Decomposition of geometric perturbations

TL;DR

This paper proves that in Riemannian manifolds, scalar, vector, and tensor perturbations decouple only if the manifold is Einstein, with implications for homogeneous and isotropic 4D space-times in cosmology.

Contribution

It establishes a precise condition for the decoupling of perturbation modes in geometric deformations, linking it to Einstein manifolds.

Findings

Decoupling occurs if and only if the manifold is Einstein.

Four-dimensional Einstein manifolds are homogeneous and isotropic.

Implications for cosmological models are discussed.

Abstract

For an infinitesimal deformation of a Riemannian manifold, we prove that the scalar, vector, and tensor modes in decompositions of perturbations of the metric tensor, the scalar curvature, the Ricci tensor, and the Einstein tensor decouple if and only if the manifold is Einstein. Four-dimensional space-time satisfying the condition of the theorem is homogeneous and isotropic. Cosmological applications are discussed.