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

TL;DR

This paper investigates the light ray transform on tensors in Lorentzian manifolds, establishing injectivity conditions linked to magnetic vector fields and their Fourier properties, with implications for geometric analysis.

Contribution

It introduces conditions under which the light ray transform is injective on tensors in Lorentzian manifolds, connecting magnetic vector fields and Fourier analysis.

Findings

Injectivity of the light ray transform up to natural obstructions.

Explicit relationship between geodesic and magnetic vector fields.

Conditions on magnetic vector fields for finite degree property.

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.