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Minimum Resolution of the Minkowski, Schwarzschild and Kerr Differential Modules

Journal of Modern Physics 2022年第4期13卷 620-670页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech France

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 GB book of 1978, we had already used a new way for studying the 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 δ-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 a few resolutions in which the orders of the successive operators may go “up and down” surprisingly, like in the conformal situation for various dimensions.

关键词： Formal Integrability involutivity Compatibility Conditions Spencer Operator Janet Sequence Spencer Sequence Differential Module Homological Algebra Extension Module

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A Mathematical Comparison of the Schwarzschild and Kerr Metrics

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Journal of Modern Physics 2020年第10期11卷 1672-1710页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech Paris France

A few physicists have recently constructed the generating compatibility conditions (CC) of the Killing operator for the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known for M, but also third order CC for S and K. In a recent paper (DOI:10.4236/jmp.2018.910125) we have studied the cases of M and S, without using specific technical tools such as Teukolski scalars or Killing-Yano tensors. However, even if S(m) and K(m, a) are depending on constant parameters in such a way that S → M when m → 0 and K→ S when a → 0, the CC of S do not provide the CC of M when m → 0 while the CC of K do not provide the CC of S when a → 0. In this paper, using tricky motivating examples of operators with constant or variable parameters, we explain why the CC are depending on the choice of the parameters. In particular, the only purely intrinsic objects that can be defined, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the

关键词： Formal Integrability involutivity Compatibility Condition Janet Sequence Spencer Sequence Minkowski Metric Schwarzschild Metric Kerr Metric

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Generating Compatibility Conditions and General Relativity

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Journal of Modern Physics 2019年第3期10卷 371-401页

作者： J.-F. Pommaret CERMICS Ecole des Ponts Paris Tech Paris France

The search for the generating compatibility conditions (CC) of a given operator is a very recent problem met in general relativity in order to study the Killing operator for various standard useful metrics. Accordingly, this paper can be considered as a natural continuation of a previous paper recently published in JMP under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of formal exactness of an operator sequence on the jet level, the lack of formal exactness of its corresponding symbol sequence and the lack of formal integrability (FI) of the initial operator is of a purely homological nature as it is based on the long exact connecting sequence provided by the so-called snake lemma in homological algebra. It is therefore quite difficult to grasp it in general and even more difficult to use it on explicit examples. It does not seem that any one of the results presented in this paper is known as most of the other authors who studied the above problem of computing the total number of generating CC are confusing this number with the degree of generality introduced by A. Einstein in his 1930 letters to E. Cartan. One of the motivating examples that we provide is so striking that it is even difficult to imagine that such an example could exist. We hope this paper could be used as a source of testing examples for future applications of computer algebra in general relativity and, more generally, in mathematical physics.

关键词： Formal Theory of Systems of Partial Differential Equations Compatibility Conditions Acyclicity Formal Integrability involutivity Differential Sequence Janet Sequence Spencer Sequence General Relativity Killing Systems

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