A correspondence between Maxwell--Einstein theory and superfluidity

A planar superfluid is considered and interpreted in terms of electromagnetism and gravity. It has previously been suggested that the superfluid flow can be regarded as analogous to an electromagnetic field and that a non-vanishing density gradient give rise to a gravity-like force. The present work seeks to reconcile these hitherto distinct pictures into a unified exposition of vortex electrodynamics in a curved analogue space-time. By constructing a theory in which the planar Maxwell's equations are coupled to a (d+1)-dimensional conformally curved space-time, we expose a correspondence between the resulting equations of motion of the embedded fields and the dynamics of a quantum fluid. Finally, an effective vortex metric whose connection components are torsional is studied and its effect on the superfluid Maxwell's equations is elucidated upon.