Therefore the characteristic equation of the system (73) of field equations, viz.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[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.
© Max Planck Society and the author(s)