The dynamics of metric-affine gravity

TL;DR

This paper investigates the conditions under which the connection in metric-affine gravity theories becomes dynamical, showing that most simple actions do not produce dynamics, but higher order terms can introduce new degrees of freedom.

Contribution

It analyzes the dynamical nature of the connection in metric-affine gravity and identifies how higher order invariants can induce new degrees of freedom.

Findings

Most minimal actions make the connection auxiliary and non-dynamical.

Including higher order curvature and torsion terms can activate the connection's dynamics.

f(R) theories form a special class with unique properties, not representative of general metric-affine theories.

Abstract

Metric-affine theories of gravity provide an interesting alternative to General Relativity: in such an approach, the metric and the affine (not necessarily symmetric) connection are independent quantities. Furthermore, the action should include covariant derivatives of the matter fields, with the covariant derivative naturally defined using the independent connection. As a result, in metric-affine theories a direct coupling involving matter and connection is also present. The role and the dynamics of the connection in such theories is explored. We employ power counting in order to construct the action and search for the minimal requirements it should satisfy for the connection to be dynamical. We find that for the most general action containing lower order invariants of the curvature and the torsion the independent connection does not carry any dynamics. It actually reduces to the role of an auxiliary field and can be completely eliminated algebraically in favour of the metric and the matter field, introducing extra interactions with respect to general relativity. However, we also show that including higher order terms in the action radically changes this picture and excites new degrees of freedom in the connection, making it (or parts of it) dynamical. Constructing actions that constitute exceptions to this rule requires significant fine tuned and/or extra a priori constraints on the connection. We also consider f(R) actions as a particular example in order to show that they constitute a distinct class of metric-affine theories with special properties, and as such they cannot be used as representative toy theories to study the properties of metric-affine gravity.