Abstract
In the literature different concepts of compatibility between a projective structure {mathscr {P}} and a conformal structure {mathscr {C}} on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. In an often cited paper (Ehlers et al. in: O’Raifertaigh (ed) General Relativity, Papers in Honour of J.L. Synge, Clarendon Press, Oxford, 2012) Ehlers/Pirani/Schild introduce still another criterion which is natural from the physical point of view: every light like geodesics of {mathscr {C}} is a geodesics of {mathscr {P}}. Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned (Trautman in Gen Relativ Gravit 44:1581–1586, 2012); (Vladimir in Commun Math Phys 329:821–825, 2014); as reported by Scholz (in: A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection, 2019). Here it is proved that the conjecture of EPS is correct.
Highlights
In a widely read paper [7] J
Schild (EPS) argued that a projective structure P and a conformal structure C on a differentiable manifold M determine a Weylian metric on M, if only the geodesics of P are light like geodesics of C
Weyl generalized the concept of a Riemannian metric in order to avoid the possibility of direct metrical comparison of vectors or other fields at finitely distant points [20,21]. He introduced a scale gauge connection in addition to a conformal structure, defining what later would be called a Weylian metric on a differentiable manifold (see below, Definition 1(ii))
Summary
In a widely read paper [7] J. Schild (EPS) argued that a projective structure P and a conformal structure C on a differentiable manifold M determine a Weylian metric on M, if only the geodesics of P are light like geodesics of C. This statement complements a proposal of H. EPS claimed that the above mentioned light-cone condition for P and C is sufficient for securing the existence of a Weylian metric which Weyl had assumed from the outset; but the argument given to substantiate the statement remains vague and is far from a mathematical proof [19]. The aim of the present paper is to fill in the gap and to prove the central statement of EPS.
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