# Quasi-normal modes of non-separable perturbation equations: the scalar non-Kerr case

## TL;DR

This paper presents a method to approximate quasi-normal mode frequencies for non-separable perturbation equations near Kerr black holes, simplifying calculations in modified gravity theories by reducing the problem to an ordinary differential equation.

## Contribution

It introduces a leading-order approximation technique for computing quasi-normal modes in non-Kerr geometries, applicable when deviations from Kerr are small.

## Key findings

- Allows calculation of quasi-normal modes via ODE in near-Kerr spacetimes
- Applicable to scalar perturbations with anomalous quadrupole moments
- Simplifies analysis of black-hole perturbations in modified gravity theories

## Abstract

Scalar, vector and tensor perturbations on the Kerr spacetime are governed by equations that can be solved by separation of variables, but the same is not true in generic stationary and axisymmetric geometries. This complicates the calculation of black-hole quasi-normal mode frequencies in theories that extend/modify general relativity, because one generally has to calculate the eigenvalue spectrum of a two-dimensional partial differential equation (in the radial and angular variables) instead of an ordinary differential equation (in the radial variable). In this work, we show that if the background geometry is close to the Kerr one, the problem considerably simplifies. One can indeed compute the quasi-normal mode frequencies, at least at leading order in the deviation from Kerr, by solving an ordinary differential equation subject to suitable boundary conditions. Although our method is general, in this paper we apply it to scalar perturbations on top of a Kerr black hole with an anomalous quadrupole moment, or on top of a slowly rotating Kerr background.