3.5 Rotating Black Holes with 3 Beyond Einstein-Maxwell3.3 The Birkhoff Theorem

3.4 The Staticity Problem 

Going back one step further on the left half of the classification scheme displayed in Fig. 1, one is led to the question whether all black holes with non-rotating horizon are static. For the EM system this issue was settled only recently [167Jump To The Next Citation Point In The Article], [168Jump To The Next Citation Point In The Article], Popup Footnote whereas the corresponding vacuum problem was solved quite some time ago [84Jump To The Next Citation Point In The Article]. Using a slightly improved version of the argument given in [84Jump To The Next Citation Point In The Article], Popup Footnote . the staticity theorem can be generalized to self-gravitating stationary scalar fields and scalar mappings [88Jump To The Next Citation Point In The Article] as, for instance, the Einstein-Skyrme system. (See also [94], [85], [96], for more information on the staticity problem.)

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 [43]. 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 [21Jump To The Next Citation Point In The Article].



3.5 Rotating Black Holes with 3 Beyond Einstein-Maxwell3.3 The Birkhoff Theorem

image Stationary Black Holes: Uniqueness and Beyond
Markus Heusler
http://www.livingreviews.org/lrr-1998-6
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