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 byBy setting
and solving for we obtain the associated oneparameter, , family of real slicings, Thus, the values of that yield real values of are given byPerturbatively, using Eq. (3.19) and writing , we find to first order:
Continuing, with small values for the imaginary part of , ( both real analytic functions) and hence small , it is easy to see that (for fixed value of ) is a bounded smooth function on the sphere, with maximum and minimum values, and . Furthermore on the () sphere, there are a finite linesegments worth of curves (circles) that lie between and such that is a monotonically increasing function on the family of curves. Hence there will be a family of circles on the sphere where the value of is a constant, ranging between and .
Summarizing, we have the result that in the complex plane there is a ribbon or strip given by all values of and line segments parametrized by between and such that the complex lightcones from each of the associated points, , all have some null geodesics that intersect real . More specifically, for each of the allowed values of there will be a circle’s worth of complex null geodesics leaving the point reaching real . It is the union of these null geodesics, corresponding to the circles on the sphere from the line segment, that produces the real family of cuts, Eq. (3.26).
The real structure associated with a complex world line is then this oneparameter family of slices (cuts) Eq. (3.26).
Remark 7. We saw earlier that the shearfree 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.
There are a pair of important (dual) results that arise from the considerations of the good cuts [7, 8]. From the stereographic angle field, i.e., from Eqs. (3.31) and (3.32), one can form two different conjugate fields, (1) the complex conjugate of :
and (2) the holomorphic conjugate, , given byThe two different pairs, the complex conjugate pair () and the holomorphic pair () determine two different null vector direction fields at , the real vector field, , and the complex field, , via the relations
and Both generate, in the spacetime interior, shearfree null geodesic congruences: the first is a real twisting shearfree congruence while the latter is a complex twistfree congruence that consists of the lightcones from the world line, , i.e., they focus on . It is this fact that they focus on the world line, that is of most relevance to us.The twist of the real congruence, , which comes from the complex divergence,
is proportional to the imaginary part of the complex world line and consequently we have the real structure associated with the complex world line coming from two (dual) places, the real cuts, Eq. (3.26) and the twist.It is the complex point of view of the complex lightcones coming from the complex world line that dominates our discussion.
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Living Rev. Relativity 15, (2012), 1
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