The Extremal Reissner-Nordström Geometry

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

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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