Importance of torsion and invariant volumes in Palatini theories of gravity

TL;DR

This paper investigates how the order of setting torsion to zero affects the field equations in Palatini gravity theories, revealing the role of invariant volumes and the decomposition of the connection.

Contribution

It demonstrates the impact of a priori versus a posteriori torsion removal on the dynamical equations and clarifies the connection decomposition involving invariant volumes in Palatini theories.

Findings

Different second-order equations arise depending on torsion removal order.

The connection decomposes into Levi-Civita plus a vector field related to invariant volume.

The antisymmetric Ricci tensor component does not influence dynamics.

Abstract

We study the field equations of extensions of General Relativity formulated within a metric-affine formalism setting torsion to zero (Palatini approach). We find that different (second-order) dynamical equations arise depending on whether torsion is set to zero i) a priori or ii) a posteriori, i.e., before or after considering variations of the action. Considering a generic family of Ricci-squared theories, we show that in both cases the connection can be decomposed as the sum of a Levi-Civita connection and terms depending on a vector field. However, while in case i) this vector field is related to the symmetric part of the connection, in ii) it comes from the torsion part and, therefore, it vanishes once torsion is completely removed. Moreover, the vanishing of this torsion-related vector field immediately implies the vanishing of the antisymmetric part of the Ricci tensor, which therefore plays no role in the dynamics. Related to this, we find that the Levi-Civita part of the connection is due to the existence of an invariant volume associated to an auxiliary metric $h_{\mu\nu}$, which is algebraically related with the physical metric $g_{\mu\nu}$.