5 Relativistic Thermodynamics of Gases. 4 Finite Speeds in Relativistic 4.3 Moments as Four-Fluxes and

4.4 Upper and Lower Bounds for the Pulse Speed 

We recall the form of the field equations (34Popup Equation)

  equation1121

which is still valid in the relativistic case, albeit with tex2html_wrap_inline4173 as the Lorentz vector of the atomic four-momentum rather than tex2html_wrap_inline4215 as in Chapter  3 . We already know that tex2html_wrap_inline4217 holds. Also tex2html_wrap_inline4173 is a time-like vector so that we have

  equation1131

for all time-like co-vectors tex2html_wrap_inline3467 .

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

  equation1140

implies that tex2html_wrap_inline4161 is space-like, or light-like and therefore - by (12Popup Equation) - all characteristic speeds are smaller than c . We conclude that the speed of light is an upper bound for the pulse speed tex2html_wrap_inline3875 .

[Recall that the requirement (65Popup Equation) of symmetric hyperbolicity did not require speeds tex2html_wrap_inline4235 . 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 (65Popup Equation) is satisfied for all co-vectors. Therefore all speeds are tex2html_wrap_inline4235 .]

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

  equation1156

and this holds in particular for tex2html_wrap_inline3875 . Obviously tex2html_wrap_inline4243 is symmetric and tex2html_wrap_inline4245 is positive definite and symmetric. Therefore it follows from linear algebra (see footnote (3)) that

  equation1171

In very recent papers, Boillat & Ruggeri [3, 4] have used this knowledge to prove lower bounds for tex2html_wrap_inline3875 . The lower bounds depend on n, the number of fields, and for the number of fields tending to infinity the lower bound of tex2html_wrap_inline3875 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.



5 Relativistic Thermodynamics of Gases. 4 Finite Speeds in Relativistic 4.3 Moments as Four-Fluxes and

image Speeds of Propagation in Classical and Relativistic Extended Thermodynamics
Ingo Müller
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