Time Separation and Scattering Rigidity for Analytic Lorentzian Manifolds

TL;DR

This paper establishes new rigidity results for real-analytic Lorentzian manifolds, showing how time separation and scattering data uniquely determine the manifold's geometry without boundary convexity assumptions.

Contribution

It introduces novel rigidity theorems for Lorentzian manifolds using time separation and scattering data, relaxing boundary convexity and causality assumptions.

Findings

Time separation function determines unknown compact sets up to isometry.

Boundary time separation function determines the entire spacetime up to isometry.

Interior and scattering relations near the light cone determine the manifold uniquely.

Abstract

In this work, we prove the following three rigidity results: (i) in a real-analytic globally hyperbolic spacetime $(M,g)$ without boundary, the time separation function restricted to a thin exterior layer of a unknown compact subset $K \subset M$ determines $K$ up to an analytic isometry, assuming no lightlike cut points in $K$; (ii) in a real-analytic globally hyperbolic spacetime $(M,g)$ with timelike boundary, the boundary time separation function determines $M$ up to an analytic isometry, assuming no lightlike cut points near $M$ and lightlike geodesics are non-trapping; (iii) in a real-analytic Lorentzian manifold $(M,g)$ with timelike boundary, the interior and complete scattering relations near the light cone, each determines $M$ up to an analytic isometry, assuming that lightlike geodesics are non-trapping. We emphasize in all of these three cases we do not assume the convexity of the boundary of the subset or the manifold. Moreover, in (iii) we do not assume causality of the Lorentzian manifold, and allow the existence of cut points. Along the way, we also prove some boundary determination results, the connections between the interior and complete scattering relations, and the connections between the lens data and the scattering relation, for Riemannian manifolds and Lorentzian manifolds with boundaries.