Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations; revised and improved version

In this report, the regularity conditions at the center for static spherically symmetric (SSS) solutions of the Einstein equations coupled to nonlinear electrodynamics (NLE) with Lagrangian $\mathcal{L}= \mathcal{L}(\mathcal{F})$, depending on the electromagnetic invariant $\mathcal{F}=F_{\mu\nu}\,F^{\mu\nu}/4$, are established. The traceless Ricci (TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue $\Psi_2$ and the scalar curvature $R$ characterize the independent Riemman tensor invariants of SSS metrics. The necessary and sufficient regularity conditions for electric NLE SSS solutions require $\lim_{r\rightarrow 0}\{\Psi_2,S,R\}\rightarrow \{0,0,(0,4\Lambda+ 4\mathcal{L}(0))\}$, such that the metric function $Q(r)$ and the electric field $q_0F_{rt}=:\mathcal{E}$ behave as $\{Q,\dot Q,\ddot Q\}\rightarrow\{0,0,2\}$ and \{\mathcal{E},\dot\mathcal{E},\ddot\mathcal{E}\}\rightarrow\{0,0,0\}, as $r\rightarrow 0$. The general linear integral representation of the electric NLE SSS metric in terms of an arbitrary electric field $\mathcal{E}$, together with $\{\Psi_2,S,R\}$, is explicitly given. Moreover, beside the regular or singular behavior at the center, these solutions may exhibit different asymptotic behavior at spatial infinity such as the Reissner--Nordtr\"om (Maxwell) asymptotic, or present the dS--AdS or other kind of asymptotic.