How (Not) to Palatini

Abstract

We revisit the problem of defining non-minimal gravity in the first order formalism. Specializing to scalar-tensor theories, which may be disguised as `higher-derivative' models with the gravitational Lagrangians that depend only on the Ricci scalar, we show how to recast these theories as Palatini-like gravities. The correct formulation utilizes the Lagrange multiplier method, which preserves the canonical structure of the theory, and yields the conventional metric scalar-tensor gravity. We explain the discrepancies between the naïve Palatini and the Lagrange multiplier approach, showing that the naïve Palatini approach really swaps the theory for another. The differences disappear only in the limit of ordinary General Relativity, where an accidental redundancy ensures that the naïve Palatini works there. We outline the correct decoupling limits and the strong coupling regimes. As a corollary we find that the so-called `Modified Source Gravity' models suffer from strong coupling problems at very low scales, and hence cannot be a realistic approximation of our universe. We also comment on a method to decouple the extra scalar using the chameleon mechanism.

18 pages, LaTeX; added references and minor improvements in sec 4

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• 6+1 lessons from f(R) gravity
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• Microscopic and Macroscopic Behaviors of Palatini Modified Gravity Theories
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• Weakening Gravity on Redshift-Survey Scales with Kinetic Matter Mixing
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• On stability of a neutron star system in Palatini gravity
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• Palatini f(R) gravity as a fixed point
• Structure of Compact Stars in R-squared Palatini Gravity
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• Gravity with Perturbative Constraints: Dark Energy Without New Degrees of Freedom
• Birth of baby universes from gravitational collapse in a modified-gravity scenario
• Structure formation in $f(R)$ gravity: A distinguishing probe between the dark energy and modified gravity
• Invariant quantities of Scalar-Tensor Theories for stellar structure
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• Hamiltonian analysis of metric-affine-$R^2$ theory
• Some aspects of modified theory of gravity in Palatini formalism unveiled
• Extended Theories of Gravity in cosmological and astrophysical applications