The Birkhoff theorem implies that the domain of outer
communication of a spherically symmetric black hole solution to
the vacuum or the EM equations is static. Like its counterpart,
the Israel theorem, the Birkhoff theorem admits no
straightforward extension to arbitrary matter models, such as
non-Ableian gauge fields: Numerical investigations have revealed
spherically symmetric solutions of the EYM equations which
describe the explosion of a gauge boson star or its collapse to a
Schwarzschild black hole [185], [186]. A systematic study of the problem for the EYM system with
arbitrary gauge groups was performed by Brodbeck and Straumann [22]. Extending previous results due to Künzle [119] (see also [120], [121]), the authors of [22] 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.