Tolman–Oppenheimer–Volkoff equation

${\textstyle m(r)}$ is the total mass contained inside radius ${\textstyle r}$, as measured by the gravitational field felt by a distant observer. It satisfies ${\textstyle m(0)=0}$.[1]

${\displaystyle {\frac {dm}{dr}}=4\pi r^{2}\rho }$

Here, ${\textstyle M}$ is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at ${\textstyle r=R}$, continuity of the metric and the definition of ${\textstyle m(r)}$ require that

${\displaystyle M=m(R)=\int _{0}^{R}4\pi r^{2}\rho \,dr}$

Computing the mass by integrating the density of the object over its volume, on the other hand, will yield the larger value

${\displaystyle M_{1}=\int _{0}^{R}{\frac {4\pi r^{2}\rho }{\sqrt {1-{\frac {2Gm}{rc^{2}}}}}}\,dr}$

The difference between these two quantities,

${\displaystyle \delta M=\int _{0}^{R}4\pi r^{2}\rho \left(1-{\frac {1}{\sqrt {1-{\frac {2Gm}{rc^{2}}}}}}\right)\,dr}$

will be the gravitational binding energy of the object divided by ${\textstyle c^{2}}$ and it is negative.

Let us assume a static, spherically symmetric perfect fluid. The metric components are similar to those for the Schwarzschild metric:[2]

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

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

${\displaystyle T_{0}^{0}=\rho c^{2}}$

and

${\displaystyle T_{i}^{j}=-P\delta _{i}^{j}}$

Where ${\textstyle \rho (r)}$ is the fluid density and ${\textstyle P(r)}$ 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 ${\textstyle G_{00}}$ component:

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

Integrating this expression from 0 to ${\textstyle r}$, we obtain

${\displaystyle e^{-\lambda }=1-{\frac {2Gm}{rc^{2}}}}$

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

${\displaystyle -{\frac {8\pi G}{c^{4}}}Pe^{\lambda }={\frac {-r\nu '+e^{\lambda }-1}{r^{2}}}}$

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

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

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

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

Rearranging terms yields:[3]

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

This gives us two expressions, both containing ${\textstyle d\nu /dr}$. Eliminating ${\textstyle d\nu /dr}$, we obtain:

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

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

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

Richard C. Tolman analyzed spherically symmetric metrics in 1934 and 1939.[4][5] 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. Using gravitational wave observations from binary neutron star mergers (like GW170817) and the subsequent information from electromagnetic radiation (kilonova), the data suggest that the maximum mass limit is close to 2.17 solar masses.[6][7][8][9][10] Earlier estimates for this limit range from 1.5 to 3.0 solar masses.[11]

In the post-Newtonian approximation, i.e., gravitational fields that slightly deviates from Newtonian field, the equation can be expanded in powers of ${\textstyle 1/c^{2}}$. In other words, we have

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

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