There exist no 4-dimensional geodesically equivalent metrics with the same stress-energy tensor

TL;DR

This paper proves that in four dimensions, geodesically equivalent Einstein metrics with the same stress-energy tensor are either affinely equivalent or flat, extending to higher dimensions under certain conditions.

Contribution

It establishes a rigidity result for Einstein metrics with shared geodesic structures and stress-energy tensors, showing they are essentially the same or flat.

Findings

No non-trivial 4D geodesically equivalent Einstein metrics with the same stress-energy tensor exist.

The result extends to higher dimensions if the metrics are complete or the manifold is closed.

Such metrics are either affinely equivalent or flat.

Abstract

We show that if two 4-dimensional metrics of arbitrary signature on one manifold are geodesically equivalent (i.e., have the same geodesics considered as unparameterized curves) and are solutions of the Einstein field equation with the same stress-energy tensor, then they are affinely equivalent or flat. Under the additional assumption that the metrics are complete or the manifold is closed, the result survives in all dimensions >2.