Trapped surface

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 $k_{+},k_{-}$ be the two future null vector normals and $q_{ab}$ the induced metric tensor. Their null expansions areHere $dA$ is an infinitesimal transverse area element and ${\lambda }_{\pm }$ an affine parameter along the respective normal null geodesics. A future trapped surface has ${\theta }_{+}<0$ and ${\theta }_{-}<0$ 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.

A closed spacelike two-surface is marginally outer trapped when its future outgoing null expansion is zero. For the future black-hole type, the ingoing null expansion is negative. An apparent horizon on a chosen spatial slice is ordinarily its outermost such surface under suitable regularity assumptions. The notion depends on the slice and is not the global definition of an event horizon.

An apparent horizon is the outer boundary of the future-trapped region on a chosen spatial slice, under the usual regularity conditions. Its location depends on the slice; it is not the globally defined event horizon. At a regular boundary the outward null expansion vanishes while the inward null expansion is negative. Its area cannot be substituted into every version of the Penrose inequality without the appropriate area and initial-data hypotheses.

An anti-trapped surface is the time reverse of a trapped surface: both future-directed orthogonal null expansions are positive.