Recall that , respectively , is the set covered by past-directed timelike, respectively causal, curves originating from , while denotes the boundary of , etc. The sets , etc., are defined as , etc., after changing time-orientation. See [143, 16, 256, 236, 266, 66] and references therein for details of causality theory.
Consider an asymptotically-flat, or -asymptotically-flat, spacetime with a Killing vector , which is timelike on the asymptotic end . The exterior region and the domain of outer communications , for which we will also use the abbreviation d.o.c., are then defined as (see Figure 1)black-hole region and the black-hole event horizon are defined as
The white-hole region and the white-hole event horizon are defined as above after changing time orientation:
By standard causality theory, an event horizon is the union of Lipschitz null hypersurfaces. It turns out that event horizons in stationary spacetimes satisfying energy conditions are as smooth as the metric allows [76, 69]; thus, smooth if the metric is smooth, analytic if the metric is.
Living Rev. Relativity 15, (2012), 7
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