Weyl Geometries and Timelike Geodesics

In view of Ehlers-Pirani-Schild formalism, since 1972 Weyl geometries should be considered to be the most appropriate and complete framework to represent (relativistic) gravitational fields. We shall here show that in any given Lorentzian spacetime (M,g) that admits global timelike vector fields any such vector field u determines an essentially unique Weyl geometry ([g], \Gamma) such that u is \Gamma-geodesic (i.e. parallel with respect to \Gamma).