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Killing Operator f.r the Kerr Metric

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j.urnal of.Modern Physics 2023年第1期14卷 31-59页

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

When D: E →f.is a linear dif.erential operator of.order q between the sections of.vector bundles over a manif.ld X of.dimension n, it is def.ned by a bundle map Φ: j.(E) →f.f. that may depend, explicitly or implici... 详细信息

When D: E →f.is a linear dif.erential operator of.order q between the sections of.vector bundles over a manif.ld X of.dimension n, it is def.ned by a bundle map Φ: j.sub>q(E) →f.f.sub>0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to f.nd the generating compatibility conditions (CC) in the f.rm of.an operator D1: f.sub>0 →f.sub>1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive f.rst order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adj.int operators, then ad (D) may not generate the CC of.ad (D1) and measuring such “gaps” led to introduce extension modules in dif.erential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/proj.ction (PP) procedure allows in general to f.nd integers r, s such that the image of.the proj.ction at order q+r of.the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the dif.erential sequence used f.r their def.nition. The purpose of.this paper is to study the above problems f.r the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of.the inclusions f.r the respective Killing operators. Other striking motivating examples are also presented.

关键词： Dif.erential Operator Adj.int Operator Dif.erential Sequence Einstein Equations Kerr Metric Dif.erential Module Extension Module Contact Structure

Minimum Resolution of.the Minkowski, Schwarzschild and Kerr Dif.erential Modules

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j.urnal 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 f.rst GB book of.1978, we had already used a new way f.r studying the compatibility conditions (CC) of.an operator that may not be necessarily f.rmally integrable (f.) in order to construct canonical f.rmally exact dif.erential sequences on the j.t level. The purpose of.this paper is to prove that the combination of.these two f.cts clearly shows the specif.c importance of.the Spencer operator and the Spencer δ-cohomology, totally absent f.om mathematical physics today. The results obtained are unavoidable because they only depend on elementary combinatorics and diagram chasing. They also provide f.r the f.rst time the purely intrinsic interpretation of.the respective numbers of.successive f.rst, 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 f.r the linearized Killing operator over the Schwarzschild and Kerr metrics. Many similar examples are illustrating these new techniques, providing in particular a f.w resolutions in which the orders of.the successive operators may go “up and down” surprisingly, like in the conf.rmal situation f.r various dimensions.

关键词： f.rmal Integrability Involutivity Compatibility Conditions Spencer Operator j.net Sequence Spencer Sequence Dif.erential Module Homological Algebra Extension Module

Nonlinear Conf.rmal Electromagnetism

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j.urnal of.Modern Physics 2022年第4期13卷 442-494页

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

In 1909 the brothers E. and f. Cosserat discovered a new nonlinear group theoretical approach to elasticity (EL), with the only experimental need to measure the EL constants. In a modern language, their idea has been to use the nonlinear Spencer sequence instead of.the nonlinear j.net sequence f.r the Lie groupoid def.ning the group of.rigid motions of.space. f.llowing H. Weyl, our purpose is to compute f.r the f.rst time the nonlinear Spencer sequence f.r the Lie groupoid def.ning the conf.rmal group of.space-time in order to provide the mathematical f.undations of.electromagnetism (EM), with the only experimental need to measure the EM constant in vacuum. With a manif.ld of.dimension n, the dif.iculty is to deal with the n nonlinear transf.rmations that have been called “elations” by E. Cartan in 1922. Using the f.ct that dimension n=4 has very specif.c properties f.r the computation of.the Spencer cohomology, we prove that there is thus no conceptual dif.erence between the Cosserat EL f.eld or induction equations and the Maxwell EM f.eld or induction equations. As a byproduct, the well known f.eld/matter couplings (piezzoelectricity, photoelasticity, streaming biref.ingence, …) can be described abstractly, with the only experimental need to measure the corresponding coupling constants. The main consequence of.this paper is the need to revisit the mathematical f.undations of.gauge theory (GT) because we have proved that EM was depending on the conf.rmal group and not on U(1), with a shif. by one step to the lef. in the physical interpretation of.the dif.erential sequence involved.

关键词： Nonlinear Dif.erential Sequences Linear Dif.erential Sequences Lie Groupoids Lie Algebroids Conf.rmal Group Spencer Cohomology Maxwell Equations Cosserat Equations

A Mathematical Comparison of.the Schwarzschild and Kerr Metrics

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j.urnal 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 f.w physicists have recently constructed the generating compatibility conditions (CC) of.the Killing operator f.r the Minkowski (M), Schwarzschild (S) and Kerr (K) metrics. They discovered second order CC, well known f.r M, but also third order CC f.r S and K. In a recent paper (DOI:10.4236/j.p.2018.910125) we have studied the cases of.M and S, without using specif.c 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 obj.cts that can be def.ned, namely the extension modules, may change drastically. As the algebroid bracket is compatible with the

关键词： f.rmal Integrability Involutivity Compatibility Condition j.net Sequence Spencer Sequence Minkowski Metric Schwarzschild Metric Kerr Metric

Dif.erential Correspondences and Control Theory

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Advances in Pure Mathematics 2021年第11期11卷 835-882页

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

When a dif.erential f.eld K having n commuting derivations is given together with two f.nitely generated dif.erential extensions L and M of.K, an important problem in dif.erential algebra is to exhibit a common dif.erential extension N in order to def.ne the new dif.erential extensions L∩M and the smallest dif.erential f.eld (L,M ) ⊂ N containing both L and M. Such a result allows to generalize the use of.complex numbers in classical algebra. Having now two f.nitely generated dif.erential modules L and M over the non-commutative ring D = K [d1,...,dn] = K [d] of.dif.erential operators with coef.icients in K, we may similarly look f.r a dif.erential module N containing both L and M in order to def.ne L∩M and L+M. This is exactly the situation met in linear or non-linear OD or PD control theory by selecting the inputs and the outputs among the control variables. However, in many recent books and papers, we have shown that controllability was a built-in property of.a control system, not depending on the choice of.inputs and outputs. The purpose of.this paper is thus to revisit control theory by showing the specif.c importance of.the two previous problems and the part plaid by N in both cases f.r the parametrization of.the control system. An important tool will be the study of.dif.erential correspondences, a modern name

关键词： Dif.erential Modules Dif.erential Extensions Dif.erential Elimination Controllability Kähler Dif.erentials Bäcklund Problem

Minimum Parametrization of.the Cauchy Stress Operator

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j.urnal of.Modern Physics 2021年第4期12卷 453-482页

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

When D: ξ→η is a linear dif.erential 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, a result f.rst f.und by M. j.net as early as in 1920, in a f.otnote. However, the link between this “j.net sequence” and the “Spencer sequence” f.rst f.und by the author of.this paper in 1978 is still not acknowledged. Conversely, when D1 is given, a more dif.icult “inverse problem” is to look f.r an operator D:

关键词： Dif.erential Operator Dif.erential Sequence Killing Operator Riemann Operator Bianchi Operator Cauchy Operator Electromagnetism Elasticity General Relativity Gravitational Waves

Airy, Beltrami, Maxwell, Einstein and Lanczos Potentials Revisited

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j.urnal of.Modern Physics 2016年第7期7卷 699-728页

作者： j.-f. pommaret CERMICS Ecole des Ponts ParisTech Marne-la-Vallée France

The purpose of.this paper is to revisit the well known potentials, also called stress f.nctions, needed in order to study the parametrizations of.the stress equations, respectively provided by G.B. Airy (1863) f.r 2-dimensional elasticity, then by E. Beltrami (1892), j.C. Maxwell (1870) f.r 3-dimensional elasticity, f.nally by A. Einstein (1915) f.r 4-dimensional elasticity, both with a variational procedure introduced by C. Lanczos (1949, 1962) in order to relate potentials to Lagrange multipliers. Using the methods of.Algebraic Analysis, namely mixing dif.erential geometry with homological algebra and combining the double duality test involved with the Spencer cohomology, we shall be able to extend these results to an arbitrary situation with an arbitrary dimension n. We shall also explain why double duality is perf.ctly adapted to variational calculus with dif.erential constraints as a way to eliminate the corresponding Lagrange multipliers. f.r example, the canonical parametrization of.the stress equations is j.st described by the f.rmal adj.int of.the components of.the linearized Riemann tensor considered as a linear second order dif.erential operator but the minimum number of.potentials needed is equal to f.r any minimal parametrization, the Einstein parametrization being “in between” with potentials. We provide all the above results without even using indices f.r writing down explicit f.rmulas in the way it is done in any textbook today, but it could be strictly impossible to obtain them without using the above methods. We also revisit the possibility (Maxwell equations of.electromagnetism) or the impossibility (Einstein equations of.gravitation) to obtain canonical or minimal parametrizations f.r various equations of.physics. It is nevertheless important to notice that, when n and the algorithms presented are known, most of.the calculations can be achieved by using computers f.r the corresponding symbolic computations. f.nally, though the paper is mathemati

关键词： Stress Equations Stress f.nctions Elasticity Theory Lagrange Multipliers f.rmal Adj.int Control Theory General Relativity Einstein Equations Lanczos Potentials Algebraic Analysis Riemann Tensor Weyl Tensor

Why Gravitational Waves Cannot Exist

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j.urnal of.Modern Physics 2017年第13期8卷 2122-2158页

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

The purpose of.this short but dif.icult paper is to revisit the mathematical f.undations of.both General Relativity (GR) and Gauge Theory (GT) in the light of.a modern approach to nonlinear systems of.ordinary or partial dif.erential equations, using new methods f.om Dif.erential Geometry (D.C. Spencer, 1970), Dif.erential Algebra (j.f. Ritt, 1950 and E. Kolchin, 1973) and Algebraic Analysis (M. Kashiwara, 1970). The main idea is to identif. the dif.erential indeterminates of.Ritt and Kolchin with the j.t coordinates of.Spencer, in order to study Dif.erential Duality by using only linear dif.erential operators with coef.icients in a dif.erential f.eld K. In particular, the linearized second order Einstein operator and the f.rmal adj.int of.the Ricci operator are both parametrizing the 4 f.rst order Cauchy stress equations but cannot themselves be parametrized. In the f.amework of.Homological Algebra, this result is not coherent with the vanishing of.a certain second extension module and leads to question the proper origin and existence of.gravitational waves. As a byproduct, we also prove that gravitation and electromagnetism only depend on the second order j.ts (called elations by E. Cartan in 1922) of.the system of.conf.rmal Killing equations because any 1-f.rm with value in the bundle of.elations can be decomposed uniquely into the direct sum (R, f. where R is a section of.the Ricci bundle of.symmetric covariant 2-tensors and the EM f.eld f.is a section of.the vector bundle of.skew-symmetric 2-tensors. No one of.these purely mathematical results could have been obtained by any classical approach. Up to the knowledge of.the author, it is also the f.rst time that dif.erential algebra in a modern setting is applied to study the specif.c algebraic f.ature of.most equations to be f.und in mathematical physics, particularly in GR.

关键词： General Relativity Riemann Tensor Weyl Tensor Ricci Tensor Einstein Equations Elastic Waves Gravitational Waves Lie Groups Lie Pseudogroups Dif.erential Galois Theory Spencer Operator j.net Sequence Spencer Sequence Dif.erential Modules Algebraic Analysis Homological Algebra Extension Modules Split Exact Sequence

Generating Compatibility Conditions and General Relativity

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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