The Dirichlet-to-Neumann map for Lorentzian Calder\'on problems with data on disjoint sets

TL;DR

This paper demonstrates that the restricted Dirichlet-to-Neumann map for the wave equation on Lorentzian manifolds allows for the reconstruction of the boundary metric and magnetic potential at certain points, using microlocal analysis and gliding rays.

Contribution

It introduces a method to recover the boundary metric and magnetic potential from partial boundary data without prior information, advancing inverse problems in Lorentzian geometry.

Findings

Reconstruction of the boundary conformal class of the metric.

Determination of the magnetic potential at boundary points.

Extension of metric knowledge from known boundary regions.

Abstract

We consider the restricted Dirichlet-to-Neumann map $\Lambda^{U,V}_{g,A,q}$ for the wave equation with magnetic potential $A$ and scalar potential $q$, on an admissible Lorentzian manifold $(M, g)$ of dimension $n \geq 3$ with boundary. Here $U$ and $V$ are disjoint open subsets of $\partial M$, where we impose the Dirichlet data on $U$ and measure the Neumann-type data on $V$. We use the gliding rays and microlocal analysis to show that, without any a priori information, one can reconstruct the conformal class of the boundary metric $g|_{T\partial M \times T\partial M}$ and the magnetic potential $A|_{T\partial M}$ at recoverable boundary points from $\Lambda^{U,V}_{g,A,q}$. In particular, the conformal factor and the jet of the metric at those points are determined up to gauge transformations. Moreover, if the metric and the time orientation are known on $U$ (or $V$), then the metric on a larger portion of $V$ (or $U$) can be reconstructed, up to gauge.