  A formula is given for the variation of the Hawking energy along any
one-parameter foliation of compact spatial 2-surfaces. A surface for which one
null expansion is positive and the other negative has a preferred orientation,
with a spatial or null normal direction being called outgoing or ingoing as the
area increases or decreases respectively. A natural way to propagate such a
surface through a hypersurface is to choose the foliation such that the null
expansions are constant over each surface. For such uniformly expanding
foliations, the Hawking energy is non-decreasing in any outgoing direction, and
non-increasing in any ingoing direction, assuming the dominant energy
condition. It follows that the Hawking energy is non-negative if the foliation
is bounded at the inward end by either a point or a marginal surface, and in
the latter case satisfies the Penrose-Gibbons isoperimetric inequality. The
Bondi-Sachs energy may be expressed as a limit of the Hawking energy at
conformal infinity, and the energy-variation formula reduces at conformal
infinity to the Bondi-Sachs energy-loss formula.
