Tolman–Oppenheimer–Volkoff equation (TOV equation)

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 ratio of enclosed gravitational mass to areal radius in units $G=c=1$. In a spherical star the local value is $m(r)/r$, and the surface value is $M/R$. Bounds on this ratio constrain relativistic stellar models; it is unrelated to topological compactness.

For a smooth centre with density ${\rho }_c$ and pressure $p_c$, the Tolman–Oppenheimer–Volkoff equation gives $m(r)=4\pi {\rho }_c{r}^3/3+O(r^5)$ and $p(r)=p_c-2\pi ({\rho }_c+p_c)({\rho }_c/3+p_c)r^2+O(r^4)$. A fixed barotropic stellar equation of state leaves one physical central parameter; the additive lapse constant fixes time normalization.

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

The Interior Schwarzschild metric has central pressureIt diverges when $M/R=4/9$, the limiting compactness in Buchdahl's theorem.

The dominant energy condition for the Interior Schwarzschild metric reduces to its largest pressure satisfying $p_c\le \rho$. Substitution of the exact central pressure gives $\sqrt{1-2M/R}\ge 1/2$, hence $M/R\le 3/8$. Equality has finite central pressure and is allowed by the non-strict condition.

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 $M/R<4/9$.

Let ${\rho }_0>0$ be a density threshold and let $a,\mu$ be the radius and enclosed mass where a monotonically decreasing stellar density reaches that threshold. Then $\mu \ge 4\pi {\rho }_0{a}^3/3$. A local compactness bound $\mu /a<4/9$ implies $a<(3\pi {\rho }_0{)}^{-1/2}$ and $\mu <4/(9\sqrt{3\pi {\rho }_0})$, independently of the equation of state above the threshold. A total-mass bound also requires suitable control of the known low-density envelope.