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 Lie pseudogroups of transformations of an n-dimensional manifold, let Ω = (Ω ij = Ω ji) be a perturbation of the non-degenerate metric ω = (ω ij = ω ji) with det(ω) = 0 and call ω −1 = (ω ij = ω ji) the inverse matrix appearing in the Dalembertian operator ✷ = ω ij d ij. The first idea is to introduce the linear transformation ¯ Ω ij = Ω ij − 1 2 ω ij tr(Ω) where tr(Ω) = ω ij Ω ij is the trace of Ω, which is invertible when n ≥ 3. The second important idea is to notice that the composite second order linearized Einstein operator ¯ Ω → Ω → E = (E ij = R ij − 1 2 ω ij tr(R)) where Ω → R = (R ij = R ji) is the linearized Ricci operator with trace tr(R) = ω ij R ij is reduced to ✷ ¯ Ω ij when ω rs d ri ¯ Ω sj = 0. The purpose of this short but striking paper is to revisit these two results in the light of the 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 ¯ Ω → E is nothing else than the formal adjoint of the Ricci operator Ω → R and that the map Ω → ¯ Ω is just the formal adjoint (transposed) of the defining tensor map R → 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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