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

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

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