Trapped Surface

A trapped surface is a closed, spacelike two-surface in general relativity with the property that both outward-pointing and inward-pointing families of null geodesics — light rays emitted orthogonally from the surface — converge toward each other. In ordinary spacetime, a spherical shell of light emitted outward expands; a trapped surface is a region where gravity has become so intense that even the 'outward' light front is pulled inward. It is the geometric signature that a region of spacetime has crossed the threshold from which classical escape is impossible.

The concept is minimal and irreducible. Unlike an event horizon, which is a global property of spacetime requiring knowledge of the entire future, a trapped surface is defined locally. Roger Penrose's 1965 singularity theorem proved that the existence of a trapped surface, combined with the strong energy condition and global hyperbolicity, is sufficient to guarantee the formation of a black hole singularity. The trapped surface is therefore the seed from which the full machinery of the Penrose-Hawking singularity theorems grows.

In astrophysical collapse, a trapped surface forms when a star has compressed within its Schwarzschild radius — though the precise relationship between trapped surfaces and horizons depends on the global structure of the spacetime. The trapped surface is the point of no return made geometrically explicit.

The trapped surface is often presented as a technical condition in singularity theorems — a step on the way to proving that collapse is irreversible. This misses its conceptual centrality: it is the exact geometric point where spacetime stops being an arena of free movement and becomes a prison. The event horizon is the warden. The trapped surface is the lock.