The Weyl-Lanczos Equations and the Lanczos Wave Equation in 4 Dimensions as Systems in Involution

TL;DR

This paper reformulates the Weyl-Lanczos and Lanczos wave equations as exterior differential systems, analyzes their involution properties using Janet-Riquier theory, and compares their Cartan characters across various spacetimes.

Contribution

It presents a novel formulation of these equations as systems in involution and provides a detailed Cartan character analysis for different spacetime metrics.

Findings

Lanczos wave equation forms a system in involution.

Cartan characters computed for various spacetimes.

Comparison of Cartan characters with Weyl-Lanczos equations.

Abstract

Using the work by Bampi and Caviglia, we write the Weyl-Lanczos equations as an exterior differential system. Using Janet-Riquier theory, we compute the Cartan characters for all spacetimes with a diagonal metric and for the plane wave spacetime since all spacetimes have a plane wave limit. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet-Riquier theory, we find that it forms a system in involution. This result can be derived from the scalar wave equation itself. We compute its Cartan characters and compare them with those of the Weyl-Lanczos equations.