Spectrum of area in the Faddeev formulation of gravity

TL;DR

This paper investigates the spectrum of elementary areas in the Faddeev formulation of gravity, revealing a discrete spectrum proportional to a Faddeev-specific parameter, and discusses implications for black hole entropy.

Contribution

It introduces a first-order minisuperspace model with piecewise constant tetrads and evaluates the discrete area spectrum in Faddeev gravity, linking it to black hole entropy.

Findings

The area spectrum is discrete and proportional to the Faddeev Barbero-Immirzi parameter.

Elementary areas can be summed over surfaces due to tetrad discontinuities.

Estimate of the Faddeev parameter consistent with black hole entropy calculations.

Abstract

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed first-order representation of the minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${\rm R}^4$ can be decomposed, an analogue of the Cartan-Weyl connection-type form of the Hilbert-Einstein action in the usual continuum GR. In the Hamiltonian formalism, the tetrad bilinears are canonically conjugate to the orthogonal connection matrices. We evaluate the spectrum of the elementary areas, functions of the tetrad bilinears. The spectrum is discrete and proportional to the Faddeev analog $\gamma_{\rm F}$ of the Barbero-Immirzi parameter $\gamma$. The possibility of the tetrad and metric discontinuities in the Faddeev gravity allows to consider any surface as consisting of a set of virtually independent elementary areas and its spectrum being the sum of the elementary spectra. Requiring consistency of the black hole entropy calculations known in the literature we are able to estimate $\gamma_{\rm F}$.