## E Shear-Free Congruences from Complex World Lines

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

with
The is taken so that is real via Eq. (3.27): , and is the arbitrary origin
for the affine parameter along each geodesic of the congruence.
Beginning with the world line, , we add to it a specific complex null displacement (to be
constructed)

parametrized by the real variable . We will show that the curve (complex null geodesic) given
by
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

and replacing the by the identity
This yields
Using the relations, Eq. (3.34), etc.,
Eq. (E.4) becomes
By adding the complex null vector (displacement),

to both sides of Eq. (E.5), we obtain

To complete our task we now restrict the values of to those that produce a real , namely
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.