GRAVITATIONAL WAVES AND PARAMETRIZATIONS OF LINEAR DIFFERENTIAL OPERATORS

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 D1:η→ζ such that Dξ=η implies D1η=0. Similarly, D1η=ζ may imply D2ζ=0 and so on. Conversely, when D1 is given, a much more difficult “inverse problem” is to look for an operator D:ξ→η with generating CC D1η=0. If this is possible, one shall say that the operator D1 is parametrized by D. The parametrization is “minimum” if the differential module defined by D does not contain any free differential submodule. The systematic use of the adjoint of a differential operator provides a constructive test. The parametrization of the Cauchy stress operator in arbitrary dimension n has attracted 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). We prove that all these works are already explicitly using the self-adjoint Einstein operator, which cannot be parametrized, and are thus all based on a confusion between the Cauchy operator, (adjoint of the Killing operator D), and the div operator induced from the Bianchi operator D2 CC of the Riemann operator D1 parametrized by D. This purely mathematical result deeply questions the origin and existence of gravitational waves that are solutions of the adjoint of the Ricci operator. We do believe that Einstein was aware of these previous works as the comparison needs no comment. The same methods are also used in order to revisit the mathematical foundations of electromagnetism.