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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 hole19. (The converse of this statement is not true: An arbitrary (spacelike) slice through a black hole need not contain any apparent horizon20.) 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-tube21. 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-tube22.


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