# Teukolsky

Install this package!

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 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 = 0.932;
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" -> 0.000022273000551180497057, "FluxHor" -> -5.9836792133833631984*10^-8, "FluxTotal" -> 0.000022213163759046663425|>


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.

## Authors and contributors

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