Dimensional reduction and gauge group reduction in Bianchi-Type cosmology

TL;DR

This paper analyzes the implementation of gauge fixing and Killing conditions in Bianchi-Type cosmologies, clarifying their equivalence in Lagrangian and Hamiltonian formulations and exploring residual gauge transformations.

Contribution

It demonstrates the equivalence of Lagrangian and Hamiltonian formulations and clarifies the role of residual gauge transformations in Bianchi-Type cosmologies.

Findings

Proved the equivalence of Lagrangian and Hamiltonian formulations.

Clarified the role of Homogeneity Preserving Diffeomorphisms.

Identified residual gauge transformations as Poincaré transformations.

Abstract

In this paper we examine in detail the implementation, with its associated difficulties, of the Killing conditions and gauge fixing into the variational principle formulation of Bianchi-Type cosmologies. We address problems raised in the literature concerning the Lagrangian and the Hamiltonian formulations: We prove their equivalence, make clear the role of the Homogeneity Preserving Diffeomorphisms in the phase space approach, and show that the number of physical degrees of freedom is the same in the Hamiltonian and Lagrangian formulations. Residual gauge transformations play an important role in our approach, and we suggest that Poincar\'e transformations for special relativistic systems can be understood as residual gauge transformations. In Appendices, we give the general computation of the equations of motion and the Lagrangian for any Bianchi-Type vacuum metric and for spatially homogeneous Maxwell fields in a nondynamical background (with zero currents). We also illustrate our counting of degrees of freedom in an Appendix.