Uniqueness of collinear solutions for the relativistic three-body problem

Continuing work initiated in an earlier publication [Yamada, Asada, Phys. Rev. D 82, 104019 (2010)], we investigate collinear solutions to the general relativistic three-body problem. We prove the uniqueness of the configuration for given system parameters (the masses and the end-to-end length). First, we show that the equation determining the distance ratio among the three masses, which has been obtained as a seventh-order polynomial in the previous paper, has at most three positive roots, which apparently provide three cases of the distance ratio. It is found, however, that, even for such cases, there exists one physically reasonable root and only one, because the remaining two positive roots do not satisfy the slow-motion assumption in the post-Newtonian approximation and are thus discarded. This means that, especially for the restricted three-body problem, exactly three positions of a third body are true even at the post-Newtonian order. They are relativistic counterparts of the Newtonian Lagrange points $L_1$, $L_2$, and $L_3$. We show also that, for the same masses and full length, the angular velocity of the post-Newtonian collinear configuration is smaller than that for the Newtonian case. Provided that the masses and angular rate are fixed, the relativistic end-to-end length is shorter than the Newtonian one.

Figure 1

Top panel: The seventh-order polynomial in the left-hand side of Eq. (13). The horizontal axis is chosen as $z$. We take ${\nu }_1=1/7$, ${\nu }_2=5/7$, ${\nu }_3=1/7$, $a/M={10}^4$ ($v\sim {10}^{-2}$) in order to exaggerate small effects in these figures. Clearly such a symmetric choice of the mass ratios produces a trivial root as $z=1$, which makes it easy to check numerical calculations. $M_2$ is relatively large so that the centrifugal force can be large. Middle panel: The seventh-order polynomial around the smallest positive root $z_S$. Bottom panel: The polynomial around the moderate positive root.