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Example: eigenvalues program

This example uses the function qn_eigenvalues() to compute the eigenvalues ${}_s\hat{\omega}_{\ell m}$ and  ${}_sA_{\ell m}$ for the $s$ spin-weighted quasinormal mode specified by $\ell$ and $m$, and for a range of values of the dimensionless angular momentum parameter, $\hat{a}$. To invoke the program, type:

eigenvalues $s$ $\ell$ $m$
for the desired (integer) values of $s$, $\ell$, and $m$. Make sure that $\ell\ge\vert s\vert$ and $0\le m\le\ell$ (the eigenvalues for negative values of $m$ can be inferred from the symmetries discussed in subsection [*]). The output of the program is five columns of data: the first column is the value of $\hat{a}$ running from just greater than $-1$ to just less than $1$ (or between $0$ and $1$ if $m=0$), the second and third columns are the real and imaginary parts of the eigenfrequency $\hat{\omega}$, and the fourth and fifth columns are the real and imaginary parts of the angular separation eigenvalue $A$. For the values of $\hat{a}<0$, the eigenvalues correspond to the mode with azimuthal index $-m$ so that the real part of the eigenfrequency is positive. A plot of the eigenfrequency output of the program eigenvalues for several runs with $s=-2$ is shown in figure [*]. The blue curves in figure [*] can be compared to figure 5 of reference [28] keeping in mind the conversion factors between Leaver's convention (which is also used in [28]) and the convention used here (see subsection [*]).

Figure: The real and imaginary parts of the eigenfrequencies, $\hat{\omega}$, as computed by the program eigenvalues with $s=-2$. Each curve corresponds to a range of values of $\hat{a}$ from $-0.9$ (left) to $+0.9$ (right) for a single mode $\ell$ and $\vert m\vert$. The open circles are placed at the values $\hat{a}=-0.9$, $-0.6$, $-0.3$, $0$, $+0.3$, $+0.6$, and $+0.9$ except when $m=0$ in which case there are no negative values of $\hat{a}$ plotted. The blue curves correspond to the $\ell=2$ modes and the red curves correspond to the $\ell=3$ modes.
\begin{figure}\index{colorpage}
\begin{center}
\epsfig{file=Figures/eigen.eps,height=7.5cm}\end{center}\end{figure}

Includes/eigenvalues.tex

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


next up previous contents
Next: Function: sw_spheroid() Up: GRASP Routines: Black hole Previous: Function: qn_eigenvalues()   Contents
Bruce Allen 2000-11-19