 Perfect fluid in general relativity

A perfect fluid in general relativity has isotropic rest-frame pressure and no rest-frame heat flux or anisotropic stress. In units $c=1$ and signature $(-,+,+,+)$, let $u^a{u}_a=-1$, let $\rho$ be rest-frame energy density and $p$ rest-frame pressure. Its stress-energy tensor isThe energy measured with the fluid is $T_{ab}{u}^a{u}^b=\rho$. The orthogonal projection $h^a{}_b={\delta }^a{}_b+u^a{u}_b$ isolates spatial directions and gives isotropic projected stress $p{h}_{ab}$. A fluid moving relative to another observer can have nonzero momentum density even though its own rest-frame stress is isotropic.

Projecting stress-energy conservation along and perpendicular to the four-velocity givesThese are the comoving energy balance and relativistic Euler equation. The inertial factor is enthalpy density $\rho +p$. An equation of state is needed to close the evolution; relativistic dust sets $p=0$, while an isotropic radiation fluid has $p=\rho /3$. Perfect-fluid behavior does not require irrotational motion: the potential-flow action below describes a restricted subset.