A general solution in the Newtonian limit of f(R)- gravity

TL;DR

This paper demonstrates that in metric f(R)-gravity, the weak field limit universally includes a Yukawa-like correction to the Newtonian potential, highlighting that only f(R)=R reproduces standard gravity.

Contribution

It provides a general proof that all analytic f(R) models in the metric approach have a Yukawa correction in the Newtonian limit, a result previously known only for specific cases.

Findings

Standard Newtonian potential is recovered only for f(R)=R.

All other analytic f(R) models exhibit Yukawa-like corrections.

Implications for physical tests of gravity theories are discussed.

Abstract

We show that any analytic $f(R)$-gravity model, in the metric approach, presents a weak field limit where the standard Newtonian potential is corrected by a Yukawa-like term. This general result has never been pointed out but often derived for some particular theories. This means that only $f(R)=R$ allows to recover the standard Newton potential while this is not the case for other relativistic theories of gravity. Some considerations on the physical consequences of such a general solution are addressed.