The Linearized Einstein Equations with Sources on Compact Riemannian Manifolds With Boundary

Abstract: We consider compact Riemannian manifolds with arbitrary interior geometry, assuming only that the Ricci tensor vanishes on the boundary. In this setting, we prove that, under the linear Bianchi gauge and extended Cauchy data, the linearized Einstein equations with sources are globally solvable if and only if the source lies in the kernel of a lifted divergence operator and vanishes on the boundary; the solution is unique. This is the first result of its kind without rigid structural restrictions on topology, geometry, or source. The proof combines generalized Hodge theory with unique continuation results.