Choreographic Solution to the General-Relativistic Three-Body Problem

We reexamine the three-body problem in the framework of general relativity. The Newtonian $N$-body problem admits choreographic solutions, where a solution is called choreographic if every massive particle moves periodically in a single closed orbit. One is a stable figure-eight orbit for a three-body system, which was found first by Moore (1993) and rediscovered with its existence proof by Chenciner and Montgomery (2000). In general relativity, however, the periastron shift prohibits a binary system from orbiting in a single closed curve. Therefore, it is unclear whether general-relativistic effects admit choreography such as the figure eight. We examine general-relativistic corrections to initial conditions so that an orbit for a three-body system can be choreographic and a figure eight. This illustration suggests that the general-relativistic $N$-body problem also may admit a certain class of choreographic solutions.

Figure 1

A schematic figure for a binary orbit in general relativity. The orbit is not closed any more, because of the periastron shift.

Figure 2

Figure-eight orbits starting at the Newtonian initial condition. The solid curve denotes a figure-eight orbit in the Newtonian gravity. The dashed curve denotes a trajectory of one mass following the EIH equation of motion under the same Newtonian initial condition.

Figure 3

Figure-eight orbits. The solid curve denotes a figure-eight orbit in the Newtonian gravity. The dashed curve denotes a figure-eight orbit at the 1PN order of general relativity.

Figure 4

Relativistic figure-eight orbit. The principal axes of the orbit are chosen as $x$ and $y$ axes.