While the vacuum and the scalar staticity theorems are based on differential identities and Stokes’ law, the new approach due to Sudarsky and Wald takes advantage of the ADM formalism and a maximal slicing property . Along these lines, the authors of [167, 168] were also able to extend the staticity theorem to 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. As the asymptotic value of a Lie algebra valued scalar is not a gauge freedom in the non-Abelian case, the EYM staticity theorem leaves some room for stationary black holes which are non-rotating – but not static. Moreover, the theorem implies that these configurations must be charged. On the perturbative level, the existence of these charged, non-static black holes with vanishing total angular momentum was recently established by rigorous means .
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