f(R) gravity, torsion and non-metricity

TL;DR

This paper demonstrates that in f(R) gravity theories with independent connections, whether torsion, non-metricity, or both are present, the connection can be algebraically eliminated, resulting in a consistent scalar-tensor equivalent theory.

Contribution

It generalizes previous results by showing that even with both torsion and non-metricity, the independent connection does not carry dynamical degrees of freedom in f(R) gravity.

Findings

Independent connection can be algebraically eliminated in all cases.

All cases lead to the same scalar-tensor theory equivalent.

f(R) actions cannot support dynamical independent connections without matter coupling.

Abstract

For both f(R) theories of gravity with an independent symmetric connection (no torsion), usually referred to as Palatini f(R) gravity theories, and for f(R) theories of gravity with torsion but no non-metricity, called U4 theories, it has been shown that the independent connection can actually be eliminated algebraically, as long as this connection does not couple to matter. Remarkably, the outcome in both case is the same theory, which is dynamically equivalent with an \omega_0=-3/2 Brans--Dicke theory. It is shown here that even for the most general case of an independent connection with both non-metricity and torsion one arrives at exactly the same theory as in the more restricted cases. This generalizes the previous results and explains why assuming that either the torsion or the the non-metricity vanishes ultimately leads to the same theory. It also demonstrates that f(R) actions cannot support an independent connection which carries dynamical degrees of freedom, irrespectively of how general this connection is, at least as long as there is no connection-matter coupling.