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

Abstract

Ehlers, Pirani, and Schild argued that measurements of null and timelike geodesics yield Weyl and projective connections, respectively, with compatibility in the lightlike limit giving an integrable Weyl connection. Their conclusions hold only for a 4-dim representation of the conformal connection on the null cone, and by restricting reparameterizations of timelike geodesics to yield a torsion-free, affine connection. An arbitrary connection gives greater freedom. A linear connection for the conformal symmetry of null geodesics requires the SO(4,2) representation. The enlarged class of projective transformations of timelike geodesics changes Weyl’s projective curvature, and we find invariant forms of the torsion and nonmetricity, along with a new, invariant, second rank tensor field generalizing the dilatational curvature without requiring a metric. We show that either projective or conformal connections require a monotonic, twice differentiable function on a spacetime region foliated by order isomorphic, totally ordered, twice differentiable timelike curves in a necessarily Lorentzian geometry. We prove that the conditions for projective and conformal Ricci flatness imply each other and gauge choices within either can reduce the geometry to the original Riemannian form. Thus, measurements of null and timelike geodesics lead to an SO(4,2) connection, with no requirement for a lightlike limit. Reduction to integrable Weyl symmetry can only follow from the field equations of a gravity theory. We show that the simplest quadratic spacetime action leads to this reduction.