### 7.2 General properties

Given certain technical assumptions (including energy conditions), the existence of any
trapped surface (and hence any apparent horizon) implies that the slice contains a black
hole. (The
converse of this statement is not true: An arbitrary (spacelike) slice through a black hole need not contain any apparent
horizon.)
However, if an apparent horizon does exist, it necessarily coincides with, or is contained in, an event
horizon. In a stationary spacetime the event and apparent horizons coincide.
It is this relation to the event horizon which makes apparent horizons valuable for numerical
computation: An apparent horizon provides a useful approximation to the event horizon in a slice, but
unlike the event horizon, an apparent horizon is defined locally in time and so can be computed “on the fly”
during a numerical evolution.

Given a family of spacelike 3 + 1 slices which foliate part of spacetime,
the union of the slices’ apparent horizons (assuming they exist) forms a
world-tube.
This world-tube is necessarily either null or spacelike. If it is null, this world-tube is slicing-independent
(choosing a different family of slices gives the same world-tube, at least so long as each slice still
intersects the world-tube in a surface with 2-sphere topology). However, if the world-tube is
spacelike, it is slicing-dependent: Choosing a different family of slices will in general give a different
world-tube.