The Riemann-Lanczos Problem as an Exterior Differential System with Examples in 4 and 5 Dimensions

TL;DR

This paper investigates the Riemann-Lanczos problem within exterior differential systems, demonstrating its non-involution in 4 and 5 dimensions, providing examples of singular solutions in specific spacetimes, and discussing its properties in higher dimensions.

Contribution

It analyzes the involution properties of the Riemann-Lanczos problem in various dimensions and provides explicit examples of singular solutions in specific spacetime models.

Findings

Riemann-Lanczos problem is not in involution in 4 and 5 dimensions.

Singular solutions exist for certain spacetime metrics.

The 5-dimensional case is neither in involution nor admits a Vessiot involution.

Abstract

The key problem of the theory of exterior differential systems (EDS) is to decide whether or not a system is in involution. The special case of EDSs generated by one-forms (Pfaffian systems) can be adequately illustrated by a 2-dimensional example. In 4 dimensions two such problems arise in a natural way, namely, the Riemann-Lanczos and the Weyl-Lanczos problems. It is known from the work of Bampi and Caviglia that the Weyl-Lanczos problem is always in involution in both 4 and 5 dimensions but that the Riemann-Lanczos problem fails to be in involution even for 4 dimensions. However, singular solutions of it can be found. We give examples of singular solutions for the Goedel, Kasner and Debever-Hubaut spacetimes. It is even possible that the singular solution can inherit the spacetime symmetries as in the Debever-Hubaut case. We comment on the Riemann-Lanczos problem in 5 dimensions which is neither in involution nor does it admit a 5-dimensional involution of Vessiot vector fields in the generic case.