Newtonian limit of $f(R)$ gravity

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 terms 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.