The Lanczos potential for Weyl-candidate tensors exists only in four dimensions

TL;DR

This paper demonstrates that a Lanczos potential for Weyl-candidate tensors exists only in four-dimensional spacetimes, by analyzing integrability conditions and showing non-existence in higher dimensions.

Contribution

It provides a proof that Lanczos potentials do not generally exist for Weyl tensors in dimensions higher than four.

Findings

Lanczos potential exists only in four dimensions

Integrability conditions prevent existence in higher dimensions

Weyl tensors in higher dimensions cannot be expressed via Lanczos potential

Abstract

We prove that a Lanczos potential L_abc for the Weyl candidate tensor W_abcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another non-trivial differential system for L_abc and W_abcd, then this system's integrability conditions should be checked; and so on. When we find a non-trivial condition involving only W_abcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential L_abc.