Massless scalar fields at null and spatial infinity in the Schwarzschild space-time

Abstract

It is known that massless scalar, Maxwell, and linearized metric fields (in an appropriate gauge) having data of compact support will evolve to be asymptotically flat on any asymptotically flat background space-time. However, little is known about the evolution of data that is reasonably well behaved but has nontrivial falloff at spatial infinity. Is the set of such data that evolves to be asymptotically flat at null infinity in a curved asymptotically flat space-time of the same size as, and does it consist of elements with falloff rates similar to the set of such data in Minkowski space-time? Stewart and Schmidt analyzed massless scalar fields on both the Minkowski and Schwarzschild space-times. Their calculations indicated that the set of Schwarzschild data in question was much smaller than the Minkowski set. In this paper, this problem is reexamined and it is determined, contrary to the indications of Stewart and Schmidt, that the Schwarzschild set is of the same size, and its elements have falloff rates similar to the corresponding Minkowski set. This result supports the ability of the definition of asymptotic flatness to admit a large class of space-times.