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Uniqueness of the Trautman--Bondi mass
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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.
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Logarithmic matching between past infinity and future infinity: The massless scalar field
Massless scalar fields with dominant logarithmic terms at null infinity obey an antipodal matching condition with a minus sign, opposite to the standard no-log matching.
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