Boundary determination and local rigidity of analytic metrics in the Lorentzian scattering rigidity problem

TL;DR

This paper investigates the problem of determining Lorentzian metrics from scattering data, demonstrating local recovery of the metric and its derivatives near lightlike convex points, especially under real analyticity assumptions.

Contribution

It introduces new results on boundary determination and local rigidity of Lorentzian metrics in scattering problems, including analytic recovery near lightlike convex points.

Findings

Recovery of the metric's jet up to gauge near lightlike convex points

Analytic metrics can be fully recovered up to gauge near such points

Establishment of local boundary rigidity results in Lorentzian scattering geometry

Abstract

We study the scattering rigidity problem in Lorentzian geometry: recovery of a Lorentzian metric from the scattering relation known on a lateral timelike boundary. We show that one can recover the jet of the metric up to a gauge transformation near a lightlike strictly convex point. Assuming that the metric is real analytic, we show that one can recover the metric up to a gauge transformation as well near such a point.