Trapped Surface

A trapped surface is a closed, spacelike two-dimensional surface (topologically a sphere) with the defining property that both families of light rays emitted orthogonally from it — the outgoing and the ingoing — are converging. Ordinarily, a flash of light from a sphere produces an outgoing wavefront that grows in area and an ingoing one that shrinks. On a trapped surface, gravity is so strong that even the outgoing wavefront has decreasing area: in the language of the Raychaudhuri equation, both null expansion scalars θ₊ and θ₋ are negative. 'Outward' no longer means 'toward larger spheres.'

Inside a black-hole event horizon, every 2-sphere is trapped: there is no escaping wavefront because all future-directed null rays head to smaller radius. The boundary case, where the outgoing expansion is exactly zero, is a marginally trapped surface, and the locus of such surfaces is the apparent horizon. Roger Penrose introduced the concept in 1965 precisely because it is defined by strict inequalities and is therefore an 'open' condition: a trapped surface present in an idealized symmetric collapse survives under small perturbations, so it cannot be dismissed as a special, measure-zero case.

This robustness is the engine of the singularity theorem. The existence of a single trapped surface, together with an energy condition and global hyperbolicity, guarantees that the spacetime is geodesically incomplete — that a singularity must form — with no appeal to symmetry. It is the physical hypothesis that turns 'collapse looks bad' into 'collapse is provably catastrophic.'

Introduced by Roger Penrose in his 1965 paper as the central concept allowing a symmetry-free proof of singularity formation; the related notions of apparent and dynamical horizons were developed later by Hawking, Hayward and Ashtekar.