### 3.1 The flat-space good-cut equation and good-cut functions

In Section 2, we saw that in the NP formalism, two of the complex spin coefficients, the optical parameters and of Eqs. (2.23) and (2.24), play a particularly important role in their description of an NGC; namely, they carry the information of the divergence, twist and shear of the congruence.

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

where is the twist of the congruence. A more detailed and much deeper understanding of the shear-free congruences can be obtained by first looking at the explicit coordinate expression, Eq. (2.19), for all flat-space NGCs:
where is an arbitrary complex function of the parameters ; , also an arbitrary function of , determines the origin of the affine parameter; and can be chosen freely. Most frequently, to simplify the form of , is chosen as
At this point, Eq. (3.2) 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 shear-free. The tetrad () is given by Eqs. (1.11.2), see [40].

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 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 [12] that the condition on the stereographic angle field for the NGC to be shear-free is that

Our task is now to find the regular solutions of Eq. (3.5). The key to doing this is via the introduction of a new complex variable and complex function [39, 40],
is related to by the CR equation (related to the existence of a CR structure on ; see Appendix B):

Remark 5. The following ‘gauge’ freedom becomes useful later. , with analytic, leaving Eq. (3.7) unchanged. In other words,

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

Though we are interested in real values for , from Eq. (3.9) we see that for arbitrary it may 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 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):

while the derivative is found by applying to Eq. (3.9),
When Eqs. (3.10) and (3.11) are substituted into Eq. (3.7), one finds that is given implicitly in terms of the cut function by

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.,

using Eq. (3.7), the shear-free condition (3.5) becomes
This equation will be referred to as the homogeneous Good-Cut Equation and its solutions as flat-space Good-Cut Functions (GCFs). In the next Section 4, an inhomogeneous version, the Good-Cut Equation, will be found for asymptotically shear-free NGCs. Its solutions will also be referred to as GCFs.

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]:

The angle field then has the form

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.

Remark 6. We point out that this construction of regular shear-free NGCs in Minkowski space is a special example of the Kerr theorem (cf. [67]). Writing Eqs. (3.16) and (3.17) as

where the () are simple combinations of the , we then find that
Noting that the right-hand side of both equations are functions only of and , we can eliminate the from the two equations, thereby constructing a function of three variables of the form

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.,