Symmetries of Cosmological Cauchy Horizons with Non-Closed Orbits

TL;DR

This paper investigates the symmetries of vacuum spacetimes with compact Cauchy horizons, showing that non-ergodic horizons typically admit multiple Killing vectors, extending previous results to cases with non-closed, densely filling null geodesics.

Contribution

It extends the analysis of symmetries in vacuum spacetimes to include non-closed horizon generators, proving the existence of multiple Killing vectors in such cases under analyticity assumptions.

Findings

Non-closed horizon generators typically fill a 2-torus in the horizon.

Vacuum spacetimes with non-ergodic horizons admit at least two commuting Killing vectors.

Ergodic horizons are conjectured to be constructible from Kasner solutions via irrational toroidal compactifications.

Abstract

We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all \textit{closed} curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such `generalized Taub-NUT' spacetimes and show that, generically, they admitted \textit{only} the single (horizon-generating) Killing field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as `ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the \textit{conjectures} that every such spacetime with an \textit{ergodic} horizon is trivially constructable from the flat Kasner solution by making certain `irrational' toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case.