On the "Legendre scalarization" of nonlinear gravity theories

TL;DR

This paper critically examines a new method for transforming f(R) gravity theories into Einstein-like equations, clarifying existing literature and highlighting issues with the proposed approach's equivalence to the original theory.

Contribution

The paper clarifies the proper use of Legendre transformations in f(R) gravity and critiques the recent proposal for its lack of equivalence to the original theory.

Findings

The Legendre transformation recasts f(R) gravity into Einstein equations without conformal rescaling.

The new method proposed does not produce equations equivalent to the original f(R) theory.

Existing literature already clarifies the transformation of f(R) theories into Einstein form.

Abstract

We discuss the proposal of a new method to transform a f(R) metric gravity theory into a general relativistic theory including an auxiliary scalar field, recently introduced by S. Cotsakis et al. We argue that (i) the fact that the fourth order equations of f(R) metric gravity can be recast (via a Legendre transformation) into Einstein equations without any conformal rescaling has been thoroughly clarified in the previous literature, and (ii) the newly proposed method produces a set of equations that are not equivalent to the original theory. In the conclusion, a comment is added on another aspect of the Legendre transformation which seems to be generally overlooked.