The Newtonian Limit of F(R) gravity

TL;DR

This paper develops an analytic method to analyze the Newtonian limit of $f(R)$ gravity, comparing it with the post-Newtonian limit, and explores the implications for gravitational potentials, Birkhoff theorem, and gravitational waves.

Contribution

It provides a general analytic framework for the Newtonian and post-Newtonian limits of $f(R)$ gravity without redefining degrees of freedom or changing frames.

Findings

The gravitational potential in $f(R)$ gravity is always corrected relative to the Newtonian case.

Birkhoff theorem does not hold generally in $f(R)$ gravity, allowing time-dependent spherically symmetric solutions.

Massive gravitational wave solutions can emerge in the post-Minkowskian limit.

Abstract

A general analytic procedure is developed to deal with the Newtonian limit of $f(R)$ gravity. A discussion comparing the Newtonian and the post-Newtonian limit of these models is proposed in order to point out the differences between the two approaches. We calculate the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom, in particular, without adopting some scalar fields and without any change from Jordan to Einstein frame. Considering the Taylor expansion of a generic $f(R)$ theory, it is possible to obtain general solutions in term of the metric coefficients up to the third order of approximation. In particular, the solution relative to the $g_{tt}$ component gives a gravitational potential always corrected with respect to the Newtonian one of the linear theory $f(R)=R$. Furthermore, we show that the Birkhoff theorem is not a general result for $f(R)$-gravity since time-dependent evolution for spherically symmetric solutions can be achieved depending on the order of perturbations. Finally, we discuss the post-Minkowskian limit and the emergence of massive gravitational wave solutions.