### 4.4 Upper and lower bounds for the pulse speed

We recall the form of the field equations (34)
which is still valid in the relativistic case, albeit with as the Lorentz vector of the atomic
four-momentum rather than as in Section 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 [6, 3] have used this knowledge to prove lower bounds for .
The lower bounds depend on , 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.