### 3.2 Real cuts from the complex good cuts, I

Though our discussion of shear-free NGCs has relied, in an essential manner, on the use of the
complexification of and the complex world lines in complex Minkowski space, it is the real structures
that are of main interest to us. We want to find the intersection of the complex GCF with real , i.e.,
what are the real points and real cuts of , (), and what are the values of that
yield real .
To construct an associated family of real cuts from a GCF, we begin with

and write
with and real. The cut function can then be rewritten
with real and . The and are easily calculated
from by
By setting

and solving for
we obtain the associated real slicing,
Thus, the values of that yield real values of are given by
As an example, using Equation (90), we find to first order in

An Important Remark: We saw earlier that the shear-free angle field was given by
where real values of should be used. If the real cuts, , were used instead
to calculate , the results would be wrong. The restriction of to yield real , does not
commute with the application of the operator, i.e.,
The differentiation must be done first, holding constant, before the reality of is used. In
other words, though we are interested in real , it is essential that we consider its (local)
complexification.