4.1 Concavity of a privileged entropy density

We recall the arguments of Section 2.2 concerning concavity in the relativistic case and choose the fields u to mean the privileged densities F ¯ζ = F A¯ζA. The privileged entropy density is assumed by (5View Equation) to be concave with respect to the privileged fields F ¯ ζ. The privileged co-vector ¯ ζA will be chosen so that the concavity of h ¯ζ implies symmetric hyperbolicity of the field equations.

From (8View Equation) we obtain after multiplication by ¯ζA

¯ Λ = ∂h-¯ζ-+ h′A ∂-ζA, (54 ) ∂F ¯ζ ∂F ¯ζ
( ) ∂Λ ∂2hζ¯ ∂ ∂ ¯ζA -----= ---------+ ----- h′A ----- . (55 ) ∂F ¯ζ ∂F ¯ζ∂F ζ¯ ∂F ¯ζ ∂F ¯ζ
h ′A is still defined as Λ ⋅ F A − hA, as in (19View Equation). From (55View Equation) it follows that the concavity of ( ) h¯ζ F ¯ζ – the negative definiteness of 2 -∂--h¯ζ--- ∂F ¯ζ∂F ¯ζ – implies global invertibility between the field vector F ¯ ζ and the Lagrange multipliers Λ, provided that the privileged co-vector ¯ζA is chosen as co-linear to the vector potential ′A h. We set
A h′A ¯ζ = − ∘--′A--′. (56 ) h hA
Indeed, in that case we have
′A ∂ζ¯A-- h ∂F ¯ = 0, (57 ) ζ
A ∂2¯ζA ∂¯ζA ∂ ¯ζA ¯ζ ---------= − ----------∼ positive semi -definite (58 ) ∂F ¯ζ∂F ¯ζ ∂F ¯ζ∂F ¯ζ
so that, by (57View Equation), the second term on the right hand side of (55View Equation) vanishes and ∂Λ--- ∂F ¯ζ is definite. Equation (58View Equation) will be used later.

With Λ as a field vector, instead of F ¯ζ, we may rephrase (8View Equation) in the form

dh′A = F A ⋅ dΛ, (59 )
∂h ′A F A = -----, (60 ) ∂Λ
F ¯ζ = ∂h¯ζ, (61 ) ∂Λ
where h′¯ = h′A¯ζA = Λ ⋅ F ¯ζ − h ¯ζ ζ. Thus h′¯ ζ is the Legendre transform of h ¯ζ with respect to the map F ⇐ ⇒ Λ ¯ζ. It follows that h′ ¯ζ is concave in Λ, since h ¯ζ is concave in F ¯ζ; thus we have
∂2h′ ----¯ζ--∼ negative definite. (62 ) ∂Λ ∂Λ

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