From Equations (27) and (28), the radial behavior of the optical parameters for general shearfree NGCs, in Minkowski space, is given by
where is the twist of the congruence. A more detailed and much deeper understanding of the shearfree congruences can be obtained by first looking at the explicit coordinate expression, Equation (23), for all flatspace NGCs: where is an arbitrary complex function of the parameters ; , also an arbitrary function of , determines the origin of the affine parameter; can be chosen freely. Most frequently, to simplify the form of , is chosen as At this point, Equation (72) describes an arbitrary NGC with labeling the geodesics and the affine distance along the individual geodesics; later will be chosen so that the congruence is shearfree.The tetrad () is given by [27]
There are several important comments to be made about Equation (72). The first is that there is a simple geometric meaning to the parameters : they are the values of the Bondi coordinates of , where each geodesic of the congruence intersects .The second concerns the geometric meaning of . At each point of , consider the past light cone and its sphere of null directions. Coordinatize that sphere (of null directions) with stereographic coordinates. The function is the stereographic angle field on that describes the null direction of each geodesic intersecting at the point . The values and represent, respectively, the direction along the Bondi and vectors. This stereographic angle field completely determines the NGC.
The twist, , of the congruence can be calculated in terms of directly from Equation (72) and the definition of the complex divergence, Equation (24), leading to
We now demand that be a regular function of its arguments, i.e., have no infinities, or, equivalently, that all members of the NGC come from the interior of the spacetime and not lie on itself.It has been shown [6] that the condition on the stereographic angle field for the NGC to be shearfree is that
Our task is now to find the regular solutions of Equation (76). The key to doing this is via the introduction of a new complex variable and complex function [26, 27], is related to by the CR equation (related to the existence of a CR structure on ; see Appendix B):Remark: The following ‘gauge’ freedom becomes useful later. , with analytic, leaving Equation (78) unchanged. In other words,
leads toWe assume, in the neighborhood of real , i.e., near the real and , that is analytic in the three arguments . The inversion of Equation (77) yields the complex analytic cut function
Though we are interested in real values for , from Equation (80) we see that for arbitrary in general it would take complex values. Shortly, we will also address the important issue of what values of are needed for real .Returning to the issue of integrating the shearfree condition, Equation (76), using Equation (77), we note that the derivatives of , and can be expressed in terms of the derivatives of by implicit differentiation. The derivative of is obtained by taking the derivative of Equation (80):
while the derivative is found by applying to Equation (80), When Equations (81) and (82) are substituted into Equation (78), one finds that is given implicitly in terms of the cut function byThus, we see that all information about the NGC can be obtained from the cut function .
By further implicit differentiation of Equation (83), i.e.,
From the properties of the operator, the general regular solution to Equation (85) is easily found: must contain only and spherical harmonic contributions; thus, any regular solution will be dependent on four arbitrary complex parameters, . If these parameters are functions of , i.e., , then we can express any regular solution in terms of the complex world line [26, 27]:
The angle field then has the formThus, we have our first major result: every regular shearfree NGC in Minkowski space is generated by the arbitrary choice of a complex world line in what turns out to be complex Minkowski space. See Equation (70) for the connection between the harmonics in Equation (86) and the Poincaré translations. We see in the next Section 4 how this result generalizes to regular asymptotically shearfree NGCs.
Remark: We point out that this construction of regular shearfree NGCs in Minkowski space is a special example of the Kerr theorem (cf. [51]). Writing Equations (87) and (88) as

This is a special case of the general solution to Equation (76), which is the Kerr theorem.
In addition to the construction of the angle field, , from the GCF, another quantity of great value in applications, obtained from the GCF, is the local change in as changes, i.e.,
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