Zero rest-mass fields including gravitation: asymptotic behaviour

Abstract

A zero rest-mass field of arbitrary spinsdetermines, at each event in space-time, a set of 2sprincipal null directions which are related to the radiative behaviour of the field. These directions exhibit the characteristic ‘peeling-off' behaviour of Sachs, namely that to orderr-k-1(k= 0, . . . , 2s), 2s-kof them coincide radially,rbeing a linear parameter in any advanced or retarded radial direction. This result is obtained in part I for fields of any spin in special relativity, by means of an inductive spinor argument which depends ultimately on the appropriate asymptotic behaviour of a very simple Hertz-type complex scalar potential. Spin (s- ½) fields are used as potentials for spinsfields, etc. Several examples are given to illustrate this, In particular, the method is used to obtain physically sensible singularity-free waves for each spin which can be of any desired algebraic type. In part II, a general technique is described, for discussing asymptotic properties of fields in curved space-times which is applicable to all asymptotically flat or asymptotically de Sitter space-times. This involves the introduction of ‘points at infinity’ in a consistent way. These points constitute a hypersurface boundaryIto a manifold whose interior is conformally identical with the original space-time. Zero rest-mass fields exhibit an essential conformal invariance, so their behaviour at ‘infinity’ can be studied at this hypersurface. Continuity atIfor the transformed field implies that the ‘peeling-off’ property holds. Furthermore, if the Einstein empty-space equations hold nearIthen continuity atIfor the transformed gravitational field is a consequence. This leads to generalizations of results due to Bondi and Sachs. The case when the Einstein-Maxwell equations hold nearIis also similarly treated here. The hypersurfaceIis space-like, time-like or null according as the cosmological constant is positive, negative or absent. The technique affords a covariant approach to the definition of radiation fields in general relativity. IfIis not null, however, the radiation field concept emerges as necessarily origin dependent. Further applications of the technique are also indicated.