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 Eq. (4.18
).
Roughly or intuitively one can see how the four complex parameters enter the solution from the
following argument. We can write Eq. (4.17) 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 10. It is of considerable interest that on -space there is a natural quadratic complex
metric – as constructed in Appendix D – that is given by the surprising relationship [49, 32
]
Returning to the issue of the solutions to the shear-free condition, i.e., Eq. (4.12),
, 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.,
Using the gauge freedom, , as in the Minkowski-space case, we impose the simple
condition
A 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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Living Rev. Relativity 15, (2012), 1
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