ON AN INTEGRAL FORMULA ON HYPERSURFACES IN GENERAL-RELATIVITY

We derive a general integral formula on an embedded hypersurface for g eneral relativistic spacetimes. Suppose the hypersurface is foliated b y two-dimensional compact 'sections' S-s. Then the formula relates the rate of change of the divergence of outgoing light rays integrated ov er S-s under change of section to geometric (convexity and curvature) properties of S-s and the energy-momentum content of the spacetime. We derive this formula using the Sparling-Nester-Witten identity for spi nor fields on the hypersurface by appropriate choice of the spinor fie lds. We discuss several special cases which have been discussed in the literature before, most notably the Bondi mass loss formula.