New black hole solutions of second and first order formulations of nonlinear electrodynamics

Inspired by the so-called Palatini formulation of general relativity and of its modifications and extensions, we consider an analogous formulation of the dynamics of a self-interacting gauge field which is determined by nonlinear extension of Maxwell's theory, usually known as nonlinear electrodynamics. In this first order formalism the field strength and the gauge potential are treated, a priori as independent, and, as such, varied independently in order to produce the field equations. Accordingly we consider within this formalism alternative and generalized nonlinear Lagrangian densities, some of them of a new kind which gives up the restriction of equivalence to second order Lagrangians. Several new spherically symmetric objects are constructed analytically and their main properties are studied. The solutions are obtained in flat spacetime ignoring gravity and for the self-gravitating case with emphasis on black holes. As a background for comparison between the first and second order formalisms, some of the solutions are obtained by the conventional second order formalism, while for others a first order formalism is applied. Among the self-gravitating solutions we find new families of black holes and study their main characteristics. Some of the flat space solutions can regularize the total energy of a point charge and a subset of them exhibit also finite field strength and energy density, although their black hole counterparts are not regular.