The exponential law: Monopole detectors, Bogoliubov transformations, and the thermal nature of the Euclidean vacuum in RP^3 de Sitter spacetime

摘要

We consider scalar field theory on the RP^3 de Sitter spacetime (RP3dS),which is locally isometric to de Sitter space (dS) but has spatial topologyRP^3. We compare the Euclidean vacua on RP3dS and dS in terms of threequantities that are relevant for an inertial observer: (i) the stress-energytensor; (ii) the response of an inertial monopole particle detector; (iii) theexpansion of the Euclidean vacuum in terms of many-particle states associatedwith static coordinates centered at an inertial world line. In all thesequantities, the differences between RP3dS and dS turn out to fall offexponentially at early and late proper times along the inertial trajectory. Inparticular, (ii) and (iii) yield at early and late proper times in RP3dS theusual thermal result in the de Sitter Hawking temperature. This conforms towhat one might call an exponential law: in expanding locally de Sitterspacetimes, differences due to global topology should fall off exponentially inthe proper time.