= Weak-field slow-motion equations for a relativistic fluid
For $ds^2=(1+2\phi)dt^2-d\mathbf x^2$ with static weak $\phi$, normalization gives $u^0=1-\phi+O(\phi^2,v^2)$ and $\Gamma^i{}_{00}=\partial_i\phi$. Around a homogeneous state, the first-order equations are $\partial_t\delta n+n_0\nabla\cdot\mathbf v=0$, $\partial_t\delta\rho+(\rho_0+p_0)\nabla\cdot\mathbf v=0$, and $(\rho_0+p_0)\partial_t\mathbf v=-\nabla\delta p-(\rho_0+p_0)\nabla\phi$. The inertial density is enthalpy density even though the flow is slow. Only a further nonrelativistic equation of state, $p\ll\rho\simeq mn$, replaces it by mass density.
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