| Experiments in the past have demonstrated that there is a phase difference at a Josephson junction in a rotating superconducting loop, and that a rotating bulk superconductor is filled with a uniform magnetic field, called the London field. The present work shows that these phenomena are manifestations of the Sagnac effect in the de Broglie waves of the superconducting electrons. The existence of a Sagnac effect in superconductors suggests the idea of a superconducting inertial guidance gyroscope. In the interest of investigating this possibility, the present work develops a frame-independent model of a rotating superconductor and uses it to derive an appropriate generalization of the Ginzburg-Landau equations. It also provides suggestions concerning how to solve the Ginzburg-Landau equations numerically; in particular, it shows that rewriting the equations in terms of the amplitude and phase of the order parameter eliminates gauge ambiguity from the problem while reducing the number of unknown real-valued functions from five to four. If one assumes that the condensate density is spatially uniform, one can prove that: (a) imposing an angular velocity on a superconductor produces the same current density as imposing a uniform magnetic field everywhere on the boundary of an insulating region that surrounds the superconductor; (b) the current density in a superconductor rotating inside a co-rotating superconducting shield equals the current density in the same superconductor at rest with no applied magnetic field. Together, these two results nearly exclude the possibility of a superconducting inertial guidance gyroscope with no net charge and no moving parts. The analytical results are confirmed and illustrated by numerical results from a computer program that calculates the current density and the magnetic field in and around a superconducting cylinder centered inside a cylindrical shell. |
