On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism

TL;DR

This paper explores the non-uniqueness of the Levi-Civita connection in the Palatini formalism, revealing a family of solutions sharing key physical properties, and argues their physical indistinguishability.

Contribution

It identifies a family of affine connections generalizing Levi-Civita, characterized by an arbitrary vector field, and discusses their mathematical and physical implications.

Findings

Family of solutions characterized by a non-dynamical vector field

Shared geodesics and Einstein equations among solutions

Physical effects of different connections are indistinguishable

Abstract

We study the most general solution for affine connections that are compatible with the variational principle in the Palatini formalism for the Einstein-Hilbert action (with possible minimally coupled matter terms). We find that there is a family of solutions generalising the Levi-Civita connection, characterised by an arbitrary, non-dynamical vector field ${\cal A}_\mu$. We discuss the mathematical properties and the physical implications of this family and argue that, although there is a clear mathematical difference between these new Palatini connections and the Levi-Civita one, both unparametrised geodesics and the Einstein equation are shared by all of them. Moreover, the Palatini connections are characterised precisely by these two properties, as well as by other properties of its parallel transport. Based on this, we conclude that physical effects associated to the choice of one or the other will not be distinguishable, at least not at the level of solutions or test particle dynamics. We propose a geometrical interpretation for the existence and unobservability of the new solutions.