General Relativity and Dance Steps

Choreographic Solution to the General Relativistic Three Body Problem

T. Imai, T. Chiba, and H. Asada

PRL 98, 201102 (2007)

URL: http://link.aps.org/abstract/PRL/v98/e201102

This group from Hirosaki University in Japan has found a periodic solution of the 3-body problem for Newtonian gravity plus first order corrections from general relativity. Though the paper is not well-written and the background is not sufficient for understanding the group's work, I found the result interesting enough to read through all four pages.

I know a little about the three-body problem. My main impression is that there is no good way to separate coordinates and find a general solution. It admits both regular and chaotic solutions.

The authors provided some interesting historical background. Euler found one of the first solutions to the 3-body problem, with all three masses in a line. Though the distance between masses may change, the ratio of the distances does not. Euler found his solution in 1765. Seven years later, Laplace found a second, highly symmetric solution in which the three masses are at the corners of an equilateral triangle. According to the authors, these are the only solutions in which each particle's orbit is an ellipse.

Later, Poincare showed that not all solutions can be obtained analytically. In 1993, Christopher Moore found a solution in which all three particles move along a figure eight --- a very interesting solution to the problem! Moore was looking at general potentials in 2+1 dimensions and the types of braids the three bodies would weave in spacetime.

This figure eight, along with the solutions of Euler and Laplace, are examples of a more general class of solutions called choreographic orbits. In such an orbit, each of the three particles moves in a single closed orbit. It's as if all three particles are performing a dance routine. Some of Moore's braids are quite complicated indeed!

One of the best known effects of general relativity is the precession of Mercury's perhelion. Mercury's orbit around the sun is an ellipse with rotating axes. Therefore, it is not periodic (at least on short time scales). Do the effects of general relativity make choreographic orbits impossible?

The authors show that the first order corrections to Newtonian gravity do admit a figure eight solution, so at least some choreographic orbits are permitted. The authors had to choose the initial conditions carefully to obtain a periodic orbit.

In the conclusion, the authors point out that the higher order effects of general relativity might not have solutions. It seems to me that once gravitational waves are included, no periodic solutions will be possible. If a system is radiating away its energy, it can never return to its initial condition, thus the orbits of its constituent particles are not closed. I don't believe any 3-body system is symmetric enough that it would not emit gravitational wave.