Upper and lower bounds for the pulse speed

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 $pA$ as the Lorentz vector of the atomic four-momentum rather than $A a p = (mc, p )$ as in Section 3. We already know that $d2F- dχ2 < 0$ holds. Also $pA$ is a time-like vector so that we have

for all time-like co-vectors $ζA$ .

Therefore the characteristic equation of the system (73 ) of field equations, viz.

implies that $ϕ,A$ 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 $Vmax$ .

[Recall that the requirement (65 ) of symmetric hyperbolicity did not require speeds $≤ c$ . 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 $≤ c$ .]

More explicitly, by (11 ), the characteristic equation (75 ) reads

and this holds in particular for $Vmax$ . Obviously $∫ d2F pApαpβ --2-dP dχ$ is symmetric and $∫ d2F − p0pαpβ---2 dP d χ$ 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 $Vmax$ . The lower bounds depend on $n$ , the number of fields, and for the number of fields tending to infinity the lower bound of $Vmax$ 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.