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^[1] is

${\frac {dP(r)}{dr}}=-{\frac {G}{r^{2}}}\left[\rho (r)+{\frac {P(r)}{c^{2}}}\right]\left[M(r)+4\pi r^{3}{\frac {P(r)}{c^{2}}}\right]\left[1-{\frac {2GM(r)}{c^{2}r}}\right]^{{-1}}\;$

Here, r is a radial coordinate, and ρ(r₀) and P(r₀) are the density and pressure, respectively, of the material at r = r₀.

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]

$ds^{2}=e^{{\nu (r)}}c^{2}dt^{2}-(1-2GM(r)/rc^{2})^{{-1}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\;$

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

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

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 exp[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:

$ds^{2}=(1-2GM_{0}/rc^{2})c^{2}dt^{2}-(1-2GM_{0}/rc^{2})^{{-1}}dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta d\phi ^{2})\;$

Total mass

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.^[1]

${\frac {dM(r)}{dr}}=4\pi \rho (r)r^{2}\;$

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 = r_B, 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,

${\displaystyle \delta M=\int _{0}^{r_{B}}4\pi \rho (r)r^{2}(1-(1-2GM(r)/rc^{2})^{-1/2})\;dr\;}$

will be the gravitational binding energy of the object divided by c² and it is negative.

Derivation from general relativity

Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric:

${\displaystyle c^{2}d\tau ^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }=e^{\nu (r)}c^{2}dt^{2}-e^{\lambda (r)}dr^{2}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta d\phi ^{2}\;}$

By the perfect fluid assumption, the stress-energy tensor is diagonal (in the central spherical coordinate system), with eigenvalues of energy density and pressure:

${\displaystyle T_{0}^{0}=\rho (r)c^{2}\;}$

and

${\displaystyle T_{i}^{j}=-P(r)\delta _{i}^{j}\;}$

Where $\rho (r)$ is the fluid density and $P(r)$ is the fluid pressure.

To proceed further, we solve Einstein's field equations:

${\displaystyle {\frac {8\pi G}{c^{4}}}T_{\mu \nu }=G_{\mu \nu }\;}$

Let us first consider the ${\displaystyle G_{00}}$ component:

${\displaystyle {\frac {8\pi G}{c^{4}}}\rho c^{2}e^{\nu }={e^{\nu } \over r^{2}}\left(1-{\frac {d}{dr}}re^{-\lambda }\right).\;}$

Integrating this expression from 0 to $r$, we obtain

${\displaystyle e^{-\lambda }=1-{2GM(r) \over rc^{2}},\;}$

where $M(r)$ is as defined in the previous section. Next, consider the $G_{{11}}$ component. Explicitly, we have:

$-{\frac {8\pi G}{c^{4}}}P(r)e^{{\lambda (r)}}={\frac {-r\nu '(r)+e^{{\lambda (r)}}-1}{r^{2}}}\;$

which we can simplify (using our expression for ${\displaystyle e^{\lambda }}$) to

${\displaystyle {d\nu \over dr}={1 \over r}\left(1-{2GM(r) \over c^{2}r}\right)^{-1}\left({2GM(r) \over c^{2}r}+{8\pi G \over c^{4}}r^{2}P\right).\;}$

We obtain a second equation by demanding continuity of the stress-energy tensor: ${\displaystyle \nabla _{\mu }T_{\nu }^{\mu }=0}$. Observing that ${\displaystyle \partial _{t}\rho =\partial _{t}P=0}$ (since the configuration is assumed to be static) and that ${\displaystyle \partial _{\phi }P=\partial _{\theta }P=0}$ (since the configuration is also isotropic), we obtain in particular

${\displaystyle 0=\nabla _{\mu }T_{1}^{\mu }=-{\mathrm {d} P \over \mathrm {d} r}-{1 \over 2}\left(P+\rho c^{2}\right){\mathrm {d} \nu \over \mathrm {d} r}\;}$

Rearranging terms yields:^[2]

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

This gives us two expressions, both containing ${\displaystyle {\mathrm {d} \nu \over \mathrm {d} r}}$. Eliminating ${\displaystyle {\mathrm {d} \nu \over \mathrm {d} r}}$, we obtain:

${\displaystyle {\frac {dP(r)}{dr}}=-{1 \over r}\left({\frac {\rho (r)c^{2}+P(r)}{2}}\right)\left({2GM(r) \over c^{2}r}+{8\pi G \over c^{4}}r^{2}P\right)\left(1-{2GM(r) \over c^{2}r}\right)^{-1}\;}$

Pulling out a factor of ${\displaystyle G \over r}$ and rearranging factors of 2 and $c^{2}$ results in the Tolman–Oppenheimer–Volkoff equation:

${\frac {dP(r)}{dr}}=-{\frac {G}{r^{2}}}\left(\rho (r)+{\frac {P(r)}{c^{2}}}\right)\left(M(r)+4\pi r^{3}{\frac {P(r)}{c^{2}}}\right)\left(1-{\frac {2GM(r)}{c^{2}r}}\right)^{{-1}}\;$

History

Richard C. Tolman analyzed spherically symmetric metrics in 1934 and 1939.^[3][4] The form of the equation given here was derived by J. Robert Oppenheimer and George 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.^[5]

See also

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

References

1. J.R. Oppenheimer & G.M. Volkoff (1939). "On Massive Neutron Cores". Physical Review. 55 (4): 374–381. Bibcode:1939PhRv...55..374O. doi:10.1103/PhysRev.55.374.
2. Tolman, R. C., 1934, Relativity Thermodynamics and Cosmology(Oxford Press.) pp 243-244
3. R.C. Tolman (1934). "Effect of Inhomogeneity on Cosmological Models". Proceedings of the National Academy of Sciences. 20 (3): 169–176. Bibcode:1934PNAS...20..169T. doi:10.1073/pnas.20.3.169.
4. R.C. Tolman (1939). "Static Solutions of Einstein's Field Equations for Spheres of Fluid". Physical Review. 55 (4): 364–373. Bibcode:1939PhRv...55..364T. doi:10.1103/PhysRev.55.364.
5. I. Bombaci (1996). "The Maximum Mass of a Neutron Star". Astronomy and Astrophysics. 305: 871–877. Bibcode:1996A&A...305..871B.