Symmetries of quasinormal modes of Kerr black hole

Question

I have been reading this following paper on numerical evolution of the Teukolsky equation (see e.g. Eq 1 in their paper) for spin -2 fields about a spinning black hole (Kerr) solution.

As the Teukolsky equation is axially symmetric, the authors can factor out $\phi$ dependence by writing the solution in the form

\begin{equation} \Psi_4(t,r,\theta,\phi)=\psi_m(t,r,\theta) e^{im\phi} \end{equation}

and solve the reduced Teukolsky equatoin for each angular number $m$.

The authors in Sec IVB of their paper extract the quasinormal ringdown from their numerical simulations, which are obtained again by solving for a particular $\psi_m$. I am confused about the following two paragraphs in their paper about the ringdown signal they extract in their code (QNM==quasinormal modes)

Interestingly, we find that the numerically extracted QNM frequencies for non-zero m do not depend on the sign ofm, i.e. we get the same values for the QNM frequencies fromevolutions for e.g.m=±1. At first sight this is surprising, since it seems to contradict well established results. According to for example Detweiler [8], the imaginary parts of the m= +1 and m=−1 modes should become quite different as a increases.

Fortunately, the answer is simple: The frequencies of both the m and the −m QNMsare present in a typical evolution.

They then go on to say tha for the Teukolsky equation, the QNM eigenfunctions have the following symmetry (which is easily verified to be true by looking at the form of the Teukolsky equation; here the first argument is the quasinormal frequency)

\begin{equation} \Psi_{l,m}(\omega,r,\theta)=\left[\Psi_{l,-m}(-\omega^*,r,\theta)\right]^* \end{equation}

In particular then, a mode with angular numbers $(l,m)$ is equal to the negative complex conjugate of a mode with angular numbers $(l,-m)$.

I do not understand why the code these authors write seems to mainly excite those modes though, and why for example their code, with $m=-|m|$, does not appreciably excite the other quasinormal modes with $m=-|m|$.

EDIT:

What I am asking is if there is a prescription for computing initial data for $\psi_m$ for which $\psi_m$ is compactly supported on the initial data surface, and that preferentially excites certain quasinormal modes as opposed to others. I suppose I could convolve an exact quasinormal mode solution to the Teukolsky equation with a bump function on the initial data surface, but I am wondering if there is a more elegant set of initial data one could use.

Answer 1

In general there is no easy way to construct initial data that pre-dominantly excites a particular quasi-normal mode. Certainly not without essentially inputting what the quasinormal mode is from the beginning.

Note that the code in the paper you cited, certainly does not do anything like this. In general their initial data while excite most quasinormal modes to some degree. They explicitly point this out in Fig. 1, where they highlight that at first the signal is dominated by one QNM, and later it is dominated by another more slowly decaying QNM.

This points to a general feature. If you wait long enough the signal will always be dominated by the QNM with the longest dampening time, i.e. the least damped quasi-normal mode.