%0 Journal Article
%T The Mathematical Foundations of General Relativity Revisited
%A Jean-Francois Pommaret
%J Journal of Modern Physics
%P 223-239
%@ 2153-120X
%D 2013
%I Scientific Research Publishing
%R 10.4236/jmp.2013.48A022
%X <p class=\"MsoNormal\">
	<span lang=\"EN-US\"><span style=\"font-family:Verdana;\">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 successively. </span><b><span style=\"font-family:Verdana;\">1) VESSIOT VERSUS CARTAN</span></b><b><span style=\"font-family:Verdana;\">:</span></b><span style=\"font-family:Verdana;\"> 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. In particular, ¡°curvature</span></span><span lang=\"EN-US\"> </span><span lang=\"EN-US\" style=\"font-family:Verdana;\">+</span><span lang=\"EN-US\"> </span><span lang=\"EN-US\"><span style=\"font-family:Verdana;\">torsion¡± (Cartan) must not be considered as a generalization of ¡°curvature alone¡± (Vessiot). </span><b><span style=\"font-family:Verdana;\">2) JANET VERSUS SPENCER</span></b><b><span style=\"font-family:Verdana;\">:</span></b><span style=\"font-family:Verdana;\"> 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¨¦ subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined 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 and contrary to the ¡°Janet sequence¡±, the ¡°Spencer sequence¡± for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. </span><b><span style=\"font-family:Verdana;\">3) ALGEBRA VERSUS GEOMETRY</span></b><b><span style=\"font-family:Verdana;\">:</span></b><span style=\"font-family:Verdana;\"> Using the powerful methods of ¡°Algebraic Analysis¡±, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy
%K General Relativity
%K Riemann Tensor
%K Weyl Tensor
%K Ricci Tensor
%K Einstein Equations
%K Lie Groups
%K Lie Pseudogroups
%K Differential Sequence
%K Spencer Operator
%K Janet Sequence
%K Spencer Sequence
%K Differential Module
%K Homological Algebra
%K Extension Modules
%K Split Exact Sequence
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=36688