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I'm trying to apply the Newman-Penrose formalism to a spacetime I know is Petrov type D to calculate its Teukoulsky equation.

In the papers I've read, it is standard to use a Kinnersley null tetrad to apply the NP formalism. However, it is never explained how to obtain this tetrad or why we choose it.

I would like to know the importance of Kinnersley null tetrad in this context, if there is a method to obtain it given an arbitrary Petrov type D spacetime, and if there is any good resource to learn more about this subject.

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In Kinnersley tetrad null vectors $l^\mu$ and $n^\mu$ of the NP formalism lie along the principal null directions (PND) of the Weyl tensor. Recall, that type D spacetime has two double PND. Chandrasekhar's book The Mathematical Theory of Black Holes is the standard reference on the subject.

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  • $\begingroup$ But what about the third vector from the tetrad? Will any null complex vector that is orthogonal to those two suffice? $\endgroup$
    – Yudi
    Commented Apr 14 at 18:12
  • $\begingroup$ I am not sure. Does your spacetime have Killing vector field with orbits orthogonal to PND (something like axial symmetry of Kerr)? $\endgroup$
    – A.V.S.
    Commented Apr 15 at 20:00
  • $\begingroup$ No, it does not. $\endgroup$
    – Yudi
    Commented Apr 15 at 22:38
  • $\begingroup$ Then you are outside the scope of Kinnersley's work. You can probably check papers like gr-qc/0601094 but those are for non-type D spacetimes so may not translate to your problem. $\endgroup$
    – A.V.S.
    Commented Apr 16 at 5:05

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