Spinors, Spin Coefficients and Lanczos Potentials

TL;DR

This paper investigates the relationship between spin coefficients, Lanczos potentials, and spinor dyads in general relativity, showing simple definitions in special cases and exploring connections to curvature-free metrics.

Contribution

It demonstrates how Lanczos potentials can be derived from spinor dyads in specific situations and discusses their relation to curvature-free connections.

Findings

Lanczos potentials can be directly defined from spinor dyads in special cases

The generalization of these definitions is limited

A link to metric, curvature-free connections is explored

Abstract

It has been demonstrated, in a number of special situations, that the spin coefficients of a canonical spinor dyad can be used to define a Lanczos potential of the Weyl curvature spinor. In this paper we explore some of these potentials and show that they can be defined directly from the spinor dyad in a very simple way, but that the results do not generalize significantly, in any direct manner. A link to metric, asymmetric, curvature-free connections, which suggests a more natural relationship between the Lanczos potential and spin coefficients, is also considered.