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 [36], the classic by Hawking and Ellis [84
], Carter's review [32
], and chapter 12 of Wald's book [178
]. 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 (Sect. 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 (Sect. 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 (Sect.
4). For a variety of matter models, such as self-gravitating
Abelian gauge fields, the reduction yields a
-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 (Sect.
5).
The third part is devoted to stationary and axisymmetric black hole space-times (Sect. 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.
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Stationary Black Holes: Uniqueness and Beyond
Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |