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 . 16 There may be multiple MOTSs in a slice, either nested within each other or intersecting17. 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  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 space18. Hawking and Ellis  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.
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