order to avoid a ghost, we need to impose $\alpha>0$, which means ${\cal G}>0$ and the condition to avoid Laplacian instabilities is $\beta<0$ or ${\cal F}>0$, which corresponds to real sound speed.

To arrive at the final result, we redefine the variable $q$ as $$q=\sqrt{\frac{A {\cal F}}{ BC {\cal H}^2}} Q\,,
\label{Qq}$$ and introduce the tortoise coordinate, ${\rm d}r=\sqrt{AB}{\rm d}r^{*}$. The action (\[eq:final2\]) finally takes the form $$\begin{aligned}
\label{eq:final}
S^{(2)}=\frac{l(l+1)}{4(l-1)(l+2)}\int {\rm d} t\,{\rm d} r^{*}\,\Bigl[\frac{{\cal F}}{{\cal G}} \dot Q^2-\Bigl(\frac{dQ}{dr^{*}}\Bigr)^2-V(r) Q^2 \Bigr]\,,\end{aligned}$$ where the potential is defined as $$\begin{aligned}
\label{waveeq}
V&=l(l+1)\frac{A}{C}\frac{{\cal F}}{{\cal H}}
-\frac{C^2}{4C'}\Bigl(\frac{ABC'^2}{C^3}\Bigr)'
-\frac{C^2{\cal F}^2}{4{\cal F}'}\Bigl(\frac{AB{\cal F}'^2}{C^2{\cal F}^3}\Bigr)'\nonumber\\
&-\frac{2A{\cal F}}{C{\cal H}}\,.\end{aligned}$$ Even if the final result is structurally similar to [@Kobayashi:2012kh], we differ in the form of the potential.

Even if the action is not defined for $l=0$ and $l=1$, these modes are not dynamical. For instance, $l=0$ is spherically symmetric and therefore obey the Birkhoff theorem, while $l=1$ corresponds to an infinitesimal shift of the position of the black hole [@Regge:1957td].

Since $\ell (\ell+1)$ corresponds to the two-dimensional Laplacian in the real space, the first term in Eq. (\[waveeq\]) represents wave propagation along the angular direction. Therefore, we also impose ${\cal H}>0$, which gives positive squared propagation speeds along 