The Palatini formalism for higher-curvature gravity theories

TL;DR

This paper compares the metric and Palatini formalisms in higher-curvature gravity theories, showing they are generally not equivalent except for Lovelock gravities where they coincide, highlighting differences in solution sets.

Contribution

It demonstrates the non-equivalence of the metric and Palatini formalisms in general higher-curvature theories and explains their equivalence specifically for Lovelock gravities.

Findings

Palatini formalism solutions form a subset of metric formalism solutions

Equivalence of formalisms holds exactly for Lovelock gravities

Provides explanation for the special case of Lovelock theories

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 in general the two formalisms are not equivalent and that the set of solutions of the Palatini equations is a non-trivial subset of the solutions of the metric equations. However we also argue that for Lovelock gravities, the equivalence of the two formalism holds completely and give an explanation of why it holds precisely for these theories.