Regular black holes sourced by nonlinear electrodynamics

The paper is a brief review on the existence and basic properties of static, spherically symmetric regular black hole solutions of general relativity, where the source of gravity is represented by nonlinear electromagnetic fields with the Lagrangian function $L$ depending on the single invariant $f = F_{\mu\nu}F^{\mu\nu}$ or on two variables: either $L(f, h)$, where $h = {^*}F_{\mu\nu} F^{\mu\nu}$, where ${^*}F_{\mu\nu}$ is the Hodge dual of $F_{\mu\nu}$, or $L(f, J)$, where $J = F_{\mu\nu}F^{\nu\rho} F_{\rho\sigma} F^{\sigma\mu}$. A number of no-go theorems are discussed, revealing the conditions under which the space-time cannot have a regular center, among which the theorems concerning $L(f,J)$ theories are probably new. These results concern both regular black holes and regular particlelike or starlike objects (solitons) without horizons. Thus, a regular center in solutions with an electric charge $q_e\ne 0$ is only possible with nonlinear electrodynamics (NED) having no Maxwell weak field limit. Regular solutions with $L(f)$ and $L(f, J)$ NED, possessing a correct (Maxwell) weak-field limit, are possible if the system contains only a magnetic charge $q_m \ne 0$. It is shown, however, that in such solutions the causality and unitarity as well as dynamic stability conditions are inevitably violated in a neighborhood of the center. Some particular examples are discussed.