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Recovery of a matrix valued potential for the wave equation on stationary spacetimes

Spyridon Filippas, Lauri Oksanen, Miika Sarkkinen

Abstract

We study the problem of recovering a time dependent matrix valued potential on a globally hyperbolic manifold from the knowledge of the source to solution map of a wave equation including a connection 1-form term. We exhibit sufficient conditions for solving this inverse problem under the assumption that the the manifold is stationary and that the connection term is time independent. The proof is based on two ingredients. The first is reduction of the problem to the study of a non-Abelian light ray transform and holds assuming global hyperbolicity only. The second is the study of this transform and establishing a link with a Riemannian analogue.

Recovery of a matrix valued potential for the wave equation on stationary spacetimes

Abstract

We study the problem of recovering a time dependent matrix valued potential on a globally hyperbolic manifold from the knowledge of the source to solution map of a wave equation including a connection 1-form term. We exhibit sufficient conditions for solving this inverse problem under the assumption that the the manifold is stationary and that the connection term is time independent. The proof is based on two ingredients. The first is reduction of the problem to the study of a non-Abelian light ray transform and holds assuming global hyperbolicity only. The second is the study of this transform and establishing a link with a Riemannian analogue.
Paper Structure (14 sections, 19 theorems, 127 equations, 2 figures)

This paper contains 14 sections, 19 theorems, 127 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Facts from Lorentzian geometry and setting of the problem
  3. Main results
  4. Previous literature
  5. Outline of the paper
  6. Reduction to a non-Abelian light ray transform
  7. The Minkowski case
  8. The case of globally hyperbolic geometries
  9. Stationary geometry
  10. Non-Abelian light ray transform
  11. General connection
  12. Time independent connection
  13. Injectivity of the non-Abelian light ray transform is conformally invariant
  14. Further details on stationary geometry

Key Result

Theorem 1.1

The following are equivalent for $(\mathcal{M}, \overline{g})$.

Figures (2)

  • Figure 1: We assume that the support of $Q$ is contained in some spacetime cylinder with the time coordinate given by the flow of the Killing vector field.
  • Figure 2: The null geodesic $\gamma$ and the time slices giving the compact interval in time where the Gaussian beam decay lemma is applied.

Theorems & Definitions (33)

  • Definition 1.1
  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.1
  • Corollary 1.5
  • Corollary 1.6
  • proof
  • Proposition 2.1
  • ...and 23 more