In this report, for static spherically symmetric (SSS) solutions of the Einstein equations coupled to nonlinear electrodynamics (NLE) the regularity conditions at the center are established. The NLE is derived from a Lagrangian $\mathcal{L}=\mathcal{L}(\mathcal{F})$, depending on the electromagnetic invariant $\mathcal{F}=F_{\mu\nu}\,F^{\mu\nu}/4$. Regular solutions are characterized by the finite behavior at the center of the curvature invariants of the Riemman tensor. For regular SSS metrics, the traceless Ricci (TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue $\Psi_2$ and the scalar curvature $R$ are singular--free at the center. Regular NLE SSS electric solutions, which are characterized by the ${\mathcal{F}}(r=0)=0$, approach to the flat or conformally flat de Sitter--Anti de Sitter (regular) spacetimes at the center; moreover, this family of solutions may exhibit different asymptotic behavior at spatial infinity such as the Reissner--Nordtrom (Maxwell) asymptotic, or present the dS--AdS or other kind of asymptotic. Pure magnetic NLE SSS solutions shear the single magnetic invariant $2\mathcal{F}_m= h_0^2/r^4$, thus they are singular in the magnetic field and may exhibit a regular flat or (A)dS behavior at the center.

中文翻译：

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爱因斯坦非线性电动力学方程球对称解的正则条件

在本报告中，对于耦合到非线性电动力学 (NLE) 的爱因斯坦方程的静态球对称 (SSS) 解，在中心建立了规律性条件。NLE 源自拉格朗日 $\mathcal{L}=\mathcal{L}(\mathcal{F})$，取决于电磁不变量 $\mathcal{F}=F_{\mu\nu}\,F ^{\mu\nu}/4$。正则解的特征在于黎曼张量曲率不变量中心的有限行为。对于常规 SSS 度量，无迹 Ricci (TR) 张量特征值 $S$、Weyl 张量特征值 $\Psi_2$ 和标量曲率 $R$ 在中心是奇异的。常规 NLE SSS 电解，其特征在于 ${\mathcal{F}}(r=0)=0$，逼近平面或共形平面 de Sitter--Anti de Sitter (regular) 时空在中心；而且，这一系列解可能在空间无穷远处表现出不同的渐近行为，例如 Reissner--Nordtrom (Maxwell) 渐近，或呈现 dS--AdS 或其他类型的渐近。纯磁 NLE SSS 解决方案剪切单个磁不变量 $2\mathcal{F}_m= h_0^2/r^4$，因此它们在磁场中是奇异的，并且可能在中心表现出规则的平坦或 (A)dS 行为.