Teukolsky

Overview

The Teukolsky package computes solutions to the Teukolsky equation, which governs perturbations of the Kerr spacetime for fields of spin-weight s (scalar, electromagnetic and gravitational). It provides the homogeneous radial solutions — the “In” and “Up” functions defined by their boundary conditions at the horizon and at infinity — and point-particle mode data such as amplitudes and energy fluxes for a source on a bound geodesic.

It builds on two other Toolkit packages: SpinWeightedSpheroidalHarmonics for the angular dependence, and KerrGeodesics for the orbital motion of the source.

Further details are available in the source repository and the package page.

Installation

Teukolsky is distributed as a paclet. Once the BHPToolkit paclet server is set up (see Get started), install it by name and load it:

PacletInstall["Teukolsky"]

Usage

Load the package:

<< Teukolsky`

Examples

The package computes solutions to:

where

$\Delta = r^2 - 2Mr + a^2$
$K=(r^2 + a^2)\omega - a m$
$s$ is the spin-weight of the perturbing field
$\lambda$ is the spin-weighted spheroidal eigenvalue
$\omega$ is the mode frequency
$\mathcal{T}$ is the source

The source has been implemented for a point particle moving along a generic bound orbit in Kerr spacetime for perturbations with spin-weight $s=\{0,\pm 1, \pm 2\}$. As an example, the gravitational wave flux for the $l=2,m=2$ mode for a circular, equatorial orbit is easily computed using:

With[{a = 0.9, p = 10.0, e=0, x=1, s = -2, l = 2, m = 2},
  orbit = KerrGeoOrbit[a, p, e, x];
  ψ4 = TeukolskyPointParticleMode[s, l, m, orbit];
  ψ4["Fluxes"]
]

This returns an association with the results:

<|"Energy" -> <|"ℐ" -> 0.000022273, "ℋ" -> -5.9836*10^-8|>,
  "AngularMomentum" -> <|"ℐ" -> 0.00072438, "ℋ" -> -1.94603*10^-6|>|>

Homogeneous solutions

The homogeneous solutions are also easily computed. They can be extracted from the ψ4 object above using R = ψ4["RadialFunctions"]. This returns a pair TeukolskyRadialFunction objects which can be evaluated at a given radius, i.e., R["In"][10.]. The homogeneous solutions can also be computed directly via the TeukolskyRadial[s, l, m, a, ω] function.

Renormalized angular momentum

Under the hood the Teukolsky package uses the MST method for computing homogeneous solutions. A key part of the MST method is the calculation of the renormalized angular momentum, $\nu$. This can be computed directly via

 ν = RenormalizedAngularMomentum[s, l, m, a, ω]

Compute a homogeneous radial solution and evaluate it at a Boyer-Lindquist radius:

Needs["Teukolsky`"]

s = -2; l = 2; m = 2; a = 0.5; ω = 0.3;
R = TeukolskyRadial[s, l, m, a, ω];

R["In"][10.0]
R["Up"]["RadialDerivative"][10.0]

For a point particle on a bound orbit, build the orbit with KerrGeodesics and pass it to TeukolskyPointParticleMode to obtain the mode amplitudes and fluxes:

Needs["KerrGeodesics`"]

orbit = KerrGeoOrbit[a, 10.0, 0, 1];           (* circular, equatorial *)
mode  = TeukolskyPointParticleMode[s, l, m, 0, 0, orbit];

mode["Fluxes"]
mode["Amplitudes"]

Further examples

See the Mathematica Documentation Centre for a tutorial and documentation on individual functions.

Authors and contributors

Barry Wardell, Niels Warburton, Marc Casals, Adrian Ottewill, Chris Kavanagh, Leanne Durkan, Ben Leather, Theo Torres, Jakob Neef

Citing Teukolsky

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

Teukolsky also requests the following citation:

• Teukolsky

See how to cite for further guidance.