What does this research mean for the field?

A Lorentzian metric in $\mathbb{R} \times \mathbb{R}^n$ can be uniquely recovered at a given point from the lengths of its space-time geodesics starting at that point, demonstrating the rigidity of Lorentzian metrics. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to recover the Lorentzian metric from the lengths of geodesics or null-geodesics in space-time.

March 29, 2026

Remarks on the Determination of the Lorentzian Metric by the Lengths of Geodesics or Null-Geodesics

Key Points

The aim is to recover the Lorentzian metric from the lengths of geodesics or null-geodesics in space-time.
Consider a Lorentzian metric in ℝ x ℝⁿ
Analyze geodesics starting at (0,y,η) when t=0
Prove rigidity of Lorentzian metrics
Examine conditions for metrics with equal Euclidean lengths of null-geodesics
Established that knowing geodesic lengths at a point allows recovery of the metric there
Proved rigidity for Lorentzian metrics, indicating uniqueness
Demonstrated that close metrics with equal lengths of null-geodesics must be identical

Abstract

We consider a Lorentzian metric in R R ⁿ. We show that, if we know the lengths of the space-time geodesics starting at (0, y, ) 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 the corresponding null-geodesics have equal Euclidian lengths, then the metrics are equal.

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Remarks on the Determination of the Lorentzian Metric by the Lengths of Geodesics or Null-Geodesics

Key Points

Abstract

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