Constant-density profile has unequal pressures at equal energy density

For this item, a differentiable barotropic pressure law is a $C^1$ map $P:J_{ϵ}\to \mathbb{R}$, where $J_{ϵ}\subset (0,\mathrm{\infty })$ is an open interval of energy-density values containing ${ϵ}_0$, both input energy density and output pressure have units $\mathrm{J}{\mathrm{m}}^{-3}$, and $p(r)=P(ϵ(r))$ for each interior radius $0\le r<R$. The finite causal-slope criterion under consideration is $0\le P^{\mathrm{\prime }}(x)\le 1$ for $x\in J_{ϵ}$, so $P^{\mathrm{\prime }}$ is dimensionless. For units only, set the formal parameter $c_s^2:=c^2{P}^{\mathrm{\prime }}(x)$, with units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$; this item assigns no fluid-dynamical meaning to it and makes no claim about an observed signal speed.

The claim refuted is that every regular static constant-density solution supplies such a map $P$ satisfying this slope criterion.

The interior profile assigns the same energy density ${ϵ}_0$ to every interior radius while the pressure varies. Thus pressure is not a single-valued function of energy density along this profile; this does not imply an infinite or superluminal signal speed.