# PV criticality of the second order quantum corrected Hořava–Lifshitz black hole

## Abstract

In this paper, higher order quantum gravity effects on the thermodynamics of Hořava–Lifshitz black hole investigated. Both Kehagius–Sfetsos and Lu–Mei–Pop solutions of Hořava–Lifshitz black hole considered and higher order corrected thermodynamics quantities obtained. The first order correction is logarithmic and second order correction considered proportional to the inverse of entropy. These corrections are due to the thermal fluctuation and interpreted as quantum loop corrections. Effect of such quantum corrections on the stability and critical points of Horava–Lifshitz black holes studied. We find that higher order correction affects critical point and stability of Lu–Mei–Pop solution and yield to the second order phase transition for the case of Kehagius–Sfetsos solution.

## 1 Introduction

Black holes are gravitational objects with maximum entropy [1, 2] which is related to the black hole event horizon area *A*. In that case, statistical mechanics help us to study the black hole thermodynamics and it is one of the important field of theoretical physics [3, 4]. Hence, the thermodynamics study of several kinds of black hole is important subject of the recent researches. Lower dimensional black holes are interesting and usually considered as a toy model to obtain gravitational solutions. For example, two-dimensional black holes may be considered to study thermodynamics [5, 6, 7, 8, 9, 10]. Higher dimensional black holes are also interesting in several theories like near-extremal solutions of Einstein-Maxwell-scalar theory [11], or hyperscaling violation backgrounds [12, 13, 14].

Due to the thermal fluctuations, the black hole entropy modified and correction terms interpreted as quantum effect, because the quantum gravity modified the manifold structure of space-time at Planck scale [15, 16]. Almost all methods of quantum gravity indicated that the leading order correction is logarithmic [17, 18] and it has been argued that the general structure of the correction terms is a universal. In that case, leading order quantum corrections to the geometry of large AdS black holes and their effects on the thermodynamics given by the Ref. [19]. Hence, the thermal fluctuations in a hyperscaling violation background considered by the Ref. [20]. Thermodynamics of Kerr–Newman–AdS black holes already studied by the Ref. [21]. Similar black hole (uncharged but *d*-dimensional) considered under the effect of the thermal fluctuations [22] and found that logarithmic correction becomes important when the size of the black hole becomes small due to the Hawking radiation [23]. Also, statistical mechanics of charged black holes considered by the Ref. [24]. Black hole thermodynamics in modified theories of gravity like *f*(*r*) [25, 26, 27, 28] also studied by the Ref. [29].

It is also possible to obtain higher order corrections [30, 31] and such corrected terms has the same universal shape as expected from the quantum gravitational effects [17, 18, 30]. Black holes in Gödel universes already introduced by the Ref. [32], in that case Kerr–Gödel black hole thermodynamics investigated by the Ref. [33]. Logarithmic correction to the Gödel black hole has been studied by Ref. [34]. Higher order correction to the Kerr–Newman–Gödel black hole recently given by the Ref. [35] and demonstrated that second order correction is proportional to the inverse of entropy. Thermodynamics of higher order entropy corrected Schwarzschild–Beltrami–de Sitter black hole investigated by the Ref. [36] and found that higher order corrections may affect the black hole stability.

STU black holes [37] are other interesting kinds of black hole which has been studied by the Ref. [38] from statistical point of view [39]. This kind of black hole is interesting in AdS/CFT correspondence [40], for example it is possible to compute hydrodynamics and thermodynamics properties of quark-gluon plasma [41, 42, 43, 44] in presence of quantum correction. Other properties of quark-gluon plasma like drag force [45, 46, 47, 48], jet-quenching [49, 50, 51, 52] and shear viscosity to entropy ratio [53] may also affected by thermal fluctuations, and this can observed by experiments [54].

It is important to note that corrections to the black hole thermodynamics can study using the non-perturbative quantum general relativity. Moreover, it is possible to investigate black hole thermodynamics under effects of thermal fluctuations by using the effect of matter fields surrounding a black hole [55, 56, 57]. The corrected thermodynamics of a dilatonic black hole from Cardy formula have been already studied by the Ref. [58] and show the same universal form of correction term as previous studies. This universality can be understood in the Jacobson formalism [59, 60]. Corrected thermodynamics of a black hole can be obtained by using the partition function [61]. Such corrections have already considered for different black objects. For example, the first order correction of the AdS charged black hole has been studied, and modified thermodynamics obtained [62], which is extended to the case of AdS charged rotating black hole [63]. In the Ref. [62] it is argued that leading order correction is logarithmic its coefficient can be considered as free parameter of the theory. Thermal fluctuations effect of a black saturn have been studied [64]. It was observed by considering charged dilatonic black saturn that the thermal fluctuations can be obtained either using a conformal field theory or by analyzing the fluctuations in the energy of this system and both yields to the similar results for a charged dilatonic black saturn [65].

An interesting kind of regular black holes is Hayward black hole [66] which can be modified [67, 68]. In that case, the logarithmic corrections of a modified Hayward black hole considered to calculate some thermodynamics quantities, and found that this leading order correction affect the pressure and internal energy by decreasing value of them [69]. Thermal fluctuations of charged black holes in gravity’s rainbow investigated by the Ref. [70], and quantum corrections to thermodynamics of quasitopological black holes studied by the Ref. [71].

Investigation of thermal fluctuation in gravitational systems may help us to test the quantum gravity effects, for examples, on dumb holes [72] or graphene [73]. Logarithmic and higher order corrections may affect the critical behaviors of black objects, for example in the Ref. [74] a dyonic charged anti-de Sitter black hole thermodynamics considered and show that holographic picture (a van der Waals fluid) is still valid. In that case logarithmic corrected van der Waals black holes in higher dimensional AdS space investigated by the Ref. [75]. Also, thermodynamics of higher dimensional black holes with higher order thermal fluctuations studied in [76].

BTZ black holes [77] also considered to investigate effects of thermal fluctuations [78] including higher order corrections [79, 80, 81]. P-V criticality of black holes also may affected by thermal fluctuations [82, 83, 84].

Hořava–Lifshitz (HL) black holes are important kind of black holes in theoretical physics [85]. The HL gravity is also an interesting theory of quantum gravity [86, 87, 88, 89] which considered in particle physics and cosmological literatures [90, 91]. We expect that the HL black hole solutions, asymptotically, become Einstein gravity solutions. In that case, Refs. [92, 93, 94, 95] investigated thermodynamics quantities of HL black holes. It has been reported some instabilities in HL black holes. Hence, such thermodynamics using logarithmic corrected entropy investigated to find quantum gravity effects [96] and found that some instabilities removed due to thermal fluctuations.

In the Ref. [96], only LMP solutions of HL black hole considered and first order correction (logarithmic correction) investigated. Now, in this paper, we would like to consider both LMP and KS solutions and investigate effects of higher order quantum corrections.

This paper organized as follows. In the Sect. 2 we review higher order correction and propose a general form with free correction coefficients. In Sect. 3, Hořava–Lifshitz (HL) black holes properties called. General thermodynamics relations introduced in Sect. 4. In Sect. 5, effects of higher order corrections on the thermodynamics of Lu–Mei–Pop solution investigated for three different cases of flat, spherical and hyperbolic spaces. In Sect. 6 we consider Kehagius–Sfetsos solution of HL black hole and study corrected thermodynamics. Finally, in Sect. 7 we give conclusion.

## 2 Higher order corrections

*N*particles given by the following expression [97, 98],

*E*is the average energy in the canonical ensemble. Having partition function, one can write entropy as,

## 3 Hořava–Lifshitz black hole

*N*is lapse function, also the cosmological constant is given by \(\Lambda =\frac{3}{2}\Lambda _{W}\) [106], where \(\Lambda _{W}\) is a constant negative parameter (we consider \(\Lambda _{W}=-l\)).

*f*(

*r*) is given by the following relation [100],

*B*is an integration constant.

## 4 Thermodynamics

*H*denotes the enthalpy and interpreted as the black hoe mass (\(H=M\)) [110]. Also temperature of black hole is given by the following relation [110],

Thermodynamical quantities of the HL black hole for LMP and KS solutions are different for spherical space \((k=1)\), flat space \((k=0)\), and hyperbolic space \((k=-1)\).

## 5 Lu–Mei–Pop solution

LMP solution given by the Eq. (13) and there are three different cases corresponding to space curvature.

### 5.1 Spherical space

Now, we can use the Eq. (5) to obtain corrected thermodynamics due to higher order quantum corrections.

*P*and

*V*, we can study \(P-V\) critical points where

Without the logarithmic correction we saw some instabilities corresponding to the small horizon radius (see dotted blue line of Fig. 3). It has been found that the magnitude of the cosmological constant increases the black hole heat capacity. The zero heat capacity limit obtained by \(r_{h}=1/\sqrt{3l}\). It means that there are some instabilities for small \(r_{h}\) where *C* is negative. Although, quantum corrections have not any important effect on this point but may make other stable points at very small radius. In all cases there is no any asymptotic behavior corresponding to phase transition. Solid cyan line of the Fig. 3 is corresponding to the case of \(\alpha =\gamma =-1\) which shows a turning point near \(r_{h}\approx 0.3\) (critical point and maximum of Helmholtz free energy). Some cases of corrected heat capacity (dashed lines) show that there is some stable region for small radius, then the black hole is unstable in a finite rage of \(r_{h}\). It means that, when the black hole size reduced due to the Hawking radiation, and thermal fluctuations become important, then the black hole is stable at quantum scales. Hence, we can’t neglect thermal fluctuations of small black holes.

### 5.2 Hyperbolic space

As we can see from the Fig. 5, Helmholtz free energy of this case is completely negative with an extremum for the uncorrected entropy and two extrema for the higher order corrected entropy.

### 5.3 Flat space

Now, we can use the Eq. (5) to obtain corrected thermodynamics due to higher order quantum corrections.

In the Fig. 9 we can see behavior of Helmholtz free energy of the LMP solution in flat space and find important effect of quantum corrections. Neglecting thermal fluctuations, the Helmholtz free energy is completely negative. In presence of higher order corrections (solid cyan line of the Fig. 9) the Helmholtz free energy has a maximum in positive regions.

Finally, we give graphical analysis of internal energy by Fig. 11. Effect of quantum corrections are important which may reduce or increase value of internal energy as illustrated by the Fig. 11.

## 6 Kehagius–Sfetsos solution

*M*from (45) into the temperature (46) one can obtain,

## 7 Conclusion

In this paper, we considered Hořava–Lifshitz black hole to study its thermodynamics in presence of quantum correction by consideration of thermal fluctuations. We considered two special case of Kehagius–Sfetsos and Lu–Mei–Pop solutions to study critical points and thermodynamics stability separately. In the case of KS solution, we consider general case and found that logarithmic correction may yield to the second order phase transition. Higher order correction with positive coefficient may yield to some instabilities at small event horizon radius (see Fig. 13b). We found that there are also critical points in presence of higher order correction as well as in absence of quantum corrections.

In the case of LMP solution, we considered three cases of flat, spherical and hyperbolic spaces and studied modified thermodynamics separately. Both analytical and numerical study on the thermodynamics quantities under effects of quantum corrections show that quantum corrections are important when the black hole size reduced due to the Hawking radiation. In all cases we found that Helmholtz free energy is decreasing function of correction parameters. Then, we obtained pressure and volume to study critical points. In the case of LMP solution with spherical space we found that higher order corrections are important to have critical point. Fig. 2a shows that only higher order corrections with negative coefficients yields to critical point which is opposite to hyperbolic space. It means that LMP solution in hyperbolic space have critical point for higher order corrections with positive coefficients. In flat space there is no critical point and quantum corrections have no any effects on critical points.

We study black hole stability by using heat capacity. We found that first order correction increases value of heat capacity while the second order correction reduces its value. In the case of spherical space, we found some stable points at very small radius due to higher order corrections and black hole is stable at quantum scales. In the case of hyperbolic space, we found that there is phase transition in all cases of corrected and uncorrected entropy. It means that higher order corrections have no any effects on the black hole phase transition. In the case of flat space heat capacity is completely positive without higher order corrections. Logarithmic correction reduces value of heat capacity to make it negative for the small black hole, when event horizon radius grows up, then the effect of thermal fluctuation is negligible. Higher order correction with positive coefficient also decrease value of the heat capacity, hence change the black hole stability.

In this paper, we considered only presence or absence of corrections by choosing \(\alpha \) and \(\gamma \) as 0 or \(\pm 1\). It is interesting to calculate exact values of these parameters which may be yields to experimental setup to test quantum gravity. It may be aim of our future study.

In summary, this paper is extension of previous work to include higher order correction to LMP solution, while logarithmic and higher order corrections of KS solution of HL black holes. It is interesting to consider higher order correction of a new regular black hole [114], Myerse–Perry black holes [115], rotating charged hairy black hole [116, 117], or five dimensional AdS black hole at \(N=2\) supergravity [118, 119].

Recently, Thermodynamic of a black hole surrounded by perfect fluid in Rastall theory have been studied [120]. Now, it is interesting to obtain thermal fluctuation effects on such kind of solutions.

## Notes

### Acknowledgements

Author would like to thanks Iran Science Elites Federation.

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