### 5.4 The staticity problem

Going back one step further in the left half of the classification scheme displayed in
Figure 3, 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 [302, 303, 84],
while the corresponding vacuum problem was solved quite some time ago [143]; 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 [152] as,
for instance, the Einstein–Skyrme system. (See also [158, 149, 160], for more information on
the staticity problem). It should also be noted that the proof given in [152] 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 [302, 303] 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 [38].