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On the interplay between the light ray and the magnetic X-ray transforms

Lauri Oksanen, Gabriel P. Paternain, Miika Sarkkinen

TL;DR

The paper addresses the injectivity of the light ray transform on symmetric tensors over stationary Lorentzian manifolds by reducing null geodesics to magnetic geodesics on the base manifold and recasting the transform as a transport equation on $SM$. A central contribution is the finite degree property of the magnetic flow $G$, which ensures injectivity up to natural obstructions when satisfied, via a link to the $s$-injectivity of the magnetic X-ray transform $I_m$. The authors provide geometric conditions (e.g., simple magnetic systems, convexity) that guarantee the finite degree property and thus injectivity, with extensions to attenuated settings. These results connect inverse problems in Lorentzian geometry and cosmology (e.g., Calderón-type problems and the CMB perturbation context) to concrete transport-theoretic and harmonic-analytic criteria for tensor recovery from light-ray data.

Abstract

We study the light ray transform acting on tensors on a stationary Lorentzian manifold. Our main result is injectivity up to the natural obstruction as long as the associated magnetic vector field satisfies a finite degree property with respect to the vertical Fourier decomposition on the unit tangent bundle. This is based on an explicit relationship between the geodesic vector field of the Lorentzian manifold and the magnetic vector field.

On the interplay between the light ray and the magnetic X-ray transforms

TL;DR

The paper addresses the injectivity of the light ray transform on symmetric tensors over stationary Lorentzian manifolds by reducing null geodesics to magnetic geodesics on the base manifold and recasting the transform as a transport equation on . A central contribution is the finite degree property of the magnetic flow , which ensures injectivity up to natural obstructions when satisfied, via a link to the -injectivity of the magnetic X-ray transform . The authors provide geometric conditions (e.g., simple magnetic systems, convexity) that guarantee the finite degree property and thus injectivity, with extensions to attenuated settings. These results connect inverse problems in Lorentzian geometry and cosmology (e.g., Calderón-type problems and the CMB perturbation context) to concrete transport-theoretic and harmonic-analytic criteria for tensor recovery from light-ray data.

Abstract

We study the light ray transform acting on tensors on a stationary Lorentzian manifold. Our main result is injectivity up to the natural obstruction as long as the associated magnetic vector field satisfies a finite degree property with respect to the vertical Fourier decomposition on the unit tangent bundle. This is based on an explicit relationship between the geodesic vector field of the Lorentzian manifold and the magnetic vector field.

Paper Structure

This paper contains 12 sections, 10 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Applications of light ray transform
  3. Cosmological perturbation theory
  4. Lorentzian Calderón problem
  5. From null geodesics to magnetic geodesics
  6. Light ray transform and transport equation
  7. Main theorem
  8. Finite degree property
  9. When does $G$ have the finite degree property?
  10. Proofs of Lemmas \ref{['deg-m-prop']} and \ref{['I_m-inj']}
  11. Properties of the stationary Lorentzian metric
  12. Regularity results for transport equations

Key Result

Theorem 1.1

Assume $G$ has the finite degree property and $\alpha \in C^\infty(S^m(T^*\overline{M}))$ has compact support in $\overline{M}$. If $({\mathcal{L}}_m \alpha)(\gamma)=0$ for all null geodesics $\gamma$, then $\alpha$ must have the form given by eq:kernel.

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • proof
  • ...and 14 more