Perturbative solutions of the $f(R)$-theory of gravity in a central gravitational field and some applications

TL;DR

This paper develops perturbative methods to find approximate solutions in non-static central gravitational fields within $f(R)$-gravity, and applies these to planetary motion, light propagation, and the SgrA*--S2 system, highlighting potential observable effects.

Contribution

It introduces a perturbative approach for solving $f(R)$-gravity in dynamic central fields, extending previous static solutions and enabling practical applications.

Findings

Vacuum solutions for specific $f(R)$ forms are derived.

Applications to planetary motion and light propagation are demonstrated.

Estimated effects on the SgrA*--S2 system suggest potential observability.

Abstract

Exact solutions of an $ f(R) $-theory (of gravity) in a static central (gravitational) field have been studied in the literature quite well, but, to find and study exact solutions in the case of a non-static central field are not easy at all. There are, however, approximation methods of finding a solution in a central field which is not necessarily static. It is shown in this article that an approximate solution of an $f(R)$-theory in a general central field, which is not necessary to be static, can be found perturbatively around a solution of the Einstein equation in the general theory of relativity. In particular, vacuum solutions are found for $f(R)$ of general and some special forms. Further, applications to the investigation of a planetary motion and light's propagation in a central field are presented. An effect of an $f(R)$-gravity is also estimated for the SgrA*--S2 system. The latter gravitational system is much stronger than the Sun--Mercury system, thus the effect could be much stronger and, thus, much more measurable.