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EDIT: I can't seem to delete this question, so I've posted the solution below (I must have made an algebra error-someone checking all this would still be appreciated!). I've left the question as is.

The Bardeen-Press (i.e. Teukolsky equation for a Schwarzschild black hole) in Schwarzschild coordinates reads:

\begin{equation} \frac{r^4}{r^2-2Mr}\partial_t^2\psi -2s\left(\frac{Mr^2}{r^2-2Mr}-r\right)\partial_t\psi -\left(r^2-2Mr\right)^{-s}\partial_r\left(\left(r^2-2Mr\right)^{s+1}\partial_r\psi\right) +(l-s)(l+s+1)\psi = 0 \end{equation}

This equation is not regular at the horizon (r=2M). I want to transform this equation to horizon penetrating (more specifically, ingoing Eddington-Finkelstein) coordinates. I apply

\begin{equation} v=t+r+2M\mathrm{log}\left(r-2M\right) , \end{equation}

use the chain rule, and get an equation of the form

\begin{equation} -2r^2\partial_v\partial_r\psi +\frac{2r\left(\left(s-3\right)M+\left(2+s\right)r\right)}{2M-r}\partial_v\psi +\frac{q(r)}{(2M-r)r}\psi +\cdots = 0 , \end{equation}

where $q(r)$ is a long function and $\cdots$ contains terms regular in $r-2M$. I have read sources that seem to suggest that if I perform the field redefinition

\begin{equation} \psi\equiv r^{-1}\left(r^2-2Mr\right)^{-s}\Psi , \end{equation}

and rewrite everything in terms of $\Psi$ the equation becomes regular at the horizon (still at $r=2M$). I have tried this and for the life of me I cannot seem to get a regular equation at the horizon. When I plug in the above transformation I get

\begin{equation} -2r^2\partial_v\partial_r\Psi -\frac{2r\left(\left(1-3s\right)M+\left(s-1\right)r\right)}{2M-r}\partial_v\Psi +\cdots = 0 , \end{equation}

the coefficient for $\partial_v\Psi$ is still singular! I feel like I may be making a algebraic error but I cannot find where the problem is.

I am looking for a reference or an explicit calculation that the above manipulations lead to a wave equation regular on the black hole horizon.

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  • $\begingroup$ This is discussed in section V of arxiv.org/pdf/gr-qc/9907085.pdf, but they don't go into much detail for the manipulations. (It does seem that the $r^{-1}$ shouldn't be there in your field redefinition. $\endgroup$ – mmeent Sep 24 at 6:30
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I must have made an algebraic error-the answer I believe is:

I use the coordinate \begin{equation} v\equiv t+2M \mathrm{ln}\left(r-2M\right) , \end{equation} and

\begin{equation} \psi\equiv \left(r^2-2Mr\right)^{-s}r^{-1}\Psi , \end{equation}

and the Bardeen-Press (Teukolsky) equation is then both horizon penetrating and regular on the horizon:

\begin{equation} r\left(r+2M\right)\partial_v^2\Psi -4Mr\partial_v\partial_r\Psi -r\left(r-2M\right)\partial_r^2\Psi +2\left(sr+\left(1+s\right)M\right)\partial_v\Psi +2\left(sr-\left(1+s\right)M\right)\partial_r\Psi +\left(\left(l-s\right)\left(l+s+1\right)+\frac{2(1+s)M}{r}\right)\Psi = 0 . \end{equation}

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