The Lanczos potential and Chern-Simons theory

TL;DR

The paper introduces a new tensor derived from the Lanczos potential, resembling the Weyl tensor, and explores its invariants, including a Chern-Simons term, with analysis through exact examples.

Contribution

It presents a novel tensor constructed from the Lanczos potential, linking it to Chern-Simons theory and exploring its invariants and properties.

Findings

Defined a new tensor D from the Lanczos potential.

Identified invariants C·D and D² related to the tensor.

Analyzed properties using exact spacetime examples.

Abstract

A new tensor $D$ is introduced which is constructed from the Lanczos potential and is of the same form as that of the Weyl tensor $C$ expressed in terms of the Lanczos potential except that covariant differentiation is replaced by transvection with a vector $v$. The new tensor has associated invariants $C\cdot D$ and $D^2$, the first of these can be interpreted as a Chern-Simons term for Weyl $C^2$ gravity. Both invariants allow various tensors to be constructed and some of the properties of these are investigated by using exact examples.