The casimir effect for a massless scalar field in the n dimensional Einstein universe with Dirichlet boundary conditions on a sphere

We use the representation theory of SO(2; n) to determine the renormalized vacuum energy for a massless scalar field in the n-dimensional Einstein universe subject to Dirichlet boundary conditions on a sphere of maximum radius. The problem is an exactly solvable one. This is in remarkable contrast to the analogous problem in flat n dimensional Minkowski space where, except for the lowest dimensional case (n = 2), there is no known exactly solvable method of solution for any radius of the spherical boundary. For n = 4 our results agree with those of Bayen and Özcan, Class. Quant. Grav., 10 (1993) L115-L121. We use our results to obtain some qualitative information about the Casimir effect for spherical boundaries of smaller radii, and we comment on how one may apply these results to obtain information about the corresponding problem in Minkowski space.