The second was via the complex cut function, , that satisfied
They determined the that satisfies Equation (163) by the parametric relations
where was an arbitrary complex world line in complex Minkowski space.
It is this pair of equations, (163) and (164), that will now be generalized to asymptotically-flat spacetimes.
In Section 2, we saw that the asymptotic shear of the (null geodesic) tangent vector fields, , of the out-going Bondi null surfaces was given by the free data (the Bondi shear) . If, near , a second NGC, with tangent vector , is chosen and then described by the null rotation from to around byUpdate and the (new) asymptotic shear of the null vector field is given by [6, 26]
By requiring that the new congruence be asymptotically shear-free, i.e., , we obtain the generalization of Equation (163) for the determination of , namely,
Again we introduce the complex potential that is related to by
In Section 4.2, we will discuss how to construct solutions of Equation (177) of the form, ; however, assuming we have such a solution, it determines the angle field by the parametric relations
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