Masses at null infinity for Einstein's equations in harmonic coordinates

TL;DR

This paper provides a clear method to define the mass at null infinity in Einstein's equations using harmonic coordinates, demonstrating that these definitions satisfy the Bondi mass loss law through three approaches.

Contribution

It introduces a novel, straightforward way to define mass at null infinity in harmonic coordinates, including an original method involving special characteristic coordinates.

Findings

All three definitions satisfy the Bondi mass loss law.

The methods depend only on the ADM mass and asymptotics of the metric.

The approach simplifies understanding of mass at null infinity in Einstein's equations.

Abstract

In this work we give a complete picture of how to in a direct simple way define the mass at null infinity in harmonic coordinates in three different ways that we show satisfy the Bondi mass loss law. The first and second way involve only the limit of metric (Trautman mass) respectively the null second fundamental forms along asymptotically characteristic surfaces (asymptotic Hawking mass) that only depend on the ADM mass. The last in an original way involves construction of special characteristic coordinates at null infinity (Bondi mass). The results here rely on asymptotics of the metric derived in [24].