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Extremal Reissner-Nordström geometry
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Extremal Reissner-Nordström spacetime diagram
A Reissner-Nordström black hole is extremal if the outer and inner horizons coincide, \[ r_+ = r_- = M \ . \] |
Free-fall spacetime diagram for the extremal Reissner-Nordström geometry
Watch extremal Reissner-Nordström morph into free-fall (46K GIF); or same morph, double-size on screen (same 46K GIF). |
Finkelstein spacetime diagram of the extremal Reissner-Nordström geometry
Watch extremal Reissner-Nordström morph into Finkelstein (38K GIF); or same morph, double-size on screen (same 38K GIF). Watch Finkelstein morph into free-fall (32K GIF); or same morph, double-size on screen (same 32K GIF). |
Penrose diagram of the extremal Reissner-Nordström geometry
Introduce a cutoff at
\(| r^\ast | < r^\ast_c\).
Define Penrose coordinates by
\begin{align}
r_\textrm{P} + t_\textrm{P}
&=
\mbox{sign} ( r^\ast + t ) \ln \left( 1 + | r^\ast + t | \right)
+
r^\ast_c \ln ( 1 + r^\ast_c )
\ ,
\\
r_\textrm{P} - t_\textrm{P}
&=
\pm \left[
- \, \mbox{sign} ( r^\ast - t ) \ln \left( 1 + | r^\ast - t | \right)
+
r^\ast_c \ln ( 1 + r^\ast_c )
\right]
\ ,
\end{align}
where the overall sign in the last equation is positive (\(+\))
outside the horizon,
\(r > r_+\),
and negative (\(-\)) inside the horizon,
\(r < r_+\).
Watch Finkelstein morph into Penrose (51K GIF); or same morph, double-size on screen (same 51K GIF). |
Penrose diagram of the complete extremal Reissner-Nordström geometry |
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Updated 19 Apr 2001; converted to mathjax 3 Feb 2018