On the Cauchy Problem for the Linearised Einstein Equation
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 wellknown 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 wellposedness 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 TTtensors 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.
 Publication:

Annales Henri Poincaré
 Pub Date:
 December 2019
 DOI:
 10.1007/s00023019008575
 arXiv:
 arXiv:1802.06028
 Bibcode:
 2019AnHP...20.3849L
 Keywords:

 Mathematics  Differential Geometry;
 General Relativity and Quantum Cosmology;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 83C35;
 35L15
 EPrint:
 31 pages