The Inverse Problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

TL;DR

This paper studies the inverse problem of recovering geometric and physical properties of a Lorentzian manifold from the Dirichlet-to-Neumann map, demonstrating stable recovery of boundary jets, lens relations, and light ray transforms, with applications to magnetic and potential fields.

Contribution

It establishes stable recovery of the metric, magnetic field, and potential from the Dirichlet-to-Neumann map on Lorentzian manifolds, including invariants like lens relations and light ray transforms, and explores applications in Minkowski space.

Findings

Stable recovery of boundary jets of g, A, q.

Recovery of lens relation and light ray transforms of A and q.

Lambda is a Fourier integral operator with canonical relation given by the lens relation.

Abstract

We consider the Dirichlet-to-Neumann map $\Lambda$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, a magnetic field $A$ and a potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary from $\Lambda$ up to a gauge transformation in a stable way. We also show that $\Lambda$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $\Lambda$ is an FIO away from the diagonal with a canonical relation given by the lens relation. We present applications for recovery of $A$ and $q$ in a logarithmically stable way in the Minkowski case, and uniqueness with partial data.