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 $$\mathcal{L}$= $\mathcal{L}$($\mathcal{F}$)$,dependingontheelectromagneticinvariant$$\mathcal{F}$=F_{\mu\nu},F^{\mu\nu}/4$, are established. The traceless Ricci (TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue ${\mathrm{\Psi }}_2$ 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}$(0))}$,suchthatthemetricfunction$Q®$andtheelectricfield$q_0F_{rt}=:$\mathcal{E}$$behaveas${Q,\dot Q,\ddot Q}\rightarrow{0,0,2}$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 $\{{\mathrm{\Psi }}_2,S,R\}$, 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.