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
Bruno
I find the statement itself weird – the hydrodynamics equations break down, full stop. What happens to the speed of sound and whether it is even still defined depends on how to approach the vacuum limit. For example, one can argue that the temperature will increase if one goes to vacuum (it does in space), and thus the speed of sound would increase. I would remove any statement about the speed of sound, and maybe add the physical reason why the limit is not defined, with the mean free path length becoming too large (hence we don't have thermodynamics any more).
Apart from this your calculation looks correct to me. If you take \rho -> 0 with k = const and \gamma = const then cs -> 0 .
-erik
On Fri, Feb 9, 2024 at 10:16 AM Bruno Giacomazzo bruno.giacomazzo@unimib.it wrote:
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
--
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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Erik Schnetter