A look on the internal structure of Teleparallel Equivalent of General Relativity

TL;DR

This paper explores the internal gauge structure of Teleparallel Equivalent of General Relativity (TEGR), revealing anisotropic responses in space, establishing analogies with Yang-Mills and Electromagnetic gauge transformations, and clarifying the physical significance of gravitoelectric and gravitomagnetic fields.

Contribution

It provides a detailed analysis of gauge transformations in TEGR using the Dirac procedure, highlighting anisotropy and establishing analogies with well-known gauge theories.

Findings

No isotropy in physical space under gauge transformations.

A gauge transformation for tetrads analogous to Yang-Mills theory.

In asymptotically flat spacetime, transformations resemble Electromagnetism.

Abstract

Using the Dirac procedure to treat constraints dynamical sistems applied to gravitation, as described in the context of Teleparallel Equivalent of General Relativity (TEGR), we investigate, from the first class constraints, the gauge transformations in the fundamental field: the components of tetrads. We have shown that there is no an isotropy in physical space with respect to gauge transformations, i.e., given an arbitrary gravitational field, coming from a gauge transformation in the internal space, physical space reacts differently in the spatial and temporal components. By making an appropriate choice, we have found a gauge transformation for the components of tetrad field that allows a direct analogy with the gauge transformations of the Yang-Mills theory. In addition, to the asymptotically flat case in which the algebra index is fixed, we recover the transformations of Electromagnetism. Moreover, still considering the asymptotically flat case, the dependence of the gauge transformation parameter in the space-time variables is periodic, just like in the Electromagnetism. Furthermore, the gravitoelectric and gravitomagnetic fields as have recently been defined, make sense since they allow a direct relation with the momenta, which is analogous to what occurs in other gauge theories.