Dilaton black holes with logarithmic source in gravity’s rainbow

Abstract

Starting from the suitable action of four-dimensional Einstein gravity, which is coupled to a scalar dilaton field and logarithmic nonlinear electrodynamics, the related field equations have been solved in a spherically symmetric and energy-dependent spacetime. It has been found that the scalar potential, as the exact solution of the scalar field equations, must be written as the linear combination of two Liouville-type potentials. Three families of novel exact black holes have been introduced as the exact solutions to the gravitational equations, which are asymptotically non-flat and non-AdS. It has been illustrated that our solutions, recover their corresponding quantities in the Einstein–Maxwell-dilaton gravity theory when the nonlinearity parameter of electrodynamics is chosen very large. The thermodynamic and conserved quantities of the black holes have been calculated under the influence of the rainbow function. Then, through calculation of the Smarr mass formula, we have shown that the first law of black hole thermodynamics remain valid for all of the new dilatonic black holes. Thermal stability or phase transition of the black holes have been investigated by use of the canonical ensemble and thermodynamic geometry approaches, separately. By comparison of the results of these two alternative approaches, it has been found that they lead to the same results provided that the thermodynamic metric of HPEM is used.