 Gravitational singularity theorem

A gravitational singularity theorem gives sufficient conditions for geodesic incompleteness: some freely falling worldline or light-ray geodesic cannot continue through an unlimited proper time or affine parameter within the stated spacetime. The conclusion concerns the failure of complete geodesic evolution; it does not by itself establish a curvature singularity.

A typical argument combines a focusing assumption such as the null energy condition, the focusing of a geodesic congruence, and assumptions on the global causal structure. The Penrose singularity theorem uses a trapped surface to show that collapse need not preserve exact spherical symmetry to force null incompleteness under its other hypotheses. This is useful because highly symmetric exact solutions alone cannot establish what happens during generic collapse.^[ref]