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Wikipedia / 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 modeled by general relativity. The equation is

, 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

is determined by the constraint

When supplemented with an equation of state 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.

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:

Total mass

.

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 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:

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

is the fluid pressure.

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

component:

, we obtain

is as defined in the previous section.

component. Explicitly, we have

) to

(since the configuration is also isotropic), we obtain in particular

Rearranging terms yields:

, we obtain:

results in the Tolman–Oppenheimer–Volkoff equation:

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History

Richard C. Tolman analyzed spherically symmetric metrics in 1934 and 1939. The form of the equation given here was derived by J. Robert Oppenheimer and George 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 ~0.7solar 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. Earlier estimates for this limit range from 1.5 to 3.0 solar masses.

Post-Newtonian approximation

In the post-Newtonian approximation, i.e., gravitational fields that slightly deviates from Newtonian field. In other words, we have

See also

References