Relativistic Euler equation

The spatial projection of stress-energy conservation is $(\rho +p)u^b{\nabla }_b{u}^a=-(g^{ab}+u^a{u}^b){\nabla }_{b}p$.

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.

For metric signature $(+---)$, a stationary perfect fluid follows a timelike Killing vector field $K$ with four-velocity $U=K/N$. Its Killing-flow four-acceleration is $U^b{\nabla }_b{U}_a=-{\nabla }_a\log N$. The relativistic Euler equation and stationarity of the pressure give the displayed hydrostatic equilibrium equation. This uses stationarity along $K$ and does not require hypersurface orthogonality.

The equation of state of isotropic blackbody radiation is $p=\rho /3$. The barotropic hydrostatic first integral gives $\rho N^4=C$. The Stefan–Boltzmann law $\rho \propto T^4$ then yields the Tolman–Ehrenfest relation $TN=\mathrm{constant}$. Here $N=\sqrt{|K^a{K}_a|}$ for the stationary timelike Killing vector field; its normalization only changes the constants.

A connected nonvacuum perfect fluid with equation of state $p=\rho /3$ obeys the stationary radiation density law $\rho =C/N^4$ with $C>0$. On a compact region of a regular spacetime, a smooth Killing vector field has bounded norm. Consequently $\rho$ cannot tend to zero at a finite regular free surface. A vanishing timelike norm instead makes the density diverge. A radiation-filled cavity can have a material wall with nonzero boundary pressure; this is a different boundary condition.

For a barotropic fluid in hydrostatic equilibrium along a Killing vector, assume $\rho +p>0$. Integrating $dp/(\rho +p)=-d\log N$ gives the displayed first integral, constant on each connected fluid region. For the equation of state $p=w\rho$ with constant $w>0$, it gives $\rho N^{(1+w)/w}=C_1$.