A static axisymmetric exact solution of $f(R)$-gravity

TL;DR

This paper derives an exact static, axisymmetric vacuum solution in $f(R)$ gravity, analyzes its properties, and shows it features a naked singularity when deviating from Einstein's General Relativity.

Contribution

It provides a new explicit exact solution in $f(R)$ gravity and explores its geometric and physical properties, including the nature of singularities.

Findings

The solution generalizes the Schwarzschild metric in $f(R)$ gravity.

The explicit form of $f(R)$ consistent with the solution is derived.

The solution exhibits a naked singularity for $f(R) eq R$.

Abstract

We present an exact, axially symmetric, static, vacuum solution for $f(R)$ gravity in Weyl's canonical coordinates. We obtain a general explicit expression for the dependence of $df(R)/dR$ upon the $r$ and $z$ coordinates and then the corresponding explicit form of $f(R)$, which must be consistent with the field equations. We analyze in detail the modified Schwarzschild solution in prolate spheroidal coordinates. Finally, we study the curvature invariants and show that, in the case of $f(R)\neq R$, this solution corresponds to a naked singularity.