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

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.

35 pages, 22 figures

References in corpus (10)

• Spectra of Length and Area in 2+1 Lorentzian Loop Quantum Gravity
• Canonical Quantization of Gravitating Point Particles in 2+1 Dimensions
• The Phase Space Structure of Multi Particle Models in 2+1 Gravity
• New Loop Representations for 2+1 Gravity
• Hamiltonian structure and quantization of 2+1 dimensional gravity coupled to particles
• $2+1$ Covariant Lattice Theory and t'Hooft's Formulation
• Quantisation of 2+1 gravity for genus 2
• The Torus Universe in the Polygon Approach to 2+1-Dimensional Gravity
• Independent Loop Invariants for 2+1 Gravity
• 't Hooft's Polygon Approach Hyperbolically Revisited

Cited by in corpus (5)

• Quantum Gravity in 2+1 Dimensions: The Case of a Closed Universe
• Canonical simplicial gravity
• Boundary conditions and symplectic structure in the Chern-Simons formulation of (2+1)-dimensional gravity
• Polygon model from first order gravity
• The polygon model for 2+1D gravity: the constraint algebra and problems of quantization