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 Chapter
3
. There are essential differences, however
is now the Lorentz vector of atomic four-momentum with
and
. Accordingly
now stands for polynomials in the components of four-momentum.
Thus instead of (27
) we have
of phase space is now equal to
instead of
.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 n 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.]
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
