# Zero Rest-Mass Fields Including Gravitation: Asymptotic Behaviour

R. Penrose

## Abstract

A zero rest-mass field of arbitrary spin s determines, at each event in space-time, a set of 2s principal 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 order r$^{-k-1}$ (k = 0, \ldots, 2s), 2s-k of them coincide radially, r being 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 - $\frac{1}{2}$) fields are used as potentials for spin s fields, 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 boundary $\mathscr{J}$ to 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 at $\mathscr{J}$ for the transformed field implies that the 'peeling-off' property holds. Furthermore, if the Einstein empty-space equations hold near $\mathscr{J}$ then continuity at $\mathscr{J}$ for 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 near $\mathscr{J}$ is also similarly treated here. The hypersurface $\mathscr{J}$ is 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. If $\mathscr{J}$ is not null, however, the radiation field concept emerges as necessarily origin dependent. Further applications of the technique are also indicated.