Effects of quantum corrections on the criticality and efficiency of black holes surrounded by a perfect fluid
We study some properties of the extended phase space of a quantum-corrected Schwarzschild black hole surrounded by a perfect fluid. In particular we demonstrate that, due to the quantum correction, there exist first and second order phase transitions for a certain range of the state parameter of the perfect fluid, and we explicitly analyze some cases. Besides that, we describe the efficiency of this system as a heat engine and the effect of quantum corrections for different surrounding fluids.
In the early seventies, Bekenstein proposed the idea that black holes have entropy and that this entropy is proportional to its area [1, 2]. At that time this idea was received with incredulity, since classical black holes cannot emit radiation, which means that they cannot have a well-defined temperature. It was Hawking who showed that, when quantum effects are taken into account, a black hole can indeed emit radiation, and this radiation can be understood as the radiation of the black hole as a perfect black body . Since then, several authors have studied such implications, and one of the most interesting is the possibility that black holes undergo phase transitions in the same way some thermodynamic systems do.
In daily life, the most usual transition we are aware is the phase transition of water becoming ice or vapor. Such transition occurs for some fixed temperature (isothermal curves) where the volume of the system varies for a fixed pressure. It can be modeled, as a first approximation, by the Van der Wall equation of state. One can be led to ask if some analogue procedure can also happen for a black hole, and for such task it is compulsory to properly define what is the pressure of the black hole. This issue is not straightforward, since for the most simple black hole, based on the Schwarzschild metric, a proper definition of pressure appears to be lacking. This issue was solved by Kastor et al.  by the use of the cosmological constant as a pressure term and the identification of the black hole’s mass with the enthalpy. Recently, Kubiznak and Mann  were able to study the P–V diagram of such thermodynamic black hole. Following these lines, one is able to identity the Schwarzschild anti-de Sitter spacetime as a full thermodynamic system.
As understood by Hawking, to properly study the thermodynamics of black holes one needs to take into account quantum effects, and to be consistent with this premise one should also take into account the back-reaction of these quantum effects on the metric. This procedure can be performed in a different number of ways, and a particular deformation of the Schwarzschild metric has been studied by Kazakov and Solodukhin , and recently extended for the Schwarzschild metric surrounded by quintessence by Shahjalal , extending the solution found by Kiselev .
In fact, Kiselev’s solution can be applied to the case of any perfect fluid surrounding the black hole. Using this technique, we shall recover the quantum corrected version of this metric found in , and study the extended phase space of this spacetime as a thermodynamic system, in which we follow the procedure of  and treat the perfect fluid as generator of the pressure term. This choice is consistent when the field is identified with a negative cosmological constant, but one should take care when one is dealing with arbitrary fields, such as quintessence, phantom matter and others.
As we shall see, the introduction of the quantum correction is important since, usually, to obtain phase transitions on a non-rotating black hole in General Relativity one needs to introduce charges in the black hole, so that the spacetime is described by the Reissner–Nordström anti-de Sitter metric instead of the Schwarzschild anti-de Sitter metric. If, instead, one introduces quantum corrections, then phase transitions can occur even in the absence of charges. Such phase transitions have been explored in several contexts and gravitational theories, for instance assuming non-linear electrodynamics [9, 10], a cloud of strings environment [11, 12, 13, 14], the effect of quark-gluon matter , alternative gravitational theories [16, 17, 18, 19, 20, 21, 22, 23], quantum gravity phenomenology , accelerating black holes  and other alternative formulations . Besides these, some properties of the thermodynamics of a quantum-corrected Schwarzschild black hole has been studied in .
Also, the presence of a pressure term allows us to consider thermodynamic cycles and the behavior of such black hole as a heat engine. We shall analyze the effect of quantum corrections on the efficiency of some processes for different perfect fluids surrounding the black hole. This framework was first idealized by Johnson  and has been explored in different contexts [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47].
This paper is organized in the following way. In Sect. 2 we will briefly review the metric for the quantum-corrected black hole surrounded by a perfect fluid. In Sect. 3 we will study the criticality of such objects, for different choices of matter fields. We will look for phase transitions of the first and second kinds. In Sect. 4 we will consider these black hole as heat engines, and describe their efficiency for different fluids. Finally, in Sect. 5, we will make some comments about the results that we found in this work.
2 Quantum corrected black hole surrounded by quintessence
2.1 Quantum corrected Schwarzschild metric
2.2 Quantum corrected black hole surrounded by a perfect fluid
3 Criticality of quantum corrected black hole
Considering the quantum corrected spherically symmetric black hole surrounded by a perfect fluid, we now move the description of the thermodynamic system in the so called extended phase space, in which the black hole’s mass is the enthalpy, the area of the event horizon is the entropy and the constant, c, of the perfect fluid plays the role of a pressure term. This approach has also been named as black hole chemistry , and allows us to study phenomena like phase transitions and criticality, besides to treat the black hole as a heat engine with a certain efficiency.
Considering these redefinitions, for some fluids surrounding the black hole, we depict \(p-x\) and \(g-t\) diagrams for fixed temperature and pressure, respectively.
3.1 Radiation fluid
3.2 Exotic fluids
The late-time cosmic acceleration of the universe bounds the equation of state of fluid responsible for this acceleration to have \(\omega <-1/3\) , but as we stated before, we only verify criticality for the opposite brach \(\omega >-1/3\). However, in order to illustrate the critical behavior of our black hole for exotic fluids, we analyze the latter case by assuming \(\omega =-1/6\) in Figs. 3a and 4a. This particular choice captures the overall nature of the \(p-x\) and \(g-t\) diagrams for any case in which \(\omega \in (-1/3,0)\). For negative values of \(\omega \), the critical pressure is always negative as can be verified from Eq. (26).
This is similar to a Hawking–Page phase transition that separates two phases of the black hole [52, 53]. The concave curve represents smaller values of the horizon radius in comparison to the convex curve. And since for the anti-de Sitter case, the mass grows with the horizon radius, the first phase (concave portion) corresponds to a lower mass and the second one (convex portion) to a higher mass black hole, respectively.
3.3 Heat capacity and second order phase transitions
4 Black hole as heat engine
Thermodynamics have started as a theoretical tool to study the efficiency of heat engines, and it is remarkable that the second law of thermodynamics, as developed by Carnot around 1820, has been first idealized in the framework of the old caloric theory of heat. But the thermodynamics of black holes took more than forty years, since its development in the seventies, to study black holes as thermal machines capable of producing work as the consequence of absorbing heat.
where \(T_C\) and \(T_H\) are the temperatures of the cold and hot reservoirs, respectively, which means that the efficiency cannot be equal to unity, since a reservoir cannot have zero temperature.
The efficiency for the Carnot cycle is depicted in Fig. 10 for several values of the parameter \(\omega \), and as function of the parameter for the quantum correction, a. As one can clearly see, the efficiency is improved for all fluids and approaches its maximum as a approaches some fixed value. This fixed value is not absolute but depends on the choices for the entropy and pressure of both reservoirs. The main concern here is that the introduction of a quantum correction improves the performance of the heat engine.
5 Concluding remarks
In this paper, we studied the effect of quantum corrections on the criticality and efficiency of black holes surrounded by a perfect fluid. The main idea was to verify if the system presents first or second order phase transitions due to the introduction of quantum corrections, as compared with the classical black hole.
Although the Schwarzschild metric does not present phase transitions, when one includes quantum corrections this feature is modified and phase transitions can occur. This is a nice feature by itself, since in general, to study phase transitions in a non-rotating black hole, one usually includes electric charge, and so the quantum-corrected black hole is an uncharged black hole where phase transitions can occur.
We showed that first order phase transitions occur for \(\omega >-1/3\) and \(w\ne 0\) verified from the existence of saddle points of isotherms in the \(p-x\) plane. We also verify a Hawking–Page-like second order phase transition for the cosmological constant case, that is found due to a discontinuity in the derivative of the Gibbs free energy with respect to the temperature, which is a manifestation of a transition that separates low and high mass black holes at a minimum temperature. Second order phase transitions were further explored for other specific fluids and also for a general state parameter.
The study of the black hole as a heat engine also show some interesting features. When we consider the most efficient cycle, the Carnot cycle, the efficiency increases as we increase the parameter of the quantum correction, independently of the equation of state we used. This does not guarantee that the same feature will happens for any equation of state, but at least for radiation, cosmological constant, some other exotic fields, it happens.
When one deals with some arbitrary cycle, the same cannot be stated. For the square cycle, for example, the efficiency of the heat engine does indeed depend on the equation of state, and our study indicates that, for non-exotic fluids, the efficiency increases as we increase the parameter related to the quantum correction, but for the exotic fluids the opposite occurs and the efficiency decreases.
The authors would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico - Brazil) for financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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