On stationary solutions to the vacuum Einstein field equations

TL;DR

This paper proves that certain stationary vacuum solutions to Einstein's equations in four and higher dimensions are flat or split into simpler geometric components, revealing rigidity properties of such spacetimes.

Contribution

It establishes flatness for 4D geodesically complete vacuum spacetimes with a timelike Killing field and extends splitting results to higher dimensions under static assumptions.

Findings

4D geodesically complete vacuum spacetimes with timelike Killing fields are flat.

Higher-dimensional static vacuum spacetimes' universal covers split as a product of Ricci flat manifold and a line.

Abstract

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.