### 3.2 Symmetric hyperbolicity

Using the new variables we may write the field equations in the form
or, by (20):
We observe that the coefficient matrices in (23) are Hessian matrices derived from the vector potential
. Therefore the matrices are symmetric.
Also the matrix is negative definite on account on the concavity (4) of with respect to
. This is so, because the defining equation of , viz.

represents the Legendre transformation from to connected with the map between
dual fields. Indeed, we have by (20, 21) and (8)
Such a transformation preserves convexity – or concavity – so that is a concave function of , since
is a concave function of .
A quasilinear system of the type (23) with symmetric coefficient matrices, of which the temporal one is
definite, is called symmetric hyperbolic. We conclude that symmetric hyperbolicity of the equations (23) for
the fields is equivalent to the concavity of the entropy density in terms of the fields of densities
.

Hyperbolicity implies finite characteristic speeds, and symmetric hyperbolic systems guarantee the
well-posedness of initial value problems, i.e. existence and uniqueness of solutions – at least in the
neighbourhood of an event – and continuous dependence on the data.

Thus without having actually calculated a single characteristic speed, we have resolved Cattaneo’s
paradox of infinite speeds. The structure of extended thermodynamics guarantees that all speeds are finite;
no paradox can occur!

The fact that a system of balance-type field equations is symmetric hyperbolic, if it is compatible with
the entropy inequality and the concavity of the entropy density was discovered by Godunov [19] in the
special case of Eulerian fluids. In general this was proved by Boillat [1]. Ruggeri & Strumia [43] have found
that the symmetry is revealed only when the Lagrange multipliers are chosen as variables; these authors
were strongly motivated by Liu’s results of 1972 and by a paper by Friedrichs & Lax [18] which appeared a
year earlier.