Computer Algebra and Lanczos Potential

TL;DR

This paper explores the mathematical structure of conformal Killing sequences using computer algebra, revealing new insights into gravitational waves, differential operators, and their historical context in general relativity.

Contribution

It introduces novel computer algebra methods to analyze extension modules in differential sequences, clarifying longstanding doubts about gravitational waves and parametrizations of Riemann and Weyl operators.

Findings

New results on rank and order changes of differential operators for various dimensions.

Identification of the 'missing link' regarding the origin of gravitational waves.

Dependence of extension modules on structure constants in Vessiot equations.

Abstract

We found in 2016 a few results on the mathematical structure of the conformal Killing differential sequence in arbitrary dimension $n$, in particular the rank and order changes of the successive differential operators for $n=3,n=4$ or $n\geq 5$. They were so striking that we did not dare to publish them before our former PhD student A. Quadrat (INRIA) could confirm them while using new computer algebra packages that he developped for studying extension modules in differential homological algebra. In the meantime, as a complementary result, we found in 2017 the "missing link" justifying the doubts we had since a long time on the origin and existence of Gravitational Waves in General Relativity. In both cases, the main tool is the explicit computation of certain extension modules for the classical or conformal Killing differential sequences. These results therefore lead to revisit the work of C. Lanczos and successors on the existence of a parametrization of the Riemann or Weyl operators and their respective formal adjoint operators. We also provide an example showing how these extension modules are depending on the structure constants appearing in the Vessiot structure equations (1903), still not acknowledged after one century even though they generalize the constant Riemannian curvature integrability condition of L.P. Eisenhart (1926) for the Killing equations. The present paper is written from a lecture gven at the recent 24 th conference on Applications of Computer Algebra (ACA 2018) held in Santiago de Compostela, Spain, june 18-22, 2018.