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:
For a Kerr black hole of a given dimensionless angular momentum parameter, , with a perturbation of spin-weight and mode and , there is a spectrum of quasinormal modes which are specified by the eigenvalues and . As discussed in the previous subsection, the eigenvalue 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 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 () mode. To convert the dimensionless eigenvalue to the (complex) frequency of the ringdown of a Kerr black hole of mass , one simply computes . The eigenfrequency is computed using the method of Leaver . Note that Leaver adopts units in which , so one finds that and in our notation. The eigenvalues satisfy the following symmetry: if and are the eigenvalues for an azimuthal index , then and are the eigenvalues for the azimuthal index .