I am reading a review on superradiance, specifically in black hole spacetimes. I wish to compute the amplification factor for a complex scalar field in the RN spacetime. There is a program given by the authors to do this numerically, however, I have some difficulty in understand the form of the differential equation that needs to be solved.
The Klein-Gordon equation for a massless complex scalar field in RN spacetime is given as:
$$ D_\mu D^\mu \Psi = 0 $$
where $D_\mu = (\nabla_\mu - iqA_\mu)$. $\nabla_\mu$ is the covariant derivative and $A_\mu = \delta^0_\mu(-\frac{Q}{r}) $ with $Q$ being the charge of the black hole.
The line element is
$$ds^2 = -f(r)dt^2 + \frac{1}{f}dr^2 + r^2d\Omega^2$$ with $f = (1 - \frac{2M}{r} + \frac{Q^2}{r^2})$.
If we consider an ansatz for the solution of the form:
$$\Psi_{lm} = e^{-i\omega t}R(r)Y_{lm}(\theta \phi)$$ then the radial equation can be written down as (see for example eq 5 of this paper):
\begin{equation} \Delta \frac{d}{dr}(\Delta \frac{dR}{dr}) + U(R) = 0 \end{equation}
where $\Delta = r^2 - 2Mr + Q^2 = r^2f$ and $U = r^4[(\omega - \frac{qQ}{r})^2 - f(l(l+1))]$. I will eventually ask another question about showing this by hand, but currently expanding this out I get:
\begin{equation*} \Delta \frac{d}{dr}(\Delta \frac{dR}{dr}) = r^2f \frac{d}{dr}(r^2f\frac{dR}{dr}) = r^2f(2rf\frac{dR}{dr} + r^2\frac{df}{dr}\frac{dR}{dr} + r^2f\frac{d^2R}{dr^2}) \end{equation*}
Using this:
\begin{equation*} r^4[\frac{2f^2}{r}R^{\prime} + ff^{\prime}R^{\prime} + f^2R^{\prime \prime}] + r^4[(\omega - \frac{qQ}{r})^2 - f(l(l+1))]R = 0 \end{equation*}
\begin{equation*} f^2R^{\prime \prime} + ff^{\prime}R^{\prime} + (\omega - \frac{qQ}{r})^2R - f(l(l+1)+ \frac{f^{\prime}}{r})R + [\frac{2f^2}{r}R^{\prime} + \frac{ff^{\prime}}{r}R] = 0 \end{equation*}
That is,
\begin{equation*} f^2R^{\prime \prime} + ff^{\prime}R^{\prime} + V(r)R + [\frac{2f^2}{r}R^{\prime} + \frac{ff^{\prime}}{r}R] = 0 \end{equation*}
where $V(r) = (\omega - \frac{qQ}{r})^2R - f(l(l+1)+ \frac{f^{\prime}}{r})$.
However, the program written by the authors starts by stating the differential equation to be solved as $$f^2R^{\prime \prime} + ff^{\prime}R^{\prime} + V(r)R = 0$$
They then proceed to solve it numerically to compute the amplification factors from the solution. So in essence, I am getting two extra terms. I am unable to understand how I'm getting two terms extra, or if for some reason the sum of those two terms go to zero.
