Hamiltonian structures for compact homogeneous universes

TL;DR

This paper develops Hamiltonian frameworks for compact homogeneous universes, incorporating Teich parameters to describe global geometry, and demonstrates how these models interpret all degrees of freedom as geometric deformations.

Contribution

It introduces a Hamiltonian formulation that includes Teich parameters for compact homogeneous universes, clarifying the geometric nature of dynamical degrees of freedom.

Findings

Hamiltonian structures are established for compact homogeneous universes.

Teich parameters effectively parameterize the global geometry.

All dynamical degrees of freedom are interpretable as geometric deformations.

Abstract

Hamiltonian structures for spatially compact locally homogeneous vacuum universes are investigated, provided that the set of dynamical variables contains the \Teich parameters, parameterizing the purely global geometry. One of the key ingredients of our arguments is a suitable mathematical expression for quotient manifolds, where the universal cover metric carries all the degrees of freedom of geometrical variations, i.e., the covering group is fixed. We discuss general problems concerned with the use of this expression in the context of general relativity, and demonstrate the reduction of the Hamiltonians for some examples. For our models, all the dynamical degrees of freedom in Hamiltonian view are unambiguously interpretable as geometrical deformations, in contrast to the conventional open models.