Dilaton black holes coupled to nonlinear electrodynamic field

The theory of nonlinear electrodynamics has received a lot of attention in recent years. It was shown that Born-Infeld nonlinear electrodynamics is not the only modification of the linear Maxwell's field that keeps the electric field of a charged point particle finite at the origin; other types of nonlinear Lagrangian modifications such as exponential and logarithmic nonlinear electrodynamics can play the same role. In this paper, we generalize the study of exponential nonlinear electrodynamics by adding a scalar dilaton field to the action. By suitably choosing the coupling of the matter field to the dilaton field, we vary the action and obtain the corresponding field equations. Then, by making a proper ansatz, we construct a new class of charged dilaton black hole solutions coupled to the exponential nonlinear electrodynamics field in the presence of two Liouville-type potentials for the dilaton field. Due to the presence of the dilaton field, the asymptotic behavior of these solutions is neither flat nor (A)dS. In the limiting case where the nonlinear parameter beta(2) goes to infinity, our solution reduces to the Einstein-Maxwell dilaton black holes. We obtain the mass, temperature, entropy, and electric potential of these solutions. We also study the behavior of the electric field, as well as the electric potential of these black holes near the origin. We find that the electric field has a finite value near the origin, which is the same as the electric field of Born-Infeld nonlinear electrodynamics, but it can diverge exactly at r = 0 depending on the model parameters. We also investigate the effects of the dilaton field on the behavior of the electric field and electric potential. Finally, we check the validity of the first law of black hole thermodynamics on the black hole horizon.