In the analog gravity framework, the acoustic disturbances in a moving
fluid can be described by an equation of motion identical to a
relativistic scalar massless field propagating in curved space-time.
This description is possible only when the fluid under consideration is
barotropic, inviscid, and irrotational. In this case, the propagation of
the perturbations is governed by an acoustic metric that depends
algebrically on the local speed of sound, density, and the background
flow velocity, the latter assumed to be vorticity-free. In this work we
provide a straightforward extension in order to go beyond the
irrotational constraint. Using a charged-relativistic and
nonrelativistic-Bose- Einstein condensate as a physical system, we show
that in the low-momentum limit and performing the eikonal approximation
we can derive a d'Alembertian equation of motion for the charged phonons
where the emergent acoustic metric depends on flow velocity in the
presence of vorticity.
