Transport along Null Curves

TL;DR

This paper introduces a new transport law for polarization vectors along non-geodesic null curves, extending the concept of Fermi Transport to null world lines and exploring its geometric and quantum mechanical implications.

Contribution

It presents a novel transport law for polarization vectors on null curves, derived from null geometry, and analyzes its spinorial form and topological features.

Findings

Transport law for vectors is integrable and depends only on local properties.

Transport law for spinors is non-integrable with a global sign topological effect.

The law does not smoothly limit to the geodesic case as null curves approach geodesics.

Abstract

Fermi Transport is useful for describing the behaviour of spins or gyroscopes following non-geodesic, timelike world lines. However, Fermi Transport breaks down for null world lines. We introduce a transport law for polarisation vectors along non-geodesic null curves. We show how this law emerges naturally from the geometry of null directions by comparing polarisation vectors associated with two distinct null directions. We then give a spinorial treatment of this topic and make contact with the geometric phase of quantum mechanics. There are two significant differences between the null and timelike cases. In the null case (i) The transport law does not approach a unique smooth limit as the null curve approaches a null geodesic. (ii) The transport law for vectors is integrable, i.e the result depends only on the local properties of the curve and not on the entire path taken. However, the transport of spinors is not integrable: there is a global sign of topological origin.