Canonical Structure of Locally Homogeneous Systems on Compact Closed 3-Manifolds of Types $E^3$, Nil and Sol

TL;DR

This paper analyzes the canonical structure of locally homogeneous spacetimes on compact 3-manifolds of types E^3, Nil, and Sol, providing a comprehensive algorithm and explicit results for their phase spaces and Hamiltonians.

Contribution

It introduces a general algorithm to express the phase space and canonical structure of locally homogeneous systems in terms of homogeneous systems and moduli spaces, and fully determines these structures for specific Thurston types.

Findings

Canonical structures are explicitly determined for E^3, Nil, and Sol types.

In many cases, the canonical structure becomes degenerate in the moduli sectors.

Locally homogeneous systems are generally not canonically closed in the full phase space.

Abstract

In this paper we investigate the canonical structure of diffeomorphism invariant phase spaces for spatially locally homogeneous spacetimes with 3-dimensional compact closed spaces. After giving a general algorithm to express the diffeomorphism-invariant phase space and the canonical structure of a locally homogeneous system in terms of those of a homogeneous system on a covering space and a moduli space, we completely determine the canonical structures and the Hamiltonians of locally homogeneous pure gravity systems on orientable compact closed 3-spaces of the Thurston-type $E^3$, $\Nil$ and $\Sol$ for all possible space topologies and invariance groups. We point out that in many cases the canonical structure becomes degenerate in the moduli sectors, which implies that the locally homogeneous systems are not canonically closed in general in the full diffeomorphism-invariant phase space of generic spacetimes with compact closed spaces.