Poincare gauge gravity from nonmetric gravity

TL;DR

This paper demonstrates how nonmetricity in a general gauge theory of gravity can be absorbed into torsion, leading to a formulation equivalent to Poincare gauge theory with conformal symmetry.

Contribution

It introduces a method to incorporate nonmetricity into torsion, reducing the structure equations to Poincare gauge theory and revealing a conformal Lie algebra structure.

Findings

Mixed symmetry nonmetricity can be absorbed into torsion.

The extended system's maximal Lie algebra is conformal.

The difference in field strengths relates to nonmetricity.

Abstract

We consider general linear gauge theory, with independent solder form and connection. These spaces have both torsion and nonmetricity. We show that the Cartan structure equations together with the defining equation for nonmetricity allow the mixed symmetry components of nonmetricity to be absorbed into an altered torsion tensor. Field redefinitions reduce the structure equations to those of Poincare gauge theory, with local Lorentz symmetry and metric compatibility. In order to allow recovery the original torsion and nonmetric fields, we replace the definition of nonmetricity by an additional structure equation and demand integrability of the extended system. We show that the maximal Lie algebra compatible with the enlarged set is isomorphic to the conformal Lie algebra. From this Lorentzian conformal geometry, we establish that the difference between the field strength of special conformal transformations and the torsion and is given by the mixed symmetry nonmetricity of an equivalent asymmetric system.