We prove that any 4-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation Ric(gM) = λgM 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.

中文翻译：

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关于真空爱因斯坦场方程的平稳解

我们证明，任何具有类时杀戮场的 4 维测地完整时空满足真空爱因斯坦场方程 Ric(gM) = λgM 且非负宇宙常数 λ ≥ 0 是平坦的。当dim ≥ 5 时，如果另外假设时空是静态的，我们证明它的通用覆盖等距分裂为Ricci 扁平黎曼流形和实线的乘积。