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The mathematical foundations of general relativity revisited

The mathematical foundations of general relativity revisited

The purpose of this paper is to present for the first time an elementarysummary of a few recent results obtained through the application of the formaltheory of partial differential equations and Lie pseudogroups in order torevisit the mathematical foundations of general relativity. Other engineeringexamples (control theory, elasticity theory, electromagnetism) will also beconsidered in order to illustrate the three fundamental results that we shallprovide. The paper is therefore divided into three parts corresponding to thedifferent formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic termsappearing in the " Riemann tensor " according to the " Vessiot structureequations " must not be identified with the quadratic terms appearing in thewell known " Cartan structure equations " for Lie groups and a similar commentcan 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 downto 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 followedby Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUSSPENCER: 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 + 1dilatation) and the conformal group of space-time (15 parameters). It can bedefined by a canonical splitting, that is to say without using the indicesleading to the standard contraction or trace of the Riemann tensor. Meanwhile,we shall obtain the number of components of the Riemann and Weyl tensorswithout any combinatoric argument on the exchange of indices. Accordingly, theSpencer sequence for the conformal Killing system and its formal adjoint fullydescribe the Cosserat/Maxwell/Weyl theory but General Relativity is notcoherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to otherequations of physics (Cauchy equations, Cosserat equations, Maxwell equations),the Einstein equations cannot be " parametrized ", that is the generic solutioncannot be expressed by means of the derivatives of a certain number ofarbitrary potential-like functions, solving therefore negatively a 1000 $challenge proposed by J. Wheeler in 1970. Accordingly, the mathematicalfoundations of mathematical physics must be revisited within this formalframework, though striking it may look like for certain apparently wellestablished theories such as electromagnetism and general relativity. We insiston the fact that the arguments presented are of a purely mathematical natureand are thus unavoidable.

Accession: 036932030

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