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

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.