Palatini versus metric formulation in higher curvature gravity

TL;DR

This paper compares the metric and Palatini formalisms in higher curvature gravity, identifying conditions under which they are equivalent or one contains the other, especially in theories including Lovelock and related Lagrangians.

Contribution

It characterizes the relationship between metric and Palatini formalisms in higher curvature theories, highlighting classes of equivalence and solution correspondence.

Findings

Identifies a class of theories where the two formalisms are equivalent.

Shows that Palatini solutions are contained within metric solutions under certain conditions.

Provides conditions for metric solutions to also solve Palatini equations.

Abstract

We compare the metric and the Palatini formalism to obtain the Einstein equations in the presence of higher-order curvature corrections that consist of contractions of the Riemann tensor, but not of its derivatives. We find that there is a class of theories for which the two formalisms are equivalent. This class contains the Palatini version of Lovelock theory, but also more Lagrangians that are not Lovelock, but respect certain symmetries. For the general case, we find that imposing the Levi-Civita connection as an Ansatz, the Palatini formalism is contained within the metric formalism, in the sense that any solution of the former also appears as a solution of the latter, but not necessarily the other way around. Finally we give the conditions the solutions of the metric equations should satisfy in order to solve the Palatini equations.