Lovelock Gravity at the Crossroads of Palatini and Metric Formulations

TL;DR

This paper investigates the relationship between Palatini and metric formulations of extended gravity theories, demonstrating that Lovelock gravity uniquely satisfies the physical equivalence criterion rooted in the Equivalence Principle.

Contribution

It proves that among a broad class of modified gravity theories, only Lovelock gravity maintains the equivalence of Palatini and metric formulations at the equations of motion level.

Findings

Lovelock gravity uniquely satisfies the equivalence criterion.

The equivalence of formulations is linked to the Equivalence Principle.

Other modified gravity theories do not satisfy this criterion.

Abstract

We consider extensions of the Einstein-Hilbert Lagrangian to a general functional of metric and Riemann curvature tensor. A given such Lagrangian describes two different theories depending on considering connection and metric (Palatini formulation), or only the metric (metric formulation) as independent dynamical degrees of freedom. Equivalence of the Palatini and metric formulations at the level of equations of motion, which as we will argue is a manifestation of the Equivalence Principle, is the physical criterion that restricts form of the Lagrangians of modified gravity theories. We prove that within the class of modified gravity theories we consider, only the Lovelock gravity satisfies this requirement.