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 Incoming links: Cauchy horizon

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.

The globally hyperbolic maximal Cauchy development of appropriate Reissner-Nordstrom spacetime data obeys the Penrose singularity theorem when it contains a closed trapped surface. Its resulting null geodesic incompleteness can include generators reaching a smoothly extendible inner Cauchy horizon in finite affine parameter. The full analytic extension is not globally hyperbolic and cannot be substituted into that version of the theorem. Incompleteness alone does not establish curvature blowup at every endpoint.

The four-dimensional charged spherically symmetric electrovacuum solution has radial function

f = 1 - 2 M / r + Q^{2} / r^{2}

. For

0 < ∣ Q ∣ < M

, its outer and inner horizon radii are

r_{\pm} = M \pm M^{2} - Q^{2}

. A complete two-ended spatial bridge has a maximal Cauchy development ending at smoothly extendible inner Cauchy horizons in the exact solution.

The strong cosmic censorship conjecture asserts that the maximal Cauchy development of generic admissible initial data is inextendible in a specified regularity class. The matter model, topology on initial data, and extension regularity are part of the assertion. An exceptional exact solution with a smooth Cauchy horizon need not violate this generic claim.