From Equations (27) and (28
), the radial behavior of the optical parameters for general shear-free
NGCs, in Minkowski space, is given by
The tetrad () is given by [27
]
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
Equation (72
) and the definition of the complex divergence, Equation (24
), leading to
It has been shown [6] that the condition on the stereographic angle field
for the NGC to be
shear-free is that
Remark: The following ‘gauge’ freedom becomes useful later. , with
analytic,
leaving Equation (78
) 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 Equation (77
) yields the complex analytic cut
function
Returning to the issue of integrating the shear-free condition, Equation (76), using Equation (77
), 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 Equation (80
):
Thus, we see that all information about the NGC can be obtained from the cut function
.
By further implicit differentiation of Equation (83), i.e.,
From the properties of the operator, the general regular solution to Equation (85
) 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
[26
, 27
]:
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
Equation (70) for the connection between the
harmonics in Equation (86
) and the Poincaré
translations. We see in the next Section 4 how this result generalizes to regular asymptotically shear-free
NGCs.
Remark: We point out that this construction of regular shear-free NGCs in Minkowski
space is a special example of the Kerr theorem (cf. [51]). Writing Equations (87
) and (88
) as
![]() |
This is a special case of the general solution to Equation (76), 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.,
http://www.livingreviews.org/lrr-2009-6 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |