Later in this section, by choosing an arbitrary complex analytic world line in -space, , we describe how to construct the shear-free angle field, . First, however, we discuss properties and the origin of Equation (180).
Roughly or intuitively one can see how the four complex parameters enter the solution from the following argument. We can write Equation (179) as the integral equation
It should be noted again that the is composed of the harmonics,
We note that using this form of the solution implies that we have set stringent coordinate conditions on the -space by requiring that the first four spherical harmonic coefficients be the four -space coordinates. Arbitrary coordinates would just mean that these four coefficients were arbitrary functions of other coordinates. How these special coordinates change under the BMS group is discussed later.
Remark: It is of considerable interest that on -space there is a natural quadratic complex metric – demonstrated in Appendix D – that is given by the surprising relationship [34, 20]Update Remarkably this turns out to be a Ricci-flat metric with a nonvanishing anti-self-dual Weyl tensor, i.e., it is intrinsically a complex vacuum metric. For vanishing Bondi shear, -space reduces to complex Minkowski space (i.e., ).
Returning to the issue of the solutions to the shear-free condition, i.e., Equation (174), , we see that they are easily constructed from the solutions to the good-cut equation, . By choosing an arbitrary complex world line in the -space, i.e.,[55, 28] is the basic variable for the construction of the metric.)
Using the gauge freedom, , in a slightly different way than in the Minkowski-space case, we impose the simple conditionA Brief Summary: The description and analysis of the asymptotically shear-free NGCs in asymptotically-flat spacetimes is remarkably similar to that of the flat-space regular shear-free NGCs. We have seen that all regular shear-free NGCs in Minkowski space and asymptotically-flat spaces are generated by solutions to the good-cut equation, with each solution determined by the choice of an arbitrary complex analytic world line in complex Minkowski space or -space. The basic governing variables are the complex GCF, , and the stereographic angle field on , , restricted to real . In every sense, the flat-space case can be considered as a special case of the asymptotically-flat case.
In Sections 5 and 6, we will show that in every asymptotically flat spacetime a special complex-world line (along with its associated NGC and GCF) can be singled out using physical considerations. This special GCF is referred to as the (gravitational) UCF, and is denoted by
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