Birkhoff's theorem and Einstein's equations

I examine Einstein's gravitational field equations ${G}_{j}^{i}=k{T}_{j}^{i}$ in a spherically symmetric space-time. I find the necessary and sufficient conditions under which the solutions of the field equations will be static for the most general spherically symmetric energy-momentum tensors. That is, the conditions for the validity of Birkhoff's theorem are presented.