On Effective Constraints for the Riemann-Lanczos System of Equations

TL;DR

This paper investigates the constraints on the Riemann-Lanczos system, confirming the existence of effective constraints in certain dimensions and challenging previous claims that all Riemann tensors can have Lanczos potentials.

Contribution

The paper provides a direct derivation of a key constraint equation, verifies it with known solutions, and refutes prior conclusions that no constraints exist in lower dimensions.

Findings

Confirmed the existence of effective constraints in 3 and 4 dimensions.

Demonstrated that known solutions satisfy the derived constraint.

Refuted the claim that all Riemann tensors have Lanczos potentials in dimensions less than 5.

Abstract

There have been conflicting points of view concerning the Riemann--Lanczos problem in 3 and 4 dimensions. Using direct differentiation on the defining partial differential equations, Massa and Pagani (in 4 dimensions) and Edgar (in dimensions n > 2) have argued that there are effective constraints so that not all Riemann tensors can have Lanczos potentials; using Cartan's criteria of integrability of ideals of differential forms Bampi and Caviglia have argued that there are no such constraints in dimensions n < 5, and that, in these dimensions, all Riemann tensors can have Lanczos potentials. In this paper we give a simple direct derivation of a constraint equation, confirm explicitly that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue that the Bampi and Caviglia conclusion must therefore be flawed. In support of this, we refer to the recent work of Dolan and Gerber on the three dimensional problem; by a method closely related to that of Bampi and Caviglia, they have found an 'internal identity' which we demonstrate is precisely the three dimensional version of the effective constraint originally found by Massa and Pagani, and Edgar.