2.3 Stationary metrics
An asymptotically-flat, or -asymptotically-flat, spacetime will be called stationary if
there exists on a complete Killing vector field , which is timelike in the asymptotic region
; such a Killing vector will be sometimes called stationary as well. In fact, in most of the
literature it is implicitly assumed that stationary Killing vectors satisfy for
some and for all large enough. This uniformity condition excludes the possibility of a
timelike vector, which asymptotes to a null one. This involves no loss of generality in well-behaved
asymptotically-flat spacetimes: indeed, this uniform timelikeness condition always holds for Killing vectors,
which are timelike for all large distances if the conditions of the positive energy theorem are
met [17, 77].
In electrovacuum, as part of the definition of stationarity it is also required that the Maxwell field be
invariant with respect to , that is
Note that this definition assumes that the Killing vector is complete, which means that for every
the orbit of is defined for all . The question of completeness of
Killing vectors is an important issue, which needs justifying in some steps of the uniqueness
arguments [57, 59].
In regions where is timelike, there exist local coordinates in which the metric takes the form
Such coordinates exist globally on asymptotically-flat ends, and if the Einstein–Maxwell equations hold, one
can also obtain there [58, Section 1.3], in dimension 3+1,
where the infinity symbol means that (2.3) holds for arbitrary .