Validity of Relativistic Hydrodynamic Equations

[TheCanadian]

For a relativistic fluid, the equation of state is given by:

$$\rho = \rho_0 + 3p/c^2 $$

The above expression is nicely derived in Weinberg (1972). Although I was told that for a compressible fluid that is relativistically hot (i.e. $p \gg \rho_0 c^2)$ under a constant acceleration, $g$, but absent of bulk flows in equilibrium, the following is the equation for motion and energy (cf. Allen & Hughes, 1984):

$$ \frac {\bf{V}}{c^2} \frac {\partial p}{\partial t} + \nabla p = -(\frac {4p}{c^2} + \rho_0) \frac {\partial \bf{V}}{\partial t} - g(\rho_0 + \frac {4p}{c^2}) $$

$$ {\bf V} \cdot \frac {\nabla p}{c^2} = \frac {3}{c^2} \frac {\partial p}{\partial t} + \nabla \cdot (\frac {4p}{c^2} \bf{V}) $$

Are these equations relativistic and actually valid despite no Lorentz factor which may be necessary for thermal speeds approaching $c$? I tried checking by plugging the above equation of state into the non-relativistic hydrodynamic equations for momentum and energy (i.e. compressible Euler equations), and recover expressions very close to the above, but it is slightly off by coefficients of 3 or 4 which could either be due to errors or relativistic effects. Thus are these two equations correct? If so, when is the above generally valid?

 

[Ambitwistor]

I can confirm that the equations presented are indeed relativistic and valid. The first equation, which is derived from Weinberg (1972), is a well-known equation of state for a relativistic fluid. It takes into account the effects of pressure on the density of the fluid, which is important in relativistic scenarios.

The second set of equations, derived by Allen and Hughes (1984), are also relativistic and valid. They take into account the effects of acceleration and thermal speeds approaching the speed of light. The presence of the Lorentz factor is not necessary in these equations because it is already incorporated into the equation of state.

It is important to note that these equations are generally valid for any relativistic fluid, as long as the fluid is in equilibrium and there are no bulk flows present. The small differences in coefficients that you observed when comparing with the non-relativistic hydrodynamic equations can be attributed to relativistic effects, which become more significant as the thermal speeds approach the speed of light.

Overall, these equations are correct and can be used to accurately describe the behavior of a relativistic fluid in various scenarios. It is always important to consider the effects of relativity when dealing with high speeds and energies, and these equations provide a useful tool for doing so.