1. Total mass

2. Derivation from general relativity

3. History

4. Post-Newtonian approximation

5. See also

6. References

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

Here, {{mvar|r}} is a radial coordinate, and {{math|ρ(r₀)}} and {{math|P(r₀)}} are the density and pressure, respectively, of the material at {{math|r {{=}} r₀}}. The quantity {{math|m(r₀)}}, the total mass within

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 {{math|ν(r)}} is determined by the constraint^[1]

When supplemented with an equation of state, {{math|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 {{math|{{sfrac|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 {{math|P(r) {{=}} 0}} and the condition {{math|e^ν {{=}} 1 − {{sfrac|2Gm|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:

Total mass

Here, {{math|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 {{math|r {{=}} R}}, continuity of the metric and the definition of {{math|m(r)}} 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 {{math|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:

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 {{math|ρ(r)}} is the fluid density and {{math|P(r)}} is the fluid pressure.

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

Let us first consider the {{math|G₀₀}} component:

Integrating this expression from 0 to {{mvar|r}}, we obtain

where {{math|M(r)}} is as defined in the previous section. Next, consider the {{math|G₁₁}} component. Explicitly, we have

which we can simplify (using our expression for {{math|e^λ}}) to

We obtain a second equation by demanding continuity of the stress-energy tensor: {{math|∇_μ T{{su|p=μ|b=ν}} {{=}} 0}}. Observing that {{math|∂_t ρ {{=}} ∂_t P {{=}} 0}} (since the configuration is assumed to be static) and that {{math|∂_φ P {{=}} ∂_θ P {{=}} 0}} (since the configuration is also isotropic), we obtain in particular

Rearranging terms yields:^[2]

This gives us two expressions, both containing {{math|{{sfrac|dν|dr}}}}. Eliminating {{math|{{sfrac|dν|dr}}}}, we obtain:

Pulling out a factor of {{math|{{sfrac|G|r}}}} and rearranging factors of 2 and {{math|c²}} results in the Tolman–Oppenheimer–Volkoff equation:

History

Post-Newtonian approximation

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

See also

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

References

• Cycling at the 1986 Commonwealth Games
• Cycling at the 1988 Summer Olympics - Men's individual road race
• Cycling at the 1988 Summer Olympics - Men's points race
• Cycling at the 1988 Summer Olympics - Women's individual road race
• Cycling at the 1988 Summer Olympics - Women's road race
• Cycling at the 1988 Summer Paralympics
• Cycling at the 1990 Commonwealth Games
• Cycling at the 1992 Summer Olympics - Men's individual road race
• Cycling at the 1992 Summer Olympics - Men's points race
• Cycling at the 1992 Summer Olympics - Men's road race
• Cycling at the 1992 Summer Olympics - Women's individual road race
• Cycling at the 1992 Summer Olympics - Women's road race
• Cycling at the 1992 Summer Olympics – Men's road race
• Cycling at the 1992 Summer Paralympics
• Cycling at the 1994 Commonwealth Games
• Cycling at the 1996 Summer Olympics - Men's cross-country
• Cycling at the 1996 Summer Olympics - Men's points race
• Cycling at the 1996 Summer Olympics - Men's road race
• Cycling at the 1996 Summer Olympics - Men's time trial
• Cycling at the 1996 Summer Olympics - Women's cross-country
• Cycling at the 1996 Summer Olympics - Women's road race
• Cycling at the 1996 Summer Olympics - Women's time trial
• Cycling at the 1996 Summer Paralympics
• Cycling at the 1998 Commonwealth Games
• Cycling at the 2000 Summer Olympics - Men's cross-country