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

Abstract: We study the Lorentzian metric independent of the time variable in the cylinder $\R\times\Omega$ where $x_0\in\R$ is the time variable and $\Omega$ is a bounded smooth domain in $\R^n$.

We consider forward null-geodesics in $\R\times \Omega$ starting on $\R\times\partial\Omega$ at $t=0$ and leaving $\R\times\Omega$ 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 in $(x_0,x)$ space then the metrics are equal.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine