Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics

2022/08/03 by Gregory Eskin, Eskin, Gregory
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.2208.01842

openalex publication_date 2022/08/03 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

We consider a Lorentzian metric in ℝ×ℝⁿ. We show that if we know the lengths of the space-time geodesics starting at (0,y,η) when t=0, then we can recover the metric at y. We prove the rigidity of Lorentzian metrics. We also prove a variant of the rigidity property for the case of null-geodesics: if two metrics are close and if corresponding null-geodesics have equal Euclidian lengths then the metrics are equal.

Related

• Rigidity for Lorentzian metrics with the same length of null-geodesics
• Characterizing intrinsic Lorentzian length spaces via τ-midpoints
• Spacetime distances: an exploration
• Broken causal lens rigidity and sky shadow rigidity of Lorentzian manifolds
• Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds