The Weyl–Lanczos relations and the four-dimensional Lanczos tensor wave equation and some symmetry generators

Abstract

We examine symmetry generators for exterior differential systems and for systems of partial differential equations and apply the Cartan theory of exterior differential systems to the Weyl-Lanczos equations and to the Lanczos wave equation in four dimensions. We look at a number of examples of symmetries for the Weyl-Lanczos equations in four dimensions and give examples of isovectors when the solution manifold is the Schwarzschild, Kasner or Gödel space-time. Solutions of the Weyl-Lanczos system are automatically solutions of the Lanczos wave equation.We give examples of symmetry generators for the Lanczos wave equation and find that they are not automatically symmetry generators for the Weyl-Lanczos equations.

The Curvature Homogeneity Bound for Lorentzian Four-Manifolds

“…In this paper we have also illustrated the applicability of exterior differential systems to the study of the curvature homogeneity problem. EDS has applications in the study of the Weyl-Lanczos problem [46] and in the analysis of vacuum solutions [47]. It is natural to expect that the use of EDS in the study of exact solutions could provide some further insights in relativity.…”

Lanczos Potential for the van Stockung Space-Time

“…Through the covariant derivative of Lanczos potential we can obtain the conformal contribution C abcd of the metric curvature. The Weyl tensor can be expressed as first order equations in terms of the Lanczos tensor L abc [3][4][5][6]. The task of generating the spacetime Weyl tensor from a tensor potential is known as the Weyl-Lanczos problem and the analogous process for the Riemann curvature tensor is called the Riemann-Lanczos problem.…”