In Weinberg: Gravitation and Cosmology chapter 2.10 (Relativistic Hydrodynamics) the speed of sound is derived as
$v_s^2 = \left(\frac{\partial p}{\partial \rho}\right)$
and the equation of state is given as
$p=(\gamma - 1) (\rho - nm)$
with $\gamma = 4/3$ for the case of a relativistic gas and $\gamma=5/3$ for the case of a nonrelativistic monoatomic gas ($nm$ is the rest mass density and $\rho$ the energy density). Then Weinberg computes the sound speed for the relativistic case as
$v_s= \sqrt{\frac{1}{3}}$
in natural units and states that this is still safely less than unity (the speed of light). However what he doesn't show is the same calculation for the nonrelativistic case. This would yield
$v_s^2 = \frac{\partial ((5/3 - 1) (\rho - nm))}{\partial \rho} = 2/3$
and so the speed of sound in the non-relativistic case would actually be $v_s =\sqrt{\frac{2}{3}}$. So the speed of sound for a non-relativistic gas is actually higher than the speed of sound for an relativistic gas!? However one usually computes the non-relativistic speed of sound with another equation of state
$p=K\rho^{\gamma}$
which yields
$v_s^2=\frac{p}{\rho} = K \gamma \rho^{\gamma-1} = \gamma \frac{p}{\rho}$
This is a completely different result compared to the one we get from the equation of state given by Weinberg. So is the equation of state given by Weinberg wrong? If so, what is the correct equation of state for a relativistic gas and what is the real maximum speed of sound for a relativistic gas? If not, what is wrong with my calculation of the non-relativistic sound speed based on the equation of state given by Weinberg?
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