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.