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 [185186]. A systematic study of the problem for the EYM system with arbitrary gauge groups was performed by Brodbeck and Straumann [23Jump To The Next Citation Point]. Extending previous results due to Künzle [119] (see also [120121Jump To The Next Citation Point]), the authors of [23Jump 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.19
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