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

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 $f(R)\ne R$, this solution corresponds to a naked singularity.

To be published in Physics Letters B. Replaced with minor changes to match published version

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• On static and spherically symmetric solutions of Starobinsky model
• Charged spherically symmetric Taub-NUT black hole solutions in $f(R)$ gravity
• Phase transitions in geometrothermodynamic model of charged generalized-NUT black holes
• Scheme of integration of vacuum $F(R)$ gravity in a travelling wave variable