Rigidity in the Lorentzian Calder\'on problem with formally determined data

TL;DR

This paper proves that the Lorentzian Calderón problem with boundary measurements uniquely determines the metric as Minkowski, even with minimal data, by introducing a novel method using distorted plane waves and geometric arguments.

Contribution

It introduces a new approach for geometric hyperbolic inverse problems, solving a formally determined Lorentzian Calderón problem with fewer measurements than previously required.

Findings

Unique determination of Minkowski metric from boundary data

Solution of a formally determined inverse problem with minimal measurements

Development of a new method using distorted plane waves

Abstract

We study the Lorentzian Calder\'on problem, where the objective is to determine a globally hyperbolic Lorentzian metric up to a boundary fixing diffeomorphism from boundary measurements given by the hyperbolic Dirichlet-to-Neumann map. This problem is a wave equation analogue of the Calder\'on problem on Riemannian manifolds. We prove that if a globally hyperbolic metric agrees with the Minkowski metric outside a compact set and has the same hyperbolic Dirichlet-to-Neumann map as the Minkowski metric, then it must be the Minkowski metric up to diffeomorphism. In fact we prove the same result with a much smaller amount of measurements, thus solving a formally determined inverse problem. To prove these results we introduce a new method for geometric hyperbolic inverse problems. The method is based on distorted plane wave solutions and on a combination of geometric, topological and unique continuation arguments.