SpinWeightedSpheroidalHarmonics.jl

SpinWeightedSpheroidalHarmonics.jl computes spin-weighted spheroidal harmonics and eigenvalues using a spectral decomposition method. As a natural by-product, it also computes spherical-spheroidal mixing coefficients.

The two main features are implemented as

• spin_weighted_spheroidal_harmonic for computing the harmonic $\,{}_{s} S_{\ell m}(\theta, \phi; c \equiv a \omega)$, and
• spin_weighted_spheroidal_eigenvalue for computing the eigenvalue and both supporting complex spheroidicity $c$ (and hence complex frequency $\omega$).

See Quick-start below for some simple examples.

In particular, we use the following normalization convention:

    \int_0^{\pi} \left[ _{s} S_{\ell m}(\theta; c) \right]^{2} \sin(\theta) \, d\theta = \frac{1}{2\pi} \; ,

identical to the convention used in the Mathematica package SpinWeightedSpheroidalHarmonics from the Black Hole Perturbation Toolkit.

Additionally, we provide two similar functions

• spin_weighted_spherical_harmonic $\,{}_{s} Y_{\ell m}(\theta, \phi)$, and
• spin_weighted_spherical_eigenvalue

that return the exact harmonic and eigenvalue respectively.

Exact partial derivatives (with respect to either $\theta$ and/or $\phi$) can be evaluated by specifying the derivative order with theta_derivative and phi_derivative respectively when calling the functions for a harmonic.

Installation

To install the package using the Julia package manager, simply type the following in the Julia REPL:

using Pkg
Pkg.add("SpinWeightedSpheroidalHarmonics")

Quick-start

Computing the spin-weighted spheroidal eigenvalue

For example, to compute the spin-weighted spheroidal eigenvalue $\lambda$ for the mode $s = -2, \ell = 2, m = 2, a = 0.7, \omega = 0.5$, simply do

using SpinWeightedSpheroidalHarmonics
s=-2; l=2; m=2; a=0.7; omega=0.5;
spin_weighted_spheroidal_eigenvalue(s, l, m, a*omega)

Computing the spin-weighted spheroidal harmonic

For example, to compute the spin-weighted spheroidal harmonic for the mode $s = -2, \ell = 2, m = 2, a = 0.7, \omega = 0.5$ at $\theta = \pi/6, \phi = \pi/3$, simply do

using SpinWeightedSpheroidalHarmonics
s=-2; l=2; m=2; a=0.7; omega=0.5;
theta=π/6; phi=π/3;
# Construct the SpinWeightedSpheroidalHarmonicFunction
swsh = spin_weighted_spheroidal_harmonic(s, l, m, a*omega)
swsh(theta, phi)

Computing the spherical-spheroidal mixing coefficients

For example, to compute the spherical-spheroidal mixing coefficients for the mode $s = -2, \ell = 2, m = 2, a = 0.7, \omega = 0.5$, simply do

using SpinWeightedSpheroidalHarmonics
s=-2; l=2; m=2; a=0.7; omega=0.5;
# Construct the SpinWeightedSpheroidalHarmonicFunction
swsh = spin_weighted_spheroidal_harmonic(s, l, m, a*omega)
# swsh.spherical_harmonics_l lists which spherical harmonic \ell modes were used in the decomposition
# swsh.coeffs lists the mixing coefficients for each of the modes
swsh.spherical_harmonics_l, swsh.coeffs

How to cite

If you have used this code in your research that leads to a publication, please cite the following article:

@article{Lo:2023fvv,
    author = "Lo, Rico K. L.",
    title = "{Recipes for computing radiation from a Kerr black hole using Generalized Sasaki-Nakamura formalism: I. Homogeneous solutions}",
    eprint = "2306.16469",
    archivePrefix = "arXiv",
    primaryClass = "gr-qc",
    month = "6",
    year = "2023"
}

License

The package is licensed under the MIT License.