4.1 Killing HorizonsStationary Black Holes: Uniqueness and 3.5 Rotating Black Holes with

4 Stationary Space-Times 

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 [18Jump To The Next Citation Point In The Article] 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 HorizonsStationary Black Holes: Uniqueness and 3.5 Rotating Black Holes with

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
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