### 11.2 Two-constituent, single fluid case

Now consider the case when there are the two constituents with densities and , two conserved
density currents and , two chemical potential covectors and , but still only one
four-velocity . The master function depends on both and meaning that
where
and is as in Equation (190) except that each is replaced with . We have chosen to
use the letter to represent what is a true multi-constituent effect, which arises due to
composition gradients in the system. An alternative would have been to use since the effect
is due to the presence of different constituents. However, in his papers Carter tends to use
, referring to the bulk entropy contribution as “caloric”. Our chosen notation is intended to
avoid confusion. It is also the case that the presence of the composition term has not
been emphasized in previous work. This may be unfortunate since a composition variation
is known to affect the dynamics of a system, e.g. by giving rise to the g-modes in a neutron
star [98].
The fact that is parallel to implies that it is only the magnitude of the entropy density
current that is independent. One can show that the condition of transverse propagation, as applied to both
currents, implies

Now, we proceed as in the previous example, noting that the equation of motion is , which
reduces to
where
The speed of sound is thus