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
].)
http://www.livingreviews.org/lrr-2012-7 |
Living Rev. Relativity 15, (2012), 7
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