# Gravitational perturbations of a Kerr black hole in $f(R)$ gravity

## TL;DR

This paper investigates how Kerr black holes respond to perturbations within $f(R)$ gravity theories, revealing that the perturbation equations decouple similarly to general relativity, which aids in understanding gravitational responses in modified gravity.

## Contribution

It demonstrates that the linear perturbation equations for Kerr black holes in $f(R)$ gravity decouple, extending the analysis of black hole stability to modified theories of gravity.

## Key findings

- Perturbation equations in $f(R)$ gravity decouple similarly to GR.
- Kerr black holes in $f(R)$ theories exhibit comparable gravitational responses.
- The quadratic term of $f(R)$ influences the perturbation structure.

## Abstract

Modified theories of gravity are often built such that they contain general relativity as a limiting case. This inclusion property implies that the Kerr metric is common to many families of theories. For example, all analytic $f(R)$ theories with vanishing constant term admit the Kerr solution. In any given theory, however, the response of the gravitational field to astrophysical disturbances is tied to the structure of the field equations. As such, even if black holes are Kerr, the underlying theory can, in principle, be probed through gravitational distortions. In this paper, we study linear perturbations of a Kerr black hole in $f(R)$ gravity using the Newman-Penrose formalism. We show that, as in general relativity, the equations governing the perturbed metric, which depend on the quadratic term of the function $f$, completely decouple.