# The Extremal Reissner-Nordström Geometry

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 Extremal Reissner-Nordström geometry

 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

 Finkelstein spacetime diagram of the extremal Reissner-Nordström geometry

 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, $$r < r_+$$.

 Penrose diagram of the complete extremal Reissner-Nordström geometry

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Updated 19 Apr 2001; converted to mathjax 3 Feb 2018