Twisting null geodesic congruences, scri, H-space and spin-angular momentum

Abstract

The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat spacetime with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twistiing) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world-line in a four-parameter complex space. Surprisingly this parameter space turns out to be the H-space that is associated with the real physical space-time under consideration. The main development in this work is the demonstration of how this complex world-line can be made both unique and also given a physical meaning. More specifically by forcing or requiring a certain term in the asymptotic Weyl tensor to vanish, the world-line is uniquely determined and becomes (by several arguments) identified as the 'complex center-of-mass'. Roughly, its imaginary part becomes identified with the intrinsic spin-angular momentum while the real part yields the orbital angular momentum.One should think of this work as developing a generalization of the properties of the algebraically special space-times in the sense that the term that is forced here to vanish, is automatically vanishing (among many other terms) for all the algebraically special metrics. This is demonstrated in one of several given examples. It was, in fact, an understanding of the algebraically special metrics and their associated shearfree null congruence that led us to this construction of the asymptotically shear-free congruences and the unique complex world-line.The Robinson-Trautman metrics and the Kerr and charged Kerr metrics with their properties are explicit examples of the construction given here.

Angular momentum, spinors and twistors

“…Differing responses to it have resulted in very different approaches to angular momentum. Some workers (notably Newman and followers -see [10][11][12][13][14] and references therein), in effect, drop the requirement that the space of origins be affine. Others drop the fourdimensionality, and indeed it is most common nowadays to pass to infinite-dimensional spaces of origins (as in the BMS-based approaches of Ashtekar and Streubel [15,16], and of Dray and Streubel [17,18]; the idea using the BMS group goes back to Winicour and Tamburino [19]).…”

“…This asymptotically constant vector is (by definition) the same as the field (1/2)o 1 o 1 + ι 1 ι 1 . That field has the constant value unity, 11 reflecting the normalization of the spin-frame to the Bondi system.…”

“…A. Twistors at I + Twistors in Minkowksi space were defined as solutions to the twistor equation (11). This equation is conformally invariant, and so extends meaningfully to I + , but it is generally overdetermined there and has only the trivial solution.…”

Asymptotic Flatness and Algebraically Special Metrics

“…Such metrics can play an important role in a search for explicit solutions describing gravitational radiation from bounded sources, as well as in general approach to such systems (see [6] and references therein. )…”

Null infinity, H-space and equations of motion

“…Recently [7][8][9][10] we returned to the study of asymptotically flat Einstein/ Einstein-Maxwell fields and discovered a hidden structure. Though shearfree ngc's cannot be found in arbitrary space-times, their idea can be generalized to asymptotically shear-free ngc's.…”