Nonlinear electromagnetic generalization of the Kerr-Newman solution with a cosmological constant

We present the two exact solutions of the Einstein-nonlinear electrodynamics equations that generalize the Kerr-Newman solution. We determined the generalized electromagnetic potentials using the alignment between the tetrad vectors of the metric and the eigenvectors of the electromagnetic field tensor. It turns out that there are only two possible nonlinear electromagnetic generalizations of the Kerr-Newman geometry, corresponding to different electromagnetic potentials. The new solutions possess horizons and satisfy physical energy conditions such that they can represent black holes with nonlinear electromagnetic charges, characterized by the parameters of mass, angular momentum, charge, and one nonlinear parameter; the nonlinear parameter resembles the effect of a cosmological constant, negative or positive, such that the solutions are asymptotically anti--de Sitter or de Sitter. The canonical form of the electromagnetic nonlinear energy-momentum tensor is analyzed in relation with the energy conditions; it is shown that the conformal symmetry is broken by the electromagnetic nonlinear matter; the corresponding nonlinear electromagnetic Lagrangian as a function of the coordinates is presented as well.