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Generating Compatibility Conditions and General Relativity

j.urnal 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 f.r 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 f.r various standard usef.l metrics. Accordingly, this paper can be considered as a natural continuation of.a previous paper recently published in j.P under the title Minkowski, Schwarschild and Kerr metrics revisited. In particular, we prove that the intrinsic link existing between the lack of.f.rmal exactness of.an operator sequence on the j.t level, the lack of.f.rmal exactness of.its corresponding symbol sequence and the lack of.f.rmal integrability (f.) 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 theref.re quite dif.icult to grasp it in general and even more dif.icult 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 conf.sing 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 dif.icult to imagine that such an example could exist. We hope this paper could be used as a source of.testing examples f.r f.ture applications of.computer algebra in general relativity and, more generally, in mathematical physics.

关键词： f.rmal Theory of Systems of Partial Dif.erential Equations Compatibility Conditions Acyclicity f.rmal Integrability Involutivity Dif.erential Sequence j.net Sequence Spencer Sequence General Relativity Killing Systems

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Minkowski, Schwarzschild and Kerr Metrics Revisited

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j.urnal of.Modern Physics 2018年第10期9卷 1970-2007页

作者： j.-f. pommaret CERMICS Ecole des Ponts ParisTech Paris France

In recent papers, a f.w physicists studying Black Hole perturbation theory in General Relativity (GR) have tried to construct the initial part of.a dif.erential sequence based on the Kerr metric, using methods similar to the ones they already used f.r studying the Schwarzschild geometry. Of.course, such a dif.erential sequence is well known f.r the Minkowski metric and successively contains the Killing (order 1), the Riemann (order 2) and the Bianchi (order 1 again) operators in the linearized f.amework, as a particular case of.the Vessiot structure equations. In all these cases, they discovered that the compatibility conditions (CC) f.r 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 f.miliar with the f.rmal theory of.systems of.partial dif.erential equations and dif.erential modules, nor with the f.rmal theory of.Lie pseudogroups. Hence, even if.they discovered a link between these dif.erential 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 f.ct, being limited by the technical use of.Weyl spinors, complex Teukolsky scalars or Killing-Yano tensors. The purpose of.this dif.icult computational paper is to provide dif.erential and homological methods in order to revisit and solve these questions, not only in the previous cases but also in the specif.c case of.any Lie group or Lie pseudogroup of.transf.rmations. These new tools, which are now available as computer algebra packages, question the mathematical f.undations of.GR and the origin of.gravitational waves.

关键词： General Relativity Killing Operator Riemann Tensor Weyl Tensor Bianchi Identities Lie Algebroid Dif.erential Sequence Dif.erential Module Homological Algebra Extension Modules

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The Mathematical f.undations of.Elasticity and Electromagnetism Revisited

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j.urnal of.Modern Physics 2019年第13期10卷 1566-1595页

作者： j.-f. pommaret CERMICS Ecole des Ponts Paris Tech Paris France

The f.rst purpose of.this striking but dif.icult paper is to revisit the mathematical f.undations of.Elasticity (EL) and Electromagnetism (EM) by comparing the structure of.these two theories and examining with details their known couplings, in particular piezoelectricity and photoelasticity. Despite the strange Helmholtz and Mach-Lippmann analogies existing between them, no classical technique may provide a common setting. However, unexpected arguments discovered independently by the brothers E. and f. Cosserat in 1909 f.r EL and by H. Weyl in 1918 f.r EM are leading to construct a new dif.erential sequence called Spencer sequence in the f.amework of.the f.rmal theory of.Lie pseudo groups and to introduce it f.r the conf.rmal group of.space-time with 15 parameters. Then, all the previous explicit couplings can be deduced abstractly and one must j.st go to a laboratory in order to know about the coupling constants on which they are depending, like in the Hooke or Minkowski constitutive relations existing respectively and separately in EL or EM. We f.nally provide a new combined experimental and theoretical proof.of.the f.ct that any 1-f.rm with value in the second order j.ts (elations) of.the conf.rmal group of.space-time can be uniquely decomposed into the direct sum of.the Ricci tensor and the electromagnetic f.eld. This result questions the mathematical f.undations of.both General Relativity (GR) and Gauge Theory (GT). In particular, the Einstein operator (6 terms) must be thus replaced by the adj.int of.the Ricci operator (4 terms only) in the study of.gravitational waves.

关键词： Elasticity Electromagnetism Mach-Lippman Analogy Helmholtz Analogy Piezoelectricity Photoelasticity Dif.erential Sequences Adj.int Operator Double Duality

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f.om Control Theory to Gravitational Waves

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Advances in Pure Mathematics 2024年第2期14卷 49-100页

作者： j.an-f.ancois pommaret CERMICS Ecole des Ponts Paris Tech Paris France

When D:ξ→η is a linear ordinary dif.erential (OD) or partial dif.erential (PD) operator, a “direct problem” is to f.nd the generating compatibility conditions (CC) in the f.rm of.an operator D1:η→ξ such that Dξ = η implies D1η = 0. When D is involutive, the procedure provides successive f.rst-order involutive operators D1,...,Dn when the ground manif.ld has dimension n. Conversely, when D1 is given, a much more dif.icult “inverse problem” is to look f.r an operator D:ξ→η having the generating CC D1η = 0. If.this is possible, that is when the dif.erential module def.ned by D1 is “torsion-f.ee”, that is when there does not exist any observable quantity which is a sum of.derivatives of.η that could be a solution of.an autonomous OD or PD equation f.r itself. one shall say that the operator D1 is parametrized by D. The parametrization is said to be “minimum” if.the dif.erential module def.ned by D does not contain a f.ee dif.erential submodule. The systematic use of.the adj.int of.a dif.erential operator provides a constructive test with f.ve steps using double dif.erential duality. We prove and illustrate through many explicit examples the f.ct that a control system is controllable if.and only if.it can be parametrized. Accordingly, the controllability of.any OD or PD control system is a “built in” property not depending on the choice of.the input and output variables among the system variables. In the OD case and when D1 is f.rmally surj.ctive, controllability j.st amounts to the f.rmal inj.ctivity of.ad(D1), even in the variable coef.icients case, a result still not acknowledged by the control community. Among other applications, the parametrization of.the Cauchy stress operator in arbitrary dimension n has attracted many f.mous scientists (G. B. Airy in 1863 f.r n = 2, j. C. Maxwell in 1870, E. Beltrami in 1892 f.r n = 3, and A. Einstein in 1915 f.r n = 4).

关键词： Dif.erential Operator Dif.erential Sequence Killing Operator Riemann Operator Bianchi Operator Cauchy Operator Control Theory Controllability Elasticity General Relativity

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