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.

  • $\begingroup$ Can you link to the "program written by the authors" that you mention? The paper you've linked to doesn't appear to use the version of the equation that's concerning you. $\endgroup$ Aug 26, 2021 at 16:41
  • $\begingroup$ In this link look for the Mathematica notebook titled "Amplification factors of the superradiant scattering of a charged wave off a spherically-symmetric or a slowly-rotating BH with generic metric." $\endgroup$
    – newtothis
    Aug 26, 2021 at 17:05

1 Answer 1


The function $\psi$ in the code by Brito, Cardano & Pani is not the same as the function $R$ in the spherical harmonic decomposition. Rather, it is related by $\psi = rR$. Under this substitution, it is not hard to show that $$ r \left\{ f^2R^{\prime \prime} + ff^{\prime}R^{\prime} + \left(\omega - \frac{qQ}{r}\right)^2R - f\left(l(l+1)+ \frac{f^{\prime}}{r}\right)R + \frac{2f^2}{r}R^{\prime} + \frac{f f^{\prime}}{r}R \right \} \\ = f^2 \psi'' + f f' \psi' + \left[\left(\omega - \frac{qQ}{r}\right)^2R - f\left(l(l+1)+ \frac{f^{\prime}}{r}\right)\right] \psi $$ as desired.

  • $\begingroup$ Thanks! I missed the definition of the mode expansion while working it out on my own. By any chance would you have any resources on separating the components and arriving at the radial equation by itself? I would like to get some practice explicitly expanding these equations. $\endgroup$
    – newtothis
    Aug 26, 2021 at 18:28
  • $\begingroup$ @newtothis: Not off the top of my head. The best shortcut I know (which I think only works for the KG equation) is to note that $$\nabla^2 \phi = \sum_{\mu,\nu}\frac{1}{\sqrt{|g|}}\partial_\mu\left( \sqrt{|g|} g^{\mu\nu} \partial_\nu \phi \right).$$ (This follows from the equation for the divergence of a vector field $T^\mu$ and setting $T^\mu = \partial^\mu\phi$; see, e.g., §3.4 of Wald.) For a diagonal metric this is pretty straightforward to write out explicitly. I'm unaware, however, of a straightforward generalization to higher-spin fields, other than just writing out the Christoffels. $\endgroup$ Aug 26, 2021 at 18:42
  • $\begingroup$ Beyond that, the techniques used in getting these equations are (I think) pretty much the same as in standard separation-of-variables problems from (say) E&M. $\endgroup$ Aug 26, 2021 at 18:46

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