Some minisuperspace model for the Faddeev formulation of gravity

TL;DR

This paper explores a minisuperspace model for Faddeev's formulation of gravity, where vector fields are piecewise constant, and demonstrates that as the regions become smaller, the model approaches the continuum Faddeev action.

Contribution

It introduces a minisuperspace formulation of Faddeev gravity with piecewise constant vector fields and shows its convergence to the continuum theory.

Findings

The minisuperspace action is explicitly derived.

Piecewise constant fields approximate smooth fields as regions shrink.

The model converges to the continuum Faddeev action in the limit of small regions.

Abstract

We consider Faddeev formulation of general relativity in which the metric is composed of ten vector fields or a $4 \times 10$ tetrad. This formulation reduces to the usual general relativity upon partial use of the field equations. A distinctive feature of the Faddeev action is its finiteness on the discontinuous fields. This allows to introduce its minisuperspace formulation where the vector fields are constant everywhere on ${\rm I \hspace{-3pt} R}^4$ with exception of a measure zero set (the piecewise constant fields). The fields are parameterized by their constant values {\it independently} chosen in, e. g., the 4-simplices or, say, parallelepipeds into which ${\rm I \hspace{-3pt} R}^4$ can be decomposed. The form of the action for the vector fields of this type is found. We also consider the piecewise constant vector fields approximating the fixed smooth ones. We check that if the regions in which the vector fields are constant are made arbitrarily small, the minisuperspace action and eqs of motion tend to the continuum Faddeev ones.