Exact solutions in metric $f(R)$-gravity for static axisymmetric spacetime

TL;DR

This paper derives explicit static axisymmetric vacuum solutions in metric $f(R)$ gravity, including the Schwarzschild solution, by expressing $f(R)$ in terms of Weyl coordinates and analyzing the modified field equations.

Contribution

It provides a general explicit form of $f(R)$ related to Weyl coordinates and solves the modified field equations for specific cases like the Newtonian potential and Schwarzschild spacetime.

Findings

Explicit $f(R)$ dependence on $r$ and $z$ coordinates.

Solutions for Newtonian potential in $f(R)$ gravity.

Schwarzschild solution in $f(R)$ gravity.

Abstract

Axially symmetric static vacuum exact solution (ASSVES) in Weyl coordinates are studied in $f(R)$ theories of gravity. In particular, we obtain a general explicit expression for the dependence between $df(R)/dR$ and the $r$ and $z$ coordinates and then the corresponding explicit form of $f(R)$ is obtained. Next, we present in detail the explicit solution of the modified field equations corresponding to the Newtonian potential due to a punctual mass placed at the origin of coordinates and also the Schwarzschild solution to the modified field equations.