Generation of spherically symmetric metrics in $f\left( R\right) $ gravity

TL;DR

This paper develops a method to generate spherically symmetric solutions in $f(R)$ gravity by reducing complex differential equations to simpler forms, enabling the discovery of new black hole solutions across various dimensions.

Contribution

It introduces a novel reduction technique for $f(R)$ gravity equations using a generating function, facilitating the derivation of new solutions including black holes with matter sources.

Findings

Derived a reduction to second order equations in $f(R)$ gravity.

Generated new black hole solutions in arbitrary dimensions.

Extended analysis to include matter sources like monopoles and Maxwell fields.

Abstract

In $D-$dimensional spherically symmetric $f\left( R\right) $ gravity there are three unknown functions to be determined from the fourth order differential equations. It is shown that the system remarkably integrates to relate two functions through the third one to provide reduction to second order equations accompanied with a large class of potential solutions. The third function which acts as the generator of the process is $F\left( R\right) =\frac{df\left( R\right) }{dR}.$ We recall that our generating function has been employed as a scalar field with an accompanying self-interacting potential previously which is entirely different from our approach. Reduction of $f\left( R\right) $ theory into system of equations seems to be efficient enough to generate a solution corresponding to each generating function. As particular examples, besides known ones, we obtain new black hole solutions in any dimension $D$. We further extend our analysis to cover non-zero energy-momentum tensors. Global monopole and Maxwell sources are given as examples.