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.

13 pages, latex. V2: reference added, major changes in section 3, conclusions partially corrected

References in corpus (15)

• Dark Energy Dominance and Cosmic Acceleration in First Order Formalism
• Metric-affine f(R) theories of gravity
• Dark energy and cosmological solutions in second-order string gravity
• Cosmological Solutions of Higher-Curvature String Effective Theories with Dilatons
• Brane cosmology with curvature corrections
• Lovelock Gravity at the Crossroads of Palatini and Metric Formulations
• The Cosmology of Ricci-Tensor-Squared Gravity in the Palatini Variational Approach
• Higher Dimensional Gravity, Propagating Torsion and AdS Gauge Invariance
• How (Not) to Palatini
• On the inflationary solutions in higher-derivative gravity with dilaton field
• Dark energy generated from a (super)string effective action with higher order curvature corrections and a dynamical dilaton
• On alpha'-corrections to D-brane solutions
• Variational and conformal structure of nonlinear metric-connection gravitational lagrangians
• The Effect of alpha' Corrections in String Gas Cosmology
• Universal field equations for metric-affine theories of gravity

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