The kinetic equation of state hyperbolicity and causality conjecture

Let $p$ be the pressure, $\eta$ the entropy per particle, and $\rho$ the proper energy density, related by the kinetic equation of state

$$p=k_Bn\theta=m_0c^2\frac{n}{\beta},$$ $$\rho=m_0c^2n\frac{K_1(\beta)}{K_2(\beta)}+3p,$$ $$n=4\pi e^4m_0^3c^3h^{-3}\exp\left(\frac{-\eta}{k_B}\right)\frac{K_2(\beta)}{\beta}\exp\left(\beta\frac{K_1(\beta)}{K_2(\beta)}\right).$$

Kinetic equation of state hyperbolicity and causality conjecture. Under this equation of state, there exists a smooth function $f_{\rm kinetic}$ such that

$$p=f_{\rm kinetic}(\eta,\rho),$$

and the relativistic Euler system is hyperbolic and causal. Its speed of sound

$$c_S:=c\sqrt{\left.\frac{\partial p}{\partial\rho}\right|_{\eta}}$$

is real and less than the speed of light; furthermore,

$$0<c_S<\frac{c}{\sqrt{3}}.$$

These properties were proved in certain temperature regimes in the cited prior work and are conjectured here in general. They provide the global hyperbolicity, causality, and thermodynamic closure needed for the relativistic Euler system.