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. (4.12View Equation)) 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 ℋ-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 ℋ-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 ℑ+ 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., ∗ ∗ ψ 0 = ψ1 = 0. 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 l = 1 harmonics of ψ01∗ vanish. It is precisely that choice of tetrad that led to our definition of the complex center of mass.

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