Newman-Penrose constants and the tails of self-gravitating waves

We study the properties of the decay of a self-gravitating radiation field by analyzing the relation between behavior in the weak field regime, test field behavior in a Schwarzschild background, and strong field behavior. Our model consists of a spherically symmetric scalar field incident on a reflecting barrier, which allows all these regimes to be treated on a common nonsingular manifold. Our primary conclusion, in the curved space case, is that there are two distinct types of late decay determined by whether or not the Newman-Penrose constant for the scalar field vanishes. For the nonvanishing case, the radiation tail decays as $\frac{1}{t}$, with respect to Bondi time, but there are also ln $\frac{t}{t^2}$ corrections, as well as the exponentially decaying contributions associated with quasinormal modes.