A Weyl-Type Theorem in Geometrized Newtonian Gravity, and How It May Bear on Shape Dynamics

Abstract

The Weyl Theorem states that the conformal structure and the projective structure jointly suffice to fix the metric up to a global constant. This is a powerful interpretive tool in general relativity: it says in effect that if I know the paths of light rays in vacuo and I know the images of the paths of freely falling particles (i.e., the spacetime curves they follow with no preferred parametrization), then I know the metric. It has particular relevance to Shape Dynamics, where the conformal structure is taken as the object of primitive geometrical interest, and one does not generally want a preferred parametrization of timelike geodesics. The spacetimes of geometrized Newtonian gravity share many important features with the way that Shape Dynamics approaches the construction of relativistic spacetimes, in particular the fixing of a preferred foliation of spacetime by spacelike slices. Studying the way that geometrized Newtonian gravity does and does not allow one to recover a Weyl-type Theorem may, therefore, shed light on the ways that Shape Dynamics may allow one to recover the structure of relativistic spacetimes. I show that, in geometrized Newtonian gravity, the conformal structure of the spatial and temporal metrics, in conjunction with the projective structure of timelike curves, allows one to recover the full metrical structure of a geometrized Newtonian gravity only if one also fixes an affine parametreization of at least one timelike geodesic. This suggests that, in far as the analogy between geometrized Newtonian gravity and Shape Dynamics is a physically significant one, Shape Dynamics may be right not to demand preferred parametrizations of timelike geodesics from the start. The tools I develop to prove the theorem may also be applicable to problems in Shape Dynamics itself. But I make no promises.