  We prove that any smooth vacuum spacetime containing a compact Cauchy horizon
with surface gravity that can be normalised to a non-zero constant admits a
Killing vector field. This proves a conjecture by Moncrief and Isenberg from
1983 under the assumption on the surface gravity and generalises previous
results due to Moncrief-Isenberg and Friedrich-R\'acz-Wald, where the
generators of the Cauchy horizon were closed or densely filled a 2-torus.
Consequently, the maximal globally hyperbolic vacuum development of generic
initial data cannot be extended across a compact Cauchy horizon with surface
gravity that can be normalised to a non-zero constant. Our result supports,
thereby, the validity of the strong cosmic censorship conjecture in the
considered special case. The proof consists of two main steps. First, we show
that the Killing equation can be solved up to infinite order at the Cauchy
horizon. Second, by applying a recent result of the first author on wave
equations with initial data on a compact Cauchy horizon, we show that this
Killing vector field extends to the globally hyperbolic region.
