# Derivation of Teukolsky equation

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.

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.