The role of nonmetricity in metric-affine theories of gravity

TL;DR

This paper investigates metric-affine theories of gravity, revealing that even with higher-order invariants involving nonmetricity and torsion, the affine connection remains an auxiliary field, leading to new metric-matter interactions.

Contribution

It demonstrates that nonmetricity and torsion terms do not introduce dynamical degrees of freedom in the connection, but instead produce additional interactions in the effective theory.

Findings

Connection acts as an auxiliary field, not dynamical.

Higher-order invariants generate new metric-matter couplings.

Nonmetricity and torsion do not lead to propagating degrees of freedom.

Abstract

The intriguing choice to treat alternative theories of gravity by means of the Palatini approach, namely elevating the affine connection to the role of independent variable, contains the seed of some interesting (usually under-explored) generalizations of General Relativity, the metric-affine theories of gravity. The peculiar aspect of these theories is to provide a natural way for matter fields to be coupled to the independent connection through the covariant derivative built from the connection itself. Adopting a procedure borrowed from the effective field theory prescriptions, we study the dynamics of metric-affine theories of increasing order, that in the complete version include invariants built from curvature, nonmetricity and torsion. We show that even including terms obtained from nonmetricity and torsion to the second order density Lagrangian, the connection lacks dynamics and acts as an auxiliary field that can be algebraically eliminated, resulting in some extra interactions between metric and matter fields.