### 8.1 History/background

The work reported in this document has had a very long gestation period. It began in 1965 [37] 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, i.e., 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 [6] for an NGC, in
asymptotically-flat spacetime, to have a vanishing asymptotic shear. This condition (our previously
discussed shear-free condition, Equation (174)), was closely related to Penrose’s asymptotic
twistor theory. In the flat-space case it lead to the Kerr theorem and totally shear-free NGCs.
From a different point of view, searching for shear-free complex null surfaces, the good-cut
equation was found with its four-complex parameter solution space. This lead 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 Minkowski space [26]. (These complex world lines could be thought of
as being complex analytic curves in the associated -space. The deeper meaning of this
remains a question still to be resolved; it is this issue which is partially addressed in the present
work.)

The complex world line mentioned above, associated with the spinning, charged and uncharged particle
metrics, now can be seen as just a special case of these regular 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. This was the one we first discovered in
1965, that 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 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 the 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.