The Mathematical Foundations of General Relativity Revisited

In 1880 S. Lie (1842-1899) studied the groups of transformations depending on a finite number of parameters and now called Lie groups of transformations. Ten years later he discovered that these groups are only examples of groups of transformations solutions of linear or nonlinear systems of ordinary differential (OD) or partial differential (PD) equations which may even be of high order and are now called Lie pseudogroups of transformations. During the next fifty years the latter groups have only been studied by two frenchmen, namely Elie Cartan (1869-1951) who is quite famous today, and Ernest Vessiot (1865-1952) who is almost ignored today. We have proved in many books and papers that the Cartan structure equations have nothing to do with the Vessiot structure equations still not known today. Accordingly, we prove in the first part of the paper: FIRST FUNDAMENTAL RESULT: The quadratic terms appearing in the Riemann tensor must not be identified with the quadratic terms appearing in the well known Maurer-Cartan 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). Though we consider that the first formal work on systems of PD equations is dating back to Maurice Janet (1888-1983) who introduced as early as in 1920 a differential sequence now called Janet sequence, it is only around 1970 that Donald Spencer (1912-2001) developped, in a quite independent way, the formal theory of systems of PD equations in order to study Lie pseudogroups, exactly like E. Cartan did with exterior systems. Nevertheless, all the physicists who tried to understand the only book ”Lie Equations ” that he published in 1972 with A. Kumpera, have been stopped by the fact that the examples of the Introduction (Janet sequence) have nothing to do with the core of the book (Spencer sequence). We obtain in the second part of the paper: SECOND FUNDAMENTAL RESULT: 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 Poincaré 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 (GR) is not coherent at all with this result. At the same time, mixing commutative algebra (module theory) and homological algebra (extension modules) but always supposing that the reader knows a lot about the work of Spencer, V.P. Palamodov (constant coefficients) and M. Kashiwara (variable coefficients) developped ” algebraic analysis ” in order to study the formal properties of finitely generated differential modules that do not depend on their presentation or even on a corresponding differential resolution, namely the 1 ha l-0 08 33 21 3, v er si on 1 12 J un 2 01 3 algebraic analogue of a differential sequence. Finally, we get in the third part of the paper: THIRD FUNDAMENTAL RESULT: 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. As no one of these results can be obtained without the previous difficult purely mathematical arguments and are thus unavoidable, the purpose of this paper is to present them for the first time in a rather self-contained and elementary way through explicit basic examples.