A mathematical comment on gravitational waves

Abstract

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.