Space-time and G_2

Abstract

A Weyl structure is a bundle over space-time, whose fiber at each space-time point is a space of maximally isotropic complex tangent planes. We develop the theory of Weyl connections for Weyl structures and show that the requirement that the connection be torsion-free fixes the Weyl connection uniquely. Further we show that to each such Weyl connection, there is naturally associated a (2, 3, 5)-Pfaffian system, as first analyzed by Cartan. We determine the associated G_2-conformal structure and calculate it explicitly in the cases of the Kapadia family of space-times and of the Schwarzschild solution