Minimum Parametrization of the Cauchy Stress Operator

Abstract

When D: ξ → η is a linear differential operator, a “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D 1 : η → ξ such that D ξ = η implies D 1 η =0 . When D is involutive, the procedure provides successive first order involutive operators D1, ..., D n , when the ground manifold has dimension n , a result first found by M. Janet as early as in 1920, in a footnote. However, the link between this “Janet sequence” and the “Spencer sequence” first found by the author of this paper in 1978 is still not acknowledged. Conversely, when D 1 is given, a more difficult “inverse problem” is to look for an operator D: ξ → η having the generating CC D 1 η =0 . If this is possible, that is when the differential module defined by D 1 is torsion-free, one shall say that the operator D 1 is parametrized by D and there is no relation in general between D and D 2 . The parametrization is said to be “minimum” if the differential module defined by D has a vanishing differential rank and is thus a torsion module. The solution of this problem, first found by the author of this paper in 1995, is still not acknowledged. As for the applications of the “differential double duality” theory to standard equations of physics ( Cauchy and Maxwell equations can be parametrized while Einstein equations cannot), we do not know other references. When erator in arbitrary dimension =1 as in control theory, the fact that controllability is a “built in” property of a control system, amounting to the existence of a parametrization and thus not depending on the choice of inputs and outputs, even with variable coefficients, is still not acknowledged by engineers. The parametrization of the Cauchy stress operator in arbitrary dimension n has nevertheless attracted, “separately” and without any general “guiding line”, many famous scientists (G.B. Airy in 1863 for n = 2, J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for n = 3 , A. Einstein in 1915 for n = 4 ). The aim of this paper is to solve the minimum parametrization problem in arbitrary dimension and to apply it through effective methods that could even be achieved by using computer algebra. Meanwhile, we prove that all these works are using the Einstein operator which is self-adjoint and not the Ricci operator, a fact showing that the Einstein operator, which cannot be parametrized, has already been exhibited by Beltrami more than 20 years before Einstein . As a byproduct, they are all based on the same confusion between the so-called div operator induced from the Bianchi operator D 2 and the Cauchy operator which is the formal adjoint of the Killing operator D parametrizing the Riemann operator D 1 for an arbitrary n . We prove that this purely mathematical result deeply questions the origin and existence of gravitational waves. We also present the similar motivating situation met in the study of contact structures when n = 3. Like the Michelson and Morley experiment, it is thus an open historical problem to know whether Einstein was aware of these previous works or not, but the comparison needs no comment.