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Abstract <p> We consider a Lorentzian metric in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\( \mathbb{R} \times \mathbb{R} ^n\)</EquationSource> </InlineEquation>. We show that, if we know the lengths of the space-time geodesics starting at <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((0,y,\eta)\)</EquationSource> </InlineEquation> when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(t=0\)</EquationSource> </InlineEquation>, then we can recover the metric at <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(y\)</EquationSource> </InlineEquation>. 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. </p>

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

  • 期刊论文

  • 发表日期：

    2026年3月27日

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  • G. Eskin

摘要

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 the corresponding null-geodesics have equal Euclidian lengths, then the metrics are equal.