A classification theorem for compact Cauchy horizons in vacuum spacetimes

TL;DR

This paper classifies the topology and null generator structure of compact Cauchy horizons in vacuum spacetimes, revealing four possible types and connecting them to known geometries, including flat Kasner solutions.

Contribution

It provides a complete classification theorem for compact Cauchy horizons in vacuum spacetimes, linking topological types to specific spacetime geometries and solving a longstanding problem for ergodic horizons.

Findings

Four possible topological types of horizons identified

All types correspond to known geometric structures

Flat Kasner spacetime characterized for ergodic horizons

Abstract

We establish a complete classification theorem for the topology and for the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum $3+1$-spacetimes. We show that, either: (i) all generators are closed, or (ii) only two generators are closed and any other densely fills a two-torus, or (iii) every generator densely fills a two-torus, or (iv) every generator densely fills the horizon. We then show that, respectively to (i)-(iv), the horizon's manifold is either: (i') a Seifert manifold, or (ii') a lens space, or (iii') a two-torus bundle over a circle, or, (iv') a three-torus. All the four possibilities are known to arise in examples. In the last case, (iv), (iv'), we show in addition that the spacetime is indeed flat Kasner, thus settling a problem posed by Isenberg and Moncrief for ergodic horizons. The results of this article open the door for a full parameterization of the metrics of all vacuum spacetimes with a compact Cauchy horizon. The method of proof permits direct generalizations to higher dimensions.