helps us to recognize the structure of the field equations.
Obviously with
as the field vector, instead of
we may rephrase the field equations (10
) as
or, by (60):
We observe that the coefficient matrices are Hessian matrices and therefore symmetric.
By the definition of symmetric hyperbolicity due to
Friedrichs [17] the system is symmetric hyperbolic, if
there exists at least one
co-vector
for which
In our case - with the concavity (5) of the entropy density
for
- it is clear that such a co-vector exists. It is
itself! Indeed we have
by (62) and (58). Thus symmetric hyperbolicity is implied by the concavity of
the entropy density both in the relativistic and the
non-relativistic case.
It is true that in the relativistic case we have to rely on
the privileged co-vector
in this context and therefore on a privileged Lorentz frame
whose entropy density
is concave in
.
The significance of this choice is not really understood
. Indeed, we might have preferred the privileged frame to be the
local rest frame of the body. In that respect it is reassuring
that
is often co-linear to the four-velocity
as we shall see in Section
4.3
below; but not always! A better understanding is needed.
Note that in the non-relativistic case the only time-like
co-vector is
, a constant vector. In that case all the above-mentioned
complications are absent: Concavity of the one and only entropy
density
is equivalent to symmetric hyperbolicity, see Chapter
3
above.
Also note that the requirement (65) of symmetric hyperbolicity ensures finite characteristic
speeds, not necessarily speeds smaller than
c
as we might have wished. [In this respect we may be tempted to
replace Friedrich's definition of symmetric hyperbolicity by one
of our own making, which might require (65) to be true
for all
time-like co-vectors
- instead of
at least one
. If we did that, we should anticipate the whole problem of
speeds greater than
c
. Indeed, we recall the characteristic equation (15) which - for our system (64) - reads
If (65) were to hold for all time-like co-vectors
, we could now conclude that
is space-like, or light-like, so that
holds. Thus (12) would imply
. This is a clear case of assuming the desired result in a
disguise and we do not follow this path.]
|
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 |
