### 2.4 Domains of outer communications, event horizons

For let denote the one-parameter group of diffeomorphisms generated by ;
we will write for whenever ambiguities are unlikely to occur.
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)

The 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:

It
follows that the boundaries of are included in the event horizons. We set
There is considerable freedom in choosing the asymptotic region . However, it is not too difficult to
show that , and hence , and , are independent of the choice of
whenever the associated ’s overlap.
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.