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Function: qn_eigenvalues()

void qn_eigenvalues(float eigenvalues[], float a, int s, int l, int m)
This routine computes the eigenvalues associated with the spheroidal and radial wave functions for a specified quasinormal mode. The arguments are:
eigenvalues: Output. An array, eigenvalues[0..3], which contains, on output, 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  $\texttt{eigenvalues[3]}={\mathrm{Im}}\,A$.
a: Input. The dimensionless angular momentum parameter of the Kerr black hole, $\vert\hat{a}\vert\le1$, which is negative if the black hole is spinning clockwise about the $\iota=0$ axis (see figure [*]).
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  $\vert\texttt{m}\vert\le\texttt{l}$.

For a Kerr black hole of a given dimensionless angular momentum parameter, $\hat{a}$, with a perturbation of spin-weight $s$ and mode $\ell$ and $m$, there is a spectrum of quasinormal modes which are specified by the eigenvalues $\hat{\omega}_n$ and $A_n$. As discussed in the previous subsection, the eigenvalue  $\hat{\omega}_n$ is associated with the separation of the time dependence of the perturbation, and it specifies the frequency and damping time of the radiation from the perturbation. The additional complex eigenvalue $A_n$ results from the separation of the radial and azimuthal dependence into the spheroidal and radial wave functions. Both of these eigenvalues will be necessary for the computation of the spheroidal wave function (below).

The routine qn_eigenvalues() can be used to compute the eigenvalues of the fundamental ($n=0$) mode. To convert the dimensionless eigenvalue $\hat{\omega}$ to the (complex) frequency of the ringdown of a Kerr black hole of mass $M$, one simply computes  $\omega=c^3\hat{\omega}/GM$. The eigenfrequency is computed using the method of Leaver [27]. Note that Leaver adopts units in which $2M=1$, so one finds that $\hat{\omega}=\frac{1}{2}\omega_{\mathrm{\scriptscriptstyle Leaver}}$ and $\hat{a}=2a_{\mathrm{\scriptscriptstyle Leaver}}$ in our notation. The eigenvalues satisfy the following symmetry: if $\rho_m=-i\hat{\omega}_m$ and $A_m$ are the eigenvalues for an azimuthal index $m$, then $\rho_{-m}=\rho_m^\ast$ and  $A_{-m}=A_m^\ast$ are the eigenvalues for the azimuthal index $-m$.

Author: Jolien Creighton, jolien@tapir.caltech.edu
Comment: For simplicity, we require that the spin-weight number, $s$, be an integer. Thus, the spinor perturbations $\chi_0$ and $\chi_1$, associated with $s=\pm\frac{1}{2}$ respectively [30], are not allowed.


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