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

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.

💡 Research Summary

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In this paper the authors investigate the weak‑field (Newtonian) limit of generic analytic f(R) gravity theories within the metric (Jordan‑frame) approach. Starting from the fourth‑order action
(A=\int d^{4}x\sqrt{-g},