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Function: sw_spheroid()

void sw_spheroid(float *re, float *im, float mu, int reset,
                 float a, int s, int l, int m, float eigenvalues[])
This routine computes the spin-weighted spheroidal wave function  ${}_sS_{\ell m}(\mu)$. The arguments are:
re: Output. The real part of the spin-weighted spheroidal wave function.
im: Output. The imaginary part of the spin-weighted spheroidal wave function.
mu: Input. The independent variable, $\mu=\cos\iota$ with $\iota$ being a polar angle, of the spin-weighted spheroidal wave function; $-1<\texttt{mu}<1$.
reset: Input. A flag that indicates that the function should reset ( $\texttt{reset}=1$) the internally stored normalization of the spin-weighted spheroidal wave function. The reset flag should be set if any of the following five arguments are changed between calls; otherwise, set $\texttt{reset}=0$ so that the routine does not recompute the normalization.
a: Input. The dimensionless angular momentum parameter, $-1<\texttt{a}<1$, of the Kerr black hole for which the spin-weighted spheroidal wave function is associated.
s: Input. The integer-valued spin-weight $s$, which should be set to $0$ for a scalar perturbation (e.g., a scalar field perturbation), $\pm 1$ for a vector perturbation (e.g., an electromagnetic field perturbation), or $\pm2$ for a spin two perturbation (e.g., a gravitational perturbation).
l: Input. The mode integer  $\texttt{l}\ge\vert\texttt{s}\vert$.
m: Input. The mode integer  $\vert\texttt{m}\vert\le\texttt{l}$.
eigenvalues: Input. An array, eigenvalues[0..3], which contains the real and imaginary parts of the eigenvalues $\hat{\omega}$ and $A$ (see below) as follows: $\texttt{eigenvalues[0]}={\mathrm{Re}}\,\hat{\omega}$, $\texttt{eigenvalues[1]}={\mathrm{Im}}\,\hat{\omega}$, $\texttt{eigenvalues[2]}={\mathrm{Re}}\,A$, and  $\texttt{eigenvalues[3]}={\mathrm{Im}}\,A$. These may be calculated for a quasinormal mode using the routine qn_eigenvalues().

The spin-weighted spheroidal wave function is also computed using the method of Leaver [27]. We have adopted the following normalization criteria for the spin-weighted spheroidal wave functions  ${}_sS_{\ell m}(\mu)$. First, the angle-averaged value of the squared modulus of  ${}_sS_{\ell m}(\mu)$ is unity: $\int_{-1}^{1}\vert{}_sS_{\ell m}(\mu)\vert^2d\mu=1$. Second, the complex phase is partially fixed by the requirement that ${}_sS_{\ell m}(0)$ is real. Finally, the sign is set to be $(-)^{\ell-\max(m,s)}$ for the real part in the limit that $\mu\to-1$ in order to agree with the sign of the spin-weighted spherical harmonics  ${}_sY_{\ell m}(\mu,0)$ (see [26]).

It is sufficient to compute the spin-weighted spheroidal wave functions with $s<0$ and  $a\omega=\hat{a}\hat{\omega}\ge0$ because of the following symmetries [29]:

\begin{displaymath}
{}_{-s}S_{\ell m}(\mu,a\omega) = {}_sS_{\ell m}(-\mu,a\omeg...
...}} \quad
{}_{-s}E_{\ell m}(a\omega) = {}_sE_{\ell m}(a\omega)
\end{displaymath} (8.4.175)

and
\begin{displaymath}
{}_sS_{\ell m}(\mu,-a\omega) = {}_sS_{\ell,-m}(-\mu,a\omega...
...h}} \quad
{}_sE_{\ell m}(-a\omega) = {}_sE_{\ell,-m}(a\omega)
\end{displaymath} (8.4.176)

where ${}_sE_{\ell m}={}_sA_{\ell m}+s(s+1)$.

Author: Jolien Creighton, jolien@tapir.caltech.edu


next up previous contents
Next: Example: spherical program Up: GRASP Routines: Black hole Previous: Example: eigenvalues program   Contents
Bruce Allen 2000-11-19