We study the following problem: given initial data on a compact Cauchy
horizon, does there exist a unique solution to wave equations on the globally
hyperbolic region? Our main results apply to any spacetime satisfying the null
energy condition and containing a compact Cauchy horizon with surface
gravity that can be normalised to a nonzero constant. Examples include the
Misner spacetime and the Taub-NUT spacetime. We prove an energy estimate
close to the Cauchy horizon for wave equations acting on sections of vector
bundles. Using this estimate we prove that if a linear wave equation can be
solved up to any order at the Cauchy horizon, then there exists a unique
solution on the globally hyperbolic region. As a consequence, we prove several
existence and uniqueness results for linear and nonlinear wave equations
without assuming analyticity or symmetry of the spacetime and without
assuming that the generators close. We overcome in particular the essential
remaining difficulty in proving that vacuum spacetimes with a compact Cauchy
horizon, with constant nonzero surface gravity, necessarily admit a Killing
vector field. This work is therefore related to the strong cosmic censorship
conjecture.
