  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.
