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

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. If we additionally assume that the metrics are complete or that the manifold is closed, the result remains valid in all dimensions ≥3.