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 [91] 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 [5]) 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
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 [113].
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:
Each fluid is also seen to satisfy an Euler-type equation, which ensures the conservation of total momentum. This equation can be written where i.e.
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