these stages. The linear theory of perturbations studied in this paper tackles the last two stages of this process, during which the waveform is typically identified with a quasinormal frequency $\omega$ and therefore this exponential damping of the perturbations is called quasi-normal ringing. Because the ringdown signal is given by a superposition of quasinormal modes (QNMs), we can write $$\begin{aligned}
\label{Fourier}
Q(t,r)=\sum_n e^{-{{i\mkern1mu}}\omega_n t}\psi_n(r)\,.\end{aligned}$$ Replacing in (\[eq:master\]), we have $$\begin{aligned}
\label{eq:master2}
\Bigl(-\frac{d^2}{dr^{*2}}+V\Bigr)\psi_n=\frac{\omega_n^2}{c_r^2}\psi_n\,.\end{aligned}$$ Any frequency $\omega_n$ solves this equation but only a discrete number solves it with additional boundary conditions dictated by the physical problem. The boundary conditions defining QNMs are purely ingoing at the horizon and outgoing at infinity for asymptotically flat or de Sitter (dS) spacetimes and should be null at infinity for AdS spacetime. These conditions are physically motivated, e.g., ingoing at the horizon means entering into the black hole and we should not have any perturbations outgoing at the horizon. Equivalently, we should not have a wave coming from infinity. These frequencies are usually complex and we can see from (\[Fourier\]) that the modes are not growing in time when $\operatorname{Im}{(\omega_n)}<0$. Therefore the stability of the black hole under linear perturbations dictate that all QNMs should fulfill 