The barotropic enthalpy variable r(epsilon): definition, strict monotonicity and the symmetric form of the fluid equations

Assume the Axiom of Countable Choice for the integral below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let a perfect-fluid model on a Lorentzian manifold be given with smooth equation of state $p=p(ϵ)$ defined on an open interval $J\subseteq \mathbb{R}$, with $ϵ+p(ϵ)>0$ and $p^{\mathrm{\prime }}(ϵ)>0$ for all $ϵ\in J$ (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy). Fix a base point ${ϵ}_{\ast }\in J$ and define the barotropic enthalpy variable $r:J\to \mathbb{R},r(ϵ):={\int }_{{ϵ}_{\ast }}^{ϵ}\frac{\sqrt{p^{\mathrm{\prime }}(z)}}{z+p(z)}dz\mathrm{.}$ The integrand is continuous on $J$ (A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion), so $r$ is well defined and $C^1$ with $r^{\mathrm{\prime }}(ϵ)=\sqrt{p^{\mathrm{\prime }}(ϵ)}\mathrm{/}(ϵ+p(ϵ))>0$ (Assuming Countable Choice, the integral function of a Riemann-integrable function is Lipschitz and differentiable almost everywhere, with derivative equal to the integrand there); hence $r$ maps $J$ diffeomorphically onto the open interval $r(J)$. Write $ϵ(r)$ for its inverse and $c_s:=\sqrt{p^{\mathrm{\prime }}}$ for the dimensionless sound-speed ratio (the physical sound speed is $v_s=c{c}_s$ when the SI light-speed constant is $c$). Thus $r^{\mathrm{\prime }}(ϵ)=c_s(ϵ)\mathrm{/}(ϵ+p(ϵ))$ and $c_s\in (0,1)$ when $0<p^{\mathrm{\prime }}<1$ is assumed.

For a smooth unit timelike fluid velocity $u$ with $g(u,u)=-1$, put ${\mathrm{\Pi }}^{ab}:=g^{ab}+u^a{u}^b$. The energy and Euler equations of The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy are equivalently $u^a{\mathrm{\nabla }}_{a}r+c_s{\mathrm{\nabla }}_a{u}^a=0,u^a{\mathrm{\nabla }}_a{u}^b+c_s{\mathrm{\Pi }}^{ba}{\mathrm{\nabla }}_{a}r=0.$ Indeed ${\mathrm{\nabla }}_{a}r=c_s{\mathrm{\nabla }}_aϵ\mathrm{/}(ϵ+p)$; multiplying the energy equation by $c_s\mathrm{/}(ϵ+p)$ gives the first equation, while ${\mathrm{\nabla }}_{a}p=c_s^2{\mathrm{\nabla }}_aϵ=(ϵ+p)c_s{\mathrm{\nabla }}_{a}r$ gives the second after division by $ϵ+p$. All factors divided out are positive on $J$, so both transformations are reversible. In a rest orthonormal basis the principal scalar/velocity cross coefficients are both $c_s$; the precise covector symbol and its positive time condition are proved in the following lemma. These two equations are the form in which the fluid system is fed into The barotropic fluid equations in enthalpy form are symmetric hyperbolic with positive principal symbol; no existence theorem is asserted here.