The Light ray transform for pseudo-Euclidean metrics

TL;DR

This paper investigates the properties and inversion of the light ray transform in pseudo-Euclidean spaces, providing stability estimates and analyzing its behavior as a Fourier Integral Operator, with comparisons to Minkowski space.

Contribution

It introduces an inversion formula and stability estimates for the light ray transform in pseudo-Euclidean spaces, and analyzes its ellipticity and singularities.

Findings

The normal operator is elliptic but singular at the light cone.

An explicit inversion formula for the light ray transform is derived.

The transform's behavior as a Fourier Integral Operator is characterized.

Abstract

We study the ray transform $L$ over null (light) rays in the pseudo-Euclidean space with signature $(n',n'')$, $n'\ge2$, $n''\ge2$. We analyze the normal operator $L'L$, derive an inversion formula, and prove stability estimates. We show that the symbol $p(\xi)$ is elliptic but singular at the light cone. We analyze $L$ as an Fourier Integral Operator as well. Finally, we compare this to the Minkowski case.