On variations in teleparallelism theories

TL;DR

This paper investigates the variation procedures in teleparallel theories, focusing on the non-commutativity with the Hodge dual, and introduces techniques to simplify calculations in electromagnetic and gravity models.

Contribution

It presents a new variational matrix technique and conditions for commutativity with the Hodge dual, offering a simplified approach for teleparallelism variations.

Findings

Derived formulas for variations involving the Hodge dual.

Established conditions for commutativity and anti-commutativity.

Applied techniques to electromagnetic and gravity Lagrangians.

Abstract

The variation procedure on a teleparallel manifold is studied. The main problem is the non-commutativity of the variation with the Hodge dual map. We establish certain useful formulas for variations and restate the master formula due to Hehl and his collaborates. Our approach is different and sometimes easier for applications. By introducing the technique of the variational matrix we find necessary and sufficient conditions for commutativity (anti-commutativity) of the variation derivative with the Hodge dual operator. A general formula for the variation of the quadratic-type expression is obtained. The described variational technique are used in two viable field theories: the electro-magnetic Lagrangian on a curved manifold and the Rumpf Lagrangian of the translation invariant gravity.