Comparing scalar-tensor gravity and f(R)-gravity in the Newtonian limit

TL;DR

This paper clarifies the Newtonian limit of $f(R)$-gravity, showing it differs from Brans-Dicke theory with $\,omega_{BD}=0$, and emphasizes the importance of scalar-tensor equivalence in understanding these models.

Contribution

It demonstrates that the Newtonian limit of $f(R)$-gravity is not equivalent to Brans-Dicke theory with $\,omega_{BD}=0$, highlighting the need for careful interpretation of PPN parameters.

Findings

The Newtonian limit of $f(R)$-gravity differs from Brans-Dicke with $\,omega_{BD}=0$.

Scalar-tensor equivalence reveals non-standard behavior in the Newtonian regime.

Reinterpretation of solutions in Jordan and Einstein frames links different formulations.

Abstract

Recently, a strong debate has been pursued about the Newtonian limit (i.e. small velocity and weak field) of fourth order gravity models. According to some authors, the Newtonian limit of $f(R)$-gravity is equivalent to the one of Brans-Dicke gravity with $\omega_{BD} = 0$, so that the PPN parameters of these models turn out to be ill defined. In this paper, we carefully discuss this point considering that fourth order gravity models are dynamically equivalent to the O'Hanlon Lagrangian. This is a special case of scalar-tensor gravity characterized only by self-interaction potential and that, in the Newtonian limit, this implies a non-standard behavior that cannot be compared with the usual PPN limit of General Relativity. The result turns out to be completely different from the one of Brans-Dicke theory and in particular suggests that it is misleading to consider the PPN parameters of this theory with $\omega_{BD} = 0$ in order to characterize the homologous quantities of $f(R)$-gravity. Finally the solutions at Newtonian level, obtained in the Jordan frame for a $f(R)$-gravity, reinterpreted as a scalar-tensor theory, are linked to those in the Einstein frame.