The Sasaki-Nakamura equation is an equation for studying perturbation of rotating black holes. It is a recasting of the Teukolsky equation to make it more amenable to numerical treatment. Previous authors had used only the leading or a few terms in the boundary condition expansions. For our work on rapidly rotating black holes this turned out to not be sufficient so I derived the full recursion relations which allow arbitrary number of terms to be calculated. The results are given in a compressed tar file which contains the recursion relations and a Mathematica notebook which demonstrates they satisfy the Sasaki-Nakamura equation. If you make use of these formula in your work please cite Phys. Rev. D 92:064029, arXiv:1506.08496.

Some details are given in the notebook in the tarball. I repeat the salient details now. The Sasaki-Nakamura equation takes the form:

\begin{align}

\frac{dX}{dr_*^2} -F(r)\frac{dX}{dr_*} – U(r) X(r) = 0

\end{align}

where \(r_*\) is the tortoise coordinate defined by \(dr_*/dr = (r^2+a^2)/\Delta\) with \(\Delta = r^2 – 2Mr +a^2\). The functions \(F(r)\) and \(U(r)\) are rather unwieldy and can be found in the Appendix B of Hughes 2000.

Asymptotically, as the horizon and spatial infinity are approached, the `outer’ and `inner’ radial solutions behave as

\begin{align}

X^\infty(r_*\rightarrow\infty) &\sim e^{i\omega r_*}, \\

X^H(r_*\rightarrow-\infty) &\sim e^{-i(\omega – m \Omega_H) r_*}, \label{eq:horiz_asymp_form}

\end{align}

respectively. In our numerical procedure we must work on a finite radial domain. Let us denote the boundaries of this domain by \(r_\text{in}\) and \(r_\text{out}\). In order to construct suitable boundary conditions at \(r_\text{in/out}\) we expand the above asymptotic forms with the following ansatz

\begin{align}

X^\infty &= e^{i\omega r_*}\sum_{k=0}^{k^\infty_\text{max}} a^\infty_k (\omega r_\text{out})^{-k}, \label{eq:inf_expansion}\\

X^H &= e^{-i(\omega – m \Omega_H) r_*}\sum_{k=0}^{k^H_\text{max}} a^H_k (r_\text{in} – r_+)^k .

\end{align}

The \(a_k^{\infty/H}\) coefficients of these series expansions are determined by substituting the expansions into the Sasaki-Nakamura equation and solving for the resulting recursion relations.

The resulting relations are pretty unwieldy, particularly the horizon recursion relations. No doubt they could be simplified further but they were sufficient in the form they are in for our purposes so I didn’t pursue this.