Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations

Abstract

In this report, the regularity conditions at the center for static spherically symmetric (SSS) solutions of the Einstein equations coupled to nonlinear electrodynamics (NLE) are established. The NLE is derived from a Lagrangian L=L(F) depending on the electromagnetic invariant F=FμνFμν∕4. The Einstein-NLE field equations for SSS metrics can be analyzed from the point of view of Euler equations; the general electrically charged SSS solution depending linearly on the electric field E=q0Frt is derived. SSS metrics are characterized by the following independent Riemann tensor invariants: the traceless Ricci (TR) tensor eigenvalue S, the Weyl tensor eigenvalue Ψ2 and the scalar curvature R. Regular solutions are characterized by the finite behavior of these curvature invariants in the whole spacetime. NLE SSS solutions are regular at the center r=0 with limr→0{Ψ2,S,R}→{0,0,(0,4Λ+4L(0))}, if and only if the metric function Q(r) and the electric field q0Frt≕E behave as {Q,Q̇,Q̈}→{0,0,2} and {E,Ė,Ë}→{0,0,0}, as r→0, i.e., the electric field and its first and second order derivatives vanish at the center where the metric asymptotically approaches to the flat or conformally flat de Sitter–Anti de Sitter (dS–AdS) spacetimes. Moreover, this family of solutions may exhibit different asymptotic behavior at spatial infinity such as the Reissner–Nordström (Maxwell) asymptotic, or present the dS–AdS or other kind of asymptotic. Pure magnetic NLE SSS solutions shear the single magnetic invariant 2Fm=h02∕r4, thus they are singular in the magnetic field and may exhibit a regular flat or (A)dS behavior at the center.