While the uniqueness theory for black-hole solutions of Einstein’s vacuum equations and the Einstein–Maxwell (EM) equations has seen deep successes, the complete picture is nowhere settled at the time of revising of this work. We know now that, under reasonable global conditions (see Definition 2.1), the domains of dependence of analytic, stationary, asymptotically-flat electrovacuum black-hole spacetimes with a connected non-degenerate horizon belong to the Kerr–Newman family. The purpose of this section is to review the various steps involved in the classification of electrovacuum spacetimes (see Figure 3). In Section 5, we shall then comment on the validity of the partial results in the presence of non-linear matter fields.

For definiteness, from now on we assume that all spacetimes are -regular. We note that the slightly weaker global conditions spelled-out in Theorem 3.1 suffice for the analysis of static spacetimes, or for various intermediate steps of the uniqueness theory, but those weaker conditions are not known to suffice for the Uniqueness Theorem 3.3.

The main task of the uniqueness program is to show that the domains of outer communications of sufficiently regular stationary electrovacuum black-hole spacetimes are exhausted by the Kerr–Newman or the MP spacetimes.

The starting point is the smoothness of the event horizon; this is proven in [76, Theorem 4.11], drawing heavily on the results in [69].

One proves, next, that connected components of the event horizon are diffeomorphic to . This was established in [85], taking advantage of the topological censorship theorem of Friedman, Schleich and Witt [106]; compare [141] for a previous partial result. (Related versions of the topology theorem, applying to globally-hyperbolic, not-necessarily-stationary, spacetimes, have been established by Jacobson and Venkataramani [180], and by Galloway [108, 109, 110, 112]; the strongest-to-date version, with very general asymptotic hypotheses, can be found in [73].)

3.1 Static solutions

3.2 Stationary-axisymmetric solutions

3.2.1 Topology

3.2.2 Candidate metrics

3.2.3 The reduction

3.2.4 The Robinson–Mazur proof

3.2.5 The Bunting–Weinstein harmonic-map argument

3.2.6 The Varzugin–Neugebauer–Meinel argument

3.2.7 The axisymmetric uniqueness theorem

3.3 The no-hair theorem

3.3.1 The rigidity theorem

3.3.2 The uniqueness theorem

3.3.3 A uniqueness theorem for near-Kerrian smooth vacuum stationary spacetimes

3.4 Summary of open problems

3.4.1 Degenerate horizons

3.4.2 Rigidity without analyticity

3.4.3 Many components?

3.2 Stationary-axisymmetric solutions

3.2.1 Topology

3.2.2 Candidate metrics

3.2.3 The reduction

3.2.4 The Robinson–Mazur proof

3.2.5 The Bunting–Weinstein harmonic-map argument

3.2.6 The Varzugin–Neugebauer–Meinel argument

3.2.7 The axisymmetric uniqueness theorem

3.3 The no-hair theorem

3.3.1 The rigidity theorem

3.3.2 The uniqueness theorem

3.3.3 A uniqueness theorem for near-Kerrian smooth vacuum stationary spacetimes

3.4 Summary of open problems

3.4.1 Degenerate horizons

3.4.2 Rigidity without analyticity

3.4.3 Many components?

Living Rev. Relativity 15, (2012), 7
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