Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

TL;DR

This paper explores the algebraic structure of Bianchi identities in Riemannian geometry, emphasizing the significance of Spencer cohomology and providing new intrinsic interpretations of compatibility conditions, with implications for Minkowski, Schwarzschild, and Kerr metrics.

Contribution

It introduces a novel intrinsic interpretation of compatibility conditions using Spencer cohomology, contrasting with previous approaches and applying to key spacetime metrics.

Findings

Spencer operator and cohomology are crucial in understanding CC.

New intrinsic formulas for CC numbers are derived.

Discrepancies with recent literature on Schwarzschild and Kerr metrics.

Abstract

Our recent arXiv preprints and published papers on the solution of the Riemann-Lanczos and Weyl-Lanczos problems have brought our attention on the importance of revisiting the algebraic structure of the Bianchi identities in Riemannian geometry. We also discovered in the meantime that, in our first book of 1978, we had already used a new way for studying the various compatibility conditions (CC) of an operator that may not be necessarily formally integrable (FI) in order to construct canonical formally exact differential sequences on the jet level. The purpose of this paper is to prove that the combination of these two facts clearly shows the specific importance of the Spencer operator and the Spencer $\delta$-cohomology, totally absent from mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide for the first time the purely intrinsic interpretation of the respective numbers of successive first, second, third and higher order generating CC. However, if they of course agree with the linearized Killing operator over the Minkowski metric, they largely disagree with recent publications on the respective numbers of generating CC for the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular the only symbol existing in the literature which is 2-ayclic witout being of finite type, contrary to the conformal situation.