1.2 Organization

The purpose of this text is to review some features of four-dimensional stationary asymptotically-flat black-hole spacetimes. Some black-hole solutions with non-zero cosmological constant can be found in [313, 36, 323, 286, 271, 15]. It should be noted that the discovery of five-dimensional black rings by Emparan and Reall [99Jump To The Next Citation Point] has given new life to the overall subject (see [100, 101Jump To The Next Citation Point] and references therein) but here we concentrate on four-dimensional spacetimes with mostly classical matter fields.

For detailed introductions into the subject we refer to Chandrasekhar’s book on the mathematical theory of black holes [56Jump To The Next Citation Point], the classic textbook by Hawking and Ellis [143Jump To The Next Citation Point], Carter’s review [50Jump To The Next Citation Point], Chapter 12 of Wald’s book [314Jump To The Next Citation Point], the overview [63] and the monograph [151Jump To The Next Citation Point].

The first part of this report is intended to provide a guide to the literature, and to present some of the main issues, without going into technical details. We start by collecting the significant definitions in Section 2. We continue, in Section 3, by recalling the main steps leading to the uniqueness theorem for electro-vacuum black-hole spacetimes. The classification scheme obtained in this way is then reexamined in the light of solutions, which are not covered by no-hair theorems, such as stationary Kaluza–Klein black holes (Section 4) and the Einstein–Yang–Mills black holes (Section 5).

The second part reviews the main structural properties of stationary black-hole spacetimes. In particular, we discuss the dimensional reduction of the field equations in the presence of a Killing symmetry in more detail (Section 6). For a variety of matter models, such as self-gravitating Abelian gauge fields, the reduction yields a σ-model, with symmetric target manifold, coupled to three-dimensional gravity. In Section 7 we discuss some aspects of this structure, namely the Mazur identity and the quadratic mass formulae, and we present the Israel–Wilson class of metrics.

The third part is devoted to stationary and axisymmetric black-hole spacetimes (Section 8). We start by recalling the circularity problem for non-Abelian gauge fields and for scalar mappings. The dimensional reduction with respect to the second Killing field leads to a boundary value problem on a fixed, two-dimensional background. As an application, we outline the uniqueness proof for the Kerr–Newman metric.

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