Analytically Approximation Solution to $R^{2}$ Gravity

TL;DR

This paper derives analytical approximate black hole solutions in $f(R)$ gravity without a cosmological constant, using a continued-fraction expansion and analyzing their stability through thermodynamics and quasi-normal modes.

Contribution

It introduces a novel analytical approximation method for black hole solutions in $f(R)$ gravity without constraints on Ricci scalar.

Findings

Derived near horizon and asymptotic solutions

Constructed complete solutions via continued-fraction expansion

Analyzed stability through thermodynamics and quasi-normal modes

Abstract

In this paper, we obtain analytical approximate black hole solutions in the framework of $f(R)$ gravity and the absence of a cosmological constant. In this area, we apply the equations of motion of the theory to a spherically symmetric spacetime with one unknown function and derive black hole solutions without any constraints on the Ricci scalar. To do so, first, we obtain the near horizon and asymptotic solutions and then use both of them to obtain a complete solution by utilizing a continued-fraction expansion. Finally, we investigate the stability of the solutions by employing the thermodynamics and quasi-normal modes.