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In the Newman-Penrose formalism, one rewrites the Einstein equations in terms of a system of linear transport equations for the Newman-Penrose scalars. I am considering the initial value problem for this system of equations: I want to set initial data on a spacelike hypersurface, and then evolve (a subset of) the Newman-Penrose equations in time.

I do not know though how to set up initial data for all the Newman-Penrose scalars that are consistent with the constraints. For example, if I set $\Psi_4$ and $\partial_t\Psi_4$ on the spacelike hypersurface (i.e. I treat their values as free initial data), presumably I cannot freely specify the rest of the Newman-Penrose scalars on the initial hypersurface without violating the constraint equations of the Einstein equations. (Note I want to evolve with the time function $t$, and not use the Newman-Penrose equations in terms of the characteristic initial value problem).

Is there a general procedure for determining consistent initial data in the Newman-Penrose formalism?

EDIT: to clarify, I'm considering the initial value problem in vacuum (no matter fields)

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  • $\begingroup$ Are you considering the vacuum case? $\endgroup$
    – TimRias
    Feb 16, 2020 at 23:10
  • $\begingroup$ @mmeent yes, I'm considering the vacuum case $\endgroup$ Feb 17, 2020 at 1:35

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