  We study a Lorentzian version of the well-known Calder\'{o}n problem that is
concerned with determination of lower order coefficients in a wave equation on
a smooth Lorentzian manifold, given the associated Dirichlet-to-Neumann map. In
the earlier work of the authors it was shown that zeroth order coefficients can
be uniquely determined under a two-sided spacetime curvature bound and the
additional assumption that there are no conjugate points along null or
spacelike geodesics. In this paper we show that uniqueness for the zeroth order
coefficient holds for manifolds satisfying a weaker curvature bound as well as
spacetime perturbations of such manifolds. This relies on a new optimal unique
continuation principle for the wave equation in the exterior regions of double
null cones. In particular, we solve the Lorentzian Calder\'{o}n problem near
the Minkowski geometry.
