Narrow Results

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 some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which ${ℝ}^4$ can be decomposed.

Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.

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 general relativity (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 four-simplices or, say, cuboids into which ${ℝ}^4$ 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 ${\gamma }_{\mathrm{\text{F}}}$ 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 ${\gamma }_{\mathrm{\text{F}}}$.

We consider the Faddeev formulation of general relativity (GR), which can be characterized by a kind of d-dimensional tetrad (typically d = 10) and a non-Riemannian connection. This theory is invariant w.r.t. the global, but not local, rotations in the d-dimensional space. There can be configurations with a smooth or flat metric, but with the tetrad that changes abruptly at small distances, a kind of “antiferromagnetic” structure.

Previously, we discussed a first-order representation for the Faddeev gravity, which uses the orthogonal connection in the d-dimensional space as an independent variable. Using the discrete form of this formulation, we considered the spectrum of (elementary) area. This spectrum turns out to be physically reasonable just on a classical background with large connection like rotations by $\pi$, that is, with such an “antiferromagnetic” structure.

In the discrete first-order Faddeev gravity, we consider such a structure with periodic cells and large connection and strongly changing tetrad field inside the cell. We show that this system in the continuum limit reduces to a generalization of the Faddeev system. The action is a sum of related actions of the Faddeev type and is still reduced to the GR action.