Hodge Dual for Soldered Bundles

TL;DR

This paper introduces a generalized Hodge dual for soldered bundles, leading to a new dual for torsion and connecting teleparallel gravity with general relativity through a unified Lagrangian framework.

Contribution

It defines a novel generalized Hodge dual for soldered bundles, revealing a new torsion dual and linking teleparallel gravity to Einstein's general relativity.

Findings

The generalized dual coincides with the usual for curvature.

For torsion, it provides a new dual definition.

The generalized Hodge dual reproduces the teleparallel and Einstein-Hilbert Lagrangians.

Abstract

In order to account for all possible contractions allowed by the presence of the solder form, a generalized Hodge dual is defined for the case of soldered bundles. Although for curvature the generalized dual coincides with the usual one, for torsion it gives a completely new dual definition. Starting from the standard form of a gauge lagrangian for the translation group, the generalized Hodge dual yields precisely the lagrangian of the teleparallel equivalent of general relativity, and consequently also the Einstein-Hilbert lagrangian of general relativity.