Probing spatial orientability of Friedmann--Robertson--Walker spatially
  flat spacetime
  One important global topological property of a spacetime manifold is
orientability. It is widely believed that spatial orientability can only be
tested by global journeys around the Universe to check for
orientation-reversing closed paths. Since such global journeys are not
feasible, theoretical arguments that combine universality of physical
experiments with local arrow of time, CP violation and CPT invariance are
usually offered to support the choosing of time- and space-orientable spacetime
manifolds. The nonexistence of globally defined spinor fields on a
non-orientable spacetime is another theoretical argument for orientability.
However, it is conceivable that orientability can be put to test by local
physical effects. In this paper, we show that it is possible to locally access
spatial orientability of a spatially flat Friedmann--Robertson-Walker spacetime
through quantum vacuum electromagnestic fluctuations. We argue that a putative
non-orientability of the spatial sections of spatially flat FRW spacetime can
be ascertained by the study of the stochastic motions of a charged particle or
a point electric dipole under quantum vacuum electromagnetic fluctuations. In
particular, the stochastic motions of a dipole permit the recognition of a
presumed non-orientability of $3-$space in itself.
