Tolman-Oppenheimer-Volkoff equation

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In astrophysics, the Tolman-Oppenheimer-Volkoff 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[1], (10) is

\frac{dP(r)}{dr}=-\frac{G(\rho(r)+P(r)/c^2)(M(r)+4\pi P(r) r^3/c^2)}{r^2(1-2GM(r)/rc^2)}.

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 [1], (9)

\frac{dM(r)}{dr}=4 \pi \rho(r) 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], (1)

ds2 = eν(r)c2dt2 − (1 − 2GM(r) / rc2) − 1dr2 − r2(dθ2 + sin2θdφ2),

where ν(r) is determined by the constraint[1], (7)

\frac{d\nu(r)}{dr}=-\frac{2}{P(r)+\rho(r)c^2} \frac{dP(r)}{dr}.

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

ds2 = (1 − 2GM0 / rc2)c2dt2 − (1 − 2GM0 / rc2) − 1dr2 − r2(dθ2 + sin2θdφ2).

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(r_B)=\int_0^{r_B} 4\pi \rho(r) 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^{r_B} \frac{4\pi \rho(r) r^2}{\sqrt{1-2GM(r)/rc^2}} \, dr.

The difference between these two quantities,

\delta M=\int_0^{r_B} 4\pi \rho(r) r^2((1-2GM(r)/rc^2)^{-1/2}-1)\, dr,

will be the gravitational binding energy of the object divided by c2.

[edit] History

Tolman analyzed spherically symmetric metrics in 1934 and 1939.[2],[3] The form of the equation given here was derived by Oppenheimer and Volkoff in their 1939 paper, "On Massive Neutron Cores"[1]. 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.[4]

[edit] References

  1. ^ a b c d e On Massive Neutron Cores, J. R. Oppenheimer and G. M. Volkoff, Physical Review 55, #374 (February 15, 1939), pp. 374–381.
  2. ^ Effect of Inhomogeneity on Cosmological Models, Richard C. Tolman, Proceedings of the National Academy of Sciences 20, #3 (March 15, 1934), pp. 169–176.
  3. ^ Static Solutions of Einstein's Field Equations for Spheres of Fluid, Richard C. Tolman, Physical Review 55, #374 (February 15, 1939), pp. 364–373.
  4. ^ The maximum mass of a neutron star, I. Bombaci, Astronomy and Astrophysics 305 (January 1996), pp. 871–877.

[edit] See also

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