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 to
It has been shown [12] that the condition on the stereographic angle field
for the NGC to be
shear-free is that
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
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.
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
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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.,
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Living Rev. Relativity 15, (2012), 1
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