Recovery of a matrix valued potential for the wave equation on stationary spacetimes

Abstract

We study the problem of recovering a time dependent matrix valued potential on a globally hyperbolic manifold from the knowledge of the source to solution map of a wave equation including a connection 1-form term. We exhibit sufficient conditions for solving this inverse problem under the assumption that the the manifold is stationary and that the connection term is time independent. The proof is based on two ingredients. The first is reduction of the problem to the study of a non-Abelian light ray transform and holds assuming global hyperbolicity only. The second is the study of this transform and establishing a link with a Riemannian analogue.