Massive scalar field in multiply connected flat spacetimes

The vacuum expectation value of the stress-energy tensor 〈0‖${\mathit{T}}_{\mathrm{\mu }\nu }$‖0〉 is calculated in several multiply connected flat spacetimes for a massive scalar field with arbitrary curvature coupling. We find that a nonzero field mass always decreases the magnitude of the energy density in chronology-respecting manifolds such as ${\mathit{R}}^3$×${\mathit{S}}^1$, ${\mathit{R}}^2$×${\mathit{T}}^2$, ${\mathit{R}}^1$×${\mathit{T}}^3$, the Möbius strip, and the Klein bottle. In Grant space, which contains nonchronal regions, whether or not 〈0‖${\mathit{T}}_{\mathrm{\mu }\nu }$‖0〉 diverges on a chronology horizon depends on the field mass. For a sufficiently large mass 〈0‖${\mathit{T}}_{\mathrm{\mu }\nu }$‖0〉 remains finite, and the metric back reaction caused by a massive quantized field may not be large enough to significantly change the Grant space geometry.