Gravitational Waves and Lanczos Potentials

Abstract

We found in 2016 a few results on the conformal Killing operator in dimension n , in particular the changes of the orders of the successive compatibility conditions for n = 3, 4 or n ≥ 5 . They were so striking that we did not dare to publish them before our former PhD student A. Quadrat (INRIA) could confirm them through computer algebra. Then, we found in 2017 that the gravitationl waves operator was the adjoint of the Ricci operator which is only depending on the n nonlinear elations of the conformal group of transformations, a result justifying the doubts we had since a long time on the mathematical foundations of General Relativity, in particular on the origin and existence of gravitational waves. These results led us to revisit the work of C. Lanczos and successors on the existence of a parametrization for the Riemann and Weyl operators and their respective adjoint operators. Comparing the last invited lecture given by Lanczos in 1962 with our work, we suddenly understood the confusion made by Lanczos between Hodge duality and differential duality. Our purpose is thus to revisit the mathematical framework of Lanczos potential theory in the light of this comment, getting closer to the formal theory of Lie pseudogroups through differential double duality and the construction of finite length differential sequences for Lie operators and their adjoint sequences. We also explain why these results are depending on the structure constants appearing in the Vessiot structure equations (1903), still not acknowledged after more than one century, though they generalize the constant Riemannian curvature integrability condition of L.P. Eisenhart (1926) for the Killing equations.