### 4.1 Asymptotically shear-free NGCs and the good-cut equation

We saw in Section 3 that shear-free NGCs in Minkowski space could be constructed by looking at their
properties near , in one of two equivalent ways. The first was via the stereographic angle field,
, which gives the directions that null rays make at their intersection with . The condition
for the congruence to be shear-free was that must satisfy
We required solutions that were all nonsingular (regular) on the sphere. (This equation has in the
past most often been solved via twistor methods [31].)
The second was via the complex cut function, , that satisfied

The regular solutions were easily given by
with inverse function,
They determined the that satisfies Eq. (4.1) by the parametric relations

or by
where was an arbitrary complex world line in complex Minkowski space.

It is this pair of equations, (4.1) and (4.2), 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

with an arbitrary stereographic angle field, then the asymptotic Weyl components transform
as
and the (new) asymptotic shear of the null vector field is given by [12, 39]
By requiring that the new congruence be asymptotically shear-free, i.e., , we obtain the
generalization of Eq. (4.1) for the determination of , namely,

To solve this equation we again complexify to by freeing to and allowing to take
complex values close to the real.
Again we introduce the complex potential that is related to by

with its inversion,
Eq. (4.12) becomes, after the change in the independent variable, , and implicit
differentiation (see Section 3.1 for the identical details),
This, the inhomogeneous good-cut equation, is the generalization of Eq. (4.2).
In Section 4.2, we will discuss how to construct solutions of Eq. (4.15) of the form, ;
however, assuming we have such a solution, it determines the angle field by the parametric
relations

We now turn to these solutions and their properties.