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

We consider the Dirichlet-to-Neumann map Λ 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 Λ up to a gauge transformation in a stable way. We also show that Λ recovers the following three invariants in a stable way: the lens relation of g, and the light ray transforms of A and q. Moreover, Λ 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.