Asian Journal of Mathematics

Volume 23 (2019)

Number 4

On stationary solutions to the vacuum Einstein field equations

Pages: 609 – 630



Bing-Long Chen (Department of Mathematics, Sun Yat-sen University, Guangzhou, China)


We prove that any $4$-dimensional geodesically complete spacetime with a timelike Killing field satisfying the vacuum Einstein field equation $\operatorname{Ric} (g_M) = \lambda g_M$ with nonnegative cosmological constant $\lambda \geq 0$ is flat. When $\operatorname{dim} \geq 5$, 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 $\mathbb{R}$.


stationary solutions, vacuum Einstein field equations

2010 Mathematics Subject Classification

Primary 53C50. Secondary 83C20.

Received 21 January 2017

Accepted 3 May 2018

Published 7 January 2020