Notes on the Wave Equation on Asymptotically Euclidean Manifolds

The primary object of this paper is to discuss the asymptotics of solutions of the wave equation on an asymptotically Euclidean manifold, when the initial data have compact support. By going over to an appropriate conformal metric, it is shown that (just as for the ordinary wave equation) such a solution has a forward (“future”) and a backward (“past”) radiation field. The same method is then used to define “end points” of bicharacteristics that begin and end above the boundary (the analogue of the sphere at infinity in the Euclidean case), and to derive a relation between the wave front sets of the two radiation fields. The extension of these results to solutions with finite energy is also briefly discussed.