The GRHydro thorn guide ( https://einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentation....) contains the following sentence: "However, in the vacuum limit the continuity equations describing the fluid break down. *The speed of sound tends to the speed of light* and everything fails (especially the conversion from conserved to primitive variables)."
A student of mine wrote that the sound speed tends to the speed of light if rho tends to zero citing this, but is this correct?
For example, taking a polytropic EOS P=k*rho^\gamma, the sound speed squared is (assuming c=1)
cs^2= \gamma * k * rho^\gamma / (rho*h)
with
rho*h=rho+rho*eps+P=rho+\gamma*P/(\gamma-1)
So cs^2 goes like rho^\gamma/(rho+\rho^gamma) and if \gamma>1 then cs^2->0 for rho->0 (while it goes to 1 if rho-> infinity).
Am I missing something?
Cheers, Bruno