A new look at the Bondi-Sachs energy-momentum

TL;DR

This paper develops a conformally invariant, gauge-independent framework for calculating the Bondi-Sachs energy-momentum at null infinity, addressing issues of normalization and mass aspect definition in numerical relativity.

Contribution

It introduces a conformally invariant GHP formalism to define the Bondi energy-momentum in a way suitable for numerical and arbitrary cuts of null infinity.

Findings

Provides a natural definition of the space of asymptotic translations.

Defines the Bondi news and mass-aspect in a conformally invariant manner.

Introduces the concept of co-curvature related to the Gau{ extquoteright}s curvature on cuts of null infinity.

Abstract

How does one compute the Bondi mass on an arbitrary cut of null infinity $\scri$ when it is not presented in a Bondi system? What then is the correct definition of the mass aspect? How does one normalise an asymptotic translation computed on a cut which is not equipped with the unit-sphere metric? These are questions which need to be answered if one wants to calculate the Bondi-Sachs energy-momentum for a space-time which has been determined numerically. Under such conditions there is not much control over the presentation of $\scri$ so that most of the available formulations of the Bondi energy-momentum simply do not apply. The purpose of this article is to provide the necessary background for a manifestly conformally invariant and gauge independent formulation of the Bondi energy-momentum. To this end we introduce a conformally invariant version of the GHP formalism to rephrase all the well-known formulae. This leads us to natural definitions for the space of asymptotic translations with its Lorentzian metric, for the Bondi news and the mass-aspect. A major role in these developments is played by the "co-curvature", a naturally appearing quantity closely related to the Gau{\ss} curvature on a cut of~$\scri$.