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 gtt 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.

Abstract. We search for spherically symmetric solutions of f (R) theories of gravity via the Noether Symmetry Approach. A general formalism in the metric framework is developed considering a point-like f (R) -Lagrangian where spherical symmetry is required. Examples of exact solutions are given.PACS numbers: 98.80.-k, 95.35.+x, 95.35.+d, 04.50.+h

We investigate the hydrostatic equilibrium of stellar structure by taking into account the modified Lané-Emden equation coming out from f (R)-gravity. Such an equation is obtained in metric approach by considering the Newtonian limit of f (R)-gravity, which gives rise to a modified Poisson equation, and then introducing a relation between pressure and density with polytropic index n. The modified equation results an integro-differential equation, which, in the limit f (R) → R, becomes the standard Lané-Emden equation. We find the radial profiles of gravitational potential by solving for some values of n. The comparison of solutions with those coming from General Relativity shows that they are compatible and physically relevant.

Spherical symmetry in f(R)-gravity is discussed in detail considering also the relations to the weak field limit. Exact solutions are obtained for constant Ricci curvature scalar and for Ricci scalar depending on the radial coordinate. In particular, we discuss how to obtain results which can be consistently compared with general relativity giving the well known post-Newtonian and post-Minkowskian limits. Furthermore, we implement a perturbation approach to obtain solutions up to the first order starting from spherically symmetric backgrounds. Exact solutions are given for several classes of f(R)-theories in both R= constant and R = R(r).

The Newtonian limit of the most general fourth-order gravity is performed with the metric approach in the Jordan frame with no gauge condition. The most general theory with fourth-order differential equations is obtained by generalizing the fðRÞ term in the action with a generic function containing two other curvature invariants: the Ricci square (R R ) and the Riemann square (R R ). The spherically symmetric solutions of the metric tensor present Yukawa-like spatial behavior, but now one has two characteristic lengths. At Newtonian order any function of curvature invariants gives us the same outcome like the so-called quadratic Lagrangian of gravity. From the Gauss-Bonnet invariant one will have an incomplete interpretation of the solutions and the absence of a possible third characteristic length linked to the Riemann square contribution. From the analysis of metric potentials, generated by a pointlike source, one has a constraint condition on the derivatives of f with respect to scalar invariants.