  A connection between superfluidity and gravitation is established for
physical stationary gravitational fields. We show that the spinning cosmic
string metric describes the gravitational field associated with the single
vortex in a superfluid condensate model for space-time outside the vortex core.
This metric differs significantly from the usual acoustic metric for the
Onsager-Feynman vortex. We also consider the question of what happens when many
vortices are present, and show that on large scales a G\"odel-like metric
emerges. In both the single and multiple vortex cases the presence of closed
time-like curves is attributed to the breakdown of superfluid rigidity.
