### 4.2 Symmetric hyperbolicity

The transformation 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 Section 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.]