Rigidity for Lorentzian metrics with the same length of null-geodesics

Abstract

We study the Lorentzian metric independent of the time variable in the cylinder $\mathbb{R}\times \Omega$ where $x_0\in \mathbb{R}$ is the time variable and $\Omega$ is a bounded smooth domain in ${\mathbb{R}}^n$. We consider forward null-geodesics in $\mathbb{R}\times \Omega$ starting on $\mathbb{R}\times \partial \Omega$ at $t=0$ and leaving $\mathbb{R}\times \Omega$ at some later time. We prove the following rigidity result: If two Lorentzian metrics are close enough in some norm and if corresponding null-geodesics have equal lengths then the metrics are equal.

The article is withdrawn because its results are contained in the replacement of article arXiv:2208.01842