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

Gregory Eskin

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arXiv · 2208.01842

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

Gregory Eskin

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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.

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Gregory Eskin. 2022-08-03. Remarks on the determination of the Lorentzian metric by the lengths of geodesics or null-geodesics. https://arxiv.org/abs/2208.01842

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