Two-body problem in general relativity

In general relativity, the Kepler problem involving solving for the motion of a particle of negligible mass in the gravitational field of another, more massive body, which is described by the Schwarzschild solution. The solution accounts for the anomalous precession of the planet Mercury, which was discovered prior to general relativity.

Derivation

General relativity depends on the definition of the Schwarzschild metric

Failed to parse (syntax error): {\displaystyle ds^{2} = \left( 1 - \frac{r_{s}}{r} \right) c^{2} dt^{2} - \frac{dr^{2}}{1 - \frac{r_{s}}{r}} – r^{2} d\phi^{2} }

where the coordinate frame has been aligned with the plane of the orbit, so that the altitude angle may be eliminated from consideration. From this metric, one can form the Lagrangian from an equivalent kinetic energy

Failed to parse (syntax error): {\displaystyle T = \frac{1}{2} \frac{ds^{2}}{ds^{2}} = \left( 1 - \frac{r_{s}}{r} \right) c^{2} \left( \frac{dt}{ds} \right)^{2} – \frac{dr^{2}}{1 - \frac{r_{s}}{r}} \left( \frac{dr}{ds} \right)^{2} – r^{2} d\phi^{2} \left( \frac{d\phi}{ds} \right)^{2} }

since there is no gravitational potential energy in general relativity. The first two Lagrangian equations are

$r^{2}{\frac {d\phi }{ds}}={\frac {1}{\sqrt {\beta }}}$

$\left(1-{\frac {r_{s}}{r}}\right){\frac {dt}{ds}}={\frac {-k}{c^{2}{\sqrt {\beta }}}}$

Substituting these equations into the definition of ‘’ds’’² yields the equation

Failed to parse (syntax error): {\displaystyle \frac{dr^{2}}{1 - \frac{r_{s}}{r}} = \left\{ -\beta r^{4} + \frac{k^{2}r^{4}}{c^{2} \left( 1 - \frac{r_{s}}{r} \right)} – r^{2} \right\} d\phi^{2} }

Changing variables to ‘’u = 1/r’’ gives the equation

$\left({\frac {du}{d\phi }}\right)^{2}={\frac {k^{2}}{c^{2}}}-\left(1-r_{s}u\right)\left(\beta +u^{2}\right)$

Making another change of variables

$\zeta ={\frac {r_{s}u}{4}}-{\frac {1}{12}}$

gives the soluble equation

Failed to parse (syntax error): {\displaystyle \left( \frac{d\zeta}{d\phi} \right)^{2} = 4 \zeta^{3} – g_{2} \zeta – g_{3} }

where

$g_{2}={\frac {1}{12}}-{\frac {r_{s}^{2}\beta }{4}}$

and

$g_{3}={\frac {1}{216}}+{\frac {r_{s}^{2}\beta }{24}}-{\frac {r_{s}^{2}k^{2}}{16c^{2}}}$

whose solution is an elliptic function.

${\frac {r_{s}}{4r}}={\frac {1}{12}}+J(\phi -\phi _{0})$

Quasi-elliptical orbits

There are three roots e₁, e₂, and e₃ at which the derivative of the inverse radius u with respect to theta is zero

Failed to parse (syntax error): {\displaystyle \frac{d\zeta}{d\phi} = 4\zeta^{3} – g_{2} \zeta – g_{3} = 0 }

We can define the change in angle between two such nodes (which is real)

Failed to parse (syntax error): {\displaystyle \Delta \phi = \int_{e_{1}}^{\infty} \frac{d\zeta}{\sqrt{4\zeta^{3} – g_{2} \zeta – g_{3}}} }

which is undetectably close to pi radians, except for the planet Mercury. The planet moves between a minimum radius

$r_{min}={\frac {3r_{s}}{1+12e_{3}}}$

and a maximum radius

$r_{max}={\frac {3r_{s}}{1+12e_{2}}}$

corresponding to the value of zeta at the two extrema of radius.

Transiently unstable circular orbits

If two of the three roots are equal and positive, the orbits are asymptotically circular at positive and negative infinite theta. Let the two repeated roots be called e which we can also call n²/3; the third, unrepeated root is –2e. The solution is then

$\zeta ={\frac {r_{s}}{4r}}-{\frac {1}{12}}=e-{\frac {n^{2}}{\cosh ^{2}n\phi }}$

As theta goes to positive or negative infinity, the orbit approaches asymptotically to the circle

${\frac {r_{s}}{4r}}-{\frac {1}{12}}=e$

In these cases, the radii of the orbits lie between 2alpha and 3alpha.

Stable circular orbits

If two roots are equal and negative, the orbits are circular, periodic orbits.

The radii of these orbits must be greater than 3alpha.

Further reading

• Whittaker, ET (1937). A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, with an Introduction to the Problem of Three Bodies (4th ed. ed.). New York: Dover Publications. ISBN. {{cite book}}: |edition= has extra text (help)

• Hagihara, Y (1931). "Unknown title". Japanese Journal of Astrophysics and Geophysics. 8: 67–176.