A trapped surface is a concept in the field of general relativity, specifically in the study of black holes and gravitational collapse. It refers to a two-dimensional surface in spacetime that has certain properties related to the behavior of light rays. In more technical terms, a trapped surface is defined as a surface such that all light rays emitted orthogonally (perpendicular) to the surface are converging.
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A trapped surface is a closed spacelike codimension-two surface whose two future-directed orthogonal null congruences both have negative expansion.
In four-dimensional spacetime, “closed” here means compact without boundary; “spacelike” means that the induced metric tensor is positive definite. Let be the two future null vector normals and the induced metric tensor. Their null expansions areHere is an infinitesimal transverse area element and an affine parameter along the respective normal null geodesics. A future trapped surface has and everywhere: even the outward-directed light front initially loses area. Multiplying either future normal by a positive function multiplies its initial null expansion by that function, preserving the sign. A small inward-converging patch is insufficient: the entire closed surface must satisfy both strict inequalities.