On the bundle of null cones

TL;DR

This paper explores the geometric and algebraic structure of null vector fields on Lorentzian manifolds, revealing a para-associative ternary product and a non-polynomial graded bundle structure.

Contribution

It introduces the null tangent bundle as a non-polynomial graded bundle and defines a para-associative ternary product on null vector fields.

Findings

Null vector fields form a bundle with a para-associative ternary product.

The null tangent bundle is a non-polynomial graded bundle.

Provides a new geometric framework for null vector fields on Lorentzian manifolds.

Abstract

We examine the bundle structure of the field of nowhere vanishing null vector fields on a (time-oriented) Lorentzian manifold. Sections of what we refer to as the null tangent, are by definition nowhere vanishing null vector fields. It is shown that the set of nowhere vanishing null vector fields comes equipped with a para-associative ternary partial product. Moreover, the null tangent bundle is an example of a non-polynomial graded bundle.