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:

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 circular orbit in Kerr spacetime. As an example, the flux in this case for the $l=2,m=2$ mode is easily computed using:

a = 0.9`32;
r0 = 10.`32;

orbit = KerrGeoOrbit[a, r0, 0, 1];

s = -2; l = 2; m = 2; n = 0; k = 0;
mode = TeukolskyPointParticleMode[s, l, m, n, k, orbit];

mode["Fluxes"]

This returns an association with the results:

<|"FluxInf" -> 0.000044546001102360994, "FluxHor" -> -1.1967358426766726*10^-7, "FluxTotal" -> 0.000044426327518093327|>

Note the high precision of the input values for $a$ and $r_0$. Currently this is often a requirement to get an accurate result.

Homogeneous solutions

The homogeneous solutions are also easily computed. They can be extracted from the mode object above using R = mode["Radial"]. This returns a TeukolskyRadialFunction[] which can be evaluated at a given radius, i.e., R[20.]. 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 defaults to using the MST method for computing the homogeneous solutions (Sasaki-Nakamura methods are in development). 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

Example notebooks can be found in the Mathematica Toolkit Examples repository.