In this appendix, we show that the family of complex light-cones with apex on a complex world line in
complex Minkowski space have null generators that form a real shear-free null geodesic congruence in
real Minkowski space [7].
Theorem. There exists a mapping from the arbitrary complex-analytic world line to
the real shear-free NGC in
given by complex null displacements.
Proof: We first recall from Section 3 that regular real shear-free NGCs in are parametrically given
by
Beginning with the world line, , we add to it a specific complex null displacement (to be
constructed)
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with fixed but varying
is identical to that given by Eq. (E.1
).
This is demonstrated by taking the world line, , written in terms of its components
(
) as
By adding the complex null vector (displacement),
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to both sides of Eq. (E.5), we obtain
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and restrict to the real.
We see that by adding a null ray, combinations of and
, directly to the complex world line
, we obtain a mapping of the complex world line directly to the real shear-free NGC, Eq. (E.1
).
Note that when the affine parameter,
, is chosen (complex) as
the
term drops out and we have the ‘point’
surrounded by the embedded complex sphere,
. The ray can be thought of as having its origin on this
surface.
We thus have the explicit relationship between the complex world line and the shear-free NGC, completing the proof.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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