The Lorentzian Calder\'{o}n problem on vector bundles

TL;DR

This paper extends the Calderón problem to Lorentzian manifolds, showing that a connection and potential on a vector bundle can be uniquely identified from boundary measurements under certain geometric conditions.

Contribution

It introduces a Lorentzian version of the Calderón problem for vector bundles and proves uniqueness of the connection and potential from Dirichlet-to-Neumann data.

Findings

Unique determination of connection and potential up to gauge transformations.

Applicable to Lorentzian manifolds with curvature bounds including Minkowski perturbations.

Builds on and generalizes previous scalar case results.

Abstract

In this paper we study a Lorentzian version of the Calder\'{o}n problem, which is concerned with the determination of a connection and potential on a Hermitian vector bundle over a Lorentzian manifold from the Dirichlet-to-Neumann map of the associated connection wave operator. For a class of Lorentzian manifolds satisfying a curvature bound, including perturbations of Minkowski space over strictly convex domains, the connection and potential is shown to be uniquely determined up to the natural gauge transformations of the problem. The proof is based on ideas from the earlier works arXiv:2008.07508, arXiv:2112.01663 of the second author in the scalar setting.