On generalization of Birkhoff's theorem

A theorem of G.D. Birkhoff states that a spherically symmetric space-time with vanishing Ricci tensor admits a Killing field beyond those given by the spherical action and thus it is static under an additional condition motivated by the exterior Schwarzschild space-time. Several generalizations of this result were obtained for spherically symmetric space-times and also for such more general ones where an isometric action of a 3-dimensional Lie group with orbits of maximal dimension 2 is given. It seems, however, that a complete account of those space-times where such a generalized theorem holds fails even now. A construction of all those spherically symmetric space-times is presented below where a generalization of Birkhoff's theorem holds. (c) 2006 Elsevier B.V. All rights reserved.