Uniqueness of the Trautman--Bondi mass

TL;DR

This paper proves that the Trautman--Bondi mass is the unique monotonic functional of the vacuum Einstein equations with a smooth conformal null infinity, and it remains well-defined for polyhomogeneous metrics.

Contribution

It establishes the uniqueness of the Trautman--Bondi mass as the only monotonic BMS-invariant functional in a natural class for vacuum Einstein solutions.

Findings

Trautman--Bondi mass is uniquely monotonic under specified conditions.

The energy remains well-defined for polyhomogeneous metrics.

The functional depends solely on the metric through specific asymptotic terms.

Abstract

It is shown that the only functionals, within a natural class, which are monotonic in time for all solutions of the vacuum Einstein equations admitting a smooth ``piece'' of conformal null infinity Scri, are those depending on the metric only through a specific combination of the Bondi `mass aspect' and other next--to--leading order terms in the metric. Under the extra condition of passive BMS invariance, the unique such functional (up to a multiplicative factor) is the Trautman--Bondi energy. It is also shown that this energy remains well-defined for a wide class of `polyhomogeneous' metrics.