16.1 Superfluids
Neutron star physics provides ample motivation for the need to develop a relativistic description of
superfluid systems. As the typical core temperatures (below
) are far below the Fermi temperature
of the various constituents (of the order of
for baryons) neutron stars are extremely cold on the
nuclear temperature scale. This means that, just like ordinary matter at near absolute zero temperature, the
matter in the star will most likely freeze to a solid or become superfluid. While the outer parts of the star,
the so-called crust, form an elastic lattice, the inner parts of the star are expected to be superfluid. In
practice, this means that we must be able to model mixtures of superfluid neutrons and superconducting
protons. It is also likely that we need to understand superfluid hyperons and color superconducting
quarks. There are many hard physics questions that need to be considered if we are to make
progress in this area. In particular, we need to make contact with microphysics calculations
that determine the various parameters of such multi-fluid systems. However, we will ignore
this aspect and focus on the various fluid models that have been used to describe relativistic
superfluids.
One of the key features of a pure superfluid is that it is irrotational. Bulk rotation is mimicked by the
formation of vortices, slim “tornadoes” representing regions where the superfluid degeneracy is broken. In
practice, this means that one would often, e.g. when modeling global neutron star oscillations, consider a
macroscopic model where one “averages” over a large number of vortices. The resulting model would closely
resemble the standard fluid model. Of course, it is important to remember that the vortices are present on
the microscopic scale, and that they may affect the values of various parameters in the problem. There are
also unique effects that are due to the vortices, e.g. the mutual friction that is thought to be the key agent
that counteracts relative rotation between the neutrons and protons in a superfluid neutron star
core [79].
For the present discussion, let us focus on the simplest model problem of superfluid
. We then have
two fluids, the superfluid Helium atoms with particle number density
and the entropy with particle
number density
. From the derivation in Section 6 we know that the equations of motion can be
written
and
To make contact with other discussions of the superfluid problem [25, 26
, 27, 29], we will use the notation
and
. Then the equations that govern the motion of the entropy obviously become
Now, since the superfluid constituent is irrotational we will have
Hence, the second equation of motion is automatically satisfied once we impose that the fluid is irrotational.
The particle conservation law is, of course, unaffected. This example shows how easy it is to
specify the equations that we derived earlier to the case when one (or several) components
are irrotational. It is worth emphasizing that it is the momentum that is quantized, not the
velocity.
It is instructive to contrast this description with the potential formulation due to Khalatnikov and
colleagues [62
, 69
]. We can obtain this alternative formulation in the following way [26]. First of all, we
know that the irrotationality condition implies that the particle momentum can be written as a gradient of
a scalar potential
(say). That is, we have
Here
is the mass of the Helium atom and
is what is traditionally (and somewhat
confusedly) referred to as the “superfluid velocity”. We see that it is really a rescaled momentum.
Next assume that the momentum of the remaining fluid (in this case, the entropy) is written
Here
is Lie transported along the flow provided that
(assuming that the equation of
motion (343) is satisfied). This leads to
There is now no loss of generality in introducing further scalar potentials
and
such that
, where the potentials are constant along the flow-lines as long as
Given this, we have
Finally, comparing to Khalatnikov’s formulation [62, 69] we define
and let
and
. Then we arrive at the final equation of motion
Equations (345) and (350), together with the standard particle conservation laws, are the key equations in
the potential formulation. As we have seen, the content of this description is identical to that of the
canonical variational picture that we have focussed on in this review. We have also seen how the various
quantities can be related. Of course, one has to exercise some care in using this description. After all,
referring to the rescaled momentum as the “superfluid velocity” is clearly misleading when the entrainment
effect is in action.