Reply to “Comment on ‘Linear superposition of regular black hole solutions of Einstein nonlinear electrodynamics”’

Abstract

The main objective of our article [A. A. Garcia-Diaz and G. Gutierrez-Cano, Phys. Rev. D 100, 064068 (2019)] was the establishing of the existence of an infinite class of electrically charged solutions to Einstein--nonlinear electrodynamics (NLE) equations, which is determined for any imaginable set of structural functions $Q\mathrm{s}$ without any integration. The curvature invariants (CI) ${S,{\mathrm{\ensuremath{\Psi}}}_{2},R}$, related to the quadratic Riemann invariants, are evaluated in terms of $Q$, and its derivatives $\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q}$, and $\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{Q}$. The necessary and sufficient conditions for the regularity at the center of the CI, ${\mathrm{lim}}_{r\ensuremath{\rightarrow}0}{S,{\mathrm{\ensuremath{\Psi}}}_{2},R}={0,0,4\mathrm{\ensuremath{\Lambda}}+4\mathcal{L}(0)}$, of the NLE electric $(\mathcal{E}\ensuremath{\mathrel{:=}}{q}_{0}{F}_{rt})$ solutions are: ${\mathrm{lim}}_{r\ensuremath{\rightarrow}0}{\mathcal{E},\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{\mathcal{E}},\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{\mathcal{E}}}={0,0,0}$, and ${\mathrm{lim}}_{r\ensuremath{\rightarrow}0}{Q,\stackrel{\ifmmode \dot{}\else \textperiodcentered \fi{}}{Q},\stackrel{\ifmmode\ddot\else\textasciidieresis\fi{}}{Q}}={0,0,2}$. Moreover, in the theorem on linear superposition of solutions $Q({\mathcal{E}}_{i})$, with the use of a single Kottler solution ${Q}_{K}$, $Q(r)={Q}_{K}+{\ensuremath{\sum}}_{i}{C}_{i}Q({\mathcal{E}}_{i})$, the redundancy in introducing sums of partial ${\mathrm{\ensuremath{\Lambda}}}_{i}$, ${C}_{i}$, 1's, and ${m}_{i}$, drops out.