Stability of Schwarzschild black hole in f(R) gravity with the dynamical Chern-Simons term

TL;DR

This paper investigates the stability of Schwarzschild black holes within a modified gravity framework that includes a dynamical Chern-Simons term, revealing stability conditions linked to scalar field properties.

Contribution

It introduces a combined scalar-tensor formulation of $f(R)$ gravity with a Chern-Simons term and analyzes the stability of black holes under this theory.

Findings

Schwarzschild black hole remains stable if the scalaron is free from tachyonic instabilities.

The Chern-Simons coupling influences odd-parity perturbations, while $f(R)$ gravity does not affect even-parity perturbations.

Stability depends on the scalaron mass being non-tachyonic.

Abstract

We perform the stability analysis of the Schwarzschild black hole in $f(R)$ gravity with the parity-violating Chern-Simons (CS) term coupled to a dynamical scalar field $\theta$. For this purpose, we transform the $f(R)$ gravity into the scalar-tensor theory by introducing a scalaron $\phi$, providing the dynamical Chern-Simons modified gravity with two scalars. The perturbation equation for the scalar $\theta$ is coupled to the odd-parity metric perturbation equation, providing a system of two coupled second order equations, while the scalaron is coupled to the even-parity perturbation equation. This implies that the CS coupling affects the Regge-Wheeler equation, while $f(R)$ gravity does not affect the Zerilli equation. It turns out that the Schwarzschild black hole is stable against the external perturbations if the scalaron is free from the tachyon.