The Post-Newtonian Limit of f(R)-gravity in the Harmonic Gauge

TL;DR

This paper develops a general analytic method to analyze the post-Newtonian limit of f(R) gravity in the harmonic gauge, providing solutions up to order (v/c)^4 and exploring deviations from General Relativity.

Contribution

It introduces a perturbative scheme for f(R) gravity solutions using Green functions and parameterizes them by derivatives of f, extending the understanding of gravitational corrections.

Findings

Derived solutions up to (v/c)^4 order for f(R) gravity

Calculated Yukawa and oscillating corrections to gravitational potential

Showed convergence to General Relativity when f approaches R

Abstract

A general analytic procedure is developed for the post-Newtonian limit of $f(R)$-gravity with metric approach in the Jordan frame by using the harmonic gauge condition. In a pure perturbative framework and by using the Green function method a general scheme of solutions up to $(v/c)^4$ order is shown. Considering the Taylor expansion of a generic function $f$ it is possible to parameterize the solutions by derivatives of $f$. At Newtonian order, $(v/c)^2$, all more important topics about the Gauss and Birkhoff theorem are discussed. The corrections to "standard" gravitational potential ($tt$-component of metric tensor) generated by an extended uniform mass ball-like source are calculated up to $(v/c)^4$ order. The corrections, Yukawa and oscillating-like, are found inside and outside the mass distribution. At last when the limit $f\rightarrow R$ is considered the $f(R)$-gravity converges in General Relativity at level of Lagrangian, field equations and their solutions.