Palatini versus metric formulation in higher-curvature gravity

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.

Einstein–Yang–Mills–Lorentz black holes

“…Indeed, whereas the gauge and the Palatini formalisms are found to be equivalent by requiring the presence of a metric-compatible connection [22], it is shown that the latter also implies the metric formalism but the opposite is not true for theories endowed with this type of higher order curvature corrections in the Lagrangian [23].…”

Higgs–Palatini inflation and unitarity

“…This very setup, the so-called Palatini formulation or first-order formalism, does not necessarily admit the Levi-Civita connection if the matter sector depends explicitly on the affine connection [14,15]. In this formalism, connection and metric are independent geometrodynamical variables [16], and this very fact has physically interesting consequences for inflation [16][17][18][19][20][21] as well as other phenomena like the cosmological constant problem [22,23].…”

Comparing metric and Palatini approaches to vector Horndeski theory

“…ρσ term was already established in literature [8,34]. And now we will calculate the actual difference at the level of solutions, expanding the equations in powers of λ.…”

“…There are known few more theories which exhibit such equivalence [7,8], but for an arbitrary gravity theory the equivalence is generally broken. Identically looking actions provide different equations of motion in F1 and F2 approaches [9].…”