Spherical symmetry in $f(R)$-gravity

TL;DR

This paper explores spherically symmetric solutions in $f(R)$ gravity, analyzing their relation to weak field limits, and provides exact and perturbed solutions for various $f(R)$ models, connecting them with General Relativity.

Contribution

It offers new exact and perturbative solutions for spherically symmetric $f(R)$ gravity models, clarifying their connection to GR limits.

Findings

Exact solutions for constant Ricci scalar $R$

Solutions for $R$ depending on radius $r$

Perturbative solutions up to first order

Abstract

Spherical symmetry in $f(R)$ gravity is discussed in details considering also the relations with 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 $R = R(r)$.