To construct an associated family of real cuts from a GCF, we begin with
and write withBy setting
and solving for we obtain the associated one-parameter, Perturbatively, using Eq. (3.19) and writing
, we find
to first order:
Continuing, with small values for the imaginary part of , (
both
real analytic functions) and hence small
, it is easy to see that
(for fixed value of
)
is a bounded smooth function on the
sphere, with maximum and minimum values,
and
. Furthermore on the (
) sphere, there are a finite
line-segments worth of curves (circles) that lie between
and
such that
is a monotonically increasing function on the family of curves. Hence there will be a family of
circles on the
-sphere where the value of
is a constant, ranging between
and
.
Summarizing, we have the result that in the complex -plane there is a ribbon or strip given by all
values of
and line segments parametrized by
between
and
such that the complex
light-cones from each of the associated points,
, all have some null geodesics that intersect real
. More specifically, for each of the allowed values of
there will be a circle’s worth of
complex null geodesics leaving the point
reaching real
. It is the union of these null
geodesics, corresponding to the circles on the
-sphere from the line segment, that produces the real
family of cuts, Eq. (3.26
).
The real structure associated with a complex world line is then this one-parameter family of slices (cuts)
Eq. (3.26).
Remark 7. We saw earlier that the shear-free angle field was given by
where real values of
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The differentiation must be done first, holding
constant, before the reality of
is used. In
other words, though we are interested in real
, it is essential that we consider its (local)
complexification.
There are a pair of important (dual) results that arise from the considerations of the good cuts [7, 8
].
From the stereographic angle field, i.e.,
from Eqs. (3.31
) and (3.32
), one can form two different
conjugate fields, (1) the complex conjugate of
:
The two different pairs, the complex conjugate pair () and the holomorphic pair (
)
determine two different null vector direction fields at
, the real vector field,
, and the complex
field,
, via the relations
The twist of the real congruence, , which comes from the complex divergence,
It is the complex point of view of the complex light-cones coming from the complex world line that dominates our discussion.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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