Rigidity in vacuum under conformal symmetry

TL;DR

This paper proves that certain vacuum spacetimes with a timelike conformal symmetry must split as a product space, providing partial support for the Bartnik splitting conjecture in general relativity.

Contribution

It establishes a rigidity result for vacuum spacetimes with conformal symmetries, showing they must be metric products and the conformal Killing field is actually Killing.

Findings

Vacuum spacetime with conformal symmetry splits as a product space.

The conformal Killing vector field is necessarily Killing.

Supports the Bartnik splitting conjecture in vacuum cases.

Abstract

Moitvated in part by [3], in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.