QNM

Overview

QNM is a Mathematica package for computing quasinormal mode (QNM) solutions of the Teukolsky equation in Schwarzschild and Kerr spacetime. The quasinormal mode frequencies $\omega$ are those of the radial Teukolsky equation,

with $K=(r^2+a^2)\omega - a m$ and $\Delta = r^2 - 2 M r + a^2$, subject to quasinormal boundary conditions (ingoing at the horizon and outgoing at infinity). For $s = 0$ this is equivalent to the scalar wave equation in Kerr spacetime.

The package provides two main functions:

QNMFrequency[s, l, m, n, a]
QNMRadial[s, l, m, n, a]

where

• $s$ is the spin-weight of the field,
• $\ell$ is the multipolar index,
• $m$ is the azimuthal index,
• $n$ is the overtone number,
• $a$ is the black hole spin parameter.

Frequencies can be computed with a range of methods and evaluated to arbitrary numerical precision. QNM builds on the SpinWeightedSpheroidalHarmonics and Teukolsky packages, and supersedes the older QuasiNormalModes package, which is no longer maintained. For a Python package for Kerr quasinormal modes see qnm.

Installation

QNM is distributed as a paclet. Once the BHPToolkit paclet server is set up (see Get started), install it together with its dependencies:

PacletSiteRegister["https://pacletserver.bhptoolkit.org", "Black Hole Perturbation Toolkit Paclet Server"]
PacletSiteUpdate["https://pacletserver.bhptoolkit.org"]
PacletInstall["SpinWeightedSpheroidalHarmonics"]
PacletInstall["Teukolsky"]
PacletInstall["QNM"]

Usage

Loading the package

<< QNM`

Quasinormal mode frequencies

Compute the quasinormal mode frequency for $s = -2$, $\ell = 2$, $m = 2$, $n = 0$ and $a = 0.3$:

QNMFrequency[-2, 2, 2, 0, 0.3]

For numerical values of $a$, QNMFrequency aims to produce a result with the maximum precision allowed by the precision of $a$. For example, evaluate the frequency to higher accuracy with

QNMFrequency[-2, 2, 2, 0, 0.3`64, Method -> "IncidenceAmplitude"]

Methods

The Method option selects the algorithm used:

If no initial guess is given, the "SpheroidalEigenvalue" and "IncidenceAmplitude" methods use the "Interpolation" result as their initial guess, falling back to the asymptotic expansions if no tabulated data is available.

Sub-options can be passed by giving the method as a list:

• "SpheroidalEigenvalue": "NumPoints" (size of the spectral grid), "InitialGuess" and "AccuracyCheck" (True by default; False for faster performance);
• "IncidenceAmplitude": "InitialGuess";
• "Interpolation": the options of Mathematica’s Interpolation;
• "Spectral2D": "NumAngularPoints" and "NumRadialPoints".

QNMFrequency[-2, 2, 2, 0, 0.3, Method -> {"SpheroidalEigenvalue", "NumPoints" -> 32, "InitialGuess" -> 0.4 - 0.08I, "AccuracyCheck" -> False}]
QNMFrequency[-2, 2, 2, 0, 0.3, Method -> {"IncidenceAmplitude", "InitialGuess" -> 0.4 - 0.08I}]
QNMFrequency[-2, All, 2, All, 0.3, Method -> {"Spectral2D", "NumAngularPoints" -> 12, "NumRadialPoints" -> 12}]

Radial eigenfunctions

QNMRadial[s, l, m, n, a] computes the radial eigenfunction of a quasinormal mode, returning a QNMRadialFunction that stores the solution along with the frequency, the spheroidal eigenvalue, the mode numbers and the asymptotic amplitudes at the horizon and infinity. Solutions are normalised so that the amplitude at the horizon is 1.

ψ = QNMRadial[-2, 2, 2, 0, 0.3];
ψ[10]
ψ["Amplitudes"]

Solutions are computed with a spectral discretisation on a hyperboloidal slice (Method -> "Hyperboloidal", the only method currently supported). Its sub-options are "NumPoints" and "Coordinates", which can be "Hyperboloidal" (the default), "CompactifiedHyperboloidal" or "Boyer-Lindquist":

QNMRadial[-2, 2, 2, 0, 0.3, Method -> {"Hyperboloidal", "NumPoints" -> 100, "Coordinates" -> "Boyer-Lindquist"}]

By default the frequency is computed with QNMFrequency; a precomputed value can be supplied with the "Frequency" option.

Examples

A mode as a function of spin

Plot the $(s, \ell, m, n) = (-2, 2, 2, 0)$ mode frequency as a function of $a$:

ParametricPlot[
  ReIm[QNMFrequency[-2, 2, 2, 0, a, Method -> {"SpheroidalEigenvalue", "AccuracyCheck" -> False}]], {a, 0, 0.9},
  AspectRatio -> 1/GoldenRatio, AxesLabel -> {"Re[ω]", "Im[ω]"}]

Near-extremal spins and higher modes

For very high spins the default options may not give an accurate result. In those cases use the "IncidenceAmplitude" method, or the "SpheroidalEigenvalue" method with a larger spectral grid:

QNMFrequency[-2, 2, 2, 0, 0.999, Method -> "IncidenceAmplitude"]
QNMFrequency[-2, 2, 2, 0, 0.999, Method -> {"SpheroidalEigenvalue", "NumPoints" -> 45}]

When no tabulated data is available for a mode, methods that need an initial guess fall back to asymptotic expansions, which may pick out a different mode than expected. A good initial guess can be supplied explicitly (though there is no guarantee that the mode found corresponds to the specified overtone number):

QNMFrequency[-2, 8, 8, 0, 0.999, Method -> {"SpheroidalEigenvalue", "InitialGuess" -> 3.5 - 0.01I}]

An unconventional mode

Find the “unconventional mode” of arXiv:2506.14635 with a high-precision initial guess:

With[{s = -2, l = 2, m = 2, a = 0.0``128, ωguess = 0.015`128 - 1.99`128 I},
  QNMFrequency[s, l, m, 0, a, Method -> {"IncidenceAmplitude", "InitialGuess" -> ωguess}]
]

Additional documentation, including reference pages for each function, is included in the Mathematica Documentation Centre.

Authors and contributors

Barry Wardell, Christiana Pantelidou, Brad Cownden, Jake Mac Uilliam, Conor O’Toole, Rodrigo Macedo, Jamil Assaad

Citing QNM

If you use QNM in your research, please acknowledge the Toolkit:

QNM also requests the following citations:

• QNM
  BibTeX

  @software{BHPToolkit:QNM,
    author       = {Wardell, Barry and Pantelidou, Christiana and Cownden, Brad and Mac Uilliam, Jake and O'Toole, Conor and Macedo, Rodrigo and Assaad, Jamil},
    title        = {QNM},
    month        = sep,
    year         = 2025,
    publisher    = {Zenodo},
    version      = {0.4.1},
    doi          = {10.5281/zenodo.17194969},
    url          = {https://doi.org/10.5281/zenodo.17194969},
  }

• Black Hole Perturbation Toolkit
  BibTeX

  @misc{BHPToolkit,
    title = {{Black Hole Perturbation Toolkit}},
    howpublished = {(\href{http://bhptoolkit.org/}{bhptoolkit.org})},
  }

See how to cite for further guidance.