Wave Zone in General Relativity

It is shown that in general relativity a "wave zone" may be defined for systems which are asymptotically flat. In this region, gravitational radiation propagates freely, independent of its interior sources, and obeys the superposition principle. The independent dynamical variables of the full theory which describe the radiation are shown to be coordinate invariant in the wave zone and to satisfy the linearized theory's equations there. Thus, the basic properties of free waves in linear field theories (e.g., electrodynamics) are reproduced for the gravitational case. True waves are also clearly distinguished from so-called "coordinate waves." Reduction to asymptotic form (taking leading powers of $\frac{1}{r}$), is not identical to linearization, since, for example, the Newtonian-like $\frac{1}{r}$ part of the metric begins quadratically in the linear theory's variables. The Poynting vector of the full theory, which measures energy flux in the wave zone, is correspondingly shown to be given by the linearized theory's formula. This Poynting vector is also shown to be coordinate-invariant in the wave zone. All the physical quantities may therefore be evaluated in any frame becoming rectangular sufficiently rapidly. A brief discussion of measurements of the canonical variables in the wave zone is given. The relation between the present work and other treatments of gravitational radiation is examined.