Does Birkhoff’s theorem really hold?

Abstract Birkhoff’s theorem states that the external gravitational field of any spherically-symmetric system is static and is described by Schwarzschild metric. In this paper, we demonstrate that it is inconsistent with the direct post-Newtonian solution to Einstein field equations in the weak field limit. Moreover, we identify the flaw in the derivation of Birkhoff’s theorem—the consequences of making the coordinate transformations for solving Einstein field equations are ignored, which is due to the misuse of the covariance in general relativity.