  The first purpose of this short but striking paper is to revisit Elasticity
(EL) and Electromagnetism (EM) by comparing the structure of these two theories
and examining with details their well known couplings, in particular
piezoelectricity and photoelasticity. Despite the strange Helmholtz and
Mach-Lippmann analogies existing between them, no classical technique may
provide a common setting. However, unexpected arguments discovered
independently by the brothers E. and F. Cosserat in 1909 for EL and by H. Weyl
in 1918 for EM are leading to construct a new differential sequence called
Spencer sequence in the framework of the formal theory of Lie pseudogroups and
to introduce it for the conformal group of space-time with 15 parameters. Then,
all the previous explicit couplings can be deduced abstractly and one must just
go to a laboratory in order to know about the coupling constants on which they
are depending, like in the Hooke or Minkowski constitutive relations existing
respectively in EL or EM separately. We finally provide a new combined
experimental and theoretical proof of the fact that any 1-form with value in
the second order jets (elations) of the conformal group of space-time can be
uniquely decomposed into the direct sum of the Ricci tensor R and the
electromagnetic field F. This result questions the mathematical foundations of
both General Relativity (GR) and Gauge Theory (GT). In particular, the Einstein
operator (6 terms) must be thus replaced by the formal adjoint of the Ricci
operator (4 terms only) in the study of gravitational waves.
