Equation of state for cool relativistic two-constituent superfluid dynamics

The natural relativistic generalization of Landau's two-constituent superfluid theory can be formulated in terms of a Lagrangian scrL that is given as a function of the entropy current four-vector s^ρ and the gradient ∇_ρcphi of the superfluid phase scalar. It is shown that in the ``cool'' regime, for which the entropy is attributable just to phonons (not rotons), the Lagrangian function scrL{s{→},{∇}cphi} is given by an expression of the form scrL=P-3{ψ} where P represents the pressure as a function just of ∇_ρcphi in the (isotropic) cold limit. The entropy current-dependent contribution {ψ} represents the generalized pressure of the (nonisotropic) phonon gas, which is obtained as the negative of the corresponding grand potential energy per unit volume, whose explicit form has a simple algebraic dependence on the sound or ``phonon'' speed cP that is determined by the cold pressure function P.