Geodesic-invariant equations of gravitation

If we start with Einstein’s beautiful hypothesis that test particles in gravitational field move along geodesic lines of some Riemannian spaces V4, it is natural to expect that the differential equations for finding the metric tensor gαβ(x) for a given distribution of matter also should be invariant under any transformations at which the geodesic equations remain invariant. However, the geodesic equations are invariant not only under arbitrary transformation of coordinates (it is rather obvious) but also under geodesic mappings Γβγ(x) → Γ α βγ(x) of the Christoffel symbols in any fixed coordinate system [1][4]. If Γβγ are Christoffel symbols in some coordinate system (x , x, x, x), and we use coordinate time t = x/c (c is speed of light) as a parameter along geodesic lines, then the differential equations of a geodesic line are of the form