Causal geometries, null geodesics, and gravity

TL;DR

This paper generalizes the concept of null geodesics using Legendrian dynamics, extends the Weyl tensor, and shows that Raychaudhuri--Sachs equations still describe null geodesic convergence in a broader geometric setting, deepening understanding of gravity.

Contribution

It introduces a generalized notion of null geodesics via Legendrian dynamics and extends key gravitational equations to this new framework.

Findings

Existence of a natural extension of the Weyl tensor depending only on the conical subbundle.

Raychaudhuri--Sachs equations hold in the generalized setting.

Null geodesic convergence phenomena are preserved in the extended geometric framework.

Abstract

The authors study a generalized notion of null geodesic defined by the Legendrian dynamics of a regular conical subbundle of the tangent bundle on a manifold. A natural extension of the Weyl tensor is shown to exist, and to depend only on this conical subbundle. Given a suitable defining function of the conical bundle, the Raychaudhuri--Sachs equations of general relativity continue to hold, and give rise to the same phenomenon of covergence of null geodesics in regions of positive energy that underlies the theory of gravitation.