Tolman–Oppenheimer–Volkoff equation

Remarks

The $tt$ component of the Einstein equations gives the equation for $m$, the $rr$ component gives $d\nu/dr = G\left(m + 4\pi r^3 p/c^2\right)/\left(r^2 c^2 (1 - 2Gm/(rc^2))\right)$, and the conservation law $\nabla_a T^{ab} = 0$ gives $dp/dr = -(\rho c^2 + p)\, d\nu/dr$. Combining them gives the equation for $p$.

To build a star one adds an equation of state $p = f(\rho)$, chooses a central density, and integrates outward until $p = 0$ at a radius $R$. Outside, the solution is matched to the Schwarzschild metric with mass $M = m(R)$.

When $p \ll \rho c^2$ and $Gm \ll rc^2$, the equation reduces to the Newtonian hydrostatic equilibrium $dp/dr = -G m \rho / r^2$. The three relativistic correction factors all make the pressure gradient steeper, so gravity is effectively stronger than in Newton's theory. This is one reason why neutron stars have a maximum mass.

Going deeper

For a student

The equation is the starting point for models of neutron stars and white dwarfs in the relativistic regime. Given an equation of state, it yields the mass–radius relation and the maximum mass of the family of stars. The constant-density solution is the interior Schwarzschild metric, and the general equation leads to Buchdahl's bound $R \ge \frac{9}{4} \frac{GM}{c^2}$ for stars with non-increasing density.