3.1 Static solutions

A stationary spacetime is called static if the Killing vector k is hypersurface-orthogonal: this means that the distribution of the hyperplanes orthogonal to k is integrable. Equivalently,
k ∧ dk = 0.
Here and elsewhere, by a common abuse of notation, we also write k for the one-form associated with k.

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 [176Jump To The Next Citation Point] and electrovacuum [177Jump To The Next Citation Point] 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, 245Jump To The Next Citation Point, 279Jump To The Next Citation Point, 281, 294Jump To The Next Citation Point]). A breakthrough was made by Bunting and Masood-ul-Alam [42], who showed how to use the positive energy theorem2 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 [72Jump To The Next Citation Point]. 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, 43Jump To The Next Citation Point]), 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 [72Jump To The Next Citation Point].

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 [81Jump To The Next Citation Point]. 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, 62Jump To The Next Citation Point] for previous partial results).

All this can be summarized in the following classification theorem:

Theorem 3.1 Let (M, g) be an electrovacuum, four-dimensional spacetime containing a spacelike, connected, acausal hypersurface 𝒮, such that --- 𝒮 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 M a complete hypersurface-orthogonal Killing vector, that the domain of outer communication ⟨⟨Mext⟩⟩ is globally hyperbolic, and that --- ∂𝒮 ⊂ M ∖ ⟨⟨Mext ⟩⟩. Then ⟨⟨Mext ⟩⟩ is isometric to the domain of outer communications of a Reissner–Nordström or a MP spacetime.

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