Referring particularly to
http://arxiv.org/abs/hep-th/9909056
in regard to the wave equation for Schwarzschild-AdS black holes (p.4), I'm trying to understand tortoise coordinates.
So starting with the 4-dimensionalSchwarzschild-AdS metric in the general form
$$ds^2=-f(r )dt^2+\frac{dr^2}{f(r )}+r^2(d\theta^2+sin^2 \theta d\phi^2),$$
if I want to find the wave equation $\Box \phi=0$ in the Schrodinger-like form. This is done by introducing the separation of variables
$$\phi=\frac{\psi(r ) Y(\theta,\phi)e^{-i\omega t}}{r}$$
and then using the tortoise coordinate $dr_*=\frac{dr}{f(r )}$ to get
$$(\partial_{r_*}^2+\omega^2-V(r_*))\psi=0.$$
But I don't fully understand what this tortoise coordinate really does. In fact when I go through these calculations myself, I use the transformation
$$\psi'(r ) \to \frac{\psi'(r )}{f(r )}$$
and (fortunately) get the Schrodinger like form as in the paper above. However, they never explicitly state the potential and what I find is
$$V(r_*)=\frac{-\ell(\ell+1) f(r )+rf'(r )}{r^2}.$$
where $\ell$ is the angular momentum mode. But note, in my transformation, I never mentioned $r_*$ and hence why my $V(r_*)$ doesn't actually mention an $r_*$. This is where my confusion lies.
Is my potential right if I just replace the $r$ by $r_*$? i.e
$$V(r_*)=\frac{-\ell(\ell+1) f(r_* )+r_*f'(r_* )}{r_*^2}?$$
(I highly doubt it.) And if not, how do I recover $V(r )$ from here?
P.s. It would actually also be extremely helpful if someone knew $V(r )$, i.e. potential in original coordinates, for the Schwarzschild-AdS black hole.
