There are no conformal Einstein rescalings of complete pseudo-Riemannian Einstein metrics

TL;DR

This paper proves that for complete pseudo-Riemannian Einstein metrics of indefinite signature, any conformally equivalent Einstein metric must be proportional, showing no non-trivial conformal Einstein rescalings exist.

Contribution

It establishes a rigidity result for Einstein metrics under conformal transformations in indefinite signature, extending previous understanding of conformal invariance.

Findings

Conformally equivalent Einstein metrics are proportional with a constant factor.

Light-line completeness implies no non-trivial conformal Einstein rescalings.

The result holds for closed manifolds without the completeness assumption.

Abstract

We prove the following statement: Let g be a light-line-complete pseudo-Riemannian Einstein metric of indefinite signature on a connected (n>2)-dimensional manifold M. Assume that a conformally equivalent metric is also Einstein. Then, the metrics are proportional with a constant coefficient. If in addition the manifold is closed, the assumption that the manifold is light-line-complete could be omitted.