results from
where
are equilibrium values.
A little calculation provides the first iterates for dynamic pressure, stress deviator and heat flux in the form
with
These are the relativistic analogues of the classical
phenomenological equations
of Navier-Stokes and Fourier.
,
and
are the bulk viscosity, the shear viscosity and the thermal
conductivity respectively; all three of these transport
coefficients are non-negative by the entropy inequality.
The only essential difference between the equations (97,
98
,
99
) and the non-relativistic 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 non-degenerate 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,
and
e
. This unwelcome feature results from the Maxwell iteration; it
persists to arbitrarily high iterates.
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Speeds of Propagation in Classical and Relativistic
Extended Thermodynamics
Ingo Müller http://www.livingreviews.org/lrr-1999-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |