To construct an associated family of real cuts from a GCF, we begin with
Perturbatively, 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 line-segments 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 light-cones 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 one-parameter family of slices (cuts) Eq. (3.26).
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 :
The 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
The twist of the real congruence, , which comes from the complex divergence,
It is the complex point of view of the complex light-cones coming from the complex world line that dominates our discussion.
Living Rev. Relativity 15, (2012), 1
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