The second was via the complex cut function, , that satisfied
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They determined the that satisfies Equation (163
) by the parametric relations
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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
by
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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