 Perfect fluid in general relativity

A relativistic perfect fluid has energy density $\rho$, isotropic pressure $p$ and unit timelike four-velocity $u$. Projecting ${\nabla }_a{T}^{ab}=0$ along and orthogonally to $u$ gives its energy and Euler equations.

For a perfect fluid, the relativistic stress tensor is $T_{ab}=(\rho +p)u_a{u}_b+p{g}_{ab}$ in signature $(-,+,+,+)$ with $u^a{u}_a=-1$.