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 [http://prola.aps.org/abstract/PR/v55/i4/p374_1 On Massive Neutron Cores] , J. R. Oppenheimer and G. M. Volkoff, "Physical Review" 55, #374 (February 15, 1939), pp. 374–381.] ^{, (10)} is:::::$frac\{dP(r)\}\{dr\}=-frac\{G(\; ho(r)+P(r)/c^2)(M(r)+4pi\; P(r)\; r^3/c^2)\}\{r^2(1-2GM(r)/rc^2)\}.$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 ^{, (9)}:::::$frac\{dM(r)\}\{dr\}=4\; pi\; ho(r)\; r^2.$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)}:::::$ds^2=e^\{\; u(r)\}\; c^2\; dt^2\; -\; (1-2GM(r)/rc^2)^\{-1\}\; dr^2\; -\; r^2(d\; heta^2\; +\; sin^2\; heta\; dphi^2),$where ν(r) is determined by the constraint^{, (7)}:::::$frac\{d\; u(r)\}\{dr\}=-frac\{2\}\{P(r)+\; ho(r)c^2\}\; frac\{dP(r)\}\{dr\}.$

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 e^ν(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:::::$ds^2=(1-2GM\_0/rc^2)\; c^2\; dt^2\; -\; (1-2GM\_0/rc^2)^\{-1\}\; dr^2\; -\; r^2(d\; heta^2\; +\; sin^2\; heta\; dphi^2).$Here, M₀ is the total mass of the object, again, as measured by the gravitational field felt by adistant observer. If the boundary is at r=r_B, continuity of the metric and the definition of M(r) require that:::::$M\_0=M(r\_B)=int\_0^\{r\_B\}\; 4pi\; ho(r)\; r^2,\; dr.$Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value:::::$M\_1=int\_0^\{r\_B\}\; frac\{4pi\; ho(r)\; r^2\}\{sqrt\{1-2GM(r)/rc^2\; ,\; dr.$The difference between these two quantities,:::::$delta\; M=int\_0^\{r\_B\}\; 4pi\; ho(r)\; r^2((1-2GM(r)/rc^2)^\{-1/2\}-1),\; dr,$will be the gravitational binding energy of the object divided by c².

History

Tolman analyzed spherically symmetric metrics in 1934 and 1939. [ [http://www.pnas.org/cgi/reprint/20/3/169 Effect of Inhomogeneity on Cosmological Models] , Richard C. Tolman, "Proceedings of the National Academy of Sciences" 20, #3 (March 15, 1934), pp. 169–176.] ^, [ [http://prola.aps.org/abstract/PR/v55/i4/p364_1 Static Solutions of Einstein's Field Equations for Spheres of Fluid] , Richard C. Tolman, "Physical Review" 55, #374 (February 15, 1939), pp. 364–373.] 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. [ [http://adsabs.harvard.edu/abs/1996A&A...305..871B The maximum mass of a neutron star] , I. Bombaci, "Astronomy and Astrophysics" 305 (January 1996), pp. 871–877.]

References

ee also

* Hydrostatic equation
* Tolman-Oppenheimer-Volkoff limit
* Solutions of the Einstein field equations
* Static spherically symmetric perfect fluid