Barotropic relativistic sound speed ($c_s^2=dp/d\rho$)

In units $c=1$, an isentropic perfect fluid in general relativity has sound speed $c_s^2=dp/d\rho$. Writing $n=p^{\prime }(ϵ)$ gives $d\rho =ϵdn$ and $c_s^2=n/(ϵp^{\prime \prime }(ϵ))$. Linearized particle number conservation in a relativistic fluid and the relativistic Euler equation then yield acoustic waves with this speed. Positive $c_s^2$ is needed for ordinary acoustic stability; a causal fluid model additionally requires $c_s^2\le 1$.