(2+1) gravity for higher genus in the polygon model

TL;DR

This paper explicitly constructs the initial data space for (2+1) gravity with higher genus in the polygon model, linking boost parameters to geodesic lengths on Riemann surfaces, and discusses singularities and phase space completeness.

Contribution

It provides an explicit construction of the initial data space for higher genus (2+1) gravity in the polygon model, connecting geometric parameters to the model's variables.

Findings

Initial data space has dimension 12g-12 for genus g

Singularities occur as big-bang or big-crunch in all configurations

Conjecture that the constructed space is the full physical phase space

Abstract

We construct explicitly a (12g-12)-dimensional space P of unconstrained and independent initial data for 't Hooft's polygon model of (2+1) gravity for vacuum spacetimes with compact genus-g spacelike slices, for any g >= 2. Our method relies on interpreting the boost parameters of the gluing data between flat Minkowskian patches as the lengths of certain geodesic curves of an associated smooth Riemann surface of the same genus. The appearance of an initial big-bang or a final big-crunch singularity (but never both) is verified for all configurations. Points in P correspond to spacetimes which admit a one-polygon tessellation, and we conjecture that P is already the complete physical phase space of the polygon model. Our results open the way for numerical investigations of pure (2+1) gravity.