On an integral formula on hypersurfaces in General Relativity

We derive a general integral formula on an embedded hypersurface for general relativistic space-times. Suppose the hypersurface is foliated by two-dimensional compact "sections" Ss . Then the formula relates the rate of change of the divergence of outgoing light rays integrated over Ss under change of section to geometric (convexity and curvature) properties of Ss and the energy-momentum content of the space-time. We derive this formula using the Sparling-Nester-Witten identity for spinor fields on the hypersurface by appropriate choice of the spinor fields. We discuss several special cases which have been discussed in the literature before, most notably the Bondi mass loss formula. 1 Introduction The purpose of this article is to derive and to discuss an integral formula which can be obtained from the part of the Einstein equations which is "intrinsic" to a hypersurface. This formula contains several special cases which have been discussed before in various different contexts. These..