On the Cauchy problem for the linearised Einstein equation

TL;DR

This paper proves the well-posedness of the Cauchy problem for the linearised Einstein equation on globally hyperbolic vacuum spacetimes, showing the solution map is a topological isomorphism and solving linearised constraints on closed manifolds.

Contribution

It establishes the continuous dependence of solutions on initial data and generalizes the existence of TT-tensors to manifolds with zero scalar curvature.

Findings

Solution map is an isomorphism of locally convex topological vector spaces.

Solutions depend continuously on initial data.

Linearised constraint equations are solvable on closed manifolds with zero scalar curvature.

Abstract

A classical problem in general relativity is the Cauchy problem for the linearised Einstein equation (the initial value problem for gravitational waves) on a globally hyperbolic vacuum spacetime. A well-known result is that it is uniquely solvable up to gauge solutions, given initial data on a spacelike Cauchy hypersurface. The solution map is an isomorphism between initial data (modulo gauge producing initial data) and solutions (modulo gauge solutions). In the first part of this work, we show that the solution map is actually an isomorphism of locally convex topological vector spaces. This implies that the equivalence class of solutions depends continuously on the equivalence class of initial data. We may therefore conclude well-posedness of the Cauchy problem. In the second part, we show that the linearised constraint equations can always be solved on a closed manifold with vanishing scalar curvature. This generalises the classical notion of TT-tensors on flat space used to produce models of gravitational waves. All our results are proven for smooth and distributional initial data of arbitrary real Sobolev regularity.