2018/01/01 by J.-F. Pommaret, J. -F. Pommaret · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Algebra over a field #Astrophysical Phenomena and Observations #Black Holes and Theoretical Physics #Differential operator #General relativity #Hypoelliptic operator #Kerr metric #Mathematical physics #Mathematics #Minkowski space #Physics #Pseudo-differential operator #Pure mathematics #Quantum mechanics #Schwarzschild metric #Schwarzschild radius #Semi-elliptic operator #Spacetime #Theoretical physics #msc:46M18 #msc:53B50 #msc:83C57 #physics.gen-ph
paper · pdf · doi:10.4236/jmp.2018.910125
published as Journal of Modern Physics, 9 (2018) 1970-2007 · In this v2 we improve the search of generating compatibility conditions for the Killing operators used in general relativity and the corresponding differential sequences, by means of new differential homological methods. The existence of a partition 10=4+4+2 of the Ricci tensor questions the mathematical coherence of Einstein equations with group theory and formal integrability
openalex publication_date 2018/01/01 · arxiv created 2018/10/19 · arxiv updated 2018/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.