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.Comment: latex, 33 page
