Is the concept of in-compressible fluid valid in special theory of relativity?

Question

Is the concept of incompressible fluid valid in special theory of relativity? Can anybody answer this question without going into speed of sound and fluid dynamics ?

Answer 1

In an incompressible fluid the density does not change in response to changes in the pressure. This means that the speed of sound is infinite, $$ c_s^2= c^2\left.\frac{\partial P}{\partial \rho}\right|_s=\infty . $$ Here, $P$ is the pressure and $\rho$ is is the energy density. In the non-relativistic limit $\rho=mnc^2$, where $m$ is the mass of the particles and $n$ is the particle density. This is obviously incompatible with relativity, disturbances in the fluid propagate faster than the speed of light.

Of course, non-relativistic fluids are not truly incompressible either, but the approximation is useful if the fluid velocity is much smaller than the speed of sound, $u\ll c_s$. In a relativistic fluid in which $u$ is comparable to $c$ this cannot be true.

Postscript: Note that in a non-relativistic fluid incompressibility means that $n=const$. Then $\rho=mnc^2$ is also constant. In a relativistic fluid we could either mean $\rho=const$, or $n=const$. Note that $\rho=const$ is the more natural generalization, and it is incompatible with $c_s^2<c^2$ as explained above. Constant particle density also not allowed, because $\partial P/\partial n \sim \chi/n$, where $\chi$ is the susceptibility,

Answer 2

The relativistic fluids I am familiar with are of the astrophysical context (e.g., relativistic fireballs), and in these cases one uses (or can use) the relativistic Euler equations: $$ \nabla_\mu T^{\mu\nu}=0 $$ where $T^{\mu\nu}$ is the stress-energy tensor. These still lead to conservation equations such as, $$ \frac{\partial \rho}{\partial t}+\frac{\partial}{\partial x_i}\rho u^i=0 $$ (cf. this paper by Zhang & Macfayden). The Euler equations can be used for either compressible and incompressible fluids alike (though most situations I've seen are compressible).

This paper by Moritz Reintjes (2016) explicitly derives incompressibility as an option for relativistic fluids, so I would think the answer to OPs question is a clear yes, incompressible fluids exist in relativity.