Gaussian curvature and global effects: Gravitational Aharonov-Bohm effect revisited

By using the Gauss-Bonnet formula, the integral of the Gaussian curvature over a 2-surface enclosed by a curve in the asymptotically flat region of a static spacetime was found to be a measure of a gravitational analogue of the Aharonov-Bohm effect by Ford and Vilenkin in the linearized regime. Employing the $1+3$ formulation of spacetime decomposition, we study the same effect in the context of the full Einstein field equations for stationary spacetimes. Applying our approach to static tubelike and cylindrical distributions of dust, not only do we recover their result but we also obtain an extra term which is interpreted to be representing the classical version of the Colella-Overhauser-Werner effect (the Colella-Overhauser-Werner experiment).

Figure 1

Closed path $C$ of a particle encircling the nonvacuum cylindrical region at the asymptotically flat region.

Figure 2

Closed path $C^{\prime }$ of a particle which does not encircle the nonvacuum cylindrical region.