  We prove that any compact Cauchy horizon with constant non-zero surface
gravity in a smooth vacuum spacetime is a smooth Killing horizon. The novelty
here is that the Killing vector field is shown to exist on both sides of the
horizon. This generalises classical results by Moncrief and Isenberg, by
dropping the assumption that the metric is analytic. In previous work by R\'acz
and the author, the Killing vector field was constructed on the globally
hyperbolic side of the horizon. In this paper, we prove a new unique
continuation theorem for wave equations through smooth compact lightlike
(characteristic) hypersurfaces which allows us to extend the Killing vector
field beyond the horizon. The main ingredient in the proof of this theorem is a
novel Carleman type estimate. Using a well-known construction, our result
applies in particular to smooth stationary asymptotically flat vacuum black
hole spacetimes with event horizons with constant non-zero surface gravity. As
a special case, we therefore recover Hawking's local rigidity theorem for such
black holes, which was recently proven by Alexakis-Ionescu-Klainerman using a
different Carleman type estimate.
