The results concerning static black holes are stronger than the general stationary case, and so this case deserves separate discussion. In any case, the proof of uniqueness for stationary black holes branches out at some point and one needs to consider separately uniqueness for static configurations.

In pioneering work, Israel showed that both static vacuum [176] and electrovacuum [177] black-hole
spacetimes satisfying a set of restrictive conditions are spherically symmetric. Israel’s ingenious
method, based on differential identities and Stokes’ theorem, triggered a series of investigations
devoted to the static uniqueness problem (see, e.g., [244, 245, 279, 281, 294]). A breakthrough
was made by Bunting and Masood-ul-Alam [42], who showed how to use the positive energy
theorem^{2} to exclude non-connected
configurations (compare [61]).^{3}

The annoying hypothesis of analyticity, which was implicitly assumed in the above treatments, has been removed in [72]. The issue here is to show that the Killing vector field cannot become null on the domain of outer communications. The first step to prove this is the Vishveshwara–Carter lemma (see Section 2.5.2 and [308, 43]), which shows that null orbits of static Killing vectors form a prehorizon, as defined in Section 2.5.2. To finish the proof one needs to show that prehorizons cannot occur within the d.o.c. This presents no difficulty when analyticity is assumed. Now, analyticity of stationary electrovacuum metrics is a standard property [245, 243] when the Killing vector is timelike, but timelikeness throughout the d.o.c. is not known yet at this stage of the argument. The nonexistence of prehorizons within the d.o.c. for smooth metrics requires more work, and is the main result in [72].

In the static vacuum case the remainder of the argument can be simplified by noting that there are no static solutions with degenerate horizons, which have spherical cross-sections [81]. This is not true anymore in the electrovacuum case, where an intricate argument to handle non-degenerate horizons is needed [83] (compare [284, 295, 225, 62] for previous partial results).

All this can be summarized in the following classification theorem:

Theorem 3.1 Let be an electrovacuum, four-dimensional spacetime containing a spacelike, connected, acausal hypersurface , such that is a topological manifold with boundary consisting of the union of a compact set and of a finite number of asymptotically-flat ends. Suppose that there exists on a complete hypersurface-orthogonal Killing vector, that the domain of outer communication is globally hyperbolic, and that . Then is isometric to the domain of outer communications of a Reissner–Nordström or a MP spacetime.

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