I am trying to follow the solution of the wave equation for a scalar field on Schwarzschild background from http://batteringram.org/science/gr/scalar_wave.pdf. I have a problem on page 2 where they have to show [eq (19)-(24)]

From \begin{equation} \left( -\partial_{t}^{2} + \frac{1}{r^{2}} \partial_{r_{*}} [r^{2} \partial_{r_{*}}] - f\frac{l(l+1)}{r^{2}} \right) \psi = 0 \end{equation} where \begin{equation} \frac{\partial r}{\partial r_{*}} = \frac{1}{f}, \quad f = 1 - 2\frac{m}{r} \end{equation} Using $ \psi = \phi/r $ show \begin{equation} \left[-\partial_{t}^{2} + \partial_{r_{*}}^{2} - V(r)\right] \phi(r_{*},t) = 0, \quad V(r) = f \left[ \frac{l(l+1)}{r^2} + \frac{2m}{r^3} \right] \end{equation}

My solution is: \begin{equation} \begin{split} \frac{1}{r^{2}} \partial_{r_{*}} \left[ r^{2} \partial_{r_{*}} \frac{\phi}{r} \right] &= \frac{1}{r^{2}} \partial_{r_{*}} \left[ r^{2} \left( \frac{\phi'}{r} - \frac{1}{f}\frac{\phi}{r^{2}} \right) \right], \quad \phi' = \frac{\partial \phi}{\partial r_{*}} \\ &= \frac{1}{r^{2}} \partial_{r_{*}} \left[ r \phi' - \frac{\phi}{f} \right] \\ &= \frac{1}{r^{2}} \left[ \frac{\phi'}{f} + r\phi'' - \frac{\phi}{f} + 2\frac{m}{r^{2}}\phi \frac{1}{f^{3}} \right] \\ &= \frac{1}{r} \left[ \phi'' + 2\frac{m}{r^{3}}\phi \frac{1}{f^{3}} \right] \\ \end{split} \end{equation} and so if we put it back \begin{equation} \begin{split} 0 &= \left( -\partial_{t}^{2} + \frac{1}{r^{2}} \partial_{r_{*}} [r^{2} \partial_{r_{*}}] - f\frac{l(l+1)}{r^{2}} \right) \frac{\phi}{r}, \;\; /.r \\ &= \left( -\partial_{t}^{2} + \partial_{r_{*}}^{2} - V(r) \right) \phi, \quad V(r) = f \left[ \frac{l(l + 1)}{r^{2}} + \frac{2m}{r^{3}} \left( -\frac{1}{f^{4}} \right) \right] \\ \end{split} \end{equation} And as you can see, I got extra $-\frac{1}{f^4}$ in the end. Any idea why?

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    $\begingroup$ Minor comment to the post (v2): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. $\endgroup$
    – Qmechanic
    Jul 3 '15 at 17:28

There's an error in the notes you posted. The tortoise coordinate is usually defined via $$ \frac{dr}{dr_*} = 1 - \frac{2m}{r} = f \neq \frac{1}{f}. $$ Note that the correct definition is given in eq. (42) of your link. I suspect this will fix your problem.

  • $\begingroup$ Thank you! I didn't even realized that... next time I should google more and read more carefully. $\endgroup$
    – user35780
    Jul 3 '15 at 17:12

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