We show that the Bronnikov theorem on the nonexistence of regular electrically charged black holes can be circumvented. In the frame of nonlinear electrodynamics, we present the exact regular black hole solutions of a hybrid type. They are electrically charged, but contain a “dual” core confining a polarization of magnetic charges. The considered example is based on a modification of the Ayón-Beato–García solution. It represents a very specific realization of the idea on confinement based on dual electrodynamics and dual superconductivity.

A unified approach to regular interiors of black holes with smooth matter distributions in the core region is given. The approach is based on a class of Kerr-Schild metrics representing minimal deformations of the Kerr-Newman solution, and allows us to give a common treatment for (charged and uncharged) rotating and nonrotating black holes. It is shown that the requirement of smoothness of the source constrains the structure of the core region in many respects: in particular, for Schwarzschild holes a de Sitter core can be selected, which is surrounded by a smooth shell giving a leading contribution to the total mass of the source. In the rotating, noncharged case the source has a similar structure, taking the form of a (anisotropic and rotating) de Sitter–like core surrounded by a rotating elliptic shell. The Kerr singular ring is regularized by anisotropic matter rotating in the equatorial plane, so that the negative sheet of the Kerr geometry is absent. In the charged case the sources take the form of “bags,” which can have de Sitter or anti–de Sitter interiors and a smooth domain wall boundary, with a tangential stress providing charge confinement. The Arnowitt-Deser-Misner and Tolman relations are used to calculate the total mass of the sources.

The Kerr solution to axidilaton gravity is analyzed in the Debney-Kerr-Schild formalism. It is shown that the Kerr principal null congruence retains its property to be geodesic and shear-free; however, the axidilatonic Kerr solution is not algebraically special; it is of type I in the Petrov-Pirani classification and may not be represented in the Kerr-Schild form. A limiting form of this solution is considered near the ringlike Kerr singularity. This limiting solution is similar to that for the field of fundamental heterotic string obtained by Sen.

We discuss the relation between the Kerr-Newman spinning particle and the Dirac electron and show that the Dirac equation may be naturally incorporated into the Kerr-Schild formalism as a master equation controlling the Kerr-Newman geometry. As a result, the Dirac electron acquires an extended space-time structure of the Kerr-Newman geometry: a singular ring of Compton size and a twistorial polarization of the gravitational and electromagnetic fields. The behavior of this Dirac-Kerr-Newman system in weak and slowly changing electromagnetic fields is determined by the wave function of the Dirac equation and is indistinguishable from the behavior of the Dirac electron. The wave function of the Dirac equation plays, in this model, the role of an “order parameter” which controls the dynamics, spin polarization and the twistorial structure of space-time.

The Kerr spinning particle has a remarkable analytical twistorial structure. Analyzing electromagnetic excitations of the Kerr circular string which are aligned to this structure, we obtain a simple stringy skeleton of the spinning particle which is formed by a topological coupling of the Kerr circular singular string and by an axial singular stringy system. We show that the chiral traveling waves, related to an orientifold world sheet of the axial stringy system, are described by the massive Dirac equation, so we argue that the axial string may play the part of a stringy carrier of wave function and play also a dominant role in the scattering processes. A key role of the third, complex Kerr string is discussed. We conjecture that it may be one more alternative to the Witten twistor string, and a relation to the spinor helicity formalism is also discussed.