Topological charged black holes in massive gravity's rainbow and their thermodynamical analysis through various approaches

Abstract

Violation of Lorentz invariancy in the high energy quantum gravity motivates one to consider an energy dependent spacetime with massive deformation of standard general relativity. In this paper, we take into account an energy dependent metric in the context of a massive gravity model to obtain exact solutions. We investigate the geometry of black hole solutions and also calculate the conserved and thermodynamic quantities, which are fully reproduced by the analysis performed with the standard techniques. After examining the validity of the first law of thermodynamics, we conduct a study regarding the effects of different parameters on thermal stability of the solutions. In addition, we employ the relation between cosmological constant and thermodynamical pressure to study the possibility of phase transition. Interestingly, we will show that for the specific configuration considered in this paper, van der Waals like behavior is observed for different topology. In other words, for flat and hyperbolic horizons, similar to spherical horizon, a second order phase transition and van der Waals like behavior are observed. Furthermore, we use geometrical method to construct phase space and study phase transition and bound points for these black holes. Finally, we obtain critical values in extended phase space through the use of a new method.

Effects of Planck-scale-modified dispersion relations on the thermodynamics of charged black holes

“…If we place this value of the momentum into an MDR, it gives us a dependence of the energy of the emitted photon with the mass of the black hole itself. This eliminates the energy-dependence of (27), and allows us to study Planck-scale departures induced by a rainbow metric without introducing extra degrees of freedom. Now, we shall quantify how the assumption of the MDRs chosen in the last section will modify some usual thermodynamic expressions.…”

“…The Hawking temperature is defined in the usual way, by Eq. (27). However, we now follow an approach different from the last section and let us suppose that we define the rainbow metric from measurements of photons with an average energy E = E such that we can identify the energy of the photons emitted from Hawking radiation with the black hole temperature, i.e, T = E [22].…”

“…From this identification, and the definition (27), the Hawking temperature T is implicitly defined by the equation:…”

“…Within this framework, the issue of black holes and their thermodynamics has been explored by many authors in several different contexts and background theories of gravity, see for instance [22][23][24][25][26][27][28][29][30]. Inspired by the Kiselev solution [31], the case of a quantum corrected black hole surrounded by a fluid with negative equation of state parameter was analyzed in [32], where this parameter assumed constant values for different kinds of fluids.…”

Modified fermion tunneling from higher-dimensional charged AdS black hole in massive gravity

“…where ξ α1···ακ and S are matrices and the action of the fermion, respectively [29,[43][44][45][46][47][48][49][50][51][52][53][54][55][56][57]. The angular momentum parameters and the radiation energy parameters of radiation particles are denoted as ∂ ϕ S = j and ∂ t S = −ω, respectively.…”