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-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.

Stationary Black Holes: Uniqueness and Beyond Markus Heusler http://www.livingreviews.org/lrr-1998-6 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de