Electromagnetic modes and energy production in the formation of the half Einstein universe

Abstract

The purpose of this paper is twofold. First, the electromagnetic modes in the static half Einstein universe are calculated, assuming that the spherical boundary lying at the position $\ensuremath{\chi}=\ensuremath{\pi}/2$ is perfectly conducting (\ensuremath{\chi} is the radial parameter). The half Einstein universe, in contradistinction to the full Einstein universe, enables us to impose this perfect conductivity condition, which is the natural analogue to the Dirichlet condition conventionally adopted in studies of the scalar field. Second, we calculate the average photon number $N$ and the corresponding energy produced in the lowest mode if the half Einstein universe is ``suddenly'' formed from an initial Minkowski universe. Here we use the same kind of formalism as previously used by Parker and others [Phys. Rev. Lett. 59, 1369 (1987)] when studying the sudden formation of cosmic strings. We find that the photon number is very small, $N\ensuremath{\simeq}0.01,$ and that it is independent of the magnitude of the cosmic scale factor ${a}_{0}.$