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Regularity conditions for spherically symmetric solutions of Einstein-nonlinear electrodynamics equations
Annals of Physics ( IF 3.0 ) Pub Date : 2020-11-01 , DOI: 10.1016/j.aop.2020.168323
Alberto A. Garcia-Diaz , Gustavo Gutierrez-Cano

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:

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.
更新日期:2020-11-01
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