Validity of relativistic hydrodynamic equations For a relativistic fluid, the equation of state is given by:
$$
\rho = \rho_0 + \frac{3p}{c^2}
\,.$$
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):
$$
\begin{align}
\frac {\bf{V}}{c^2} \frac {\partial p}{\partial t} + \nabla p & ~~ = ~~ -\left(\frac {4p}{c^2} + \rho_0\right) \frac {\partial \bf{V}}{\partial t} - g \left(\rho_0 + \frac {4p}{c^2} \right)
\\
\\
{\bf V} \cdot \frac {\nabla p}{c^2} & ~~ = ~~ \frac {3}{c^2} \frac {\partial p}{\partial t} + \nabla \cdot \left(\frac {4p}{c^2} \bf{V} \right)
\end{align}
$$
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?
 A: I'll sketch a derivation of the first equation, and show that it is an approximation for small speeds.
In GR if you start from the stress-energy tensor of a perfect fluid and assume a weak-field metric, you get the following equation for fluid particles:
$$(\rho+p/c^2)(\partial_\beta u^\alpha +\Gamma^\alpha_{\lambda\beta} u^\lambda )u^{\beta}+\partial^\alpha p+\partial_\beta p\, u^\beta u^\alpha / c^2=0$$
In the Newtonian limit it reduces to the usual Euler equation. Next we substitute your equation of state and write $u^\alpha \approx (c, \mathbf v)$. For $\alpha = i$ we get:
$$(\rho_0+4 p/c^2)\left(\frac{\partial \mathbf v}{\partial t}+\mathbf v \cdot \nabla \mathbf v +\Gamma^i_{\lambda\beta} u^{\beta} u^\lambda \right)+\nabla p+\left(\frac{\partial p}{\partial t}+\mathbf v \cdot \nabla  p\right) \mathbf v / c^2=0$$
In the weak-field limit the only surviving Christoffel symbol is in this case  $\Gamma^i_{00}\approx \mathbf g / c^2$, the gravitational potential. Ignoring terms $\mathcal{O}(\mathbf v^2)$:
$$\nabla p+\frac{\mathbf v}{c^2} \frac{\partial p}{\partial t}=-(\rho_0+4 p/c^2)\left(\frac{\partial \mathbf v}{\partial t} +\mathbf g\right)$$
which is the first equation you wrote down.
It is therefore valid when: 1) the speeds involved are much less than the speed of light and the gravitational field is 2) weak and 3) static. The paper you quote (Allen & Hughes 1984) explicitly states that these conditions hold for the problem they're considering. For more on fluids in GR you can check the references they quote: Weinberg 1972 and Landau & Lifshitz 1963.
