Penrose singularity theoremtheorem

In 1965, Roger Penrose used methods of global differential geometry to prove: if in spacetime the energy dominance condition holds (gravity always attracts) and there exists a closed trapped surface (a surface from which even light rays converge), then a singularity inevitably arises — the edge of the causal past where geodesics terminate. This meant that singularities are not an artifact of symmetry, but a generic feature of gravitational collapse. The theorem was the first rigorous statement about singularities in GR without assuming symmetry. Later, Hawking and Penrose extended it to the cosmological singularity.

The Penrose singularity theorem states that once a trapped surface forms inside a collapsing star (a surface from which even light cannot escape outward), the collapse inevitably ends in a singularity (a point of infinite curvature where the laws of physics break down). It's like trying to compress a sponge: if there's nowhere else to compress, it will tear. But with gravity, spacetime tears to infinite density.

The Penrose singularity theorem (1965) is the first singularity theorem in general relativity not requiring spherical symmetry. Formulation: Let spacetime \((M, g)\) satisfy: 1) the energy dominance condition (\(R_{\mu\nu} v^\mu v^\nu \ge 0\) for any timelike vector \(v^\mu\)); 2) global hyperbolicity (a Cauchy hypersurface exists); 3) contains a closed trapped surface (a two-dimensional surface on which converging null geodesics have negative divergence). Then in \(M\) there exists an inextendible timelike or null geodesic — a singularity.