For physical reasons, the black hole equilibrium states are expected to be stationary. Space-times admitting a Killing symmetry exhibit a variety of interesting features, some of which will be discussed in this section. In particular, the existence of a Killing field implies a canonical 3+1 decomposition of the metric. The projection formalism arising from this structure was developed by Geroch in the early seventies [71, 70], and can be found in Chapter 16 of the book on exact solutions by Kramer et al. [117].

A slightly different, rather powerful approach to stationary space-times is obtained by taking advantage of their Kaluza–Klein (KK) structure. As this approach is less commonly used in the present context, we will discuss the KK reduction of the Einstein–Hilbert(–Maxwell) action in some detail, (the more so since this yields an efficient derivation of the Ernst equations and the Mazur identity). Moreover, the inclusion of non-Abelian gauge fields within this framework [19] reveals a decisive structural difference between the Einstein–Maxwell (EM) and the Einstein–Yang–Mills (EYM) system. Before discussing the dimensional reduction of the field equations in the presence of a Killing field, we start this section by recalling the concept of the Killing horizon.

4.1 Killing horizons

4.2 Reduction of the Einstein–Hilbert action

4.3 The coset structure of vacuum gravity

4.4 Stationary gauge fields

4.5 The stationary Einstein–Maxwell system

4.2 Reduction of the Einstein–Hilbert action

4.3 The coset structure of vacuum gravity

4.4 Stationary gauge fields

4.5 The stationary Einstein–Maxwell system

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