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

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