### 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 [39]. Extending previous results of
Künzle [205] (see also [206, 207]), the authors of [39] were able to classify the principal bundles over
spacetime, which – for a given gauge group – admit as symmetry group, acting by bundle
automorphisms. It turns out that the Birkhoff theorem can be generalized to bundles, which admit only
-invariant connections of Abelian type. We refer to [39] for the precise formulation of
the statement in terms of Stiefel diagrams, and to [33, 34, 138, 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.