A classification theorem for compact Cauchy horizons in vacuum spacetimes

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.

Surface Gravity of Compact Non-degenerate Horizons Under the Dominant Energy Condition

“…2.13] for the vacuum case. We also argue that the horizon classifications by Rendall [41], and by Bustamante and Reiris [5] can be extended to the non-vacuum case.…”

“…Classification in the 3+1 dimensional case. By using the invariance of the Petersen metric σ under the flow ϕ, Bustamante and Reiris obtained a classification for the topology and for the orbital types of the null generators of the compact non-degenerate Cauchy horizons in smooth vacuum 3 + 1-spacetimes [5]. This classification improved a previous classification in [34].…”

“…We provide a couple of examples of vacuum spacetimes admitting a compact Cauchy horizon, and show how to conveniently calculate their surface gravity. For more examples, with classification results for topologies and dynamical behavior of generators, see [5,22].…”

“…For a non-degenerate H , chosen n such that κ = −1, as L n ω = 0 the following (Petersen) Riemannian metric 5 [5,37] is left invariant by the flow ϕ…”

“…Of course, these results have very important consequences, as they fix considerably the possible topologies and symmetries of a spacetime admitting a compact Cauchy horizon [5,18,34].…”