Gravitational Waves and Black Holes: Beyond the Mirror

Abstract

E. Beltrami introduced in 1892 the Beltrami operator acting on six stress functions in order to parametrize the Cauchy stress equations of elasticity theory in space, similarly to the single Airy stress function for plane elasticity, but this number has been then reduced to three by J.C. Maxwell and G. Morera. In 1915, A. Einstein introduced the Einstein operator for general relativity (GR) in space-time without any reference to Beltrami though the comparison needs no comment. In fact, both are using the same operator, ignoring it is self-adjoint 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. This purely mathematical result proves that the ten equations of the gravitational waves (GW) are described by the adjoint of the Ricci operator and GW cannot thus exist, not because of a problem of detection but because of a more fundamental problem of equations that we shall point out. The second purpose of this paper is to prove also that black holes (BH) 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 differential geometry but never applied to GR. After recalling the way to construct these two sequences separately through explicit examples, we apply these results to Einstein equations, proving that the important object is not a metric but its group of invariance. Indeed, we shall explain why 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 highly nonlinear elations along a way initiated by H. Weyl in 1918 for unifying electromagnetism with gravitation. Explicit motivating examples illustrate this paper at a student level, in order to introduce the new homological methods that are introduced for the first time in GR. Many among them are dealing with Lie pseudogroups that are groups of transformations solutions of systems of ordinary or partial differential equations.