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arXiv · 1911.06374

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

Alberto A. Garcia-Diaz·Gustavo Gutierrez-Cano
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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) with Lagrangian $\mathcal{L}= \mathcal{L}(\mathcal{F})$, depending on the electromagnetic invariant $\mathcal{F}=F_{μν}\,F^{μν}/4$, are established. The traceless Ricci (TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue $Ψ_2$ 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 $\lim_{r\rightarrow 0}\{Ψ_2,S,R\}\rightarrow \{0,0,(0,4Λ+ 4\mathcal{L}(0))\}$, 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 $\{Ψ_2,S,R\}$, 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--Nordtröm (Maxwell) asymptotic, or present the dS--AdS or other kind of asymptotic.

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Alberto A. Garcia-Diaz, Gustavo Gutierrez-Cano. 2020-06-23. Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations; revised and improved version. https://doi.org/10.1016/j.aop.2020.168323

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