  The duality found by Aharonov and Casher for topological phases in the
electromagnetic field is generalized to an arbitrary linear interaction. This
provides a heuristic principle for obtaining a new solution of the field
equations from a known solution. This is applied to the general relativistic
Sagnac phase shift due to the gravitational field in the interference of mass
or energy around a line source that has angular momentum and the dual phase
shift in the interference of a spin around a line mass. These topological
phases are treated both in the linearized limit of general relativity and the
exact solutions for which the gravitational sources are cosmic strings
containing torsion and curvature, which do not have a Newtonian limit.
