On the equivalence between $ f\left(R\right) $ theories and Einstein gravity

TL;DR

This paper demonstrates that perturbative off-shell $f(R)$-theories are equivalent to Einstein gravity, Brans-Dicke, and Einstein scalar field models, with potential extensions to higher-order gravity models.

Contribution

It establishes a trivial equivalence between off-shell $f(R)$-theories and Einstein gravity, providing insights into their relationships and possible generalizations.

Findings

Off-shell $f(R)$-theories are equivalent to Einstein gravity.

The equivalence extends to Brans-Dicke and Einstein scalar field models.

Potential for generalizing to higher-order gravitational models.

Abstract

In this brief note we present a somewhat trivial result. Namely, we show that perturbative off-shell $f(R)$-theory is equivalent to Einstein gravity, as well as to the Brans-Dicke theory and the Einstein scalar field model. We also discuss possible generalisation of this result to higher-order gravitational field models.