General Birkhoff’s Theorem

Space-time is spherically symmetric if it admits the group of SO(3) as a group of isometries, with the group orbits spacelike two-surfaces. These orbits are necessarily two-surface of constant positive curvature. One commonly chooses coordinate {t, r, θ, φ} so that the group orbits become surfaces {t, r = const} and the radial coordinate r is defined by the requirement that 4πr is the area of these spacelike two-surfaces with the range of zero to infinity. According to the Birkhoff’s theorem upon the above assumptions, Schwarzschild metric is the only solution of the vacuum Einstein field equations. Our aim is to reconsider the solution of the spherically symmetric vacuum Einstein field equations by regarding a weaker requirement. We admit the evident fact that in the completely empty space the radial coordinate r may be defined so that 4πr becomes the area of spacelike two-surfaces {t, r = const} with the range of zero to infinity. This is not necessarily to be true in the presence of a material point mass M. It turns out that inspite of imposing asymptotically flatness and staticness as initial conditions the equations have general classes of solutions which Schwarzschild metric is the only member of them which has an intrinsic singularity at the location of the point mass M. The area of {t, r = const} is 4π(r + αM) in one class and 4π(r + a1Mr + a2M ) in the other class while the center of symmetry is at r = 0. PACS numbers: 04.20.Jb,04.70.Bw