  The low-energy dynamics of relativistic continuous media is given by a
shift-symmetric effective theory of four scalar fields. These scalars describe
the embedding in spacetime of the medium and play the role of St\"uckelberg
fields for spontaneously broken spatial and time translations. Perfect fluids
are selected imposing a stronger symmetry group or reducing the field content
to a single scalar. We explore the relation between the field theory
description of perfect fluids to thermodynamics. By drawing the correspondence
between the allowed operators at leading order in derivatives and the
thermodynamic variables, we find that a complete thermodynamic picture requires
the four Stuckelberg fields. We show that thermodynamic stability plus the
null-energy condition imply dynamical stability. We also argue that a
consistent thermodynamic interpretation is not possible if any of the shift
symmetries is explicitly broken.
