A mathematical comment on gravitational waves

TL;DR

This paper offers a new interpretation of gravitational wave equations in General Relativity using algebraic analysis, revealing duality relations and adjoint operators that differ from traditional linearization methods.

Contribution

It introduces a differential duality perspective, showing that the linearized Einstein operator is the formal adjoint of the Ricci operator, providing a novel mathematical interpretation.

Findings

The linearized Einstein operator is the formal adjoint of the Ricci operator.

The map from metric perturbations to the Einstein tensor is the transpose of the Ricci map.

Cauchy (stress) equations can be parametrized by the adjoint of the Ricci operator.

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.