Lorentzian Calderón problem near the Minkowski geometry

Abstract

We study a Lorentzian version of the well-known Calderón problem that is concerned with determination of lower order coefficients in a wave equation on a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map on its timelike boundary. In the earlier work of the authors it was shown that zeroth order coefficients can be uniquely determined under a two-sided spacetime curvature bound and the additional assumption that there are no conjugate points along null or spacelike geodesics. In this paper we show that uniqueness for the zeroth order coefficient holds for manifolds satisfying a weaker curvature bound and for spacetime perturbations of such manifolds. This relies on a new enhanced optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calderón problem near the Minkowski geometry.