Nonlinear Conformal Electromagnetism and Gravitation

Abstract

In 1909 the brothers E. and F. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of the nonlinear Janet sequence for the Lie groupoid defining the group of rigid motions of space. Following H. Weyl, our purpose is to compute for the first time the nonlinear Spencer sequence for the Lie groupoid defining the conformal group of space-time in order to provide the physical foundations of both electromagnetism (EM) and gravitation, with the only experimental need to measure the EM constant in vacuum and the gravitational constant. With a manifold of dimension $n$, the difficulty is to deal with the $n$ nonlinear transformations that have been called "elations" by E. Cartan in 1922. Using the fact that dimension $n=4$ has very specific properties for the computation of the Spencer cohomology, we prove that there is no conceptual difference between the Cosserat EL field or induction equations and the Maxwell EM field or induction equations. As a byproduct, the well known field/matter couplings (piezzoelectricity, photoelasticity, ...) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. In the sudy of gravitation, the dimension $n=4$ also allows to have a conformal factor defined everywhere but at the central attractive mass and the inversion law of the subgroupoid made by strict second order jets transforms attraction into repulsion.