Counterexamples to inverse problems for the wave equation

<p style='text-indent:20px;'>We construct counterexamples to inverse problems for the wave operator on domains in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n \ge 2 $\end{document}</tex-math></inline-formula>, and on Lorentzian manifolds. We show that non-isometric Lorentzian metrics can lead to same partial data measurements, which are formulated in terms certain restrictions of the Dirichlet-to-Neumann map. The Lorentzian metrics giving counterexamples are time-dependent, but they are smooth and non-degenerate. On <inline-formula><tex-math id="M3">\begin{document}$ \mathbb{R}^{n+1} $\end{document}</tex-math></inline-formula> the metrics are conformal to the Minkowski metric.</p>