Determining Lorentzian manifold from non-linear wave observation at a single point

TL;DR

This paper demonstrates that the conformal structure of a Lorentzian manifold can be uniquely reconstructed from the source-to-solution map of a non-linear wave equation observed at a single point, using microlocal analysis.

Contribution

It introduces a novel method to determine the conformal structure of a Lorentzian manifold from limited observational data at one point.

Findings

Conformal structure can be recovered from single-point wave observations.

Microlocal analysis effectively handles non-linear wave interactions.

The approach advances inverse problems in Lorentzian geometry.

Abstract

We consider an inverse problem for a non-linear hyperbolic equation. We show that conformal structure of a Lorentzian manifold can be determined by the source-to-solution map evaluated along a single timelike curve. We use the microlocal analysis of non-linear wave interaction.