Lorentzian Calder\'{o}n problem under curvature bounds

TL;DR

This paper develops a new method for solving inverse boundary value problems for wave equations on Lorentzian manifolds, allowing recovery of coefficients under curvature bounds without requiring real analyticity of the metric.

Contribution

It introduces a novel approach based on a unique continuation result that generalizes the Boundary Control method to non-analytic Lorentzian metrics with curvature bounds.

Findings

Recovery of zeroth order coefficients under curvature bounds

Non-empty interior of metrics satisfying curvature bounds

Generalization of Boundary Control method to non-analytic cases

Abstract

We introduce a method of solving inverse boundary value problems for wave equations on Lorentzian manifolds, and show that zeroth order coefficients can be recovered under certain curvature bounds. The set of Lorentzian metrics satisfying the curvature bounds has a non-empty interior in the sense of arbitrary, smooth perturbations of the metric, whereas all previous results on this problem impose conditions on the metric that force it to be real analytic with respect to a suitably defined time variable. The analogous problem on Riemannian manifolds is called the Calder\'on problem, and in this case the known results require the metric to be independent of one of the variables. Our approach is based on a new unique continuation result in the exterior of the double null cone emanating from a point. The approach shares features with the classical Boundary Control method, and can be viewed as a generalization of this method to cases where no real analyticity is assumed.