Teukolsky equation

In Boyer-Lindquist coordinates \((t,r,\theta,\phi)\), the Teukolsky equation takes the form

\[\begin{align} \mathcal{O}^T_s \Psi_s = -4\pi \zeta\bar{\zeta} T_s, \end{align}\]

where \(\zeta=r-ia\cos\theta\), \(T_s\) is the (tetrad-projected) source, and the spin-weighted Teukolsky operator is given by,

\[\begin{split}\begin{align} \mathcal{O}^T_s &:=-\left(\frac{(r^2+a^2)^2}{\Delta} - a^2 \sin^2\theta\right)\partial_t^2 - \frac{4 M a r}{\Delta} \partial_t \partial_\phi - \left(\frac{a^2}{\Delta} - \frac{1}{\sin^2\theta}\right)\partial_\phi^2 \\ & \qquad \qquad + \Delta^{-s}\partial_r\left(\Delta^{s+1} \partial_r\right) + \frac{1}{\sin\theta}\partial_\theta\left({\sin\theta}\partial_\theta \right) + 2s\left(\frac{M(r^2-a^2)}{\Delta} - r-ia\cos\theta \right)\partial_t \\ \notag & \qquad \qquad \qquad \qquad +2 s\left(\frac{a(r-M)}{\Delta} - \frac{i\cos\theta}{\sin^2\theta} \right)\partial_\phi - s\left(\frac{s\cos^2\theta}{\sin^2\theta} - 1\right), \end{align}\end{split}\]

with \(\Delta = r^2-2Mr+a^2\), and \(\Psi_{0}=\Phi\) for scalar perturbations; \(\Psi_{+1}=\phi_0\) and \(\Psi_{-1}=\zeta^2\phi_2\) ; and \(\Psi_{+2}=\psi_0\) and \(\Psi_{-2}=\zeta^4\psi_4\) for gravitational perturbations.

Transforming to the frequency domain, the fields and sources can be mode-decomposed into

\[\begin{split}\begin{align} \Psi_s = \int d\omega \sum_{jm} \Psi_{sjm\omega}(r) S_{sjm\gamma}(\theta) e^{im\phi} e^{-i\omega t}, \\ T_s = \int d\omega \sum_{jm} T_{sjm\omega}(r) S_{sjm\gamma}(\theta) e^{im\phi} e^{-i\omega t}, \end{align}\end{split}\]

with \(\gamma = a\omega\), leading to separated ODEs

\[\left[\Delta^{-s} \frac{d}{dr} \left(\Delta^{s+1} \frac{d }{dr} \right) + \left(\frac{K^2-2is(r-M)K}{\Delta}+4is\omega r - \lambda_{sjm\omega} \right)\right]R_{sjm\omega} = T_{slm\omega},\]

and

\[ \left[\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta \frac{d}{d\theta} \right) - \left(\gamma^2\sin^2\theta+\frac{(m+s\cos\theta)^2}{\sin^2\theta} +2\gamma s\cos\theta-s-2m\gamma-\lambda_{sjm\gamma} \right)\right]S_{sjm\gamma} = 0, \]

with eigenvalues \(\lambda_{sjm\gamma}\) and spheroidicity \(\gamma\).

The time-averaged rate of change of the orbital energy \(\langle \dot{E}\rangle\), angular momentum \(\langle \dot{L}_z\rangle\), and Carter constant \(\langle \dot{Q}\rangle\) can then be expressed in terms of the Teukolsky amplitudes \(Z^\mathrm{Up/In}_{sjmkn}\),

\[\begin{split} \begin{align} \langle \dot{\mathcal{J}} \rangle & = \langle \dot{\mathcal{J}} \rangle^\mathrm{inf} + \langle \dot{\mathcal{J}} \rangle^\mathrm{hor}, \\ &= \sum_{jmkn} \langle \dot{\mathcal{J}} \rangle^\mathrm{inf/hor}_{jmkn}, \\ &= \sum_{jmkn} \alpha_{sjmkn}^{(\mathcal{J})\mathrm{inf/hor}}\left| Z^\mathrm{Up/In}_{sjmkn} \right|^2, \end{align}\end{split}\]

where \(\mathcal{J} = (E, L_z, Q)\), \(\alpha_{sjmkn}^{(\mathcal{J})\mathrm{inf}/\mathrm{hor}}\) are known coefficients, and \(s=0, \pm 2\) produce either scalar or gravitational fluxes, respectively.