Concavity of a privileged entropy density

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 (5

) 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 (8 ) we obtain after multiplication by $¯ζA$

hence

$h ′A$ is still defined as $Λ ⋅ F A − hA$ , as in (19

). From (55

) 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

Indeed, in that case we have

hence

so that, by (57

), the second term on the right hand side of (55

) vanishes and $∂Λ--- ∂F ¯ζ$ is definite. Equation (58

) will be used later.

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

hence

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