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

Solution

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 $κ = 0$ , the singularity remains timelike, and the extremal throat has infinite spatial length.

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