Some remarks on loop variables, holonomy transformation, and gravitational Aharonov-Bohm effect

Abstract

We show that the loop variables for cylindrically symmetric space-times are elements of the Lorentz group SO(3, 1), or more generally, they are elements of the covering group of the Lorentz group, in order to include fermions. We use our results to study the gravitational analogue of the Aharanov-Bohm effect. Some examples of stationary space-times that provide a gravitational Aharanov-Bohm effect are given. We compute the exact expression for the holonomy transformation, for a circle, that corresponds to the field of the multiple cosmic string solution, showing that only the cosmic strings encircled by the loop (circles, in our case) contribute to the phase factor acquired by a vector when parallel transported in this background space-time. We use our results to study the topological nature of the space-time corresponding to two cosmic strings moving relative to each other.