= Perfect fluid in general relativity
{title2=$T_{ab}=(\rho+p)u_au_b+pg_{ab}$}
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 $u^au_a=-1$, let $\rho$ be rest-frame <energy density> and $p$ rest-frame <pressure>. Its <stress-energy tensor> is
$$
T_{ab}=(\rho+p)u_au_b+pg_{ab}.
$$
The energy measured with the fluid is $T_{ab}u^au^b=\rho$. The <orthogonal projection> $h^a{}_b=\delta^a{}_b+u^au_b$ 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> gives
$$
u^a\nabla_a\rho+(\rho+p)\nabla_au^a=0,\qquad
(\rho+p)u^a\nabla_au_b=-h_b{}^a\nabla_ap.
$$
These 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.
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