Motivated by low energy effective action of string theory and large applications of BTZ black holes, we will consider minimal coupling between dilaton and nonlinear electromagnetic fields in three dimensions. The main goal is studying thermodynamical structure of black holes in this set up. Temperature and heat capacity of these black holes are investigated and a picture regarding their phase transitions is given. In addition, the role and importance of studying the mass of black holes is highlighted. We will see how different parameters modify thermodynamical quantities, hence thermodynamical structure of these black holes. In addition, geometrical thermodynamics is used to investigate thermodynamical properties of these black holes. In this regard, the successful method is presented and the nature of interaction around bound and phase transition points is studied.Comment: 15 pages, 8 figures, 1 table. accepted in Phys. Lett.

Motivated by the energy representation of Riemannian metric, in this paper we study different approaches toward the geometrical concept of black hole thermodynamics. We investigate thermodynamical Ricci scalar of Weinhold, Ruppeiner and Quevedo metrics and show that their number and location of divergences do not coincide with phase transition points arisen from heat capacity. Next, we introduce a new metric to solve these problems. We show that the denominator of the Ricci scalar of the new metric contains terms which coincide with different types of phase transitions. We elaborate the effectiveness of the new metric and shortcomings of the previous metrics with some examples. Furthermore, we find a characteristic behavior of the new thermodynamical Ricci scalar which enables one to distinguish two types of phase transitions. In addition, we generalize the new metric for the cases of more than two extensive parameters and show that in these cases the divergencies of thermodynamical Ricci scalar coincide with phase transition points of the heat capacity.

Motivated by recent developments in black hole thermodynamics, we investigate van der Waals phase transitions of charged black holes in massive gravity. We find that massive gravity theories can exhibit strikingly different thermodynamic behaviour compared to that of Einstein gravity, and that the mass of the graviton can generate a range of new phase transitions for topological black holes that are otherwise forbidden.PACS numbers: 04.70. Dy, 04.40.Nr, 04.20.Jb, 04.70.Bw Introduction-Understanding the quantum behaviour of gravity could be related to the possible mass of the graviton. The consistency of including this in the context of extending general relativity has been a long-standing basic physical question of classical field theory. Although the primitive linear theory of massive gravity [1] contains Boulware-Deser ghost modes [2], a nonlinear generalization that is ghost free to all orders has recently been constructed [3] by de Rham, Gabadadze and Tolley (dRGT massive theory). dRGT is stable, enjoys the absence of Boulware-Deser ghost [4], and has yielded a number of intriguing results in terms of its cosmological behaviour and black hole solutions [5]. The mass terms are produced by consideration of a reference metric, which plays a crucial role in the construction of dRGT [6]. Motivated by applications of gauge/gravity duality, Vegh proposed a new reference metric in which the graviton behaves like a lattice excitation and exhibits a Drude peak [7]. This theory is also ghost-free and stable [8], and d−dimensional (d ≥ 3) black hole solutions in the presence of linear and nonlinear electrodynamics with van der Waals like behavior have been obtained [9]. Higher curvature generalizations have also been constructed [10]. Although some classes of nonlinear massive gravity theories are Lorentz-violating and bear a close relation to Horava-Lifshitz gravity [11], it was shown that there are Lorentz invariant versions of nonlinear massive gravity as well [3]. Massive gravity is also motivated by observation. Obtaining an empirical upper limit on the mass of the graviton remains an outstanding challenge (for more details see [12]) , one that should soon be attainable once recent LIGO results [13] are improved and expanded. In this regard, one may use the results of Refs. [14] and [15] to obtain a bound on the energy flux emitted from a binary pulsar and on the propagation speed of the graviton.Here, we consider a class of dRGT theories, which we regard as the minimal modification of general relativity that yields a massive graviton [3]. We demonstrate that black holes of non-spherical topology can exhibit van der Waals phase transitions in dRGT like gravity. Such tran-

Recent astronomical observations indicate that our Universe is currently undergoing a phase of accelerated expansion [1]. In the context of standard cosmology, based on Einstein gravity, this acceleration cannot be explained unless an unknown energy component usually dubbed "dark energy" is proposed. Another way for explanation of such an acceleration is the modification of the Einstein theory of gravity. In this regards, various modifications of Einstein gravity proposed in the literatures. Among them are Lovelock gravity [2], braneworld scenario [3], scalar-tensor theories [4,5], f (R) gravity [6], etc.The studies on the black hole as a thermodynamic system date back to the work of Hawking and Bekenstein [7]. According to the black holes thermodynamics, the geometrical quantities such as horizon area and surface gravity are related to the thermodynamic quantities such as entropy and temperature. The first law of black hole thermodynamics implies that the entropy and the temperature together with the energy (mass) of the black hole satisfy dE = T dS [7]. In recent years, the investigations on the thermodynamical properties of the black holes have got a lot of interests. In particular, thermodynamic properties of black holes in anti de-Sitter (adS) spaces are improved in an extended phase space in which the cosmological constant and its conjugate variable are considered as thermodynamic pressure and volume, respectively [8][9][10]. In addition, there has been some proposals to consider constants (such as Born-Infeld nonlinearity parameter, Gauss-Bonnet parameter, Newton constant and etc.) as thermodynamical variable which contributes to thermodynamical behavior of the system [11]. It was shown that considering these constants as thermodynamical variables will enrich the phase structure of the black holes and describe new and interesting phenomena such as Van der Waals like liquid/gas behavior. In this paper, motivated by these reasons, we will extend the phase structure of dilatonic black holes by considering dilaton parameter as a thermodynamical extensive parameter.Recently, there has been several attempts in studying the phase transition in dynamical context. It was shown that the quasinormal modes of a perturbed black hole near critical point, exhibits different behaviors. In Ref. [12], it was argued that due to existence of normal modes only for massless BTZ black holes, there is a phase transition from non-rotating BTZ black holes. On the other hand, the four dimensional topological black holes with scalar hair present the signature of the phase transition in their quasi normal modes [13]. Some evidences regarding second order phase transition of a topological black hole to hairy one are given in Ref. [14]. Furthermore, a dramatic change in the slopes of quasinormal frequencies in small and large black holes near the critical point was observed for four-dimensional Reissner-Nordström-adS black holes [15].One of the interesting aspects of the black hole thermodynamics is stability of black holes. In order to...

In this paper, we consider a spherical symmetric metric to extract the hydrostatic equilibrium equation of stars in (3+1)-dimensional gravity's rainbow in the presence of cosmological constant. Then, we generalize the hydrostatic equilibrium equation to d-dimensions and obtain the hydrostatic equilibrium equation for this gravity. Also, we obtain the maximum mass of neutron star using the modern equations of state of neutron star matter derived from the microscopic calculations. It is notable that, in this paper, we consider the effects of rainbow functions on the diagrams related to the mass-central mass density (M-ρc) relation and also the mass-radius (M-R) relation of neutron star. We also study the effects of rainbow functions on the other properties of neutron star such as the Schwarzschild radius, average density, strength of gravity and gravitational redshift. Then, we apply the cosmological constant to this theory to obtain the diagrams of M-ρc (or M-R) and other properties of these stars. Next, we investigate the dynamical stability condition for these stars in gravity's rainbow and show that these stars have dynamical stability. We also obtain a relation between mass of neutron stars and Planck mass. In addition, we compare obtained results of this theory with the observational data.

In this paper, we take into account the black hole solutions of Einstein gravity in the presence of logarithmic and exponential forms of nonlinear electrodynamics. At first, we consider the cosmological constant as a dynamical pressure to study the analogy of the black hole solutions with the Van der Waals liquid-gas system in the extended phase space. We plot P − v, T − v and G − T diagrams and investigate the phase transition of adS black holes in the canonical ensemble. We study the nonlinearity effects of electrodynamics and see how the power of nonlinearity affects critical behavior and phase transition of the system. We also investigate the effects of dimensionality on the critical values and analyze its crucial role. Moreover, we show the changes in the universal ratio Pcvc/Tc for variation of different parameters. In addition, we make a comparison between linear and nonlinear electromagnetic fields and show that the lowest critical temperature belongs to Maxwell theory. Also, we make some arguments regarding to how power of nonlinearity brings the system to Schwarzschild-like and Reissner-Nordström-like limitations. Next, we study the critical behavior of the system in context of heat capacity. We show that critical behavior of system is similar to the one in phase diagrams of extended phase space. We point out that phase transition points of the extended phase space only appear as divergencies of heat capacity. We also extend the study of phase transition points through geometrothermodynamics (GTD) method. We introduce two new thermodynamical metrics for extended phase space and show that divergencies of thermodynamical Ricci scalar of the new metrics coincide with phase transition points of the system. The characteristic behavior of these divergencies, hence critical points is exactly the one that is obtained in extended phase space and heat capacity. Then, we introduce a new method for obtaining critical pressure and horizon radius by considering denominator of the heat capacity. We show that there are several benefits that make this approach favorable comparing to other ones.

The paper at hand studies the heat engine provided by black holes in the presence of massive gravity. The main motivation is to investigate the effects of massive gravity on different properties of the heat engine. It will be shown that massive gravity parameters and graviton's mass modify the efficiency of engine on a significant level. Furthermore, it will be shown that it is possible to have the heat engine for non-spherical black holes in massive gravity and we study the effects of topological factor on properties of the heat engine. Surprisingly, it will be shown that the highest efficiency for the heat engine belongs to black holes with hyperbolic horizon, while the lowest one belongs to spherical black holes.

In this paper, we study massive gravity in the presence of Born-Infeld nonlinear electrodynamics. First, we obtain metric function related to this gravity and investigate the geometry of the solutions and find that there is an essential singularity at the origin (r = 0). It will be shown that due to contribution of the massive part, the number, type and place of horizons may be changed. Next, we calculate the conserved and thermodynamic quantities and check the validation of the first law of thermodynamics. We also investigate thermal stability of these black holes in context of canonical ensemble. It will be shown that number, type and place of phase transition points are functions of different parameters which lead to dependency of stability conditions to these parameters. Also, it will be shown how the behavior of temperature is modified due to extension of massive gravity and strong nonlinearity parameter. Next, critical behavior of the system in extended phase space by considering cosmological constant as pressure is investigated. A study regarding neutral Einstein-massive gravity in context of extended phase space is done. Geometrical approach is employed to study the thermodynamical behavior of the system in context of heat capacity and extended phase space. It will be shown that GTs, heat capacity and extended phase space have consistent results. Finally, critical behavior of the system is investigated through use of another method. It will be pointed out that the results of this method is in agreement with other methods and follow the concepts of ordinary thermodynamics.