The inverse problem for the Dirichlet-to-Neumann map on Lorentzian manifolds

Abstract

We consider the Dirichlet-to-Neumann map $\mathrm{\Lambda }$ on a cylinder-like Lorentzian manifold related to the wave equation related to the metric $g$, the magnetic field $A$ and the potential $q$. We show that we can recover the jet of $g,A,q$ on the boundary from $\mathrm{\Lambda }$ up to a gauge transformation in a stable way. We also show that $\mathrm{\Lambda }$ recovers the following three invariants in a stable way: the lens relation of $g$, and the light ray transforms of $A$ and $q$. Moreover, $\mathrm{\Lambda }$ 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.