  We introduce a method of solving inverse boundary value problems for wave
equations on Lorentzian manifolds, and show that zeroth order coefficients can
be recovered under certain curvature bounds. The set of Lorentzian metrics
satisfying the curvature bounds has a non-empty interior in the sense of
arbitrary, smooth perturbations of the metric, whereas all previous results on
this problem impose conditions on the metric that force it to be real analytic
with respect to a suitably defined time variable. The analogous problem on
Riemannian manifolds is called the Calder\'on problem, and in this case the
known results require the metric to be independent of one of the variables. Our
approach is based on a new unique continuation result in the exterior of the
double null cone emanating from a point. The approach shares features with the
classical Boundary Control method, and can be viewed as a generalization of
this method to cases where no real analyticity is assumed.
