Regular electrically charged vacuum structures with de Sitter centre in nonlinear electrodynamics coupled to general relativity

We address the question of existence of regular spherically symmetric electrically charged solutions in nonlinear electrodynamics (NED) coupled to general relativity. The stress-energy tensor of the electromagnetic field has the algebraic structure T-0(0) = T-1(1). In this case, the weak energy condition leads to the de Sitter asymptotic on approaching a regular centre. In the de Sitter centre of an electrically charged NED structure, electric field, geometry and stress-energy tensor are regular without the Maxwell limit which is replaced by the de Sitter limit: energy density of a field is maximal and gives an effective cut-off on self-energy density, produced by NED coupled to gravity and related to the cosmological constant A. Regular electric solutions, satisfying WEC, suffer from one cusp in the Lagrangian L(F), which creates the problem in an effective geometry whose geodesics are world lines of NED photons. We investigate propagation of photons and show that their world lines never terminate which suggests absence of singularities in the effective geometry. To illustrate these results we present the particular example of the new exact analytic spherically symmetric electrically charged solution with the de Sitter centre.