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Pré-Publication, Document De Travail Année : 2017

A mathematical comment on gravitational waves

Résumé

In classical General Relativity, the way to exhibit the equations for the gravitational waves is based on two " tricks " allowing to transform the Einstein equations after linearizing them over the Minkowski metric. With specific notations used in the study of {\it Lie pseudogroups} of transformations of an $n$-dimensional manifold, let $\Omega=({\Omega}_{ij}={\Omega}_{ji})$ be a perturbation of the non-degenerate metric $\omega=({\omega}_{ij}={\omega}_{ji})$ with $det(\omega)\neq 0$ and call ${\omega}^{-1}=({\omega}^{ij}={\omega}^{ji})$ the inverse matrix appearing in the Dalembertian operator $\Box = {\omega}^{ij}d_{ij}$. The first idea is to introduce the linear transformation ${\bar{\Omega}}_{ij}={\Omega}_{ij}-\frac{1}{2}{\omega}_{ij}tr(\Omega)$ where $tr(\Omega)={\omega}^{ij}{\Omega}_{ij}$ is the {\it trace} of $\Omega$, which is invertible when $n\geq 3$. The second important idea is to notice that the composite second order linearized Einstein operator $\bar{\Omega} \rightarrow \Omega \rightarrow E=(E_{ij}=R_{ij} - \frac{1}{2}{\omega}_{ij}tr(R))$ where $\Omega \rightarrow R=(R_{ij}=R_{ji})$ is the linearized Ricci operator with trace $tr(R)={\omega}^{ij}R_{ij}$ is reduced to $\Box {\bar{\Omega}}_{ij}$ when ${\omega}^{rs}d_{ri}{\bar{\Omega}}_{sj}=0$. The purpose of this short but striking paper is to revisit these two results in the light of the {\it differential duality} existing in Algebraic Analysis, namely a mixture of differential geometry and homological agebra, providing therefore a totally different interpretation. In particular, we prove that the above operator $\bar{\Omega} \rightarrow E$ is nothing else than the formal adjoint of the Ricci operator $\Omega \rightarrow R$ and that the map $\Omega \rightarrow \bar{\Omega}$ is just the formal adjoint (transposed) of the defining tensor map $R \rightarrow E$. Accordingly, the Cauchy operator (stress equations) can be directly parametrized by the formal adjoint of the Ricci operator and the Einstein operator is no longer needed.
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Dates et versions

hal-01576131 , version 1 (22-08-2017)
hal-01576131 , version 2 (04-09-2017)

Identifiants

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Jean-François Pommaret. A mathematical comment on gravitational waves. 2017. ⟨hal-01576131v2⟩
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