### 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 four-tensors , and into their suggestive time-like and space-like
components. We have
and the components have suggestive meaning as follows
At least this is how through are to be interpreted in the rest frame of the gas.
We have defined and 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 ,
the heat flux and the dynamic pressure – the non-equilibrium part of the pressure –
vanish.

The equilibrium pressure is a function of and , the thermal equation of state. In
thermodynamics it is often useful to replace the variables by

because these two variables can be measured – at least in principle. Also and are the natural
variables of statistical thermodynamics which provides the thermal equation of state in the form
. The transition between the new variables and the old ones can be effected
by the relations

where and here and below denote differentiation with respect to and respectively.
If we restrict attention to a linear theory in , , and , we can satisfy the principle of
relativity with linear isotropic functions for , viz.

Note that vanishes in equilibrium so that no entropy production occurs in that state. The coefficients
and in (89, 90) are functions of and , or and . In fact, the entropy principle
determines the ’s fully in terms of the thermal equation of state as follows
The ’s in (90) are restricted by inequalities, viz.

All ’s have the dimension 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

where 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, heat-conducting
gases is quite explicit – provided we are given the thermal equation of state –
except for the coefficients . These coefficients must be measured and we proceed to show
how.