  We write a covariant transport equation for the phonon excitations of a
relativistic superfluid valid at small temperatures. The hydrodynamical
equations for this system are derived from the effective field theory
associated to the superfluid phonons. We describe how to construct the kinetic
theory for the phonon quasiparticles using a relativistic generalization of the
analogue model of gravity developed by Unruh. This gravity analogy relies on
the equivalence between the action of a phonon field moving in a superfluid
background with that of a boson propagating in a given curved space-time.
Exploiting this analogy we obtain continuity equations for the phonon current,
entropy and energy-momentum tensor in a covariant form, valid in any reference
frame. Our aim is to shed light on some aspects of transport phenomena of
relativistic superfluidity. In particular, we are interested in the low
temperature regime of the color flavor locked phase, which is a color
superconducting and superfluid phase of high density QCD that may be realized
in the core of neutron stars.
