Quasi-Localisation of Bondi-Sachs Energy Loss

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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.