The GRHydro thorn guide (https://einsteintoolkit.org/thornguide/EinsteinEvolve/GRHydro/documentation.html) 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
-- Prof. Bruno Giacomazzo
Department of Physics
University of Milano-Bicocca
Piazza della Scienza 3
20126 Milano
Italy
email: bruno.giacomazzo@unimib.it
phone: (+39) 02 6448 2321
web: http://www.brunogiacomazzo.org
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