One reason relativistic fluids are needed is that they can be used to model neutron stars. However, even though neutron stars are clearly general relativistic objects, one often starts with good old Newtonian physics when considering new applications. The main reason is that effects (such as acoustic modes of oscillation) which are primarily due to the fluids that make up the star can often be understood qualitatively from Newtonian calculations. There are also certain regimes in a neutron star where the Newtonian limit is sufficient quantitatively (such as the outer layers).
There has been much progress recently in the analysis of Newtonian multiple fluid systems. Prix  has developed an action-based formalism, analogous to what has been used here. Carter and Chamel [22, 23, 24] have done the same except that they use a fully spacetime covariant formalism. We will be somewhat less ambitious (for example, as in ) by extracting the Newtonian equations as the non-relativistic limit of the fully relativistic equations. Given the results of Prix as well as of Carter and Chamel we can think of this exercise as a consistency check of our equations.
The Newtonian limit consists of writing the general relativistic field equations to order where is the speed of light. The Newtonian equations are obtained, strictly speaking, in the limit where . To order the metric becomes[22, 23, 24].
If is the proper time as measured along a fluid element worldline, then the curve it traces out can be written
In order to write the single fluid Euler equations, keeping terms to the required order, it is necessary to explicitly break up the master function into its mass, kinetic, and “potential” energy parts, i.e. to write as
A Newtonian two-fluid system can be obtained in a similar fashion. As discussed in Section 10, the main difference is that we need two sets of worldlines, describable, say, by curves where is the proper time along a constituent’s worldline. Of course, entrainment also comes into play. Its presence implies that the relative flow of the fluids is required to specify the local thermodynamic state of the system, and that the momentum of a given fluid is not simply proportional to that fluid’s flux. This is the situation for superfluid [94, 110], where the entropy can flow independently of the superfluid Helium atoms. Superfluid can also be included in the mixture, in which case there will be a relative flow of the isotope with respect to , and relative flows of each with respect to the entropy .
Let us consider a two-fluid model like a mixture of and , or neutrons and protons in a neutron star. We will denote the two fluids as fluids and . The magnitude squared of the difference of three-velocities
The number density of each fluid obeys a continuity equation:[91, 6].
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