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\begin{document}

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\section*{Reissner-Nordström spacetime}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{metric}{Metric}\dotfill \pageref*{metric} \linebreak
\noindent\hyperlink{horizons}{Horizons}\dotfill \pageref*{horizons} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{InteriorSolutions}{Interior solutions}\dotfill \pageref*{InteriorSolutions} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The Reissner-Nordstr\"o{}m solution to the [[equations of motion]] of [[Einstein-Maxwell theory]] describes the [[spacetime]] outside of a [[electric charge|electrically charged]] (and/or [[magnetic charge|magnetically charged]]) point [[mass]], hence describes a \emph{charged} [[black hole]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{metric}{}\subsubsection*{{Metric}}\label{metric}

For [[mass]] $M$, [[electric charge]] $Q$ and [[magnetic charge]] $P$ the [[pseudo-Riemannian metric]] for the Reissner-Nordstr\"o{}m spacetime is, in [[polar coordinates]] $(t,r,\phi,\theta)$ on $\mathbb{R}^4 - \{0\} = \mathbb{R} \times \mathbb{R}_{\gt 0} \times S^2$, given by

\begin{displaymath}
ds^2_{M,Q,P} \coloneqq - H d t^2 + H^{-1} d r^2 + r^2 ds_{S^2}^2
\end{displaymath}
where (in natural units)

\begin{displaymath}
H \coloneqq 1 - \frac{M}{r} + \frac{Q^2 + P^2}{r^2}
  \,.
\end{displaymath}
The corresponding [[Faraday tensor]]/electromagnetic [[field strength]] [[curvature|curvature 2-form]] is

\begin{displaymath}
F = \frac{Q}{r^2} d t \wedge d r + P \sin(\theta) d\theta \wedge d\phi
  \,.
\end{displaymath}
\begin{quote}%
(check sign)


\end{quote}
\hypertarget{horizons}{}\subsubsection*{{Horizons}}\label{horizons}

The function $H$ above vanishes at

\begin{displaymath}
r_{\pm} 
    \coloneqq 
  \tfrac{1}{2}\left(
    M \pm \sqrt{M^2 - 4 (P^2 + Q^2)}
  \right)
  \,.
\end{displaymath}
The larger of the two solutions is an [[event horizon]], the other is a [[Cauchy horizon]]. In the case that they coincide, for $M = 2\sqrt{Q^2 + P^2}$, one speaks of an [[extremal black hole]] (at least when also $P = 0$).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{InteriorSolutions}{}\subsubsection*{{Interior solutions}}\label{InteriorSolutions}

Instead of having a mass and charge of singular support at the singular locus of the RN-spacetime, one may consider a finite mass and charge distribution, i.e. a ``charged star''. Outside of this mass/charge distribution then the metric is the RN-spacetime as above, but in the interior it becomes smoothed out to a non-singular interior solution.

Several attempts have been made to round out the Reissner metric with an interior. \hyperlink{TiwariRaoKanakamelda85}{Tiwari-Rao-Kanakamedala 85} found an interior with the condition $g_{11} g_{44} = 1$, which obviously holds for the exterior Reissner, but does not match the interior Schwarzschild solution for e = 0. \hyperlink{KyleMartin67}{Kyle-Martin 67} found a rather complicated solution. They discussed the self-energy of the fields of charged matter. \hyperlink{Wilson69}{Wilson 69} modified this solution assuming a different value for the total charge. \hyperlink{CohenCohen69}{Cohen-Cohen 69} applied this solution to the special case of a charged thin shell and showed that the energy density of the electromagnetic field contributes to the mass. \hyperlink{Boulware73}{Boulware 73} studied the time development of thin shells. \hyperlink{GravesBrill60}{Graves-Brill 60} considered a possible oscillatory character of the Reissner-Nordstr\"o{}m metric by examining the metric by Kruskal-like coordinates \hyperlink{Bekenstein71}{Bekenstein 71} studied another ansatz for the stress-energy-tensor of charged matter. \hyperlink{KroriJayantimala75}{Krori-Jayantimala 75} made use of the same ansatz, and he found a solution free of singularities. \hyperlink{GautreauHoffman73}{Gautreau-Hoffman 73} studied the sources of Weyl-type electrovac fields. They obtained the parameters for the source with the junction condition for the exterior solution. \hyperlink{Efinger64}{Efinger 64} deduced the stability of a charged particle from the self-energy of the gravitation field. In \hyperlink{Burghardt09}{Burghardt 09} is constructed another interior solution with the help of embedding the geometry in a 5-dimensional flat space.

(Literature summary taken from \hyperlink{Burghardt09}{Burghardt 09})

See also (\hyperlink{Mehra82}{Mehra 82}, \hyperlink{WX87}{WX 87}, \hyperlink{GuptaKumarPratibha12}{Gupta-Kumar-Pratibha 12}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[no-hair theorem]]

\end{itemize}
[[!include charged and rotating black holes -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Review includes

\begin{itemize}%
\item Mammadov, \emph{Reissner-Nordstr\"o{}m metric} (\href{http://gmammado.mysite.syr.edu/notes/RN_Metric.pdf}{pdf})


\item Wikipedia, \emph{}



\end{itemize}
That the [[near horizon geometry]] of the [[extremal black hole|extremal]] Reissner-Nordström black hole in 4d ins 2d [[anti de Sitter spacetime]] times the [[2-sphere]], $AdS_2 \times S^2$, was observed in

\begin{itemize}%
\item [[Gary Gibbons]], in F. del Aguila, [[J. de Azcarraga]], [[Luis Ibanez]] (eds.) \emph{Supersymmetry, Supergravity and Related Topics}(see section 11.1 here: \href{http://www.hartmanhep.net/topics2015/11-nearhorizon.pdf}{pdf} [[ExtremalReissnerNordstrom.pdf:file]])



\end{itemize}
Discussion of interior solutions includes

\begin{itemize}%
\item Graves J. C., Brill D. R., \emph{Oscillatory character of Reissner-Nordstr\"o{}m metric for an ideal charged wormehole.} Phys. Rev. 120, 1507, 1960


\item Efinger H. J., \emph{Der kugelsymmetrische Fall einer statischen Ladungsverteilung bei stark komprimierter Materie in der allgemeinen Relativit\"a{}tstheorie. Acta Phys. Austr. 17, 347, 1964.}

\emph{\"U{}ber die Selbstenergie und Ladung eines durch Gravitationswirkung stabilisierten Teilchens in einem gekr\"u{}mmten Raum}, Acta Phys. Austr. 188, 31, 1965


\item Kyle C. F., Martin A. W., \emph{Self-energy considerations in general relativity and the exact fields of charge and mass distribution}, Nuov. Cim 50, 883, 1967


\item Cohen J. M., Cohen M. D., \emph{Exact fields of charge and mass distributions in general relativity}. Nuov. Cim. 60, 241, 1969


\item Wilson, S. J., \emph{Exact solution of a static charged sphere in general relativity}. Can. Journ. Phys. 47, 2401, 1969


\item [[Jacob Bekenstein]], \emph{Hydrostatic equilibrium and gravitational collapse of relativistic charged fluid balls}, Phys. Rev. D 4, 2185, 1971


\item Boulware, D. G., \emph{Naked singularities, thin shells, and the Reissner-Nordstr\"o{}m metric}, Phys. Rev. D 8, 2363, 1973


\item Gautreau R., Hoffman R. B., The structure of Weyl-type electrovac fields in general relativity. Nouv. Cim. 16 B, 162, 1973


\item Krori K. D., Jayantimala B., \emph{A singularity-free solution for a charged fluid in general relativity}, J. Phys. A 8, 508, 1975


\item A.L. Mehra, \emph{An interior solution for a charged sphere in general relativity}, Physics Letters A Volume 88, Issue 4, 8 March 1982, Pages 159-161 (\href{http://www.sciencedirect.com/science/article/pii/0375960182905515}{publisher})


\item Tiwari R. N., Rao J. R., Kanakamedala R. R., \emph{Electromagnetic mass models in general relativity}, Phys. Rev. D 30, 489, 1984


\item Wang Xingxiang, \emph{Exact Solution of a Static Charged Sphere in General Relativity}, General Relativity and Gravitation, Vol. 19, No. 7, 1987


\item Rainer Burghardt, \emph{Reissner exterior and interior} \href{https://arxiv.org/abs/0902.0918}{arXiv:0902.0918}


\item Gupta, Y. K.; Kumar, Jitendra; Pratibha, \emph{A Class of Well Behaved Charged Analogues of Schwarzchild's Interior Solution}, International Journal of Theoretical Physics October 2012, Volume 51, Issue 10, pp 3290--3302 (\href{http://link.springer.com/article/10.1007%2Fs10773-012-1209-4}{publisher})



\end{itemize}
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[[!redirects Reissner-Nordstrom spacetime]] [[!redirects Reissner-Nordstrom spacetimes]]

[[!redirects Reissner-Nordström black hole]] [[!redirects Reissner-Nordström black holes]]

[[!redirects Reissner-Nordstrom black hole]] [[!redirects Reissner-Nordstrom black holes]]

[[!redirects charged black hole]] [[!redirects charged black holes]]



\end{document}