X-ray transforms in pseudo-Riemannian geometry

TL;DR

This paper investigates the problem of reconstructing functions on pseudo-Riemannian manifolds from integrals over null geodesics, providing uniqueness results, characterizations of non-uniqueness, and conditions for reconstruction across various geometries.

Contribution

It introduces new methods for null geodesic integral transforms, characterizes when unique reconstruction is possible, and extends understanding to Minkowski spaces and tori.

Findings

Uniqueness of reconstruction in certain signatures

Characterization of non-uniqueness cases

Conditions for reconstructability in different geometries

Abstract

We study the problem of recovering a function on a pseudo-Riemannian manifold from its integrals over all null geodesics in three geometries: pseudo-Riemannian products of Riemannian manifolds, Minkowski spaces and tori. We give proofs of uniqueness anc characterize non-uniqueness in different settings. Reconstruction is sometimes possible if the signature $(n_1,n_2)$ satisfies $n_1\geq1$ and $n_2\geq2$ or vice versa and always when $n_1,n_2\geq2$. The proofs are based on a Pestov identity adapted to null geodesics (product manifolds) and Fourier analysis (other geometries). The problem in a Minkowski space of any signature is a special case of recovering a function in a Euclidean space from its integrals over all lines with any given set of admissible directions, and we describe sets of lines for which this is possible. Characterizing the kernel of the null geodesic ray transform on tori reduces to solvability of certain Diophantine systems.