4.4 Upper and lower bounds for the pulse speed

We recall the form of the field equations (34View Equation)
[∫ ] A d2F- p pαp βdχ2 dP Λβ,A = πα (73 )
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
∫ d2F ζA pAp αpβ --2-dP ∼ negative de finite (74 ) dχ
for all time-like co-vectors ζA.

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

( ) ∫ A d2F det ϕ,A p pαpβ --2-dP = 0 (75 ) dχ
implies that ϕ,A is space-like, or light-like and therefore – by (12View Equation) – 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 (65View Equation) 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 (65View Equation) is satisfied for all co-vectors. Therefore all speeds are ≤ c.]

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

(∫ ( V ) d2F ) det pana − --p0 pαp β---2 dP = 0 (76 ) c dχ

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

∫ ( Vmax ) d2F pana − ----p0 pαpβ --2-dP ∼ negative semi -definite. (77 ) c dχ
In very recent papers, Boillat & Ruggeri [63] 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.

  Go to previous page Go up Go to next page