Counterexamples to inverse problems for the wave equation

Abstract

We construct counterexamples to inverse problems for the wave operator on domains in ${\mathbb{R}}^{n+1}$, $n\ge 2$, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On ${\mathbb{R}}^{n+1}$ the metrics are conformal to the Minkowski metric.

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