On an integral formula on hypersurfaces in General Relativity

TL;DR

This paper derives a new integral formula in General Relativity relating the change in divergence of outgoing light rays on hypersurfaces to geometric and energy-momentum properties, generalizing known results like the Bondi mass loss formula.

Contribution

It introduces a general integral formula for hypersurfaces in space-times using the Sparling-Nester-Witten identity, extending previous special-case results.

Findings

Relates divergence change to geometric and energy-momentum properties

Generalizes the Bondi mass loss formula

Uses spinor fields and the Sparling-Nester-Witten identity

Abstract

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'' $S_s$. Then the formula relates the rate of change of the divergence of outgoing light rays integrated over $S_s$ under change of section to geometric (convexity and curvature) properties of $S_s$ 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.