Tolman–Oppenheimer–Volkoff equation

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In astrophysics, the Tolman–Oppenheimer–Volkoff (TOV) equation constrains the structure of a spherically symmetric body of isotropic material which is in static gravitational equilibrium, as modeled by general relativity. The equation is

${\displaystyle {\frac {dP}{dr}}=-{\frac {Gm}{r^{2}}}\rho \left(1+{\frac {P}{\rho c^{2}}}\right)\left(1+{\frac {4\pi r^{3}P}{mc^{2}}}\right)\left(1-{\frac {2Gm}{rc^{2}}}\right)^{-1}}$

Here, ${\textstyle r}$ is a radial coordinate, and ${\textstyle \rho (r)}$ and ${\textstyle P(r)}$ are the density and pressure, respectively, of the material at radius ${\textstyle r}$. The quantity ${\textstyle m(r)}$, the total mass within ${\textstyle r}$, is discussed below.

The equation is derived by solving the Einstein equations for a general time-invariant, spherically symmetric metric. For a solution to the Tolman–Oppenheimer–Volkoff equation, this metric will take the form

${\displaystyle ds^{2}=e^{\nu }c^{2}\,dt^{2}-\left(1-{\frac {2Gm}{rc^{2}}}\right)^{-1}\,dr^{2}-r^{2}\left(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}\right)}$

where ${\textstyle \nu (r)}$ is determined by the constraint

${\displaystyle {\frac {d\nu }{dr}}=-\left({\frac {2}{P+\rho c^{2}}}\right){\frac {dP}{dr}}}$

When supplemented with an equation of state, ${\textstyle F(\rho ,P)=0}$, which relates density to pressure, the Tolman–Oppenheimer–Volkoff equation completely determines the structure of a spherically symmetric body of isotropic material in equilibrium. If terms of order ${\textstyle 1/c^{2}}$ are neglected, the Tolman–Oppenheimer–Volkoff equation becomes the Newtonian hydrostatic equation, used to find the equilibrium structure of a spherically symmetric body of isotropic material when general-relativistic corrections are not important.

If the equation is used to model a bounded sphere of material in a vacuum, the zero-pressure condition ${\textstyle P(r)=0}$ and the condition ${\textstyle e^{\nu }=1-2Gm/c^{2}r}$ should be imposed at the boundary. The second boundary condition is imposed so that the metric at the boundary is continuous with the unique static spherically symmetric solution to the vacuum field equations, the Schwarzschild metric:

${\displaystyle ds^{2}=\left(1-{\frac {2GM}{rc^{2}}}\right)c^{2}\,dt^{2}-\left(1-{\frac {2GM}{rc^{2}}}\right)^{-1}\,dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2})}$