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 to mean the privileged densities The privileged entropy density is
assumed by (5) to be concave with respect to the privileged fields . The privileged co-vector
will be chosen so that the concavity of implies symmetric hyperbolicity of the field
From (8) we obtain after multiplication by
is still defined as , as in (19). From (55) it follows that the concavity of –
the negative definiteness of – implies global invertibility between the field vector and the
Lagrange multipliers , provided that the privileged co-vector is chosen as co-linear to the vector
potential . We set
Indeed, in that case we have
so that, by (57), the second term on the right hand side of (55) vanishes and is definite.
Equation (58) will be used later.
With as a field vector, instead of , we may rephrase (8) in the form
where . Thus is the Legendre transform of with respect to the map
. It follows that is concave in , since is concave in ; thus we have