Ultra-Relativistic sound speed problem

[Barnak]

I'm trying to understand how to use the special relativistic sound formula for a perfect fluid :

$c_s = c \, \sqrt{\frac{dp}{d\rho}},$

where $p$ is the isotropic pressure and $\rho$ is the total energy density (not the internal energy density $\rho_{int}$ or mass density $\rho_{mass}$).

In the case of an ultra-relativistic fluid, we have $p(\rho) = \frac{1}{3} \rho$, so we get

$c_s = \frac{c}{\sqrt{3}} \approx 58\% \, c.$

This is given in Weinberg's book on general relativity and looks clear to me.

Now the problem is this : what about the following equation of state ?

$p(\rho) = \kappa \, \rho^{\gamma}.$

For an ultra-relativistic perfect fluid, we have the adiabatic index $\gamma = \frac{4}{3}$, so the previous formula doesn't give the same speed as Weinberg's :

$c_s = c \, \sqrt{\frac{dp}{d\rho}} = c \, \sqrt{\frac{\gamma \, p}{\rho}} \ne \frac{c}{\sqrt{3}}$

What am I doing wrong here ?

I suspected it's because the variable isn't the same here (symbol confusion ?).
I know that the total energy density, mass density and internal density are related by this relation :

$\rho = \rho_{mass} + \rho_{int},$

but then, what density should I use in the equation of state $p(\rho) = \kappa \, \rho^{\gamma}$ ?
$\rho_{tot} \equiv \rho$ ? $\rho_{mass}$ ? or $\rho_{int} = \rho - \rho_{mass}$ ? And if it's $\rho_{mass}$, how can I define $\rho_{int}$ as a function of $\rho_{mass}$ ?

And I don't understand why we have two equations of states for the perfect ultra-relativistic fluid :

$p(\rho) = \frac{1}{3} \rho$

and

$p(\rho) = \kappa \, \rho^{4/3}.$

??

 

[Barnak]

Never mind, I've found the solution to my problem.

In the equation of state

$p = \kappa \, \rho^{\gamma},$

$\rho$ is actually the proper mass energy density ; $\rho_{mass}$, not the total energy density.

We can also write this equation of state

$p = (\gamma - 1) \, \rho_{int},$

with the relation

$\rho_{tot} = \rho_{mass} + \rho_{int}.$

Here, $\rho_{int}$ is the internal energy density. Equating both equations of state gives a simple relation between $\rho_{tot}$ and $\rho_{mass}$, and allows the calculation of the sound velocity in full relativistic form :

$c_s = c \, \sqrt{\frac{dp}{d\rho}} = c \, \sqrt{\frac{\gamma \, p}{\rho \, + \, p}},$

where $\rho \equiv \rho_{tot}$ is the total energy density. That's what I was looking for.

In the ultra-relativistic regime, we have by definition $p = \frac{1}{3} \rho_{tot}$. Putting this into the previous velocity formula, I get $c_s = \frac{c}{\sqrt{3}}$ only if $\gamma = \frac{4}{3}$, which is fine !

The velocity formula given above is great since it's valid for all regimes, from non-relativistic up to the ultra-relativistic regime !
I never saw that formula elsewhere in any book or document on the web. I'm now wondering if it's already known somewhere...