Inspection of the basic features of these solutions challenges any naive attempt to generalize the classification scheme developed for spacetime dimension four: One can find black rings and Myers–Perry black holes with the same mass and angular momentum, which must necessarily fail to be isometric since the horizon topologies do not coincide. In fact there are non-isometric black rings with the same Poincaré charges; consequently a classification in terms of mass, angular momenta and horizon topology also fails. Moreover, the Black Saturns provide examples of regular vacuum multi–black-hole solutions, which are widely believed not to exist in dimension four; interestingly, there exist Black Saturns with vanishing total angular momentum, a feature that presumably distinguishes the Schwarzschild metric in four dimensions.
Nonetheless, results concerning 4-dimensional black holes either generalize or serve as inspiration in higher dimensions. This is true for landmark results concerning black-hole uniqueness and, in fact, classification schemes exist for classes of higher dimensional black-hole spacetimes, which mimic the symmetry properties of the “static or axisymmetric” alternative, upon which the uniqueness theory in four-dimensions is built.
For instance, staticity of -regular, vacuum, asymptotically-flat, non-rotating, non-degenerate black holes remains true in higher dimensions6. Also, Theorem 3.1 remains valid for vacuum spacetimes of dimension , , whenever the positive energy theorem applies to an appropriate doubling of (see , Section 3.1 and references therein). Moreover, the discussion in Section 3.1 together with the results in [282, 283] suggest that an analogous generalization to electrovacuum spacetimes exists, which would lead to uniqueness of the higher-dimensional Reissner–Nordström metrics within the class of static solutions of the Einstein–Maxwell equations, for all (see also [101, Section 8.2],  and references therein).
Rigidity theorems are also available for -dimensional, asymptotically-flat and analytic black-hole spacetimes: the non-degenerate horizon case was established in  (compare ), and partial results concerning the degenerate case were obtained in . These show that stationary rotating (analytic) black holes are “axisymmetric”, in the sense that their isometry group contains ; the factor corresponds to the action generated by the stationary vector, while the circle action provides an “extra” axial Killing vector. A conjecture of Reall , supported by the results in , predicts the existence of 5-dimensional black holes with exactly isometry group; in particular, it is conceivable that the rigidity results are sharp when providing only one “axial” Killing vector. The results in [133, 92, 166] are likely to be relevant in this context.
So we see that, assuming analyticity and asymptotic flatness, the dichotomy provided by the rigidity theorem remains valid but its consequences appear to be weaker in higher dimensions. A gap appears between the two favorable situations encountered in dimension four: one being the already discussed staticity and the other corresponding to black holes with cohomogeneity-two Abelian groups of isometries. We will now consider this last scenario, which turns out to have connections to the four dimensional case.
Living Rev. Relativity 15, (2012), 7
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