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 (174View Equation)), 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 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 Minkowski space [26]. (These complex world lines could be thought of as being complex analytic curves in the associated ℋ-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 + 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 = ψ∗1 = 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 the shear-free congruence so that the l = 1 harmonics of ψ0 ∗ 1 vanish. It is precisely that choice of tetrad that led to our definition of the complex center of mass.

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