The black hole region of an asymptotically-flat spacetime is defined [81, 82] as the set of events from which no future-pointing null geodesic can reach future null infinity (). The event horizon is defined as the boundary of the black hole region. The event horizon is a null surface in spacetime with (in the words of Hawking and Ellis [82, Page 319]) “a number of nice properties” for studying the causal stucture of spacetime.
The event horizon is a global property of an entire spacetime and is defined nonlocally in time: The event horizon in a slice is defined in terms of (and cannot be computed without knowing) the full future development of that slice.
In practice, to find an event horizon in a numerically-computed spacetime, we typically instrument a numerical evolution code to write out data files of the 4-metric. After the evolution (or at least the strong-field region) has reached an approximately-stationary final state, we then compute a numerical approximation to the event horizon in a separate post-processing pass, using the 4-metric data files as inputs.
As a null surface, the event horizon is necessarily continuous. In theory it need not be anywhere differentiable6, but in practice this behavior rarely occurs7: The event horizon is generally smooth except for possibly a finite set of “cusps” where new generators join the surface; the surface normal has a jump discontinuity across each cusp. (The classic example of such a cusp is the “inseam” of the “pair of pants” event horizon illustrated in Figures 4 and 5.)
A black hole is defined as a connected component of the black hole region in a 3 + 1 slice. The boundary of a black hole (the event horizon) in a slice is a 2-dimensional set of events. Usually this has 2-sphere () topology. However, numerically simulating rotating dust collapse, Abrahams et al.  found that in some cases the event horizon in a slice may be toroidal in topology. Lehner et al. , and Husa and Winicour  have used null (characteristic) algorithms to give a general analysis of the event horizon’s topology in black hole collisions; they find that there is generically a (possibly brief) toroidal phase before the final 2-spherical state is reached. Lehner et al.  later calculated movies showing this behavior for several asymmetric black hole collisions.
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