5.4 The staticity problem

Going back one step further in the left half of the classification scheme displayed in Figure 3View Image, one is led to the question of whether all black holes with non-rotating horizon are static. For non-degenerate EM black holes this issue was settled by Sudarsky and Wald [302Jump To The Next Citation Point, 303Jump To The Next Citation Point, 84Jump To The Next Citation Point],7 while the corresponding vacuum problem was solved quite some time ago [143Jump To The Next Citation Point]; the degenerate case remains open. Using a slightly improved version of the argument given in [143], the staticity theorem can be generalized to self-gravitating stationary scalar fields and scalar mappings [152Jump To The Next Citation Point] as, for instance, the Einstein–Skyrme system. (See also [158Jump To The Next Citation Point, 149, 160], for more information on the staticity problem). It should also be noted that the proof given in [152Jump To The Next Citation Point] works under less restrictive topological assumptions, since it does not require the global existence of a twist potential.

While the vacuum and the scalar staticity theorems are based on differential identities and integration by parts, the approach due to Sudarsky and Wald takes advantage of the ADM formalism and the existence of a maximal slicing [84]. Along these lines, the authors of [302Jump To The Next Citation Point, 303Jump To The Next Citation Point] were able to extend the staticity theorem to topologically-trivial non-Abelian black-hole solutions. However, in contrast to the Abelian case, the non-Abelian version applies only to configurations for which either all components of the electric Yang–Mills charge or the electric potential vanish asymptotically. This leaves some room for stationary black holes, which are non-rotating and not static. Moreover, the theorem implies that such configurations must be charged. On a perturbative level, the existence of these charged, non-static black holes with vanishing total angular momentum was established in [38Jump To The Next Citation Point].

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