The Light Ray transform in Stationary and Static Lorentzian geometries

TL;DR

This paper investigates the injectivity of the light ray transform on functions and tensors in Lorentzian geometries, establishing conditions under which the transform is invertible and applying results to inverse problems in hyperbolic PDEs.

Contribution

It proves injectivity of the light ray transform in stationary and static Lorentzian manifolds under specific geometric conditions, extending previous results and applying them to inverse boundary value problems.

Findings

Injectivity holds with convex foliation or real analyticity and no cut points.

Injectivity for tensor fields in static Lorentzian manifolds based on spatial geodesic transform.

Applications to inverse problems for hyperbolic PDEs with boundary data.

Abstract

Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.