What is the adiabatic index of stiff matter? 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.
 A: 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).
A: 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.
