Bianchi Cosmological Models and Gauge Symmetries

TL;DR

This paper clarifies the concept of gauge symmetries in Bianchi cosmological models, proposing a geometrical definition that aligns with dynamical invariance and addresses previous shortcomings in localizability.

Contribution

It introduces a new geometrical definition of gauge symmetries for Bianchi models that ensures local independence and matches the dynamical invariance criteria.

Findings

The new definition aligns geometrical and dynamical perspectives.

Explicit verification of phase space equivalence.

Remarks on Ashtekar variables in Bianchi models.

Abstract

We analyze carefully the problem of gauge symmetries for Bianchi models, from both the geometrical and dynamical points of view. Some of the geometrical definitions of gauge symmetries (=``homogeneity preserving diffeomorphisms'') given in the literature do not incorporate the crucial feature that local gauge transformations should be independent at each point of the manifold of the independent variables ( = time for Bianchi models), i.e, should be arbitrarily localizable ( in time). We give a geometrical definition of homogeneity preserving diffeomorphisms that does not possess this shortcoming. The proposed definition has the futher advantage of coinciding with the dynamical definition based on the invariance of the action ( in Lagrangian or Hamiltonian form). We explicitly verify the equivalence of the Lagrangian covariant phase space with the Hamiltonian reduced phase space. Remarks on the use of the Ashtekar variables in Bianchi models are also given.