### 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 [99] has given new life to the overall subject (see [100, 101] 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 [56], the classic textbook by Hawking and Ellis [143], Carter’s review [50],
Chapter 12 of Wald’s book [314], the overview [63] and the monograph [151].

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 -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.