the physical mass increases and decreases before and after the critical horizon radius respectively. Also in Fig. 1 (b), we see the critical points were shifted by the different values of .
The first law of thermodynamic for the black hole reads,

(17)

where is the thermodynamic volume, and is the electric potential. In that case, by using entropy and temperature [71] one can obtain Helmholtz free energy as,

(18)

The first order correction of Helmholtz free energy can see by Fig. 1 (c). One can see that the behavior of Helmholtz free energy is decreasing for any values of . In Fig. 1 ( c), the Helmholtz free energy is decreased when the parameter is small. Also, we can see that the behavior of Helmholtz free energy comparing to uncorrected Helmholtz free energy is decreasing for and is increasing for . From the diagram Fig. 1 (c) in case of large horizon radius, one can see that the positive correction to the entropy and temperature will increase the Helmholtz free energy.
Using definition, , the internal energy is calculated as,

(19)

We have plotted the internal energy of the charged BTZ black hole in Fig. 2 (a). From the diagram, one can see that the small correction to the entropy and temperature will increase the internal energy of such black hole when the horizon radius is large. For the large values of parameter, the internal energy is increasing when the horizon radius was large, but the internal energy was decreased when the horizon radius was small. Finally one can obtain the enthalpy as following,

(20)

The effect of first order correction to the enthalpy in terms of the horizon radius can be seen in Fig. 2 (b). One can see that the behavior of enthalpy is decreasing for small values of . But, to increase the large values of parameter, the enthalpy is decreased when the horizon radius is small. But the enthalpy is increases when the horizon radius is large.

Figure 2: (a) The internal energy, (b) enthalpy, (c) Gibbs free energy, (d) specific heat : in terms of for all values of .

We will discuss the critical points and the stability of the charged BTZ black hole by using Gibbs free energy and specific heat in the next section.

3.1 The critical points and stability of charged BTZ black bole

Now, we are going to consider stability condition for the corresponding system. In that case, we need two quantities which play an important role for the study of the stability system as Gibbs free energy and heat capacity. The critical point for the charged BTZ black hole in the phase transition with the corrected entropy and temperature obtained by the following conditions,

(21)

which yields to the following relations,

(22)

and,

(23)

Using these conditions one obtain the critical values,

(24)

and,

(25)

the expression for specific volume is given by,

(26)

Now, the critical ratio is calculated as,

(27)

As we see and were increased but and are decreased by increasing . Also we observe when is negative and don’t change but and decrease. If , the above product will be as a usual relation This show when we have charged BTZ black hole the changed to form of without correction.
There is another method for studying critical behavior which in extended phase space where calculating critical quantities is easy and straightforward [72, 73]. When the Gibbs free energy is negative , the system has global stability. In order to discuss such global stability of the black hole, we need to calculate the Gibbs free energy in presence of quantum corrected entropy and corrected temperature which is given by,

(28)

The graphical analysis of the Gibbs free energy in the case quantum corrected entropy and temperature for charged BTZ black hole can be seen in Fig. 2 (c). As we see in Fig. 2 (c), parameter plays an important role in global stability for certain and parameters. We observe that the correction terms decrease the value of Gibbs free energy for such a black hole. For small values parameter, the correction terms do not effect the Gibbs free energy. The specific heat is an important measurable physical quantity and also is determine the local thermodynamic stability of the system. The mentioned specific heat has the following relation with corrected entropy and temperature :

(29)

in the cases of and the black hole is in stable and unstable phases respectively. So, corresponds to the phase transition of the van der Waals fluid. By using of the above equation, we can obtain specific heat as,

(30)

In Fig. 2 (d), we observe the behavior of the heat capacity at constant pressure for a few correction parameter in the positive values of and . Here, we can see the effects of the logarithm corrected entropy and corrected temperature on the stability of charged BTZ black hole. Also, we can see that the small values have no significant effect on the graph () and we choose large values of to see the changes. We can see that the stability of the black hole whit the large values parameter decreases ( increases ) in the negative ( positive ) of the region, respectively. The increasing charges help to obtain a smaller minimum size. Therefore we have more of a change to see the effects of thermal fluctuations. For the case of , one may have . We notice that the heat capacity with the correction parameter dose diverge for the values of constant Then there is the second-order phase transition.

4 Corrected thermodynamics for uncharged hairy AdS black hole

Now, if we assume in equation (6), in that case, the metric function reduced to the following expression,

(31)

where the physical mass is,

(32)

The thermodynamic volume can be calculated by,

(33)

So, the corresponding temperature will be,

(34)

the entropy density of Uncharged hairy black holes is also given by (12). Utilizing the relations (3) and (4), the first-order corrected entropy and corrected temperature for the uncharged hairy AdS black hole is computed as,

(35)

and,

(36)
Figure 3: The uncharged hairy AdS black hole; (a) temperature, (b) physical mass, (c) Helmholtz free energy; in terms of for all values of .

The temperature of the uncharged hairy AdS black hole is plotted in Fig. 3 (a), we see that the quantum correction is effect when horizon radius is small. For small values parameter, the correction terms do not effect the temperature. Also, we see that with increase the large with negative the maximum points decreases in the critical point. But with increasing of the large of with positive we will have the minimum points that decrease in the critical point. From equation (32), the corrected physical mass for the uncharged hairy AdS black hole is,

(37)

We plot Fig. 3 (b) to see the behavior of physical mass in terms of horizon radius. We fix and parameters too see that the physical mass is shifted by any correction of in the positive region. Also, the physical mass is an increasing function for the large of correction with negative values.
The first law of thermodynamic for such black hole given by the equation (17), So, here one can obtain the Helmholtz free energy for uncharged hairy AdS black hole which is obtained by the following expression,

(38)

where .
The effect of first order correction to the Helmholtz free energy can be seen for uncharged hairy AdS black hole in Fig. 3 (c). It can be seen that with increases parameter the behavior of Helmholtz free energy in construct uncorrected Helmholtz free energy is decreasing when horizon radius is small. For small values of parameter, the correction terms do not effect the Helmholtz free energy. The energy can be obtained as,

(39)
Figure 4: The uncharged hairy AdS black hole; (a) internal energy, (b) enthalpy, (c) Gibbs free energy, in terms of for all values of .

Finally one can obtain the enthalpy as following,

(41)

We have plotted the internal energy and enthalpy of the uncharged hairy AdS black hole in Fig. 4. From this diagram, one can see that the small corrections to the entropy and temperature do not effect on the internal energy and enthalpy of such black hole. Also in Fig. 4 (a) and (b) we see that the internal energy and enthalpy are shifted to positive region by any value of It can also be seen from the Fig. 4 that existence of increases value of the internal energy and enthalpy.
We will discuss the critical points and the stability of the uncharged hairy AdS black hole in the next section.

4.1 Stability of uncharged hairy AdS black hole

The critical point for the black hole in the phase transition can be obtained by the condition (21), which yields to the following critical point,

(42)

Now, we want to discuss the global stability condition for the corresponding system. We drive the Gibbs free energy in the case of quantum corrected entropy and corrected temperature which is given by,

(43)

The graphical analysis of the Gibbs free energy in case quantum corrected entropy and the corrected temperature for the uncharged hairy AdS black hole can see in Fig. 4 (c) where we fix constant and . In Fig. 4 (c), we observe that correction terms are a few effects the Gibbs free energy much. But the Gibbs free energy will have global stability.
In order to discuss the local stability of the black hole we need to calculate the specific heat at constant pressure which is given by,

(44)
Figure 5: The uncharged hairy AdS black hole, (a) heat capacity at constant pressure, ( b ) heat capacity at constant volume in terms; for ( c) heat capacity at constant volume for ; and all values of .

We draw in terms of with constant values of parameter and quantum correction coefficient for uncharged hairy AdS black hole in Fig. 5.

Also, we see in Fig. 5 (a) for the case of large with positive correction we have some minimum points (critical points) in the positive region of . But in case of large with negative correction we some minimum points in the negative region of . We notice that the heat capacity for different values of is diverge that there is the second order phase transition. As we see in figure, the heat capacity has a discontinuity for values of with negative, but by increasing this parameter, the points of discontinuity decrease and the heat capacity is negative for such black hole which suggests the black hole is thermodynamically unstable. The important consequence of quantum correction is that the black hole is unstable for and stable for . We can see that there are some stable region corresponding to . Furthermore, is a minimum value of the black hole horizon radius where the phase transition happen. For example, at phase transition happen at and at phase transition happen at . So, the black hole is completely stable for .
We notice that the heat capacity with correction parameter is diverge for small values of constant B then there is the second-order phase transition. Also, we see in the figure, the heat capacity has a discontinuity for positive values of parameter but by decreasing this parameter the points of discontinuity decrease and the heat capacity are positive for such a black hole which suggest that the black hole is thermodynamically stable.
The heat capacity at constant volume is,

(45)

We draw in terms of with different values of B parameter and quantum correction coefficient for Uncharged hairy AdS black hole in Fig. 5. Also, we can see that the small values have no significant effect on the graph , and we choose large values of to see the changes. In Fig. 5, we observe the behavior of the heat capacity at constant volume for a few correction parameter with the positive B. We see that all the points in the minimum occur in the region . We notice that the heat capacity with correction parameter dose diverge for positive values of B that with increasing parameter the slop of the graph decreases. And all of these graphs confluence at a point in the negative region until is zero.

In right plot of Fig. 5 we observe the behavior of the heat capacity at constant volume for a few correction parameter in with the negative . We see that all the points in the maximum occur in region We notice that the heat capacity with correction parameter dose diverge for negative values of B that with increasing parameter the slop of the graph increases. And all of these graphs confluence at a point in the positive region until is zero. Meanwhile we observe the mirroring behavior of the heat capacity with respect .
For the local stability equilibrium for a thermodynamic system requires that,

(46)

which yields,

(47)

Therefore we can find,

(48)

It is completely agree with choosing [74].

5 Corrected thermodynamics for conformally dressed AdS black hole

If we assume in the equation (6), we will have the conformally dressed AdS black hole solution. In this section, we will use the form for the corrections to the entropy and the temperature of a conformally dressed AdS black hole to obtain an explicit expression for various thermodynamic quantities. Then, we will use these explicit values to study phase transition in this system. So,

(49)

Because of the mass will be,

(50)

Now, exploiting relations (49) and (50), the Hawking temperature of the event horizon can be calculated by

(51)

The entropy density of conformally dressed AdS black hole is given by (12). Hence, the first-order corrected entropy and temperature for the conformally dressed AdS black hole is computed as

(52)

and,

(53)

The temperature was plotted in Fig. 6. We can observe that the effects of quantum correction increases the value of temperature.

Figure 6: The conformally dressed AdS black hole; (a) the temperature, (b) the physical mass, (c) the Helmholtz free energy ; in terms of for all values of .

From equation (50), the corrected physical mass for the conformally dressed AdS black hole is,

(54)

We plot Fig. 6 to see the behavior of physical mass in terms of horizon radius for such black hole. Also, we can see that for small values the physical mass behave like uncorrected physical mass and we choose large values of to see the changes. We see in Fig. 6 (b) that by increasing the large with positive the physical mass increases when horizon radius is small. Also, we can see that the large with negative behaves like the large with positive behavior.
The thermodynamic first law for the black hole reads

(55)

We can obtain the Helmholtz free energy for the conformally dressed AdS black hole as following,

(56)

The effect of first - order correction to the Helmholtz free energy can be seen in Fig. 6. It can be seen that with increases parameter the Helmholtz free energy is increased. Also in Fig. 6 (c) we see that the Helmholtz free energy is shifted to the positive region by any value of .
The internal energy is calculated by,

(57)

We have plotted the internal energy and enthalpy in Fig.7. From the diagram, we see that the internal energy and enthalpy are shifted to the positive region by any value of Also in Fig. 7, one can see that the small correction to the entropy and temperature will effect the internal energy of such a black hole to some extent when the horizon radius is small and positive.
Also, we can see that with increases the large of with positive in construct the negative value the internal energy and enthalpy is increases when the horizon radius is positive.

Figure 7: The conformally dressed AdS black hole: (a) the internal energy, (b) the Enthalpy (c) the Gibbs free energy (d) the heat capacity; for all values of

And,

(58)

Now, We will discuss the critical points and the stability of the conformally dressed AdS Black hole in the next subsection.

5.1 Phase transition for conformally dressed AdS black bole

We can obtain the Gibbs free energy as the following expression,

(59)

We see in Fig. 7 (c), the graphical analysis of the Gibbs free energy in case quantum corrected entropy and the temperature for conformally dressed AdS black hole. We observe that the correction terms increase the Gibbs free energy for such black hole when is positive. So, the Gibbs free energy in this case, has not global stability. Also, we can see that the Gibbs free energy will have global stability for negative the correction terms. The specific heat at constant pressure with corrected entropy and corrected temperature can obtain as follows,

(60)

In Fig. 7 (d), we observe the behavior of the heat capacity at constant pressure for any values of the parameter. In this diagram, we can see that the small values of have a few effect on the graph (). But by choosing , the corrected heat capacity is decreasing. For the large of with positive values, the effect of the corrected heat capacity with the negative is the same.
Therefore with the large of , the black hole is stable (unstable) in the negative (positive) region. Without quantum corrections, it is clear that the black hole will have not critical points. So, we can see that we will have the first-order and second-order phase transitions for such black by quantum corrections.

6 Conclusion

In this paper, we have analyzed the thermodynamics of hairy black hole that contains a scalar field coupled minimally or nonminimally to gravity. First of all, we imply different cases of the charged hairy black hole. We have studied effects from the entropy and temperature with logarithmic corrected terms on the thermodynamics of hairy black holes. Considering the corrections to the entropy and temperature, specific heat and thermodynamical quantities were calculated for charged BTZ, uncharged hairy AdS, and conformally dressed AdS black holes. Besides, we calculate internal energy, enthalpy, Helmholtz energy and Gibbs free energy for the above black holes and analyze the effects of corrected entropy and temperature.

Critical values of event horizon radii for phase transitions are shown to was shifted due to the corrections of entropy and temperature. This shifting is also indicated by physical mass, specific heat and enthalpy for charged BTZ black hole.

Our main work is finding the effect of corrections on the thermodynamics quantities. Corrections exist for any black hole, but they are important for a small black holes and negligible for the large black holes. The advantage of a charged BTZ black hole is its holographic picture, which is a van der Waals fluid. We have shown that in the presence of corrections there is still a van der Waals fluid as a dual picture. Only we should fix the black hole charge and which corresponds to the electric charges of a van der Waals fluid. We obtained some thermodynamics quantities like Gibbs and Helmholtz free energies and showed that corrections don’t have important effects on the large black hole.
On the other hand, for the small black holes, they are important and have a crucial role. Using specific heat, we found that corrections reduced stable regions of the black hole. However, there are enough stable regions to see quantum gravity effects before the phase transition of a van der Waals fluid. This means that there is a minimum radius, which we call the critical radius, where a black hole is stable in the presence of corrections, and in this region, a charged BTZ black hole in the presence of corrections is a dual of the van der Waals fluid.
It is still possible to find effect of such logarithmic correction on a Hyperscaling violation background [75] and discuss stability conditions.

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