The following chapter highlights this situation by considering a viscous heatconducting gas, a material which is fully characterized by 14 fields, viz. the density and flux of mass, energy and momentum, and stress and heat flux. With this choice of fields we shall be able to exploit the principle of relativity and the entropy inequality in explicit form and to calculate some specific pulse speeds.
It is true that much of the rigorous formal structure of the preceding chapters is lost when it comes to specific calculations. Linearization around equilibrium cannot be avoided, if we wish to obtain specific results, and that destroys global invertibility and general symmetric hyperbolicity.
These properties are now restricted to situations close to equilibrium.
5.1 Thermodynamic Processes in Viscous, HeatConducting Gases
The objective of thermodynamics of viscous, heatconducting gases is the determination of the 14 fields
$$\begin{array}{c}\begin{array}{cc}{A}^{A}:& \text{particle flux vector}\\ {A}^{AB}:& \text{energymomentum tensor}\end{array}\end{array}$$ 
(78)

in all events
${x}^{D}$
. Both
${A}^{A}$
and
${A}^{AB}$
are Lorentz tensors. The energymomentum tensor is assumed symmetric so that it has 10 independent components.
For the determination of these fields we need field equations and these are formed by the conservation laws of particle number and energymomentum, viz.
$$\begin{array}{c}{A}_{,A}^{A}=0\end{array}$$ 
(79)

$$\begin{array}{c}{A}_{,B}^{AB}=0\end{array}$$ 
(80)

and by the equations of balance of fluxes
$$\begin{array}{c}{A}_{,C}^{ABC}={I}^{AB}.\end{array}$$ 
(81)

${A}^{ABC}$
is the flux tensor – it is completely symmetric –, and
${I}^{AB}$
is its production density. We assume
$$\begin{array}{c}{I}_{A}^{A}=0\text{and}{A}_{B}^{AB}={c}^{2}{A}^{A}\end{array}$$ 
(82)

so that among the 15 equations ( 79 , 80 , 81 ) there are 14 independent ones, which is the appropriate number for 14 fields.
The components of
${A}^{A}$
and
${A}^{AB}$
have the following interpretations
$$\begin{array}{c}\begin{array}{cc}{A}^{0}:& c\cdot \text{rest mass density},\\ {A}^{a}:& \text{flux of rest mass},\\ {A}^{00}:& \text{energy density},\\ {A}^{0a}:& 1/c\cdot \text{energy flux},\\ {A}^{a0}:& c\cdot \text{momentum density},\\ {A}^{ab}:& \text{momentum flux}.\end{array}\end{array}$$ 
(83)

The motivation for the choice of equations ( 79 , 80 , 81 ), and in particular ( 81 ), stems from the kinetic theory of gases. Indeed
${A}^{A}$
and
${A}^{AB}$
are the first two moments in the kinetic theory and
${A}_{,A}^{A}=0$
and
${A}_{,B}^{AB}=0$
are the first two equations of transfer. Therefore it seems reasonable to take further equations from the equation of transfer for the third moment
${A}^{ABC}$
and these have the form ( 81 ). In the kinetic theory the two conditions ( 82 ) are satisfied.
The set of equations (
79 , 80 , 81 ) must be supplemented by constitutive equations for the flux tensor
${A}^{ABC}$
and the flux production
${I}^{AB}$
. The generic form of these relations in a viscous, heatconducting gas reads
$$\begin{array}{c}\begin{array}{ccc}{A}^{ABC}& =& {\hat{A}}^{ABC}({A}^{M},{A}^{MN})\\ {I}^{AB}& =& {\hat{I}}^{AB}({A}^{M},{A}^{MN}).\end{array}\end{array}$$ 
(84)

If the constitutive functions
$\hat{A}$
and
$\hat{I}$
are known, we may eliminate
${A}^{ABC}$
and
${I}^{BC}$
between ( 79 , 80 , 81 ) and ( 84 ) and obtain a set of field equations for
${A}^{M}$
,
${A}^{MN}$
. Each solution is called a thermodynamic process.
It is clear upon reflection that this theory, based on (
79 , 80 , 81 ) and ( 84 ), provides a special case of the generic structure explained in Chapter 2 .
5.2 Constitutive Theory
We recall the restrictive principles of the constitutive theory from Chapter 2 and adjust them to the present case

∙
entropy inequality
${h}_{,A}^{A}\ge 0$
with
${h}^{A}={\hat{h}}^{A}({A}^{M},{A}^{MN})$
,

∙
principle of relativity.
The former principle was discussed and exploited in the general scheme of Chapter 2 , but the principle of relativity was not. This principle assumes that the constitutive functions
${\hat{A}}^{ABC}$
,
${\hat{I}}^{AB}$
,
${\hat{h}}^{A}$
– generically
$\hat{C}$
– are invariant under Lorentz transformations
$${\stackrel{*}{x}}^{A}={\stackrel{*}{x}}^{A}\left({x}^{B}\right).$$
Thus the principle of relativity may be stated in the form
$$\begin{array}{c}C=\hat{C}({A}^{M},{A}^{MN})\text{and}\stackrel{*}{C}=\hat{C}({\stackrel{*}{A}}^{M},{\stackrel{*}{A}}^{MM})\end{array}$$ 
(85)

Note that
$\hat{C}$
is the same function in both equations.
It is complicated and cumbersome to exploit the constitutive theory but the results are remarkably specific, at least for nearequilibrium processes:

∙
${\hat{A}}^{ABC}$
will be reduced to the thermal equation of state.

∙
${\hat{I}}^{AB}$
will be reduced to the relaxation times of the gas, which may be considered to be of the order of magnitude of the mean time of free flight of its molecules.
For details of the calculation the reader is referred to the literature, in particular to the book by Müller & Ruggeri [
39,
40]
or the paper by Liu, Müller & Ruggeri [
31]
. Here we explain only the results.
5.3 Results of the Constitutive Theory
No matter how much a person may be conditioned to think relativistically, he will appreciate the decomposition of the fourtensors
${A}^{A}$
,
${A}^{AB}$
and
${h}^{A}$
into their suggestive timelike and spacelike components. We have
$$\begin{array}{c}\begin{array}{cc}{A}^{A}& =nm{U}^{A}\\ {A}^{AB}& ={t}^{\langle AB\rangle}+\left(p\right(n,e)+\pi ){h}^{AB}+\frac{1}{{c}^{2}}({U}^{A}{q}^{B}+{U}^{B}{q}^{A})+\frac{e}{{c}^{2}}{U}^{A}{U}^{B}\\ {h}^{A}& =h{U}^{A}+{\Phi}^{A},\end{array}\end{array}$$ 
(86)

and the components have suggestive meaning as follows
$$\begin{array}{c}\begin{array}{cc}n:& \text{number density}\\ {U}^{A}:& \text{velocity}\\ {t}^{\langle AB\rangle}:& \text{stress deviator}\\ p+\pi :& \text{pressure}\\ {q}^{A}:& \text{heat flux}\\ e:& \text{energy density}\\ h:& \text{entropy density}\\ {\Phi}^{A}:& \text{(nonconvective) entropy flux.}\end{array}\end{array}$$ 
(87)

At least this is how
$n$
through
${\Phi}^{A}$
are to be interpreted in the rest frame of the gas.
We have defined
${h}^{AB}=\frac{1}{{c}^{2}}{U}^{A}{U}^{B}{g}^{AB}$
and
$m$
is the molecular rest mass.
The decomposition (
86 ) is not only popular because of its intuitive quality but also, since it is now possible to characterize equilibrium as a process in which the stress deviator
${t}^{\langle AB\rangle}$
, the heat flux
${q}^{A}$
and the dynamic pressure
$\pi $
– the nonequilibrium part of the pressure – vanish.
The equilibrium pressure
$p$
is a function of
$n$
and
$e$
, the thermal equation of state. In thermodynamics it is often useful to replace the variables
$(n,e)$
by
$$\text{fugacity}\alpha \text{and absolute temperature}T,$$
because these two variables can be measured – at least in principle. Also
$\alpha $
and
$T$
are the natural variables of statistical thermodynamics which provides the thermal equation of state in the form
$p=p(\alpha ,T)$
. The transition between the new variables
$(\alpha ,T)$
and the old ones
$(n,e)$
can be effected by the relations
$$\begin{array}{c}nm=\frac{1}{T}\dot{p}\text{and}e={p}^{\prime}p\end{array}$$ 
(88)

where
$\dot{}$
and
${}^{\prime}$
here and below denote differentiation with respect to
$\alpha $
and
$lnT$
respectively.
If we restrict attention to a linear theory in
${t}^{\langle AB\rangle}$
,
${q}^{A}$
, and
$\pi $
, we can satisfy the principle of relativity with linear isotropic functions for
${A}^{ABC}$
,
${I}^{BC}$
viz.
$$\begin{array}{c}\begin{array}{cc}{A}^{ABC}=& ({C}_{1}^{0}+{C}_{1}^{\pi}\pi ){U}^{A}{U}^{B}{U}^{C}+\frac{{c}^{2}}{6}(nm{C}_{1}^{0}{C}_{1}^{\pi}\pi )\cdot \\ & \cdot ({g}^{AB}{U}^{C}+{g}^{BC}{U}^{A}+{g}^{CA}{U}^{B})+{C}_{3}({g}^{AB}{q}^{C}+{g}^{BC}{q}^{A}+{g}^{CA}{q}^{B})\\ & \frac{6}{{c}^{2}}{C}_{3}({U}^{A}{U}^{B}{q}^{C}+{U}^{B}{U}^{C}{q}^{A}+{U}^{C}{U}^{A}{q}^{B})+\\ & +{C}_{5}({t}^{\langle AB\rangle}{U}^{C}+{t}^{\langle BC\rangle}{U}^{A}+{t}^{\langle CA\rangle}{U}^{B}),\end{array}\end{array}$$ 
(89)

$$\begin{array}{c}{I}^{BC}={B}_{1}^{\pi}\pi {g}^{AB}\frac{4}{{c}^{2}}{B}_{1}^{\pi}\pi {U}^{A}{U}^{B}+{B}_{3}{t}^{\langle AB\rangle}+\frac{1}{{c}^{2}}{\hat{B}}^{4}({q}^{A}{U}^{B}+{q}^{B}{U}^{A}).\end{array}$$ 
(90)

Note that
${I}^{BC}$
vanishes in equilibrium so that no entropy production occurs in that state. The coefficients
$C$
and
$B$
in ( 89 , 90 ) are functions of
$e$
and
$n$
, or
$\alpha $
and
$T$
. In fact, the entropy principle determines the
$C$
's fully in terms of the thermal equation of state
$p=p(\alpha ,T)$
as follows
$$\begin{array}{c}\begin{array}{cc}{C}_{1}^{0}=& \frac{{\Gamma}_{1}^{\prime}}{2{c}^{2}T}\text{with}{\Gamma}_{1}=2{c}^{2}{T}^{6}\int \frac{\dot{p}}{{T}^{7}}dT\\ \\ {C}_{1}^{\pi}=& \frac{2}{{c}^{2}T}\frac{\left[\begin{array}{ccc}\ddot{p}& \dot{p}{\dot{p}}^{\prime}& {\dot{\Gamma}}_{1}\\ \dot{p}{\dot{p}}^{\prime}& {p}^{\prime}{p}^{\prime \prime}& {\Gamma}_{1}^{\prime}{\Gamma}_{1}\\ {\dot{\Gamma}}_{1}& {\Gamma}_{1}^{\prime}{\Gamma}_{1}& \frac{5}{3}{\Gamma}_{2}\end{array}\right]}{\left[\begin{array}{ccc}\ddot{p}& \dot{p}{\dot{p}}^{\prime}& {\dot{\Gamma}}_{1}\\ \dot{p}{\dot{p}}^{\prime}& {p}^{\prime}{p}^{\prime \prime}& {\Gamma}_{1}^{\prime}{\Gamma}_{1}\\ \dot{p}& {p}^{\prime}& \frac{5}{3}{\Gamma}_{1}\end{array}\right]}\\ \\ {C}_{3}=& \frac{1}{2T}\frac{\left[\begin{array}{cc}\dot{p}& {\dot{\Gamma}}_{1}\\ {\Gamma}_{1}& {\Gamma}_{2}\end{array}\right]}{\left[\begin{array}{cc}\dot{p}& {\dot{\Gamma}}_{1}\\ {p}^{\prime}& {\Gamma}_{1}{\Gamma}_{1}^{\prime}\end{array}\right]}\\ \\ {C}_{5}=& \frac{1}{2T}\frac{{\Gamma}_{2}}{{\Gamma}_{1}}\text{with}{\Gamma}_{2}=2{c}^{2}{T}^{8}\int \frac{1}{{T}^{3}}{\dot{\Gamma}}_{1}dT.\end{array}\end{array}$$ 
(91)

The
$B$
's in ( 90 ) are restricted by inequalities, viz.
$$\begin{array}{c}{B}_{1}^{\pi}\ge 0,{\hat{B}}_{4}\ge 0,{B}_{3}\le 0.\end{array}$$ 
(92)

All
$B$
's have the dimension
$1/\text{sec}$
and we may consider them to be of the order of magnitude of the collision frequency of the gas molecules.
In conclusion we may write the field equations in the form
$$\begin{array}{c}(nm{U}^{A}{)}_{,A}=0\end{array}$$ 
(93)

$$\begin{array}{c}({t}^{\langle BA\rangle}+(p+\pi ){h}^{BA}+\frac{1}{{c}^{2}}({q}^{B}{U}^{A}+{q}^{A}{U}^{B})+\frac{e}{{c}^{2}}{U}^{B}{U}^{A}{)}_{,A}=0\end{array}$$ 
(94)

$$\begin{array}{c}{A}_{,A}^{BCA}={B}_{1}^{\pi}\pi {g}^{BC}\frac{4}{{c}^{2}}{B}_{1}^{\pi}\pi {U}^{B}{U}^{C}+{B}_{3}{t}^{\langle BC\rangle}+\frac{1}{{c}^{2}}{\hat{B}}_{4}({q}^{B}{U}^{C}+{q}^{C}{U}^{B}),\end{array}$$ 
(95)

where
${A}^{BCA}$
must be inserted from ( 89 ) and ( 91 ). This set of equations represents the field equations of extended thermodynamics. We conclude that extended thermodynamics of viscous, heatconducting gases is quite explicit – provided we are given the thermal equation of state
$p=p(\alpha ,T)$
– except for the coefficients
$B$
. These coefficients must be measured and we proceed to show how.
5.4 The Laws of NavierStokes and Fourier
It is instructive to identify the classical constitutive relations of NavierStokes and Fourier of TIP within the scheme of extended thermodynamics. They are obtained from ( 93 , 94 , 95 ) by the first step of the socalled Maxwell iteration which proceeds as follows: The
${n}^{\text{th}}$
iterate
$${\stackrel{\left(n\right)}{I}}^{AB}\left(\text{or}\stackrel{\left(n\right)}{\pi},{\stackrel{\left(n\right)}{t}}^{AB},{\stackrel{\left(n\right)}{q}}^{A}\right)$$
results from
${A}_{,A}^{A}$

= 0

${\stackrel{(n1)}{A}}_{,B}^{AB}$

= 0

${\stackrel{(n1)}{A}}_{,C}^{ABC}$

=
${\stackrel{\left(n\right)}{I}}^{AB}$

with the

initiation

agreement

${\stackrel{\left(E\right)}{A}}_{,A}^{A}$

= 0

${\stackrel{\left(E\right)}{A}}_{,B}^{AB}$

= 0

${\stackrel{\left(E\right)}{A}}_{,C}^{ABC}$

=
${\stackrel{\left(1\right)}{I}}^{AB}$


$$\begin{array}{c}\end{array}$$ 
(96)

where
$\stackrel{\left(E\right)}{A}$
are equilibrium values.
A little calculation provides the first iterates for dynamic pressure, stress deviator and heat flux in the form
$$\begin{array}{c}\stackrel{\left(1\right)}{\pi}=\lambda \left[{U}_{,A}^{A}\right]\end{array}$$ 
(97)

$$\begin{array}{c}{\stackrel{\left(1\right)}{t}}_{\langle AB\rangle}=\mu \left[{h}_{A}^{M}{h}_{B}^{N}{U}_{\langle M,N\rangle}\right]\end{array}$$ 
(98)

$$\begin{array}{c}{q}_{A}=\kappa \left[{h}_{A}^{M}(lnT{)}_{,M}\frac{1}{{c}^{2}}\frac{d{U}_{M}}{d\tau}\right]\end{array}$$ 
(99)

with
$$\begin{array}{ccc}\lambda & =& \frac{1}{2T{B}_{1}^{\pi}}\frac{\left[\begin{array}{ccc}\ddot{p}& \dot{p}{\dot{p}}^{\prime}& {\dot{\Gamma}}_{1}\\ \dot{p}{\dot{p}}^{\prime}& {p}^{\prime}{p}^{\prime \prime}& {\Gamma}_{1}^{\prime}{\Gamma}_{1}\\ {\dot{\Gamma}}_{1}& {\Gamma}_{1}^{\prime}{\Gamma}_{1}& \frac{5}{3}{\Gamma}_{2}\end{array}\right]}{\left[\begin{array}{cc}\ddot{p}& \dot{p}{\dot{p}}^{\prime}\\ \dot{p}{\dot{p}}^{\prime}& {p}^{\prime}{p}^{\prime \prime}\end{array}\right]}\end{array}$$  
$$\begin{array}{ccc}\mu & =& \frac{1}{2T{B}_{3}}{\Gamma}_{1}\end{array}$$  
$$\begin{array}{ccc}\kappa & =& \frac{1}{2{T}^{2}{\hat{B}}_{4}}\frac{\left[\begin{array}{cc}\dot{p}& {\dot{\Gamma}}_{1}\\ {p}^{\prime}& {\Gamma}_{1}{\Gamma}_{1}^{\prime}\end{array}\right]}{\dot{p}}\end{array}$$  
These are the relativistic analogues of the classical phenomenological equations of NavierStokes and Fourier.
$\lambda $
,
$\mu $
and
$\kappa $
are the bulk viscosity, the shear viscosity and the thermal conductivity respectively; all three of these transport coefficients are nonnegative by the entropy inequality.
The only essential difference between the equations (
97 , 98 , 99 ) and the nonrelativistic phenomenological equations is the acceleration term in ( 99 ). This contribution to the Fourier law was first derived by Eckart, the founder of thermodynamics of irreversible processes. It implies that the temperature is not generally homogeneous in equilibrium. Thus for instance equilibrium of a gas in a gravitational field implies a temperature gradient, a result that antedates even Eckart. We have emphasized that the field equations of extended thermodynamics should provide finite speeds. Below in Section 5.6 we shall give the values of the speeds for nondegenerate gases. In contrast TIP leads to parabolic equations whose fastest characteristic speeds are always infinite.
Indeed, if the phenomenological equations (
97 , 98 , 99 ) are introduced into the conservation laws ( 93 , 94 ) of particle number, energy and momentum, we obtain a closed system of parabolic equations for
$n$
,
${U}_{A}$
and
$e$
. This unwelcome feature results from the Maxwell iteration; it persists to arbitrarily high iterates.
5.5 Specific Results for a NonDegenerate Relativistic Gas
For a relativistic gas Jüttner [
22,
23]
has derived the phase density for Bosons and Fermions, namely
$$\begin{array}{c}{f}_{E}=\frac{y}{exp\left[\frac{m}{k}\alpha +\frac{{U}_{A}}{kT}{p}^{A}\right]\mp 1}\text{or}{f}_{E}=\frac{y}{exp\left[\frac{m}{k}\alpha +\frac{m{c}^{2}}{kT}\sqrt{1+\frac{{p}^{2}}{{m}^{2}{c}^{2}}}\right]\mp 1}.\end{array}$$ 
(100)

The latter equation is valid in the rest frame of the gas.
${p}^{A}$
is the atomic fourmomentum and we have
${p}_{A}{p}^{A}={m}^{2}{c}^{2}$
. Jüttner has used these phase densities to calculate the equations of state. For the nondegenerate gas he found that Bessel functions of the second kind, viz.
$$\begin{array}{c}{K}_{n}\left(\frac{m{c}^{2}}{kT}\right)={\int}_{0}^{\infty}cosh\left(n\rho \right)exp\left(\frac{m{c}^{2}}{kT}cosh\rho \right)d\rho \end{array}$$ 
(101)

are the relevant special functions. The thermal equation of state
$p=p(\alpha ,T)$
reads
$$\begin{array}{c}p=nkT\text{with}n=exp\left(\frac{m}{k}\alpha \right)\cdot 4\pi y{m}^{3}{c}^{3}\frac{{K}_{2}\left(\frac{m{c}^{2}}{kT}\right)}{\frac{m{c}^{2}}{kT}},\end{array}$$ 
(102)

where
$1/y$
is the smallest phase space element. From ( 102 ) we obtain with
$G=\frac{{K}_{3}}{{K}_{2}}$
and
$\gamma =\frac{m{c}^{2}}{kT}$
$$\begin{array}{c}e=nm{c}^{2}\left(G\frac{1}{\gamma}\right),\frac{{\Gamma}_{1}}{T}=nm{c}^{2}\frac{2}{\gamma}G\end{array}$$ 
(103)

and hence
$$\begin{array}{c}\begin{array}{cc}{C}_{1}^{0}& =nm\left(1+\frac{6}{\gamma}G\right)\\ {C}_{1}^{\pi}& =\frac{6}{{c}^{2}}\frac{\left(2\frac{5}{{\gamma}^{2}}\right)+\left(\frac{19}{\gamma}\frac{30}{{\gamma}^{3}}\right)G\left(2\frac{45}{{\gamma}^{2}}\right){G}^{2}\frac{9}{\gamma}{G}^{3}}{\frac{3}{\gamma}\left(2\frac{20}{{\gamma}^{2}}\right)G\frac{13}{\gamma}{G}^{2}+2{G}^{3}}\\ {C}_{3}& =\frac{1}{\gamma}\frac{1+\frac{6}{\gamma}G{G}^{2}}{1+\frac{5}{\gamma}G{G}^{2}}\\ {C}_{5}& =\left(\frac{6}{\gamma}+\frac{1}{G}\right).\end{array}\end{array}$$ 
(104)

The transport coefficients read
$$\begin{array}{c}\begin{array}{cc}\lambda & =\frac{1}{3}\frac{nm{c}^{2}}{\gamma}\frac{1}{{B}_{1}^{\pi}}\frac{\frac{3}{\gamma}+\left(2\frac{20}{{\gamma}^{2}}\right)G+\frac{13}{\gamma}{G}^{2}+2{G}^{3}}{1\frac{1}{{\gamma}^{2}}+\frac{5}{\gamma}G{G}^{2}}\\ \mu & =nkT\frac{1}{{B}_{3}}G\\ \kappa & =nkT\frac{{c}^{2}}{{\hat{B}}_{4}}\left(\gamma +5G\gamma {G}^{2}\right).\end{array}\end{array}$$ 
(105)

It is instructive to calculate the leading terms of the transport coefficients in the nonrelativistic case
$m{c}^{2}\gg kT$
. We obtain
$$\begin{array}{c}\lambda =\frac{5}{6{B}_{1}^{\pi}}nkT\frac{1}{{\gamma}^{2}}\end{array}$$ 
(106)

$$\begin{array}{c}\mu =\frac{1}{{B}_{3}}nkT\end{array}$$ 
(107)

$$\begin{array}{c}\kappa =\frac{5}{2{\hat{B}}_{4}}\frac{n{k}^{2}T}{m}.\end{array}$$ 
(108)

It follows that the bulk viscosity does not appear in a nonrelativistic gas. Recall that the coefficients
$1/B$
are relaxation times of the order of magnitude of the meantime of free flight; so they are not in any way ”relativistically small”.
Note that
$\lambda $
,
$\mu $
and
$\kappa $
are measurable, at least in principle, so that the
$B$
's may be calculated from ( 105 ). Therefore it follows that the constitutive theory has led to specific results. All constitutive coefficients are now explicit: The
$C$
's can be calculated from the thermal equation of state
$p=p(\alpha ,T)$
and the
$B$
's may be measured.
It might seem from (
106 ) and ( 97 ) that the dynamic pressure is of order
$O\left(\frac{1}{{\gamma}^{2}}\right)$
but this is not so as was recently discovered by Kremer & Müller [
27]
. Indeed, the second step in the Maxwell iteration for
$\pi $
provides a term that is of order
$O\left(\frac{1}{\gamma}\right)$
, see also [
28]
. That term is proportional to the second gradient of the temperature
$T$
so that it may be said to be due to heating or cooling.
Specific results of the type (
104 , 105 ) can also be calculated for degenerate gases with the thermal equation of state
$p(\alpha ,T)$
for such gases. That equation was also derived by Jüttner [
23]
.
The results for 14 fields may be found in Müller & Ruggeri [
39,
40]
.
5.6 Characteristic Speeds in a Viscous, HeatConducting Gas
We recall from Section 2.4 , in particular ( 14 ), that the jumps
$\delta \text{}\mathit{u}\text{}$
across acceleration waves and their speeds of propagation are to be calculated from the homogeneous system
$$\begin{array}{c}{\phi}_{,A}\frac{\partial \text{}\mathit{F}{\text{}}^{A}}{\partial \text{}\mathit{u}\text{}}\delta \text{}\mathit{u}\text{}=0.\end{array}$$ 
(109)

In the present context, where the field equations are given by ( 79 , 80 ) this homogeneous algebraic system spreads out into three equations, viz.
$$\begin{array}{c}{\phi}_{,A}\delta {A}^{A}=0,{\phi}_{,A}\delta {A}^{AB}=0,{\phi}_{,A}\delta {A}^{ABC}=0.\end{array}$$ 
(110)

By ( 89 ) and ( 91 ) this is a fully explicit system, if the thermal equation of state
$p=p(\alpha ,T)$
is known. The vanishing of its determinant determines the characteristic speeds. Seccia & Strumia [
44]
have calculated these speeds – one transversal and two longitudinal ones – for nondegenerate gases and obtained the following results in the nonrelativistic and ultrarelativistic cases
$\frac{m{c}^{2}}{kT}\gg 1:{V}_{\text{trans}}=\sqrt{\frac{7k}{5m}T}$
,

${V}_{\text{long}}^{1}=\sqrt{\frac{4k}{3m}T}$
,

${V}_{\text{long}}^{2}=\sqrt{5.18\frac{k}{m}T}$
,


$\frac{m{c}^{2}}{kT}\ll 1:{V}_{\text{trans}}=\sqrt{\frac{1}{5}}c$
,

${V}_{\text{long}}^{1}=\sqrt{\frac{1}{3}}c$
,

${V}_{\text{long}}^{2}=\sqrt{\frac{3}{5}}c$
.




$$\begin{array}{c}\end{array}$$ 
(111)

All speeds are finite and smaller than
$c$
. Inspection shows that in the nonrelativistic limit the order of magnitude of these speeds is that of the ordinary speed of sound, while in the ultrarelativistic case the speeds come close to
$c$
.
5.7 Discussion
So as to anticipate a possible misunderstanding I remark that the equations ( 93 , 94 , 95 ) with
${A}^{ABA}$
from ( 89 ) and ( 91 ) are neither symmetric nor fully hyperbolic. Indeed the underlying symmetry of the system ( 79 , 80 , 81 ), and ( 84 ) reveals itself only when the Lagrange multipliers
$\text{}\mathbf{\Lambda}\text{}$
are used as variables. But ( 93 , 94 , 95 , 89 , 91 ) are equations for the physical variables
${A}^{A},{A}^{AB}$
or in fact
$n,T,{U}^{A},{t}^{AB}$
, and
${q}^{A}$
. Also the hyperbolicity in the whole state space is lost, because the equations ( 89 , 90 ) are restricted to linear terms. Therefore the system is hyperbolic only in the neighbourhood of equilibrium. For a more detailed discussion of these aspects, see Müller & Ruggeri [
39,
40]
.
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