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

TL;DR

This paper demonstrates that the Lorentzian metric in a spacetime can be uniquely determined from the lengths of geodesics and null-geodesics, establishing a rigidity property that links geodesic lengths to metric recovery.

Contribution

It proves the uniqueness and rigidity of Lorentzian metrics based on geodesic length data, including a variant for null-geodesics with Euclidean length equality.

Findings

Lorentzian metric can be recovered from geodesic lengths

Rigidity of Lorentzian metrics established

Null-geodesic length equality implies metric equality

Abstract

We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of the space-time geodesics starting at $(0,y,\eta)$ 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.