Symmetric hyperbolicity

4.2 Symmetric hyperbolicity

The transformation $Fζ¯⇐ ⇒ Λ$ helps us to recognize the structure of the field equations. Obviously with $Λ$ as the field vector, instead of $F ¯ζ$ 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 $ζA$ for which

In our case – with the concavity (5

) of the entropy density $h ¯ζ$ for $A h′A ¯ζ = − ∘---′A--′ h h A$ – it is clear that such a co-vector exists. It is $¯ζA$ 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 $h′A ¯ζA = − ∘------- h′Ah′A$ in this context and therefore on a privileged Lorentz frame whose entropy density $h¯ζ$ is concave in $F ¯ζ$ . 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 $′A h$ is often co-linear to the four-velocity $A U$ 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 $ζA = (1,0,0,0 )$ , a constant vector. In that case all the above-mentioned complications are absent: Concavity of the one and only entropy density $0 h$ 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 $ζ A$ – 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

( ∂2h′A ) det ϕ,A------- = 0. ∂ Λ ∂Λ

If (65 ) were to hold for all time-like co-vectors $ζA$ , we could now conclude that $ϕ,A$ is space-like, or light-like, so that $gABϕ ϕ ≤ 0 ,A ,B$ holds. Thus (12 ) would imply $V 2 ≤ c2$ . This is a clear case of assuming the desired result in a disguise and we do not follow this path.]