In relativity, "stiff matter" is described by the relation $p = \rho$, where $p$ is the fluid's pressure and $\rho$ is its total energy density. The sound velocity in stiff matter equals the velocity of light in vacuum (I'm using natural units, so $c \equiv 1$): \begin{equation}\tag{1} c_s = \sqrt{\frac{dp}{d\rho}} = 1. \end{equation} For a general polytrop fluid (pressure $p = \kappa \, \rho_{\text{mass}}^{\gamma}$), we can prove the following expression: \begin{equation}\tag{2} c_s = \sqrt{\frac{\gamma \, p}{\rho + p}}. \end{equation} I need a confirmation that the adiabatic index $\gamma$ of stiff matter is $\gamma = 2$ ($p = \rho$ in expression (2) gives $\gamma = 2$, when $c_s = 1$). I find this value puzzling, since the adiabatic index of a polytrop fluid is usually the exponent of the following state equation ($\rho_{\text{mass}}$ is the proper mass density, not the total energy density): \begin{equation}\tag{3} p = \kappa \, \rho_{\text{mass}}^{\gamma}. \end{equation} So for $\gamma = 2$, we get $p = \kappa \, \rho_{\text{mass}}^2$ for stiff matter? Is that right? I feel there's an inconsistency somewhere.
2 Answers
The equation $$ p = \kappa \rho_{\rm mass}^{\gamma},$$ with $\gamma = 2$, does represent a perfect fluid with $p = \rho$ in the limit that $\rho$ becomes very large.
This is shown for example by Chavanis (2014, see section II).
My answer follows Hawking & Ellis (section 3.3, Example 4: Isentropic perfect fluid) except for notation which was taken from the answer.
Isentropic perfect fluid is described by proper mass density $\rho_\text{mass}$ and an elastic potential $e$ (which is a function of $\rho_\text{mass}$). Then the energy density is $$ \rho = \rho_\text{mass} (1+e),$$ while the pressure is $$ p = \rho^2_\text{mass} \frac{\mathrm{d}\, e}{\mathrm{d} \rho_\text{mass}}. $$
Assuming $e = \kappa \,\rho_\text{mass} - 1$ we find that $$ \rho =\kappa \, \rho^2_\text{mass},\qquad p=\kappa \, \rho^2_\text{mass} = \rho. $$
Thus, there is no inconsistency: the equation of state corresponds to stiff matter for all values of pressure/energy.
The difference with Chavanis' paper and Rob Jeffries' answer comes from the choice of constant $A$ in Chavanis' equation $(4)$. By setting $A$ to $1$ Chavanis ensures that $\rho\approx \rho_\text{mass}$ when $\rho_\text{mass}\to 0$. But if we set $A=0$ we would ensure that $p=\rho$ for all values of pressure.
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