3.4 Summary of open problems

For the convenience of the reader, we summarize here the main open problems left in the no-hair theorem.

3.4.1 Degenerate horizons

We recall that there exist no vacuum static spacetimes containing degenerate horizons with compact spherical sections [81Jump To The Next Citation Point]. On the other hand, MP [220Jump To The Next Citation Point, 262Jump To The Next Citation Point] black holes provide the only electro-vacuum static examples with non-connected degenerate horizons. See [78Jump To The Next Citation Point, 154Jump To The Next Citation Point] and references therein for a discussion of the geometry of MP black holes.

Under the remaining hypotheses of Theorem 3.3, the only step where the hypothesis of non-degeneracy enters is the proof of existence of a maximal hypersurface 𝒮 in the black-hole spacetime under consideration, such that 𝒮 is Cauchy for the domain of outer communications. The geometry of Cauchy surfaces in the case of degenerate horizons is well understood by now [62, 79Jump To The Next Citation Point], and has dramatically different properties when compared to the non-degenerate case. A proof of existence of maximal hypersurfaces in this case would solve the problem, but requires new insights. A key missing element is an equivalent of Bartnik’s a priori height estimate [10], established for asymptotically-flat ends, that would apply to asymptotically-cylindrical ends.

3.4.2 Rigidity without analyticity

Analyticity enters the current argument at two places: First, one needs to construct the second Killing vector near the horizon. This can be done by first constructing a candidate at the horizon, and then using analyticity to extend the candidate to a neighborhood of the horizon. Next, the Killing vector has to be extended to the whole domain of outer communications. This can be done using analyticity and a theorem by Nomizu [253], together with the fact that I+-regular domains of outer communications are simply connected. Finally, analyticity can be used to provide a simple argument that prehorizons do not intersect ⟨⟨Mext⟩⟩ (but this is not critical, as a proof is available now within the smooth category of metrics [72Jump To The Next Citation Point]).

A partially-successful strategy to remove the analyticity condition has been invented by Alexakis, Ionescu and Klainerman in [2Jump To The Next Citation Point]. Their argument applies to non-degenerate near-Kerrian configurations, but the general case remains open.

The key to the approach in [2] is a unique continuation theorem near bifurcate Killing horizons proven in [174], which implies the existence of a second Killing vector field, say m, in a neighborhood of the horizon. One then needs to prove that m extends to the whole domain of outer communications. This is established via another unique continuation theorem [175] with specific convexity conditions. These lead to non-trivial restrictions, and so far the argument has only been shown to apply to near-Kerrian configurations.

A unique continuation theorem across more general timelike surfaces would be needed to obtain the result without smallness restrictions.

It follows from what has been said in [72Jump To The Next Citation Point] that the boundary of the set where two Killing vector fields are defined cannot become null within a domain of outer communications; this fact might be helpful in solving the full problem.

3.4.3 Many components?

The only known examples of singularity-free stationary electrovacuum black holes with more than one component are provided by the MP family. (Axisymmetric MP solutions are possible, but MP metrics only have one Killing vector in general.) It has been suspected for a very long time that these are the only such solutions, and that there are thus no such vacuum configurations. This should be contrasted with the five-dimensional case, where the Black Saturn solutions of Elvang and Figueras [97Jump To The Next Citation Point] (compare [71Jump To The Next Citation Point, 305Jump To The Next Citation Point]) provide non-trivial two-component examples.

It might be convenient to summarize the general facts known about four-dimensional multi-component solutions.4 In case of doubts, I+-regularity should be assumed.

We start by noting that the static solutions, whether connected or not, have already been covered in Section 3.1.

A multi-component electro-vacuum configuration with all components non-degenerate and non-rotating would be, by what has been said, static, but then no such solutions exist (all components of an MP black hole are degenerate). On the other hand, the question of existence of a multi–black-hole configuration with components of mixed type, none of which rotates, is open; what’s missing is the proof of existence of maximal hypersurfaces in such a case. Neither axisymmetry nor staticity is known for such configurations.

Analytic multi–black-hole solutions with at least one rotating component are necessarily axisymmetric; this leads one to study the corresponding harmonic-map equations, with candidate solutions provided by Weinstein or by inverse scattering techniques [198, 322, 23, 22]. The Weinstein solutions have no singularities away from the axes, but they are not known in

explicit form, which makes difficult the analysis of their behavior on the axis of rotation. The multi–black-hole metrics constructed by multi-soliton superpositions or by Bäcklund transformation techniques are obtained as rational functions with several parameters, with explicit constraints on the parameters that lead to a regular axis [222], but the analysis of the zeros of their denominators has proved intractable so far. It is perplexing that the five dimensional solutions, which are constructed by similar methods [268], can be completely analyzed with some effort and lead to regular solutions for some choices of parameters, but the four-dimensional case remains to be understood.

In any case, according to Varzugin [306, 307] and, independently, to Neugebauer and Meinel [251] (a more detailed exposition can be found in [249Jump To The Next Citation Point, 147Jump To The Next Citation Point]), the multi-soliton solutions provide the only candidates for stationary axisymmetric electrovacuum solutions. A breakthrough in the understanding of vacuum two-component configurations has been made by Hennig and Neugebauer [147, 249Jump To The Next Citation Point], based on the area-angular momentum inequalities of Ansorg, Cederbaum and Hennig [145Jump To The Next Citation Point] as follows: Hennig and Neugebauer exclude many of the solutions by verifying that they have negative total ADM mass. Next, configurations where two horizons have vanishing surface gravity are shown to have zeros in the denominators of some geometric invariants. For the remaining ones, the authors impose a non-degeneracy condition introduced by Booth and Fairhurst [25]: a black hole is said to be sub-extremal if any neighborhood of the event horizon contains trapped surfaces. The key of the analysis is the angular momentum - area inequality of Hennig, Ansorg, and Cederbaum [145Jump To The Next Citation Point], that on every sub-extremal component of the horizon it holds that

8π|J| < A, (3.1 )
where J is the Komar angular-momentum and A the area of a section. (It is shown in [7] and [146, Appendix] that κ = 0 leads to equality in (3.1View Equation) under conditions relevant to the problem at hand.) Hennig and Neugebauer show that all remaining candidate solutions violate the inequality; this is their precise non-existence statement.

The problem with the argument so far is the lack of justification of the sub-extremality condition. Fortunately, this condition can be avoided altogether using ideas of [88] concerning the inequality (3.1View Equation) and appealing to the results in [96, 6, 73Jump To The Next Citation Point] concerning marginally–outer-trapped surfaces (MOTS): Using existence results of weakly stable MOTS together with various aspects of the candidate Weyl metrics, one can adapt the argument of [145] to show [70] that the area inequality (3.1View Equation), with “less than” there replaced by “less than or equal to”, would hold for those components of the horizon, which have non-zero surface gravity, assuming an I+-regular metric of the Weyl form, if any existed. The calculations of Hennig and Neugebauer [249] together with a contradiction argument lead then to

Theorem 3.6 + I-regular two-Kerr black holes do not exist.

The case of more than two horizons is widely open.

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