Generalized Hodge dual for torsion in teleparallel gravity

TL;DR

This paper investigates the possibility of extending a generalized Hodge dual for torsion in teleparallel gravity beyond four dimensions, concluding that such a generalization is not feasible in other dimensions and highlighting the dimensional specificity of the original formulation.

Contribution

The paper demonstrates the dimensional limitations of the generalized Hodge dual for torsion, showing it cannot be straightforwardly extended beyond four dimensions and requires ad hoc definitions.

Findings

Generalized Hodge dual for torsion is specific to four dimensions.

Extension to other dimensions fails due to mathematical inconsistencies.

A consistent definition in general dimensions would require an unexpected ad hoc dual operation.

Abstract

For teleparallel gravity in four dimensions, Lucas and Pereira have shown that a generalized Hodge dual for torsion tensor can be defined with coefficients determined by mathematical consistency. In this paper, we demonstrate that a direct generalization to other dimensions fails and no new generalized Hodge dual operator could be given. Furthermore, if one enforces the definition of a generalized Hodge dual to be consistent with the action of teleparallel gravity in general dimensions, the basic identity for any sensible Hodge dual would require an \textit{ad hoc} definition for the second Hodge dual operation which is totally unexpected. Therefore, we conclude that at least for the torsion tensor, the observation of Lucas and Pereira only applies to four dimensions.