Weak-field slow-motion equations for a relativistic fluid

For $d{s}^2=(1+2\varphi )d{t}^2-d{\mathbf{x}}^2$ with static weak $\varphi$, normalization gives $u^0=1-\varphi +O({\varphi }^2,v^2)$ and ${\Gamma }^i{}_{00}={\partial }_i\varphi$. 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 $({\rho }_0+p_0){\partial }_t\mathbf{v}=-\nabla \delta p-({\rho }_0+p_0)\nabla \varphi$. 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.