  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.
