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 non-linear 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 non-linear 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.