= Gravitational singularity theorem
= Singularity theorem
{synonym}
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.<Gravitational Collapse and Space-Time Singularities by Roger Penrose>[Pages 57–59: symmetry-independent collapse argument, trapped surface and null-geodesic incompleteness]{ref}
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