Broken causal lens rigidity and sky shadow rigidity of Lorentzian manifolds

TL;DR

This paper demonstrates that the topology, smooth structure, and metric of certain Lorentzian manifolds can be uniquely reconstructed from boundary data involving broken geodesics, under specific causality and convexity conditions.

Contribution

It introduces new boundary data techniques for uniquely determining Lorentzian manifolds, including cases with broken geodesics and lightlike geodesics, without assuming conjugate points.

Findings

Unique determination of manifold's topology, smooth structure, and metric from boundary data

Reconstruction of conformal class using lightlike geodesics under no conjugate points

Results hold under strong causality and convexity assumptions

Abstract

We prove that the topology, smooth structure, and metric of a compact Lorentzian manifold with boundary is uniquely determined by data at the boundary. The data consists of the lengths and directions of future-directed once-broken geodesics connecting points on the boundary, which are first timelike and then lightlike. This requires the strong causality condition and a weak convexity assumption, but it holds without any assumptions about conjugate points. With an additional convexity assumption we prove the analogous statement for future-directed once-broken timelike geodesics. If there are no conjugate points and lightlike geodesics never refocus, the analogous data using lightlike geodesics and once-broken lightlike geodesics may be used to reconstruct the manifold up to a conformal factor. This is a corollary of a result which shows that the conformal class is determined by the collection of sets of future-directed lightlike vectors at the boundary which give geodesics which all intersect in a single point.