Null Geodesic Congruences, Asymptotically Flat Space-Times and Their Physical Interpretation

TL;DR

This paper explores the geometric properties of shear-free null geodesic congruences in asymptotically flat space-times and demonstrates their applications in defining physical quantities like mass, momentum, and angular momentum directly from asymptotic fields in general relativity.

Contribution

It develops a framework linking asymptotically shear-free congruences to the extraction of physical properties from gravitational and electromagnetic fields at infinity.

Findings

Defined asymptotic center-of-mass and equations of motion.

Provided a kinematic interpretation of Bondi momentum.

Established a method to determine intrinsic spin and angular momentum at infinity.

Abstract

Shear-free or asymptotically shear-free null geodesic congruences 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 affects. It is the purpose of this paper to develop these issues and find applications in GR. The applications center around the problem of extracting interior physical properties of an asymptotically flat space-time 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.