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If some energy conditions on the stress-energy tensor are violated, is possible construct regular black holes in General Relativity and in alternative theories of gravity. This type of solution has horizons but does not present singularities. The first regular black hole was presented by Bardeen and can be obtained from Einstein equations in the presence of an electromagnetic field. E. Ayon-Beato and A. Garcia reinterpreted the Bardeen metric as a magnetic solution of General Relativity coupled to a nonlinear electrodynamics. In this work show that the Bardeen model may also be interpreted as a solutions of Einstein equations in the presence of a electric source, whose electric field does not behaves as a Coulomb field. We analyzed the asymptotic forms of the Lagrangian for the electric case and also analyzed the energy conditions.

We study the generalization of the mass function of classes of regular spherically symmetric black holes solutions of in 4D coming from the coupling of General Relativity with Non-linear Electrodynamics, through the requirement of the energy conditions. Imposing that the solution must be regular and that the Weak and Dominant Energy Conditions are simultaneously satisfied, for models with the symmetry T 0 0 = T 1 1 of the tensor energy-momentum, we construct a new general class of regular black holes of which two cases are asymptotically Reissner-Nordstrom.

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In this work, we study the possibility of generalizing solutions of regular black holes with an electric charge, constructed in general relativity, for the f (G) theory, where G is the Gauss-Bonnet invariant. This type of solution arises due to the coupling between gravitational theory and nonlinear electrodynamics. We construct the formalism in terms of a mass function and it results in different gravitational and electromagnetic theories for which mass function. The electric field of these solutions are always regular and the strong energy condition is violated in some region inside the event horizon. For some solutions, we get an analytical form for the f (G) function. Imposing the limit of some constant going to zero in the f (G) function we recovered the linear case, making the general relativity a particular case.