### 7.1 Definition

Given a (spacelike) 3 + 1 slice, a “trapped surface” is defined as a smooth closed 2-surface in the slice
whose future-pointing outgoing null geodesics have negative expansion . The “trapped region” in the
slice is then defined as the union of all trapped surfaces, and the “apparent horizon” is defined as the outer
boundary of the trapped region.
While mathematically elegant, this definition is not convenient for numerically finding apparent horizons.
Instead, an alternate definition can be used: A MOTS is defined as a smooth (differentiable) closed
orientable 2-surface in the slice whose future-pointing outgoing null geodesics have zero expansion .
There may be multiple MOTSs in a slice, either nested within each other or
intersecting.
An apparent horizon is then defined as an outermost MOTS in a slice, i.e. a MOTS not contained in any
other MOTS. Kriele and Hayward [98] have shown that subject to certain technical conditions, this
definition is equivalent to the “outer boundary of the trapped region” one.

Notice that the apparent horizon is defined locally in time (it can be computed using only Cauchy
data on a spacelike slice), but (because of the requirement that it be closed) non-locally in
space.
Hawking and Ellis [82] discuss the general properties of MOTSs and apparent horizons in more
detail.

Except for flow algorithms (Section 8.7), all numerical “apparent horizon” finding algorithms
and codes actually find MOTSs, and hereinafter I generally follow the common (albeit sloppy)
practice in numerical relativity of blurring the distinction between an MOTS and an apparent
horizon.