Definitions

# Tolman-Oppenheimer-Volkoff equation

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 modelled by general relativity. The equation, (10) is
$frac\left\{dP\left(r\right)\right\}\left\{dr\right\}=-frac\left\{G\left(rho\left(r\right)+P\left(r\right)/c^2\right)\left(M\left(r\right)+4pi P\left(r\right) r^3/c^2\right)\right\}\left\{r^2\left(1-2GM\left(r\right)/rc^2\right)\right\}.$
Here, r is a radial coordinate, and ρ(r0) and P(r0) are the density and pressure, respectively, of the material at r=r0. M(r0) is the total mass inside radius r=r0, as measured by the gravitational field felt by a distant observer. It satisfies M(0)=0 and , (9)
$frac\left\{dM\left(r\right)\right\}\left\{dr\right\}=4 pi rho\left(r\right) r^2.$
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, (1)
$ds^2=e^\left\{nu\left(r\right)\right\} c^2 dt^2 - \left(1-2GM\left(r\right)/rc^2\right)^\left\{-1\right\} dr^2 - r^2\left(dtheta^2 + sin^2 theta dphi^2\right),$
where ν(r) is determined by the constraint, (7)
$frac\left\{dnu\left(r\right)\right\}\left\{dr\right\}=-frac\left\{2\right\}\left\{P\left(r\right)+rho\left(r\right)c^2\right\} frac\left\{dP\left(r\right)\right\}\left\{dr\right\}.$

When supplemented with an equation of state, F(ρ, 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 1/c2 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 P(r)=0 and the condition eν(r)=1−2GM(r)/rc2 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

$ds^2=\left(1-2GM_0/rc^2\right) c^2 dt^2 - \left(1-2GM_0/rc^2\right)^\left\{-1\right\} dr^2 - r^2\left(dtheta^2 + sin^2 theta dphi^2\right).$
Here, M0 is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at r=rB, continuity of the metric and the definition of M(r) require that
$M_0=M\left(r_B\right)=int_0^\left\{r_B\right\} 4pi rho\left(r\right) r^2, dr.$
Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value
$M_1=int_0^\left\{r_B\right\} frac\left\{4pi rho\left(r\right) r^2\right\}\left\{sqrt\left\{1-2GM\left(r\right)/rc^2\right\}\right\} , dr.$
The difference between these two quantities,
$delta M=int_0^\left\{r_B\right\} 4pi rho\left(r\right) r^2\left(\left(1-2GM\left(r\right)/rc^2\right)^\left\{-1/2\right\}-1\right), dr,$
will be the gravitational binding energy of the object divided by c2.

## History

Tolman analyzed spherically symmetric metrics in 1934 and 1939., The form of the equation given here was derived by Oppenheimer and Volkoff in their 1939 paper, "On Massive Neutron Cores". In this paper, the equation of state for a degenerate Fermi gas of neutrons was used to calculate an upper limit of ~0.7 solar masses for the gravitational mass of a neutron star. Since this equation of state is not realistic for a neutron star, this limiting mass is likewise incorrect. Modern estimates for this limit range from 1.5 to 3.0 solar masses.