Angular Teukolsky equation#
The spin-weighted spheroidal harmonics \(S_{sjm\gamma}(\theta)\) satisfy the equation
\[
\left[\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta \frac{d}{d\theta} \right)
- \left(\gamma^2\sin^2\theta+\frac{(m+s\cos\theta)^2}{\sin^2\theta}
+2\gamma s\cos\theta-s-2m\gamma-\lambda_{sjm\gamma} \right)\right]S_{sjm\gamma} = 0,
\]
with eigenvalues \(\lambda_{sjm\gamma}\) and spheroidicity \(\gamma\). For the Teukolsky equation, \(\gamma = a\omega\).
The spheroidal harmonics can be expressed as rapidly convergent sums of spin-weighted spherical harmonics,
\[
S_{sjm\gamma}(\theta)e^{im\phi} = \sum_{\ell = \ell_\mathrm{min}}^\infty b^{\ell}_{sjm\gamma} Y_{s\ell m}(\theta,\phi),
\]
where \(\ell_\mathrm{min} = \mathrm{max}[|m|, |s|]\) and the coupling coefficients \(b^{\ell}_{sjm\gamma}\) satisfy a five-term recursion relation. Similarly we can represent spin-weighted spherical harmonics as finite series of scalar spherical harmonics \(Y_{lm}=Y_{0lm}\)
\[
Y_{s\ell m}(\theta)
= \sin^{-|s|}\theta \sum_{l=|m|}^\infty \mathcal{A}^l_{s\ell m}
Y_{lm}(\theta),
\]
where the coefficients are given analytically in terms of the Wigner \(3j\) symbol,
\[\begin{split}\begin{align}
\mathcal{A}^l_{s\ell m} &=
(-1)^{m+s(1+\mathrm{sgn}(s))/2} \mathcal{C}_{|s|\ell l}
\left(
\begin{array}{ccc}
|s| & \ell & l
\\
0 & m & -m
\end{array}
\right)
\left(
\begin{array}{ccc}
|s| & \ell & l
\\
s & -s & 0
\end{array}
\right),
\\
\mathcal{C}_{s\ell l} &= \sqrt{\frac{4^{s} (s!)^2 (2\ell+1)(2l+1)}{(2s)!}}.
\end{align}\end{split}\]
The selection rules of the \(3j\)-symbol mean that the coefficients vanish unless \(\ell - |s| \leq l \leq \ell + |s|\).