Lorentzian Calder\'{o}n problem near the Minkowski geometry

TL;DR

This paper advances the understanding of inverse problems for wave equations on Lorentzian manifolds, demonstrating uniqueness of lower order coefficients under weaker curvature conditions, especially near Minkowski space.

Contribution

It proves uniqueness results for the Lorentzian Calderón problem with relaxed curvature bounds and introduces a new unique continuation principle for wave equations.

Findings

Uniqueness of zeroth order coefficients under weaker curvature assumptions.

Development of a new optimal unique continuation principle for wave equations.

Solution of the Lorentzian Calderón problem near Minkowski geometry.

Abstract

We study a Lorentzian version of the well-known Calder\'{o}n 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. 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 as well as spacetime perturbations of such manifolds. This relies on a new optimal unique continuation principle for the wave equation in the exterior regions of double null cones. In particular, we solve the Lorentzian Calder\'{o}n problem near the Minkowski geometry.