Aspects of nonmetricity in gravity theories

TL;DR

This paper demonstrates that certain metric-affine gravity theories can be simplified to Riemann-Cartan form by canceling nonmetricity, revealing that nonmetric degrees of freedom can decouple or act as matter fields.

Contribution

It shows how nonmetricity in metric-affine gravities can be reduced to Riemann-Cartan form, clarifying the role of nonmetric degrees of freedom.

Findings

Nonmetricity can be canceled against symmetric spin connection components.

Nonmetric degrees of freedom decouple from the geometry in the reduction.

Nonmetricity may behave as dynamical matter fields without geometric interpretation.

Abstract

In this work, we show that a class of metric-affine gravities can be reduced to a Riemann-Cartan one. The reduction is based on the cancelation of the nonmetricity against the symmetric components of the spin connection. A heuristic proof, in the Einstein-Cartan formalism, is performed in the special case of diagonal unitary tangent metric tensor. The result is that the nonmetric degrees of freedom decouple from the geometry. Thus, from the point of view of isometries on the tangent manifold, the equivalence might be viewed as an isometry transition from the affine group to the Lorentz group, $A(d,\mathbb{R})\longmapsto SO(d)$. Furthermore, in this transition, depending on the form of the starting action, the nonmetricity degrees might present a dynamical matter field character, with no geometric interpretation in the Riemann-Cartan geometry.