Collinear solution to the general relativistic three-body problem

The three-body problem is reexamined in the framework of general relativity. The Newtonian three-body problem admits Euler’s collinear solution, where three bodies move around the common center of mass with the same orbital period and always line up. The solution is unstable. Hence, it is unlikely that such a simple configuration would exist owing to general relativistic forces dependent not only on the masses but also on the velocity of each body. However, we show that the collinear solution remains true with a correction to the spatial separation between masses. Relativistic corrections to the Sun-Jupiter Lagrange points $L_1$, $L_2$, and $L_3$ are also evaluated.

Figure 1

Schematic figure for a classical configuration of three masses denoted by $M_1$ (red), $M_2$ (green) and $M_3$ (blue), which represent a Newtonian collinear solution. The filled disks denote each mass at $t=0$. Definitions of $a$, $R_I$, and $R_{IJ}$ are also indicated.

Figure 2

Orbit of each mass representing the post-Newtonian collinear solution in an inertial frame. We assume $M_1(\mathrm{red}):M_2(\mathrm{green}):M_3(\mathrm{blue})=1:2:3$, $R_{12}=1$ and $a/M=100$. The filled disks denote each mass at $t=0$, when M_1, M_2, and M_3 are located from right to left on the horizontal axis, the triangles at $t=T_N/2$, and the circles at $t=T/2$, which can be obtained also by the reflection of the filled disks with respect to the $\overline{x}=0$ line.