### 3.3 The Birkhoff theorem

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-Abelian
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 [23]. Extending previous results due to Künzle [119] (see also [120, 121]), the
authors of [23] 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.