Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.