2.3 Stationary metrics

An asymptotically-flat, or KK-asymptotically-flat, spacetime (M, g) will be called stationary if there exists on M a complete Killing vector field k, which is timelike in the asymptotic region ๐’ฎext; 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 g(k, k) < − ๐œ– < 0 for some ๐œ– and for all r 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 k, that is

LkF ≡ 0. (2.6 )

Note that this definition assumes that the Killing vector k is complete, which means that for every p ∈ M the orbit ฯ•t[k](p) of k is defined for all t ∈ โ„. The question of completeness of Killing vectors is an important issue, which needs justifying in some steps of the uniqueness arguments [57Jump To The Next Citation Point, 59].

In regions where k is timelike, there exist local coordinates in which the metric takes the form

g = − V 2(dt + ๐œƒ dxi)2 + γ dxidxj , (2.7 ) โ—Ÿiโ—โ—œโ—ž โ—Ÿij-โ—โ—œ---โ—ž =:๐œƒ =:γ
with
k = ∂t =⇒ ∂tV = ∂t๐œƒi = ∂tγij = 0. (2.8 )
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,
γij − δij = O ∞ (r−1), ๐œƒi = O∞ (r−1), V − 1 = O ∞(r− 1), (2.9 )
and
A = O (r−1), (2.10 ) μ ∞
where the infinity symbol means that (2.3View Equation) holds for arbitrary d.
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