Twisting Null Geodesic Congruences, Scri, H-Space and Spin-Angular Momentum

TL;DR

This paper explores asymptotically flat Einstein solutions by linking shear-free null geodesic congruences to a complex world-line in H-space, revealing a geometric interpretation of spin and orbital angular momentum.

Contribution

It introduces a novel method to identify a unique complex world-line associated with asymptotically flat spacetimes, connecting it to physical angular momentum properties.

Findings

Complex world-line corresponds to the physical center-of-mass.

Imaginary part of the world-line relates to spin-angular momentum.

Real part of the world-line relates to orbital 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 space-time 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.