Nonexistence of Petrov type III Space-Times on which Weyl's Neutrino Equation or Maxwell's Equations satisfy Huygens' Principle

TL;DR

This paper proves that Petrov type III space-times cannot support Weyl neutrino or Maxwell equations satisfying Huygens' principle, using computational algebra tools to show the absence of solutions.

Contribution

It extends previous results by employing Maple's NPspinor and grobner packages to rigorously demonstrate the nonexistence of such space-times.

Findings

No Petrov type III space-times satisfy Huygens' principle for these equations.

The proof involves converting necessary conditions into polynomial systems.

Computational algebra confirms the systems have no solutions.

Abstract

Extending previous results we show that there are no Petrov type III space-times on which either the Weyl neutrino equation or Maxwell's equations satisfy Huygens' principle. We prove the result by using Maple's NPspinor package to convert the five-index necessary condition obtained by Alvarez and Wunsch to dyad form. The integrability conditions of the problem lead to a system of polynomial equations. We then apply Maple's grobner package to show that this system has no admissible solutions.