F(R) gravity in purely affine formulation

TL;DR

This paper investigates the relationship between different formulations of F(R) gravity, revealing that a purely affine formulation is generally not possible unless the theory is conformally transformed to the Einstein frame, where it reduces to general relativity with a scalar field.

Contribution

It demonstrates that F(R) gravity cannot be expressed in a purely affine form in its original frame, but this equivalence is restored via conformal transformation to the Einstein frame.

Findings

Purely affine formulation is not possible for nonlinear F(R) gravity.

Conformal transformation to Einstein frame restores the affine equivalence.

F(R) gravity reduces to GR with a scalar field in the Einstein frame.

Abstract

The purely affine, metric-affine and purely metric formulation of general relativity are dynamically equivalent and the relation between them is analogous to the Legendre relation between the Lagrangian and Hamiltonian dynamics. We show that one cannot construct a dynamically equivalent, purely affine Lagrangian from a metric-affine or metric F(R) Lagrangian, nonlinear in the curvature scalar. Thus the equivalence between the purely affine picture and the two other formulations does not hold for metric-affine and metric theories of gravity with a nonlinear dependence on the curvature, i.e. F(R) gravity does not have a purely affine formulation. We also show that this equivalence is restored if the metric tensor is conformally transformed from the Jordan to the Einstein frame, in which F(R) gravity turns into general relativity with a scalar field. This peculiar behavior of general relativity, among relativistic theories of gravitation, with respect to purely affine, metric-affine and purely metric variation could indicate the physicality of the Einstein frame. On the other hand, it could explain why this theory cannot interpolate among phenomenological behaviors at different scales.