A note on the linear stability of black holes in quadratic gravity

TL;DR

This paper investigates the linear stability of black holes in a specific scale-invariant quadratic gravity model, showing that unlike in general $f(R)$-gravity, these black holes are stable at the linear level.

Contribution

It demonstrates that black holes in the $f(R)=R^2$ gravity model are linearly stable, contrasting with the known instabilities in more general $f(R)$-gravity theories.

Findings

Static and stationary black holes are linearly stable in $f(R)=R^2$ gravity.

The instability mechanism present in general $f(R)$-gravity does not occur in this scale-invariant case.

Abstract

Black holes in $f(R)$-gravity are known to be unstable, especially the rotating ones. In particular, an instability develops that looks like the classical black hole bomb mechanism: the linearized modified Einstein equations are characterized by an effective mass that acts like a massive scalar perturbation on the Kerr solution in General Relativity, which is known to yield instabilities. In this note, we consider a special class of $f(R)$ gravity that has the property of being scale-invariant. As a prototype, we consider the simplest case $f(R)=R^2$ and show that, in opposition to the general case, static and stationary black holes are stable, at least at the linear level.