the angular direction. In summary, the avoidance of ghost and Laplacian instability demands the conditions $$\begin{aligned}
\label{eq:stab}
{\cal F}>0,~~{\cal G}>0,~~{\cal H}>0\,.\end{aligned}$$

Stability
=========

From the variation of the action (\[eq:final\]), we find $$\begin{aligned}
\label{eq:master}
-\frac{\partial^2 Q}{\partial t^2}+\frac{\cal G}{\cal F}\frac{\partial^2 Q}{\partial r^{*2}}-\frac{\cal G}{\cal F} V Q=0\,.\end{aligned}$$ Therefore, we see that we can define the speed of fluctuations propagating radially as $$\begin{aligned}
c_r^2=\frac{\cal G}{\cal F}\,.\end{aligned}$$ Also, the term $\frac{\cal G}{\cal F} VQ$, includes the angular part, which from (\[waveeq\]), is $$\begin{aligned}
l(l+1)\frac{A}{C}\frac{\cal G}{\cal H}\,.\end{aligned}$$ This comes from the angular part of the Laplacian, from which we can define the speed of angular excitations $$\begin{aligned}
c_\Omega^2=\frac{\cal G}{\cal H}\,.\end{aligned}$$ We therefore ensure the positivity of the speed squared of propagation of the perturbations due to the conditions given in (\[eq:stab\]). Having the equation of perturbations, we now turn to their stability. Once a black hole is perturbed, it responds to perturbations by emitting gravitational waves (GWs). This signal can be divided into three stages

1.  An early response which depends strongly on the initial conditions.

2.  An exponentially decaying phase, known as the ringdown.

3.  A late tail.

The distortion of the solution reduces to the spherical solution after the emission of GWs during 