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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 modeled by general relativity. The equation^{ [1] } is

- Total mass
- Derivation from general relativity
- History
- Post-Newtonian approximation
- See also
- References

Here, is a radial coordinate, and and are the density and pressure, respectively, of the material at radius . The quantity , the total mass within , is discussed below.

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] }

where is determined by the constraint^{ [1] }

When supplemented with an equation of state, , 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 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 and the condition 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:

is the total mass contained inside radius , as measured by the gravitational field felt by a distant observer. It satisfies .^{ [1] }

Here, is the total mass of the object, again, as measured by the gravitational field felt by a distant observer. If the boundary is at , continuity of the metric and the definition of require that

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

The difference between these two quantities,

will be the gravitational binding energy of the object divided by 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] }

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

and

Where is the fluid density and is the fluid pressure.

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

Let us first consider the component:

Integrating this expression from 0 to , we obtain

where is as defined in the previous section. Next, consider the component. Explicitly, we have

which we can simplify (using our expression for ) to

We obtain a second equation by demanding continuity of the stress-energy tensor: . Observing that (since the configuration is assumed to be static) and that (since the configuration is also isotropic), we obtain in particular

Rearranging terms yields:^{ [3] }

This gives us two expressions, both containing . Eliminating , we obtain:

Pulling out a factor of and rearranging factors of 2 and results in the Tolman–Oppenheimer–Volkoff equation:

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 . In other words, we have

The **Navier–Stokes equations** are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

In fluid mechanics, **hydrostatic equilibrium** is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space. It is what makes heavenly bodies spherical, in general.

In fluid mechanics, the **Grashof number** is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.

The **gravitational binding energy** of a system is the minimum energy which must be added to it in order for the system to cease being in a gravitationally bound state. A gravitationally bound system has a lower gravitational potential energy than the sum of the energies of its parts when these are completely separated—this is what keeps the system aggregated in accordance with the minimum total potential energy principle.

In physics and astronomy, the **Reissner–Nordström metric** is a static solution to the Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass *M*. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric.

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In stellar physics, the **Jeans instability** causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter. For stability, the cloud must be in hydrostatic equilibrium, which in case of a spherical cloud translates to

The **Newman–Penrose** (**NP**) **formalism** is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members often asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.

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In general relativity, the **Vaidya metric** describes the non-empty external spacetime of a spherically symmetric and nonrotating star which is either emitting or absorbing null dusts. It is named after the Indian physicist Prahalad Chunnilal Vaidya and constitutes the simplest non-static generalization of the non-radiative Schwarzschild solution to Einstein's field equation, and therefore is also called the "radiating(shining) Schwarzschild metric".

In fluid mechanics, the **Rayleigh–Plesset equation** or **Besant–Rayleigh–Plesset equation** is a nonlinear ordinary differential equation which governs the dynamics of a spherical bubble in an infinite body of incompressible fluid. Its general form is usually written as

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In condensed matter physics and physical chemistry, the **Lifshitz theory of van der Waals forces**, sometimes called the **macroscopic theory of van der Waals forces**, is a method proposed by Evgeny Mikhailovich Lifshitz in 1954 for treating van der Waals forces between bodies which does not assume pairwise additivity of the individual intermolecular forces; that is to say, the theory takes into account the influence of neighboring molecules on the interaction between every pair of molecules located in the two bodies, rather than treating each pair independently.

The **Ellis drainhole** is the earliest-known complete mathematical model of a traversable wormhole. It is a static, spherically symmetric solution of the Einstein vacuum field equations augmented by inclusion of a scalar field minimally coupled to the geometry of space-time with coupling polarity opposite to the orthodox polarity :

In quantum mechanics, a **Dirac membrane** is a model of a charged membrane introduced by Paul Dirac in 1962. Dirac's original motivation was to explain the mass of the muon as an excitation of the ground state corresponding to an electron. Anticipating the birth of string theory by almost a decade, he was the first to introduce what is now called a type of Nambu–Goto action for membranes.

In astrophysics, **Chandrasekhar's white dwarf equation** is an initial value ordinary differential equation introduced by the Indian American astrophysicist Subrahmanyan Chandrasekhar, in his study of the gravitational potential of completely degenerate white dwarf stars. The equation reads as

In general relativity, **Buchdahl's theorem**, named after Hans Adolf Buchdahl, makes more precise the notion that there is a maximal sustainable density for ordinary gravitating matter. It gives an inequality between the mass and radius that must be satisfied for static, spherically symmetric matter configurations under certain conditions. In particular, for areal radius , the mass must satisfy

- 1 2 3 4 5 Oppenheimer, J. R.; Volkoff, G. M. (1939). "On Massive Neutron Cores".
*Physical Review*.**55**(4): 374–381. Bibcode:1939PhRv...55..374O. doi:10.1103/PhysRev.55.374. - ↑ Misner, Charles W.; Thorne, Kip S.; Wheeler, John Archibald (2017). "Coordinates and Metric for a Static, Spherical System".
*Gravitation*. Princeton University Press. pp. 594–595. ISBN 978-0-691-17779-3. - ↑ Tolman, R. C. (1934).
*Relativity Thermodynamics and Cosmology*. Oxford Press. pp. 243–244. - ↑ Tolman, R. C. (1934). "Effect of Inhomogeneity on Cosmological Models" (PDF).
*Proceedings of the National Academy of Sciences*.**20**(3): 169–176. Bibcode:1934PNAS...20..169T. doi: 10.1073/pnas.20.3.169 . PMC 1076370 . PMID 16587869. - ↑ Tolman, R. C. (1939). "Static Solutions of Einstein's Field Equations for Spheres of Fluid" (PDF).
*Physical Review*.**55**(4): 364–373. Bibcode:1939PhRv...55..364T. doi:10.1103/PhysRev.55.364. - ↑ Margalit, B.; Metzger, B. D. (2017-12-01). "Constraining the Maximum Mass of Neutron Stars from Multi-messenger Observations of GW170817".
*The Astrophysical Journal*.**850**(2): L19. arXiv: 1710.05938 . Bibcode:2017ApJ...850L..19M. doi: 10.3847/2041-8213/aa991c . S2CID 119342447. - ↑ Shibata, M.; Fujibayashi, S.; Hotokezaka, K.; Kiuchi, K.; Kyutoku, K.; Sekiguchi, Y.; Tanaka, M. (2017-12-22). "Modeling GW170817 based on numerical relativity and its implications".
*Physical Review D*.**96**(12): 123012. arXiv: 1710.07579 . Bibcode:2017PhRvD..96l3012S. doi:10.1103/PhysRevD.96.123012. S2CID 119206732. - ↑ Ruiz, M.; Shapiro, S. L.; Tsokaros, A. (2018-01-11). "GW170817, general relativistic magnetohydrodynamic simulations, and the neutron star maximum mass".
*Physical Review D*.**97**(2): 021501. arXiv: 1711.00473 . Bibcode:2018PhRvD..97b1501R. doi:10.1103/PhysRevD.97.021501. PMC 6036631 . PMID 30003183. - ↑ Rezzolla, L.; Most, E. R.; Weih, L. R. (2018-01-09). "Using Gravitational-wave Observations and Quasi-universal Relations to Constrain the Maximum Mass of Neutron Stars".
*Astrophysical Journal*.**852**(2): L25. arXiv: 1711.00314 . Bibcode:2018ApJ...852L..25R. doi: 10.3847/2041-8213/aaa401 . S2CID 119359694. - ↑ "How massive can neutron star be?". Goethe University Frankfurt. 15 January 2018. Retrieved 19 February 2018.
- ↑ Bombaci, I. (1996). "The Maximum Mass of a Neutron Star".
*Astronomy and Astrophysics*.**305**: 871–877. Bibcode:1996A&A...305..871B.

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