which is still valid in the relativistic case, albeit with
as the Lorentz vector of the atomic four-momentum rather than
as in Chapter
3
. We already know that
holds. Also
is a time-like vector so that we have
for
all time-like co-vectors
.
Therefore the characteristic equation of the system (73) of field equations, viz.
implies that
is space-like, or light-like and therefore - by (12
) - all characteristic speeds are smaller than
c
. We conclude that the speed of light is an upper bound for the
pulse speed
.
[Recall that the requirement (65) of symmetric hyperbolicity did not require speeds
. I have discussed that point at the end of Section
4.2
. Now, however, in extended thermodynamics of moments, because of
the specific form of the vector potential, the condition (65
) is satisfied for all co-vectors. Therefore all speeds are
.]
More explicitly, by (11), the characteristic equation (75
) reads
and this holds in particular for
. Obviously
is symmetric and
is positive definite and symmetric. Therefore it follows from
linear algebra (see footnote (3)) that
In very recent papers, Boillat & Ruggeri [3,
4] have used this knowledge to prove lower bounds for
. The lower bounds depend on
n, the number of fields, and for the number of fields tending to
infinity the lower bound of
tends to
c
from below. The strategy of proof is similar to the one employed
in Section
3.6
for the non-relativistic case.
Therefore the pulse speeds of all moment theories are smaller than c, but they tend to c as the number of moments tends to infinity. This result compares well with the corresponding result in Section 3.6 concerning the non-relativistic theory. In that case there was no upper bound so that the pulse speeds tended to infinity for extended thermodynamics of very many moments.
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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 |