Geometry from geodesics: fine-tuning Ehlers, Pirani, and Schild

Ehlers, Pirani, and Schild argued that measurements of timelike and null geodesics yield projective and Weyl connections, respectively, with compatibility giving an integrable Weyl geometry. We find greater freedom for both connections, including either a 1-parameter class of projective connections, or a conformal connection for timelike curves, and conformal symmetry for null curves. Examining the effect of every member of both the projective class and the conformal transformations on the curvature, torsion, nonmetricity, and metric, we find the changes from the original geometry. Forming invariant connections we compute both the projectively and conformally invariant forms of the curvature, torsion, and nometricity, showing that the projective and conformal tensors depend on the same gauge vector. We prove that the conditions for projective and conformal Ricci flatness imply each other and either reduces the geometry to the original form. The different transformations depend on the same monotonic, twice differentiable function on a region of spacetime. Existence of such a function requires a region foliated by order isomorphic, totally ordered, twice differentiable timelike curves in a necessarily Lorentzian geometry. To find a geometry consistent with the allowed symmetry, we develop the real, linear, 6-dim representation of the conformal connection. Taking the simplest form of the spacetime action using this connection, we show that reductions of the system lead us back to an integrable Weyl geometry.