The exponential law: monopole detectors, Bogoliubov transformations, and the thermal nature of the Euclidean vacuum in RP3 de Sitter spacetime

We consider scalar field theory on the RP3 de Sitter spacetime (RP(3)dS), w hich is locally isometric to de Sitter space (dS) but has spatial topology RP3. We compare the Euclidean vacua on RP(3)dS and dS in terms of three qua ntities that are relevant for an inertial observer: (a) the stress-energy t ensor; (b) the response of an inertial monopole particle detector; (c) the expansion of the Euclidean vacuum in terms of many-particle states associat ed with static coordinates centred at an inertial worldline. In all these q uantities, the differences between RP(3)dS and dS turn out to fall off expo nentially at early and late proper rimes along the inertial trajectory. In particular, (b) and (c) yield at early and late proper times in RP(3)dS the usual thermal result for the de Sitter Hawking temperature. This conforms to what one might call an exponential law: in expanding locally de Sitter s pacetimes, differences due to global topology should fall off exponentially in the proper time.