Teukolsky

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A Mathematica package for computing solutions to the Teukolsky equation. Note this package depends upon the SpinWeightedSpheroidalHarmonics and the KerrGeodesics package to run.

Explicitly the package computes solutions to:

\[\Delta^{-s} \dfrac{d}{dr} \bigg[\Delta^{s+1}\dfrac{d R}{dr}\bigg] + \bigg[\frac{K^2 - 2 i s (r-M)K}{\Delta} + 4 i s \omega r - \lambda \bigg]R = \mathcal{T} \nonumber\]

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

Currently 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]

Further examples

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

Citing

In addition to acknowledging the Black Hole Perturbation Toolkit as suggested on the front page we also recommend citing the specific package version you use via the citation information on the package’s Zenodo page linked from the above DOI.

Authors and contributors

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

Black Hole Perturbation Toolkit