On a new condition distinguishing Weyl and Lorentz space-times

Abstract

The paper is concerned with the derivability of a Lorentz instead of only a Weyl manifold as space-time structure from postulates about free fall and light propagation. For this purpose it identifies a property distinguishing both kinds of space-times. The property is one of a particular metric of the conformal class of the Weyl manifold. viz. that in suitably chosen locally geodesic coordinates theg _i4 components,i=1, 2, 3 vanish along the time axis. Although seemingly somewhat hidden, one is led to this property in looking for a metric which can play a distinguished role. We demonstrate that for a Lorentzian manifold such a condition is always given; thus it is a necessary one. It is sufficient since for a Weyl space it has the consequence that the metric connection of the selectedg is projectively equivalent to the Weyl connection. Thus, if a Weyl space-time complies with it, it is a reducible one. The results of this paper lay the ground for deriving in a second step this condition from a simple, empirically testable postulate about free-fall worldlines and “radar” measurements by light signals.