5.3 The Birkhoff theorem

The Birkhoff theorem shows that the domain of outer communication of a spherically-symmetric black-hole solution to the vacuum or the EM equations is static. The result does not apply to many other matter models: dust, fluids, scalar fields, Einstein-Vlasov, etc., and it is natural to raise the question for non-Abelian gauge fields. Now, the Einstein Yang–Mills equations have a well-posed Cauchy problem, so one needs to make sure that the constraint equations admit non-stationary solutions: Bartnik [11] has indeed proved existence of such initial data. The problem has also been addressed numerically in [328, 329], where spherically-symmetric solutions of the EYM equations describing the explosion of a gauge boson star or its collapse to a Schwarzschild black hole have been found. A systematic study of the problem for the EYM system with arbitrary gauge groups was performed by Brodbeck and Straumann [39Jump To The Next Citation Point]. Extending previous results of Künzle [205] (see also [206, 207Jump To The Next Citation Point]), the authors of [39Jump To The Next Citation Point] were able to classify the principal bundles over spacetime, which – for a given gauge group – admit SO (3) as symmetry group, acting by bundle automorphisms. It turns out that the Birkhoff theorem can be generalized to bundles, which admit only SO (3 )-invariant connections of Abelian type. We refer to [39Jump To The Next Citation Point] for the precise formulation of the statement in terms of Stiefel diagrams, and to [33Jump To The Next Citation Point, 34Jump To The Next Citation Point, 138Jump To The Next Citation Point, 255] for a classification of EYM solitons. The results in [104, 13] concerning particle-like EYM solutions are likely to be relevant for the corresponding black-hole problem, but no detailed studies of this exist so far.
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