  The purpose of this paper is to present for the first time an elementary
summary of a few recent results obtained through the application of the formal
theory of partial differential equations and Lie pseudogroups in order to
revisit the mathematical foundations of general relativity. Other engineering
examples (control theory, elasticity theory, electromagnetism) will also be
considered in order to illustrate the three fundamental results that we shall
provide. The paper is therefore divided into three parts corresponding to the
different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms
appearing in the " Riemann tensor " according to the " Vessiot structure
equations " must not be identified with the quadratic terms appearing in the
well known " Cartan structure equations " for Lie groups and a similar comment
can be done for the " Weyl tensor ". In particular, " curvature+torsion"
(Cartan) must not be considered as a generalization of "curvature alone"
(Vessiot). Roughly, Cartan and followers have not been able to " quotient down
to the base manifold ", a result only obtained by Spencer in 1970 through the
"nonlinear Spencer sequence" but in a way quite different from the one followed
by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS
SPENCER: The " Ricci tensor " only depends on the nonlinear transformations
(called " elations " by Cartan in 1922) that describe the "difference "
existing between the Weyl group (10 parameters of the Poincar\'e subgroup + 1
dilatation) and the conformal group of space-time (15 parameters). It can be
defined by a canonical splitting, that is to say without using the indices
leading to the standard contraction or trace of the Riemann tensor. Meanwhile,
we shall obtain the number of components of the Riemann and Weyl tensors
without any combinatoric argument on the exchange of indices. Accordingly, the
Spencer sequence for the conformal Killing system and its formal adjoint fully
describe the Cosserat/Maxwell/Weyl theory but General Relativity is not
coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other
equations of physics (Cauchy equations, Cosserat equations, Maxwell equations),
the Einstein equations cannot be " parametrized ", that is the generic solution
cannot be expressed by means of the derivatives of a certain number of
arbitrary potential-like functions, solving therefore negatively a 1000 $
challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical
foundations of mathematical physics must be revisited within this formal
framework, though striking it may look like for certain apparently well
established theories such as electromagnetism and general relativity. We insist
on the fact that the arguments presented are of a purely mathematical nature
and are thus unavoidable.
