Extension of bundles of null directions

TL;DR

This paper explores the geometric structure of the bundle of null directions over Einstein space-times, extending it to an elliptic fibration and reinterpreting the Petrov classification through this extension.

Contribution

It constructs the invariants of the G-structure on the null direction bundle and introduces a natural, though non-unique, elliptic extension that reinterprets Petrov types.

Findings

Extension of P is an elliptic fibration over space-time.

The invariants of the G-structure are explicitly constructed.

The Petrov classification is reinterpreted via the fibers of the extension.

Abstract

The geometry of P, the bundle of null directions over an Einstein space-time, is studied. The full set of invariants of the natural G-structure on P is constructed using the Cartan method of equivalence. This leads to an extension of P which is an elliptic fibration over the space-time. Examples are given which show that such an extension, although natural, is not unique. A reinterpretation of the Petrov classification in terms of the fibres of an extension of P is presented.