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.