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 (10View Equation) as
∂F--A ∂Λ Λ,A = π(Λ ), (63 )
or, by (60View Equation):
∂2h ′A ------Λ,A = π(Λ ). (64 ) ∂Λ ∂Λ
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

2 ′A ( ) -∂-h--ζ ∼ negative de finite gAB ζ ζ = 1, ζ > 0 . (65 ) ∂Λ ∂Λ A A B 0
In our case – with the concavity (5View Equation) 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
2 ′A ∂2h ′ ¯ ′A -∂-h---¯ζA = -----¯ζ-+ ∂ζA-∂h---∼ negative definite (66 ) ∂ Λ∂ Λ ∂ Λ∂ Λ ∂Λ ∂ Λ
by (62View Equation) and (58View Equation). 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 (65View Equation) 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 (65View Equation) 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 (15View Equation) which – for our system (64View Equation) – reads

( ∂2h′A ) det ϕ,A------- = 0. ∂ Λ ∂Λ

If (65View Equation) 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 (12View Equation) 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.]

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