Minkowski, Schwarzschild and Kerr Metrics Revisited

Abstract

In recent papers, a few physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of a differential sequence based on the Kerr metric, using methods similar to the ones they already used for studying the Schwarzschild geometry. Of course, such a differential sequence is well known for the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized framework, as a particular case of the Vessiot structure equations . In all these cases, they discovered that the compatibility conditions (CC) for the corresponding Killing operator were involving a mixture of both second order and third order CC and their idea has been to exhibit only a minimal number of generating ones . Unhappily, these physicists are neither familiar with the formal theory of systems of partial differential equations and differential modules, nor with the formal theory of Lie pseudogroups. Hence, even if they discovered a link between these differential sequences and the number of parameters of the Lie group preserving the background metric, they have been unable to provide an intrinsic explanation of this fact, being limited by the technical use of Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of this difficult computational paper is to provide differential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specific case of any Lie group or Lie pseudogroup of transformations. These new tools, which are now available as computer algebra packages, question the mathematical foundations of GR and the origin of gravitational waves.