Space-time and G_2

TL;DR

This paper explores Weyl structures on space-time, develops the theory of Weyl connections, and links them to G_2-conformal structures, providing explicit calculations for specific space-times.

Contribution

It establishes the uniqueness of torsion-free Weyl connections and connects them to (2,3,5)-Pfaffian systems and G_2-conformal structures, with explicit examples.

Findings

Unique torsion-free Weyl connections determined by Weyl structures

Explicit G_2-conformal structures derived for specific space-times

Connection between Weyl connections and Cartan's Pfaffian systems

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