Radial Teukolsky equation#
The radial Teukolsky equation for spin-weight \(s\), harmonic numbers \((j,m)\), frequency \(\omega\), and \(a\) the dimensionless Kerr spin parameter is given by
\[\left[\Delta^{-s} \frac{d}{dr} \left(\Delta^{s+1} \frac{d }{dr} \right) + \left(\frac{K^2-2is(r-M)K}{\Delta}+4is\omega r - \lambda_{sjm\omega} \right)\right]R_{sjm\omega} = T_{slm\omega},\]
where \(\Delta=r^2-2Mr+a^2\), \(K=(r^2+a^2)\omega-ma\), \(\lambda_{sjm\omega}\) is the spheroidal eigenvalue (separation constant), and \(T_{slm\omega}\) is the radial decomposition of the source.
Homogeneous solutions#
For \(T_{slm\omega} = 0\), we construct the homogeneous solutions \(R^\mathrm{In}_{sjm\omega}\) and \(R^\mathrm{Up}_{sjm\omega}\), which correspond to the asymptotic boundary conditions
\[\begin{split}\begin{align}
R^\mathrm{In}_{sjm\omega} (r \rightarrow r_+) &\sim A^\mathrm{trans}_s \Delta^{-s} e^{-i k r_*},
\\
R^\mathrm{In}_{sjm\omega} (r \rightarrow \infty) &\sim A^\mathrm{ref}_s r^{-(2s+1)} e^{i\omega r_*}
+ A^\mathrm{inc}_s r^{-1} e^{-i\omega r_*},
\\
R^\mathrm{Up}_{sjm\omega} (r \rightarrow r_+) &\sim B^\mathrm{ref}_s \Delta^{-s} e^{-i k r_*}
+ B^\mathrm{inc}_s e^{i k r_*},
\\
R^\mathrm{Up}_{sjm\omega} (r \rightarrow \infty) &\sim
B^\mathrm{trans}_s r^{-(2s+1)} e^{i\omega r_*},
\end{align}\end{split}\]
where \(k = \omega - m \omega_+\), \(\omega_+ = a/(2Mr_+)\), and the tortoise coordinate is given by the differential relation \(dr_{*}/dr = (r^2+a^2)/\Delta\), which can be immediately integrated, leading to
\[r_* = r + \frac{r_+}{\kappa} \ln \frac{r-r_+}{2M} - \frac{r_-}{\kappa} \ln \frac{r-r_-}{2M},\]
where \(\kappa = \sqrt{1 - q^2}\) and \(q = a/M\).
Inhomogeneous solutions#
Rather than constructing the full inhomogeneous solutions, we can instead construct the so-called extended homogeneous solutions for a point-particle source on a bound periodic geodesic,
\[\begin{split}\begin{align}
\Psi_s &= \Psi_s^+ \Theta(r-r_p) + \Psi_s^- \Theta(r_p-r),
\\
\Psi_s^\pm &= \sum_{jmkn}\Psi^\pm_{sjmkn}(r)S_{sj\gamma_{mkn}}(\theta)e^{im\phi}e^{-i\omega_{mkn}t},
\end{align}\end{split}\]
where we have the mode frequencies \(\omega_{mkn} = m\Omega_\phi + k \Omega_\theta + n \Omega_r\), the discrete spheroidicity \(\gamma_{mkn} = a\omega_{mkn}\), and the extended homogeneous radial solutions
\[
\Psi^\pm_{sjmkn}(r) = Z^{\mathrm{Up/In}}_{sjmkn} R^{\mathrm{Up/In}}_{sjmkn}(r),
\]
with Teukolsky amplitudes \(Z^{\mathrm{Up/In}}_{sjmkn}\).