Curvature invariants and the electromagnetic invariant diverge at $r=0$, so it is a genuine <curvature singularity>, not a coordinate singularity. The causal structures are the higher-dimensional analogues of the familiar four-dimensional diagrams:

* for $r_-=0$, there is one nondegenerate horizon and a spacelike singularity, giving the Schwarzschild Penrose diagram;
* for $0<r_-<r_+$, the outer event horizon and inner <Cauchy horizon> enclose a region ending at a timelike singularity, with the maximally extended Reissner-Nordström chain of blocks;
* for $r_-=r_+$, the horizon is degenerate with $\kappa=0$, the singularity remains timelike, and the extremal throat has infinite spatial length.

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 311 / 2 / d / Solution

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The future or past Cauchy horizon of a partial Cauchy hypersurface is the boundary of the region whose physical evolution is determined by data on that hypersurface. Crossing it can destroy uniqueness of the maximal globally hyperbolic development.


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 Incoming links: Cauchy horizon