Massive scalar field in multiply connected flat spacetimes

Abstract

The vacuum expectation value of the stress-energy tensor $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ 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 $R^3 \times S^1$, $R^2 \times T^2$, $R^1 \times T^3$, the M\"{o}bius strip, and the Klein bottle. In Grant space, which contains nonchronal regions, whether $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ diverges on a chronology horizon or not depends on the field mass. For a sufficiently large mass $\left\langle 0\left| T_{\mu\nu} \right|0\right\rangle$ remains finite, and the metric backreaction caused by a massive quantized field may not be large enough to significantly change the Grant space geometry.