  We prove that any 4-dimensional geodesically complete spacetime with a
timelike Killing field satisfying the vacuum Einstein field equation
$Ric(g_{M})=\lambda g_{M}$ with nonnegative cosmological constant $\lambda\geq
0$ is flat. When dim $\geq 5$, if the spacetime is assumed to be static
additionally, we prove that its universal cover splits isometrically as a
product of a Ricci flat Riemannian manifold and a real line.
