Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Lorentzian Calderón problem on vector bundles

Sean Gomes, Lauri Oksanen

TL;DR

This work addresses the inverse problem for the connection wave operator on Hermitian vector bundles over Lorentzian manifolds, showing that boundary Dirichlet-to-Neumann data determine the connection and Hermitian potential up to gauge in a causally defined recovery domain. The authors develop a bundle-appropriate adaptation of the BC/BLR framework, including a unique continuation principle, exact controllability, and propagation of singularities, and they construct Gaussian beams to compare coefficients. Their results apply to broad geometries, including Minkowski perturbations and ultrastatic manifolds with curvature, extending scalar Lorentzian Calderón-type theory to higher-rank bundles. The combination of UCP, observability, and microlocal Gaussian beams yields a robust gauge-invariant recovery mechanism for $\nabla$ and $V$ from boundary measurements in a geometrically controlled setting.

Abstract

In this paper we study a Lorentzian version of the Calderón 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.

The Lorentzian Calderón problem on vector bundles

TL;DR

This work addresses the inverse problem for the connection wave operator on Hermitian vector bundles over Lorentzian manifolds, showing that boundary Dirichlet-to-Neumann data determine the connection and Hermitian potential up to gauge in a causally defined recovery domain. The authors develop a bundle-appropriate adaptation of the BC/BLR framework, including a unique continuation principle, exact controllability, and propagation of singularities, and they construct Gaussian beams to compare coefficients. Their results apply to broad geometries, including Minkowski perturbations and ultrastatic manifolds with curvature, extending scalar Lorentzian Calderón-type theory to higher-rank bundles. The combination of UCP, observability, and microlocal Gaussian beams yields a robust gauge-invariant recovery mechanism for and from boundary measurements in a geometrically controlled setting.

Abstract

In this paper we study a Lorentzian version of the Calderón 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.
Paper Structure (14 sections, 26 theorems, 184 equations, 1 figure)

This paper contains 14 sections, 26 theorems, 184 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Examples
  3. Relation to existing literature
  4. Organisation of the paper
  5. Preliminaries
  6. Lorentzian geometry
  7. Local computations
  8. Unique continuation principle
  9. Direct problem and exact controllability
  10. Proof of Theorem \ref{['thm:main']}
  11. Propagation of singularities
  12. Observability estimate
  13. Gaussian beam construction
  14. Energy estimates

Key Result

Theorem 1.1

Let $(M,g)$ be a Lorentzian manifold of the form eq:productform, eq:metricform satisfying hypotheses (H1-H5). Let $\tilde{g}$ be a smooth metric lying in a sufficiently small $\mathcal{C}^2$ neighbourhood of $g$. Let $E$ be a rank $N$ Hermitian bundle over $M$ with the structure of a $G$-bundle for there exists $A\in \mathcal{C}^\infty(\mathcal{D};G(E))$ such that $A^*\nabla_1 A=\nabla_2$ and $A^

Figures (1)

  • Figure 2:

Theorems & Definitions (49)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 39 more