Scalar Field in Nariai's Universe with the Topology S Tilde/sub 2/ X S/sub 2/

摘要

As a requel to the previous note (Preprint RRK 84 - 19) dealing with a quantized scalar field in the de Sitter and Nariai universes (both satisfy the vacuum Einstein equations R/sub mu nu / = lambda g/sub mu nu /), a scalar field in Nariai's universe with the topology S tilde/sub 2/ x S/sub 2/ is worked out. The relating metrics specified by Eq.(2.1) are reduced not only to f(t) = e/sup t/a/ (epsilon = 0, dealt with in RRK 84 - 19), but also to f(t) = ch(t/a)(epsilon = 1) or sh(t/a)(epsilon = -1) (all of them satisfy the same equation f/f = 1/a/sup 2/), where A(t) = a ident to 1/ sqrt lambda . The differential equation (2.13) for T/sub kl/(t) (which is the time-dependent part of the scalar field phi(t,x)) is solved in the three cases epsilon = 0 and epsilon = +- 1, except for the fixation of complex constants B/sub k/ and C/sub k/absolute value of B sub K squared - absolute value of C sub k squared = 1). (ERA citation 13:025279)