### 8.1 History/background

The work reported in this document has had a very long gestation period. It began in 1965 [53] with
the publication of a paper where a complex coordinate transformation was performed on the
Schwarzschild/Reissner–Nordström solutions. This, in a precise sense, moved the ‘point source’ onto a
complex world line in a complexified spacetime. It thereby led to a derivation of the spinning and
the charged-spinning particle metrics. How and why this procedure worked was considered
to be rather mysterious and a great deal of effort by a variety of people went into trying to
unravel it. In the end, the use of the complex coordinate transformation for the derivation
of these metrics appeared as if it was simply an accident; that is, a trick with no immediate
significance. Nevertheless, the idea of a complex world line, appearing in a natural manner, was an
intriguing thought, which frequently returned. Some years later, working on an apparently
unrelated subject, we studied and found the condition for a regular NGC, in asymptotically-flat
spacetime, to have a vanishing asymptotic shear [12]. This led to the realization that a regular NGC
was generated by a complex world line, though originally there was no relationship between
the two complex world lines. This condition (our previously discussed shear-free condition,
Eq. (4.12)) was eventually shown to be closely related to Penrose’s asymptotic twistor theory, and
in the flat-space case it led to the Kerr theorem and totally shear-free NGCs. From a very
different point of view, searching for asymptotically shear-free complex null surfaces, the good-cut
equation was found with its four-complex parameter solution space, leading to the theory of
-space.
Years later, the different strands came together. The shear-free condition was found to be closely related
to the good-cut equation; namely, that one equation could be transformed into the other. The major
surprise came when we discovered that the regular solutions of either equation were generated by complex
world lines in an auxiliary space [39]. These complex world lines were interpreted as being
complex analytic curves in the associated -space. The deeper meaning of this remains a major
question still to be fully resolved; it is this issue which is partially addressed in the present
work.

At first, these complex world lines were associated with the spinning, charged and uncharged
particle metrics – type D algebraically special metrics, but now can be seen as just special
cases of the asymptotically flat solutions. Since these metrics were algebraically special, among
the many possible asymptotically shear-free NGCs there was (at least) one totally shear-free
(rather than asymptotically shear-free) congruence coming from the Goldberg–Sachs theorem.
Their associated world lines were the ones first discovered in 1965 (coming, by accident, from
the complex coordinate transformation), and became the complex center-of-mass world line
(which coincided with the complex center of charge in the charged case.). This observation
was the clue for how to search for the generalization of the special world line associated with
algebraically-special metrics and thus, in general, how to look for the special world line (and
congruence) to be identified with the complex center of mass for arbitrary asymptotically flat
spacetimes.

For the algebraically-special metrics, the null tetrad system at with one leg being the tangent null
vector to the shear-free congruence leads to the vanishing of the asymptotic Weyl tensor component, i.e.,
. For the general case, no tetrad exists with that property but one can always find a null
tetrad with one leg being tangent to an asymptotically shear-free congruence so that the harmonics
of vanish. It is precisely that choice of tetrad that led to our definition of the complex center of
mass.