### 4.3 Moments as four-fluxes and the vector potential

Just like in the non-relativistic case the most plausible – and popular – choice of the four-fluxes in
relativistic thermodynamics is moments of the phase density of the atoms, viz.
This is formally identical to the non-relativistic case that was treated in
Section 3.
There are essential differences, however
Both are important differences. But many results from the non-relativistic theory will remain formally
valid.

Thus for instance in the relativistic case we still have

with
just like (33) and (39). We conclude that the vector potential is not generally in the class of
moments. However, in the non-degenerate limit, where holds, we obtain from (69) (see
also (41))
Therefore for a non-degenerate gas reads
and that is in the class of moments. In fact is equal to the four-velocity of the gas to within a
factor. We have
where is the number density of atoms in the rest frame of the gas.
We recall the discussion – in Section 4.2 – of the important role played by in ensuring symmetric
hyperbolicity of the field equations: Symmetric hyperbolicity was due to the concavity of in the
privileged frame moving with the four-velocity . Now we see from (72) that – for the
non-degenerate gas – we have so that the privileged frame is the local rest frame of the gas.
This is quite satisfactory, since the rest frame is naturally privileged. [There remains the question of why
the rest frame is not the privileged one for a degenerate gas. This point is open and invites
investigation.]