Living Reviews in Relativity

"Stationary Black Holes: Uniqueness and Beyond"
Piotr T. ChruĊ›ciel and João Lopes Costa and Markus Heusler 

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1 Introduction
1.1 General remarks
1.2 Organization
2 Definitions
2.1 Asymptotic flatness
2.2 Kaluza–Klein asymptotic flatness
2.3 Stationary metrics
2.4 Domains of outer communications, event horizons
2.5 Killing horizons
2.6 I+-regularity
3 Towards a classification of stationary electrovacuum black hole spacetimes
3.1 Static solutions
3.2 Stationary-axisymmetric solutions
3.3 The no-hair theorem
3.4 Summary of open problems
4 Classification of stationary toroidal Kaluza–Klein black holes
4.1 Black holes in higher dimensions
4.2 Stationary toroidal Kaluza–Klein black holes
4.3 Topology of the event horizon
4.4 Orbit space structure
4.5 KK topological censorship
4.6 Classification theorems for KK-black holes
5 Beyond Einstein–Maxwell
5.1 Spherically symmetric black holes with hair
5.2 Static black holes without spherical symmetry
5.3 The Birkhoff theorem
5.4 The staticity problem
5.5 Rotating black holes with hair
6 Stationary Spacetimes
6.1 Reduction of the Einstein–Hilbert action
6.2 The coset structure of vacuum gravity
6.3 Stationary gauge fields
6.4 The stationary Einstein–Maxwell system
7 Some Applications
7.1 The Mazur identity
7.2 Mass formulae
7.3 The Israel–Wilson–Perjés class
8 Stationary and Axisymmetric Spacetimes
8.1 Integrability properties of Killing fields
8.2 Two-dimensional elliptic equations
8.3 The Ernst equations
8.4 The uniqueness theorem for the Kerr–Newman solution
9 Acknowledgments
Open References References