General relativity and the peeling-off behaviour of the gravitational field in an asymptotically flat space-time

Abstract

This thesis investigates the peeling-off property of zero rest-mass fields in asymptotically flat space-times, as described by Penrose. Unlike other literature, this work is designed to be accessible to undergraduate physics or mathematics students. It provides a more detailed derivation including numerous explicit calculations, which is appropriate for the assumed background knowledge. The thesis is structured to build from fundamental mathematical concepts to advanced applications in general relativity. The mathematical concepts introduced are topology, compactifications and differential geometry. The main body of the work applies these mathematical tools to the Minkowski space-time and extends the analysis to more general asymptotically flat space-times.

A zero rest-mass field of spin s determines at each event in space-time a set of 2s principal null directions. These are related to the radiative behaviour of the field. These directions exhibit the 'peeling-off' behaviour: to order r^-k-1 (k = 0, ..., 2s), 2s-k directions coincide radially, where r is an affine parameter on a null geodesic. Criteria for asymptotically simple and asymptotically flat space-times are given, and this peeling-off behaviour is studied in these settings. This involves the introduction of points 'at infinity' through a conformal completion. These points at infinity then become an ordinary hypersurface ℑ to the conformally completed manifold. The conformal transformations of zero rest-mass fields are investigated so that their behaviour at infinity can be studied at this hypersurface. If the transformed field is continuous at ℑ, we find that the peeling-off property holds. If the Einstein empty-space equations without cosmological constant hold near the boundary, the transformed gravitational field is found to be continuous at the boundary, so that the peeling-off property holds.