arXiv subjects

Jean-Francois Pommaret

Publications and source records attributed to Jean-Francois Pommaret.

5 recordsLinked to original sources

arXiv2024

From Kalman to Einstein and Maxwell: the Structural Controllability Revisited

In the Special Relativity paper of Einstein (1905), only a footnote provides a reference to the conformal group of space-time for the Minkowski metric $ω$. We prove that General Relativity (1915) will depend on the following {\it cornerstone} result of differential homological algebra (1990). Let $K$ be a differential field and $D=K[d_1,...,d_n]$ be the ring of differential operators with coefficients in $K$. If $M$ is the differential module over $D$ defined by the Killing operator ${\cal{D}} :T \rightarrow S_2T^*: ξ\rightarrow Ω= {\cal{L}}(ξ) ω$ and $N$ is the differential module over $D$ defined by the $Cauchy = ad(Killing)$ adjoint operator with torsion submodule $t(N)$, then $t(N) \simeq {ext}^1_D(M) = 0$ and the Cauchy operator can be thus parametrized by stress functions having strictly nothing to do with $Ω$. This result is largely superseding the Kalman controllability test in classical OD control theory and is showing that controllability is a structural "{\it built-in}" property of an OD/PD control system not depending on the choice of inputs and outputs, contrary to the engineering tradition. It also points out the {\it terrible confusion} done by Einstein (1915) while following Beltrami (1892), both of them using the Einstein operator but ignoring that it was self-adjoint in the framework of differential double duality (1995). We finally prove that the structure of electromagnetism and gravitation only depends on the nonlinear {\it elations} of the conformal group of space-time, showing thus that {\it nothing is left from the mathematical foundations of both general relativity and gauge theory}.

math-ph↗

arXiv2023

Control Theory and Parametrizations of Linear Partial Differential Operators

When ${\cal{D}}:ξ\rightarrow η$ is a linear OD or PD operator, a "direct problem" is to find compatibility conditions (CC) as an operator ${\cal{D}}_1:η\rightarrow ζ$ such that ${\cal{D}}ξ=η$ implies ${\cal{D}}_1η=0$. When ${\cal{D}}$ is involutive, the procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ in dimension $n$. Conversely, when ${\cal{D}}_1$ is given, a much more difficult " inverse problem " is to look for an operator ${\cal{D}}: ξ\rightarrow η$ having the generating CC ${\cal{D}}_1η=0$. This is possible when the differential module defined by ${\cal{D}}_1$ is " {\it torsion-free} ", one shall say that ${\cal{D}}_1$ is parametrized by ${\cal{D}}$. The systematic use of the adjoint of a differential operator provides a constructive test. A control system is controllable {\it if and only if} it can be parametrized. Accordingly, the controllability of any OD or PD control system is a " {\it built in} " property not depending on the choice of the input and output variables among the system variables. In the OD case when ${\cal{D}}_1$ is formally surjective, controllability just amounts to the injectivity of $ad({\cal{D}}_1)$. Among applications, the parametrization of the Cauchy stress operator has attracted many famous scientists from G.B. Airy in 1863 for $n=2$ to A. Einstein in 1915 for $n=4$. We prove that all these works are already explicitly using the self-adjoint Einstein operator {\it which cannot be parametrized} and are based on a confusion between the $div$ operator induced from the Bianchi operator ${\cal{D}}_2$ and the Cauchy operator, adjoint of the Killing operator ${\cal{D}}$ for an arbitrary $n$. This purely mathematical result deeply questions the origin and existence of gravitational waves.

math-ph↗

arXiv2022

Killing Operator for the Kerr Metric

When ${\cal{D}}: E \rightarrow F$ is a linear differential operator of order $q$ between the sections of vector bundles over a manifold $X$ of dimension $n$, it is defined by a bundle map $Φ: J_q(E) \rightarrow F=F_0$ that may depend, explicitly or implicitly, on constant parameters $a, b, c, ...$. A "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator ${\cal{D}}_1: F_0 \rightarrow F_1$. When ${\cal{D}}$ is involutive, that is when the corresponding system $R_q=ker(Φ)$ is involutive, this procedure provides successive first order involutive operators ${\cal{D}}_1, ... , {\cal{D}}_n$ . Though ${\cal{D}}_1 \circ {\cal{D}}=0 $ implies $ad({\cal{D}}) \circ ad({\cal{D}}_1)=0$ by taking the respective adjoint operators, then $ad({\cal{D}})$ may not generate the CC of $ad({\cal{D}}_1)$ and measuring such "gaps" led to introduce extension modules in differential homological algebra. They may also depend on the parameters. When $R_q$ is not involutive, a standard {\it prolongation/projection} (PP) procedure allows in general to find integers $r,s$ such that the image $R^{(s)}_{q+r}$ of the projection at order $q+r$ of the prolongation $ρ_{r+s}(R_q) = J_{r+s}(R_q) \cap J_{q+r+s}(E)\subset J_{r+s}(J_q(E)) $ 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 differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr $(m, a)$, Schwarzschild $(m, 0)$ and Minkowski $(0, 0)$ parameters while computing the dimensions of the inclusions $R^{(3)}_1\subset R^{(2)}_1 \subset R^{(1)}_1 =R_1 \subset J_1(T(X))$ for the respective Killing operators.

physics.gen-ph↗

arXiv2018

Generating Compatibility Conditions in Mathematical Physics

The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the 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. 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 a kind of differential transcendence degree, also called degree of generality by A. Einstein in his 1930 letters to E. Cartan. The motivating examples that we provide are among the rare ones known in the literature and could be used as testing examples for future applications of computer algebra.

math.DG↗

arXiv2017

Algebraic Analysis and Mathematical Physics

This paper aims to revisit the mathematical foundations of both General Relativity and Electromagnetism after one century, in the light of the formal theory of systems of partial differential equations and Lie pseudogroups (D.C. Spencer, 1970) or Algebraic Analysis, namely a mixture of differential geometry and homological algebra (M. Kashiwara, 1970). Among the new results obtained, we may quote: 1) In dimension 4 only, the 9 Bianchi identities that must be satisfied by the 10 components of the Weyl tensor are described by a second order operator and have thus nothing to do with the 20 first order Bianchi identities for the 20 components of the Riemann tensor. This result, not known after one century, has been recently confirmed by A. Quadrat (INRIA) using new computer algebra packages. 2) The Ricci tensor R is a section of the Ricci bundle of symmetric covariant 2-tensors which is the kernel of the canonical projection of the Riemann bundle onto the Weyl bundle, induced by the canonical inclusion of the classical Killing system (Poincare group) into the conformal Killing system (Conformal group). It has only to do with the second order jets (elations) of the conformal Killing system because any 1-form with value in the bundle of elations can be decomposed in the direct sum (R,F) where the electromagnetic field F is a section of the vector bundle of skewsymmetric covariant 2-tensors. It follows therefore that electromagnetism and gravitation have only to do with second order jets. 3) The 10 linearized second order Einstein equations are parametrizing the 4 first order Cauchy stress equations but cannot be parametrized themselves. As a byproduct of this negative result, these 4 Cauchy stress equations have nothing to do with the 4 divergence-type equations usually obtained from the 20 Bianchi identities by contraction of indices.

math-ph↗