Abstract
A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues.This work starts with a detailed exposition of the theory of shear-free and asymptotically shear-free null geodesic congruences, i.e., congruences with shear that vanishes at future conformal null infinity. A major portion of the exposition lies in the analysis of the space of regular shear-free and asymptotically shear-free null geodesic congruences. This analysis leads to the space of complex analytic curves in an auxiliary four-complex dimensional space, {mathcal H}-space. They in turn play a dominant role in the applications.The applications center around the problem of extracting interior physical properties of an asymptotically-flat spacetime directly from the asymptotic gravitational (and Maxwell) field itself, in analogy with the determination of total charge by an integral over the Maxwell field at infinity or the identification of the interior mass (and its loss) by (Bondi’s) integrals of the Weyl tensor, also at infinity.More specifically, we will see that the asymptotically shear-free congruences lead us to an asymptotic definition of the center-of-mass and its equations of motion. This includes a kinematic meaning, in terms of the center-of-mass motion, for the Bondi three-momentum. In addition, we obtain insights into intrinsic spin and, in general, angular momentum, including an angular-momentum-conservation law with well-defined flux terms. When a Maxwell field is present, the asymptotically shear-free congruences allow us to determine/define at infinity a center-of-charge world line and intrinsic magnetic dipole moment.
Highlights
Though from the very earliest days of Lorentzian geometries, families of null geodesics (null geodesic congruences (NGCs)) were obviously known to exist, it took many years for their significance to be realized
We investigate the Universal-Cut Function (UCF) associated with asymptotically-vanishing Maxwell fields and in particular the shear-free congruences associated with the Lienard–Wiechert fields
∙ The complex light-cones emanating from a timelike complex analytic curve in complex Minkowski space, za = ξa(τ ) parametrized by the complex parameter τ = s + iλ, has for each fixed value of s and λ a limited set of null geodesics that reach real I+
Summary
These revisions were done exclusively by Adamo and Newman, the author order was slightly changed. There was nothing essentially wrong in the earlier version, but we have included several new results (in the text and in appendices), corrected an error of interpretation, and (the main reason for the revision) we found much easier ways of doing some of the long calculations with very much simpler arguments. 2. In Section 3.2 we have added in a fair amount to the discussion of real structures associated with the complex world lines. We basically rewrote the entire section; i.e., the derivation of our major results, using much simpler arguments. This greatly shortened the derivation and associated argument. Two new Appendices (E, F) were added presenting new material found since the previous version
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