From Eqs. (2.23) and (2.24), the radial behavior of the optical parameters for general shear-free NGCs, in Minkowski space, is given by.
There are several important comments to be made about Eq. (3.2). 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 Eq. (3.2) and the definition of the complex divergence, Eq. (2.20), leading toregular 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  that the condition on the stereographic angle field for the NGC to be shear-free is that[39, 40],
Remark 5. The following ‘gauge’ freedom becomes useful later. , with analytic, leaving Eq. (3.7) unchanged. In other words,leads to
We assume, in the neighborhood of real , i.e., near the real and , that is analytic in the three arguments . The inversion of Eq. (3.6) yields the complex analytic cut function
Returning to the issue of integrating the shear-free condition, Eq. (3.5), using Eq. (3.6), 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 Eq. (3.9):
Thus, we see that all information about the NGC can be obtained from the cut function .
By further implicit differentiation of Eq. (3.12), i.e.,
From the properties of the operator, the general regular solution to Eq. (3.14) 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 [39, 40]:
Thus, we have our first major result: every regular shear-free 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 Eq. (2.66) for the connection between the harmonics in Eq. (3.15) and the Poincaré translations. We see in the next Section 4 how this result generalizes to regular asymptotically shear-free NGCs.
This is a special case of the general solution to Eq. (3.5), 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.,
Living Rev. Relativity 15, (2012), 1
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