Abstract
E. Beltrami introduced in 1892 the six stress functions well known by mechanicians in order to parametrize the Cauchy stress equations of elasticity theory in space, similarly to the single Airy stress function for plane elasticity. In 1915, A. Einstein introduced the Einstein operator for gen eral relativity (GR) in space-time, ignoring that it was self-adjoint and without any reference to Beltrami though the comparison (never done !) needs no comment and confusing therefore stress functions with the variation of the metric. I proved in 1995 that the Einstein equations in vacuum cannot be parametrized like the Maxwell equations, solving negatively for the first time a 1000 dollars challenge proposed by J. Wheeler in 1970 who refused to accept this result. Such a purely mathematical result also proves that the equations of the gravitational waves are just described by the adjoint of the Ricci operator and are thus not coherent with differential homological algebra. The main purpose of this paper is to prove that black holes cannot exist, not for a problem of detection but because their existence should contradict the link existing between the Janet and Spencer differential sequences existing in the literature but never applied in GR. As Einstein never proposed any way for choosing a metric among the solutions of the Einstein equations, it will follow that the important object is not a metric but its group of invariance. Indeed, the Spencer sequence is isomorphic to the tensor product of the Poincar´e sequence for the exterior derivative by a Lie algebra of dimensions 10, 4 or 2 when dealing respectively with the Minkowski (M), the Schwarzschild (S) or the Kerr (K) metrics. Therefore, instead of shrinking down the dimension of this group, the idea is rather to enlarge the dimension of the group from 10 to 11 or 15 by using respectively the Poincar´e group of space-time, the Weyl group by adding 1 dilatation or the conformal group by adding 4 elations along a way initiated by H. Weyl in 1918 for unifying electromagnetism with gravitation. Many explicit motivating examples illustrate this paper at a student level, most of them dealing with Lie groups and Lie pseudogroups of transformations
Keywords