Projective equivalence of Einstein spaces in general relativity

There has been some recent interest in the relation between two spacetimes which have the same geodesic paths, that is, spacetimes which are projectively equivalent (sometimes called geodesically equivalent). This paper presents a short and accessible proof of the theorem that if two spacetimes have the same geodesic paths and one of them is an Einstein space then (either each is of constant curvature or) their Lévi-Civitá connections are identical. It also clarifies the relationship between their associated metrics. The results are extended to include the signatures (+ + + +) and (- - + +), and some examples and discussion are given in the case of dimension n > 4. Some remarks are also made which show how these results may be useful in the study of projective symmetry.