### 4.3 Real cuts from the complex good cuts, II

The construction of real structures from the complex structures, i.e., finding the complex values of
that yield real values of and the associated real cuts, is virtually identical to the flat-space
construction of Section 3. The real structure associated with the complex Minkowski space complex world
lines is easily extended to the -space world lines associated with asymptotically flat spacetimes. The
only difference is that we start with the GCF
rather than the flat-space

Again assuming that the Bondi shear is sufficiently small and the -space complex world line is not
too far from the “real”, the solution to the good-cut equation (4.15), i.e., Eq. (4.38), with , is
decomposed into real and imaginary parts,

Setting the imaginary part to zero and solving for we obtain an expression of the form,
As in the flat case, for fixed , has values on a line segment bounded between some and
. The allowed values of are again on a ribbon in the -plane (i.e., region which is
topologically for an interval ); all values of and allowed values on the -line
segments.

Each level curve of the function constant on the -sphere (closed
curves or isolated points) determines a specific subset of the null directions and associated null
geodesics on the light-cone of the complex point that intersect the real
. These geodesics will be referred to as ‘real’ geodesics. As moves over all allowed
values of its segment, we obtain the set of -space points, and their
collection of ‘real’ geodesics. From Eq. (4.39), these ‘real’ geodesics intersect on the real cut

As varies we obtain a one-parameter family of cuts. If these cuts do not intersect with each other we
say that the complex world line is by definition ‘timelike.’ This occurs when the time
component of the real part of the complex velocity vector, , is sufficiently
large.