Volume elements and torsion

TL;DR

This paper investigates the consistency of the minimal action formulation with minimal coupling in spaces with torsion, showing that the trace of torsion must be a gradient and a modified volume element is necessary, with implications for gravity and crystalline defects.

Contribution

The paper weakens assumptions needed to establish the necessity of a modified volume element in torsion spaces, clarifying the role of torsion trace in physical theories.

Findings

Proper minimal coupling requires torsion trace to be a gradient.

Modified volume element is essential in the action formulation.

Propagating torsion theories are inconsistent with observational data.

Abstract

We reexamine here the issue of consistency of minimal action formulation with the minimal coupling procedure (MCP) in spaces with torsion. In Riemann-Cartan spaces, it is known that a proper use of the MCP requires that the trace of the torsion tensor be a gradient, $T_\mu=\partial_\mu\theta$, and that the modified volume element $\tau_\theta = e^\theta \sqrt{g} dx^1\wedge...\wedge dx^n $ be used in the action formulation of a physical model. We rederive this result here under considerably weaker assumptions, reinforcing some recent results about the inadequacy of propagating torsion theories of gravity to explain the available observational data. The results presented here also open the door to possible applications of the modified volume element in the geometric theory of crystalline defects.