Perfect fluid in general relativity ($T_{ab}=(\rho +p)u_a{u}_b+p{g}_{ab}$)

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.

The timelike projection of stress-energy conservation is $u^a{\nabla }_a\rho +(\rho +p){\nabla }_a{u}^a=0$. It is the local adiabatic first law for a perfect fluid.

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

The Tolman–Oppenheimer–Volkoff equation governs a static, spherically symmetric perfect fluid in general relativity. In geometrized units, if $m(r)$ is the mass inside areal radius $r$, then

The interior Schwarzschild metric is the exact static solution for a spherical fluid of constant density. If the star has mass $M$ and radius $R$, its pressure is

Buchdahl's theorem gives $2M/R\le 8/9$ for a static spherical perfect-fluid star whose density is nonincreasing outwards, subject to its regularity assumptions. Equality would require divergent central pressure, so every regular such star satisfies $M/R<4/9$.