Organization

1.2 Organization

The purpose of this text is to review some of the most important features of black hole space-times. Since the investigation of dynamical problems lies beyond the scope of this report, we shall mainly be concerned with stationary situations. Moreover, the concept of the event horizon requires asymptotic flatness. (Black hole solutions with cosmological constant are, therefore, not considered in this text.⁵ ) Hence, we are dealing with asymptotically flat, stationary black configurations of self-gravitating classical matter fields.

The emphasis is given to the recent developments in the field and to the fundamental concepts. For detailed introductions into the subject we refer to Chandrasekhar’s book on the mathematical theory of black holes [37 ], the classic by Hawking and Ellis [84 ], Carter’s review [33 ], and Chapter 12 of Wald’s book [177 ]. Some of the issues which are not raised in this text can be found in [87 ], others will be included in a future version.

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 recalling the main steps involved in the uniqueness theorem for electro-vacuum black hole space-times (Section 2). The classification scheme obtained in this way is then reexamined in the light of the solutions which are not covered by no-hair theorems, such as the Einstein–Yang–Mills black holes (Section 3).

The second part reviews the main structural properties of stationary black hole space-times. In particular, we recall the notion of a Killing horizon, and discuss the dimensional reduction of the field equations in the presence of a Killing symmetry in some detail (Section 4). For a variety of matter models, such as self-gravitating Abelian gauge fields, the reduction yields a $σ$ -model with symmetric target manifold, effectively coupled to three-dimensional gravity. Particular applications of this distinguished structure are the Mazur identity, the quadratic mass formulas and the Israel Wilson class (Section 5).

The third part is devoted to stationary and axisymmetric black hole space-times (Section 6). 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 yields a boundary value problem on a fixed, two-dimensional background, provided that the field equations assume the coset structure on the effective level. As an application we recall the uniqueness proof for the Kerr–Newman metric.