We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation Ric(g M ) = λg M with nonnegative cosmological constant λ ≥ 0 is flat. When dim ≥ 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.AMS Mathematics Subject Classification Numbers: Primary 53c50; Secondary 83c20.to the existence of such global R-action. Here, our usage of "stationary" is in a broader sense, it only refers to the existence of a timelike Killing field.One of the main results of this paper is the following theorem:Theorem 1.1 Let (M, g M ) be a geodesically complete spacetime of dimension 4 with a timelike Killing field X such that g M satisfies the Einstein equationHere, (M, g M ) is said to be geodesically complete if the affine parameters of any g M -geodesic on M can be extended to the whole real line R . The Einstein equation satisfied by the spacetime in Theorem 1.1 is equivalent to T = 0 and Λ ≥ 0 in (1.1). We remark that when λ < 0, the result of Theorem 1.1 is not true. The simplest counter examples are anti-De Sitter spacetimes, which are static, geodesically complete, and satisfying Ric(g M ) = λg M for λ < 0.Recall that a Lorentzian manifold (M, g M ) is said to be chronological if it contains no closed timelike curves. In [1], M. T. Anderson proved that if the spacetime (M 4 , g M ) is geodesically complete, chronological and admits an isometric timelike R-action such that g M satisfies the vacuum Einstein field equation Ric(g M ) ≡ 0, then (M 4 , g M ) must be flat. When the R-orbit space M/R is an asymptotically flat 3-manifold, the result was due to A. Lichnerowicz [13] in 1955. The previous pioneering work was due to A. Einstein and A. Einstein-W. Pauli, see [9].The asymptotic flatness on the orbit space is usually a reasonable assumption for an isolated chronological physical system. The chronological condition is used to ensure that the R-orbit space M/R (denoted by N ) is a paracompact Hausdorff and smooth manifold, see [10]. Actually, in this case, the manifold M is diffeomorphic to R × N , and the metric g M has the following global form (see [1],[10], [12])