Derivation of Teukolsky equation

Question

I have been trying to derive the Teukolsky equation via the Newman-Penrose formalism. I have derived the following formula (see Eq. (2.14) in Teukolsky's paper)

\begin{equation} (*) \left[(\Delta +3\gamma - \gamma^*+4\mu + \mu^*)(D+4\epsilon-\rho)-(\delta^*-\tau^*+\beta^*+3\alpha+4\pi)(\delta-\tau+4\beta)-3\psi_2\right]\psi_4 = 0. \end{equation}

The Greek letters (except for $\psi_4$) are all defined with respect to the Kerr background.

What I am struggling with is: deriving the actual wave equation (Eq. (4.7) in Teukolsky's paper) for $\psi_4$ using the Kinnersly tetrad (see Eq (4.4)-(4.6) in Teukolsky's paper) in Boyer-Lindquist coordinates, starting from Eq.(*). I am plugging in the defined values like $\Delta=l^{\mu}\partial_{\mu}$, etc. but I cannot recover Teukolsky's full equation: I have the wrong coefficients for the $\partial_{\theta}\psi_4$, $\partial_{\phi}\psi_4$, $\partial_r\psi_4$, and $\psi_4$ components.

I would greatly appreciate notes on deriving the Teukolsky equation, a link to a mathematica notebook, or a demonstration in someones answer on going from the Newman-Penrose Eq.(*) to the Teukolsky equation in Boyer-Lindquist coordinates.

NOTE

In case the links break: Teukolsky's paper refers to:

• Teukolsky, Saul A.; Astrophysical Journal, Vol. 185, pp. 635-648 (1973)

Newman-Penrose refers to

• Newman, Ezra and Penrose, Roger; Journal of Mathematical Physics, Volume 3, Issue 3, p.566-578

Answer 1

I figured out what my problem was: If one looks at Table 1 of Teukolsky's paper, in fact I should be setting \begin{equation} \psi_4= \rho^4\psi, \end{equation} and then one recovers the Teukoslky equation for a spin $-2$ field.