On Extracting Physical Content from Asymptotically Flat Space-Time Metrics

TL;DR

This paper develops a geometric framework to extract meaningful physical quantities like energy, momentum, and angular momentum from asymptotically flat space-times in general relativity, using null geodesic congruences and BMS symmetry.

Contribution

It generalizes shear-free null geodesic congruences to asymptotically shear-free ones, providing a geometric interpretation of the Bondi four-momentum and its evolution.

Findings

Defines a center of mass and spin vector with geometric meaning.

Derives a conservation law for angular momentum with flux.

Provides a kinematic interpretation of Bondi momentum in all asymptotically flat spacetimes.

Abstract

A major issue in general relativity, from its earliest days to the present, is how to extract physical information from any solution or class of solutions to the Einstein equations. Though certain information can be obtained for arbitrary solutions, e.g., via geodesic deviation, in general, because of the coordinate freedom, it is often hard or impossible to do. Most of the time information is found from special conditions, e.g., degenerate principle null vectors, weak fields close to Minkowski space (using coordinates close to Minkowski coordinates) or from solutions that have symmetries or approximate symmetries. In the present work we will be concerned with asymptotically flat space times where the approximate symmetry is the Bondi-Metzner-Sachs (BMS) group. For these spaces the Bondi four-momentum vector and its evolution, found from the Weyl tensor at infinity, describes the total energy-momentum of the interior source and the energy-momentum radiated. By generalizing the structures (shear-free null geodesic congruences) associated with the algebraically special metrics to asymptotically shear-free null geodesic congruences, which are available in all asymptotically flat space-times, we give kinematic meaning to the Bondi four-momentum. In other words we describe the Bondi vector and its evolution in terms of a center of mass position vector, its velocity and a spin-vector, all having clear geometric meaning. Among other items, from dynamic arguments, we define a unique (at our level of approximation) total angular momentum and extract its evolution equation in the form of a conservation law with an angular momentum flux.