2+1 gravity, chaos and time machines

TL;DR

This paper explores 2+1 dimensional gravity with torus topology, extending spacetime across Cauchy horizons, and investigates the dynamics of Lorentzian tori, revealing connections to Bianchi IX models and ergodic behavior.

Contribution

It introduces a method to extend 2+1 gravity spacetimes across Cauchy horizons into Lorentzian regions and links these models to Bianchi IX dynamics and ergodic properties of the modular group.

Findings

Extended spacetime description across Cauchy horizons.

Lorentzian tori form a geodesic family in Teichmüller space.

Modular group acts ergodically on Lorentzian tori's Teichmüller space.

Abstract

2+1 gravity for spacetimes with topology RxT^2 has been much studied. We add a description of how to extend these spacetimes across a Cauchy horizon into a region where the torus becomes Lorentzian. The result is a one parameter family of tori given by a geodesic in the "Teichmueller space" of Lorentzian tori. We describe this in detail. We also point out that if the modular group is regarded as part of the gauge group then these spacetimes offer a nice toy model for the dynamics of Bianchi IX models; in the region where the tori are spacelike the dynamics is described exactly by a hyperbolic billiard. On the other hand the modular group acts ergodically on the Teichmueller space of Lorentzian tori.