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

TL;DR

This paper investigates how the global topology of RP^3 de Sitter spacetime affects local quantum field observables, showing that differences from standard de Sitter space diminish exponentially over time, preserving thermal properties.

Contribution

It demonstrates that global topological differences in RP^3 de Sitter spacetime lead to exponentially decreasing deviations in local quantum observables compared to standard de Sitter space.

Findings

Differences in stress-energy tensor decay exponentially over time.

Detector response approaches thermal de Sitter results exponentially.

Vacuum state expansions converge exponentially at early and late times.

Abstract

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 topology RP^3. We compare the Euclidean vacua on RP3dS and dS in terms of three quantities that are relevant for an inertial observer: (i) the stress-energy tensor; (ii) the response of an inertial monopole particle detector; (iii) the expansion of the Euclidean vacuum in terms of many-particle states associated with static coordinates centered at an inertial world line. In all these quantities, the differences between RP3dS and dS turn out to fall off exponentially at early and late proper times along the inertial trajectory. In particular, (ii) and (iii) yield at early and late proper times in RP3dS the usual thermal result in the de Sitter Hawking temperature. This conforms to what one might call an exponential law: in expanding locally de Sitter spacetimes, differences due to global topology should fall off exponentially in the proper time.