Description
A classic theorem of Weyl (1921) states that a Weyl metric—a natural generalisation of a pseudo-Riemannian metric—is uniquely determined by its conformal and projective structures. This theorem forms the basis for the famous Ehlers–Pirani–Schild (EPS) axiomatic reconstruction (1972) of Lorentzian spacetime structure from light rays and the worldlines of massive particles.
An equivalent formulation of Weyl’s result is that a torsion-free linear connection compatible with a pseudo-Riemannian conformal structure is uniquely determined by its projective structure. We discuss analogous results for suitably defined notions of conformal structure for Galilei and Carroll geometry, i.e. for spacetime geometries arising as the Newtonian and ‘ultra-relativistic’ limits of Lorentzian geometry.