Compact Cauchy horizons admit constant surface gravity

TL;DR

This paper proves that compact Cauchy horizons in spacetimes satisfying the null energy condition always admit a smooth lightlike tangent vector field with constant surface gravity, extending previous results to degenerate cases.

Contribution

It establishes the existence of a smooth lightlike tangent vector field with constant surface gravity on compact Cauchy horizons under the null energy condition, solving an open problem.

Findings

Existence of a smooth lightlike tangent vector field with constant surface gravity.

Results hold in any spacetime dimension under the null energy condition.

Uses ergodic, Hodge, and Riemannian flow theories in the proof.

Abstract

We prove that in any spacetime dimension and under the null energy condition, every totally geodesic connected smooth compact null hypersurface (hence every compact Cauchy horizon) admits a smooth lightlike tangent vector field of constant surface gravity. That is, we solve the open degenerate case by showing that, if there is a complete generator, then there exists a smooth future-directed geodesic lightlike tangent field. The result can be stated as an existence result for a particular cohomological equation. The proof uses elements of ergodic theory, Hodge theory and Riemannian flow theory. We emphasize that, remarkably, these results really require only the null energy condition, whereas previous works assumed, already in the Killing or the non-degenerate case, the stronger dominant energy condition.