Spherically symmetric solution of $f(R,\mathcal{G})$ gravity at low energy

TL;DR

This paper develops the weak-field, slow-motion limit of $f(R, ext{G})$ gravity in a spherically symmetric setting, deriving vacuum solutions up to $(v/c)^4$ order and comparing static behaviors with general relativity and other modified theories.

Contribution

It provides the first detailed derivation of low-energy solutions in $f(R, ext{G})$ gravity up to fourth order, including static corrections and their comparison with $f(R)$ and general relativity.

Findings

Static Yukawa-like behavior is compatible with fourth-order theories.

At $(v/c)^4$ order, static corrections differ from general relativity.

Spatial behaviors at $(v/c)^2$ order are similar for $f(R, ext{G})$ and $f(R)$ gravity.

Abstract

The weak-field and slow-motion limit of $f(R,\mathcal{G})$ gravity is developed up to $(v/c)^{4}$ order in a spherically symmetric background. Considering the Taylor expansion of a general function $f$ around vanishing values of $R$ and $\mathcal{G}$, we present general vacuum solutions up to $(v/c)^{4}$ order for the gravitational field generated by a ball-like source. The spatial behaviors at $(v/c)^{2}$ order are the same for $f(R,\mathcal{G})$ gravity and $f(R)$ gravity, and their corresponding real valued static behaviors are presented and compared with the one in general relativity. The static Yukawa-like behavior is proved to be compatible with the previous result of the most general fourth-order theory. At $(v/c)^{4}$ order, the static corrections to the Yukawa-like behavior for $f(R,\mathcal{G})$ gravity, $f(R)$ gravity, and the Starobinsky gravity are presented and compared with the one in general relativity.