Speck–Strain conjecture on the relativistic equation of state and sound speed

Under the relations referred to in the source as equations

$$–$$

, let $p$ be the pressure, $e$ the energy density, and $S$ the entropy variable. The conjecture asserts that $p$ is a smooth positive function of $(e,S)$ on $(0,\infty)\times(0,\infty)$, so that

$$p=p(e,S)$$

is well-defined for every $(e,S)\in(0,\infty)\times(0,\infty)$. Speck–Strain's sound-speed conjecture. On this domain,

$$0<\frac{\partial p}{\partial e}\Big|_{S}(e,S)=\frac{\partial p(e,S)}{\partial e}\Big|_{S}<\frac{1}{3}.$$

This is stated as a stronger relativistic sound-speed assertion than the preceding coordinate-map conjecture. The supplied source does not provide evidence that it has been resolved.