Duality Operator in Teleparallel Gravity

TL;DR

This paper investigates the duality operator in teleparallel gravity, aiming to define it consistently, but finds that the proposed approach by Lucas and Pereira is either non-unique or non-existent, challenging previous assumptions.

Contribution

The paper provides a rigorous mathematical analysis of the duality operator in teleparallel gravity, demonstrating issues with its consistent definition as previously proposed.

Findings

Lucas and Pereira's approach does not yield a unique duality operator.

The duality operator, as defined, may not exist under certain conditions.

The study clarifies limitations in formulating a Yang-Mills-like action for teleparallel gravity.

Abstract

Teleparallel gravity is a theory of gravity which replaces the Levi-Civita connection by a teleparallel connection - a metric-compatible connection with vanishing curvature. Teleparallel equivalent of general relativity (TEGR) is a special case from this class of theories, and its dynamics is governed by equations which are equivalent to Einstein field equations of general relativity. In 2009 Lucas and Pereira presented the idea of a new operator that would allow us to express the action of TEGR in a form reminiscent of the Yang-Mills action for a gauge theory, for which torsion is the field strength. The authors constructed the operator as a duality operator acting on torsion 2-forms and curvature 2-forms, taking into account all the contractions of these forms with the volume form of the spacetime manifold. The definition of this operator for other forms, however, remained unclear. In this thesis, we seek to give a mathematically consistent definition of the operator, and we follow the results published by Lucas and Pereira. We show that the approach taken by Lucas and Pereira does not lead to a consistent definition of this operator, because it results in the operator with the desired properties either being non-unique or non-existent.