P − V Criticality of the Non-linear Charged Black Hole Solutions in Massive Gravity

Abstract

In this paper, we explore the black hole solutions with the rainbow deformed metric in the presence of the exponential form of the nonlinear electrodynamics with asymptotic Reissner- Nordstr ̈om properties. We calculate the exact solution of metric function and explore the geometri- cal properties in the background of massive gravity. From the obtained solution, the existence of the singularity is confirmed in proper limits. Using the solutions we also investigate the thermodynamic properties of the solutions by checking the validity of the first law of thermodynamics. Continuing the thermodynamic study, we investigate the conditions under which the system is thermally stable from the heat capacity and the Gibbs free energy. We also discuss the possible phase transition and the criticality of the system. It was found that the quantum gravitational effects of gravity’s rain- bow render the thermodynamic system stable in the vicinity of the singularity. Hence, we obtained a first order phase transition which is interpreted as the large/small black hole phase transition. From the equation of state it was found that after diverging at the singularity, the system evolves asymptotically into pressure-less dust as one moves away from the central singularity. We also calculated the quantum work using the change of the Helmholtz free energy.