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