Having discussed the single fluid model, and how one accounts for stratification, it is time to move on to the problem of modeling multi-fluid systems. We will experience for the first time novel effects due to the presence of a relative flow between two interpenetrating fluids, and the fact that there is no longer a single, preferred rest-frame. This kind of formalism is necessary, for example, for the simplest model of a neutron star, since it is generally accepted that the inner crust is permeated by an independent neutron superfluid, and the outer core is thought to contain superfluid neutrons, superconducting protons, and a highly degenerate gas of electrons. Still unknown is the number of independent fluids that would be required for neutron stars that have quark matter in the deep core . The model can also be used to describe superfluid Helium and heat-conducting fluids, which is of importance for incorporation of dissipation (see Section 14). We will focus on this example here. It should be noted that, even though the particular system we concentrate on consists of only two fluids, it illustrates all new features of a general multi-fluid system. Conceptually, the greatest step is to go from one to two fluids. A generalization to a system with more degrees of freedom is straightforward.
In keeping with the previous Section 9, we will rely on use of the constituent index, which for all remaining formulas of this section will range over . In the example that we consider the two fluids represent the particles () and the entropy (). Once again, the number density four-currents, to be denoted , are taken to be separately conserved, meaning
We make this happen by introducing the three-dimensional matter space, the difference being that we now need two such spaces. These will be labelled by coordinates , and we recall that . This is depicted in Figure 8, which indicates the important facts that (i) a given spatial point can be intersected by each fluid’s worldline and (ii) the individual worldlines are not necessarily parallel at the intersection, i.e. the independent fluids are interpenetrating and can exhibit a relative flow with respect to each other. Although we have not indicated this in Figure 8 (in order to keep the figure as uncluttered as possible) attached to each worldline of a given constituent will be a fixed number of particles , , etc. (cf. Figure 7). For the same reason, we have also not labelled (as in Figure 7) the “push-forwards” (represented by the arrows) from the matter spaces to spacetime.
By pulling-back each constituent’s three-form onto its respective matter space we can once again construct three-forms that are automatically closed on spacetime, i.e. let
Lagrangian displacements on spacetime for each fluid, to be denoted , will now be introduced. They are related to the variations via
Associated with each constituent’s Lagrangian displacement is its own Lagrangian variation. These are naturally defined to be.
We are now in a position to construct an action principle that will yield the equations of motion and the stress-energy tensor. Again, the central quantity is the “master” function , which is now a function of all the different scalars that can be formed from the , i.e. the scalars together with
An unconstrained variation of with respect to the independent vectors and the metric takes the form
In the general case, the momentum of one constituent carries along some mass current of the other constituents. The entrainment only vanishes in the special case where is independent of () because then we obviously have . Entrainment is an observable effect in laboratory superfluids [94, 110] (e.g. via flow modifications in superfluid and mixtures of superfluid and ). In the case of neutron stars, entrainment is an essential ingredient of the current best explanation for the so-called glitches [95, 97]. Carter  has also argued that these “anomalous” terms are necessary for causally well-behaved heat conduction in relativistic fluids, and by extension necessary for building well-behaved relativistic equations that incorporate dissipation.
In terms of the constrained Lagrangian displacements, a variation of now yields
It must be noted that Equation (183) is significantly different from the multi-constituent version of Equation (166). This is true even if one is solving for a static and spherically symmetric configuration, where the fluid four-velocities would all necessarily be parallel. Simply put, Equation (183) still represents two independent equations. If one takes entropy as an independent fluid, then the static and spherically symmetric solutions will exhibit thermal equilibrium . This explains, for instance, why one must specify an extra condition (e.g. convective stability ) to solve for a double-constituent star with only one four-velocity.
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