Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\mathcal{L}=
\mathcal{L}(\mathcal{F})$, depending on the electromagnetic invariant
$\mathcal{F}=F_{\mu\nu}\,F^{\mu\nu}/4$, are established. The traceless Ricci
(TR) tensor eigenvalue $S$, the Weyl tensor eigenvalue $\Psi_2$ 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 $\lim_{r\rightarrow 0}\{\Psi_2,S,R\}\rightarrow
\{0,0,(0,4\Lambda+ 4\mathcal{L}(0))\}$, 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 $\{\Psi_2,S,R\}$, 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.

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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: cs
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: alg-geom
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ph
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The categories of this paper are: eess