Null geodesic completeness

A spacetime is null-geodesically complete when every inextendible affinely parametrized null geodesic has an affine parameter extending without a finite endpoint in both directions.

The Penrose singularity theorem states that a globally hyperbolic spacetime with a noncompact Cauchy hypersurface, a closed trapped surface, and the null convergence condition is future null-geodesically incomplete. Time reversal replaces the trapped surface by an anti-trapped surface and gives past incompleteness.