The linear stability of the post-Newtonian triangular equilibrium in the three-body problem

Continuing work initiated in an earlier publication [Yamada, Tsuchiya, and Asada, Phys. Rev. D 91, 124016 (2015)], we reexamine the linear stability of the triangular solution in the relativistic three-body problem for general masses by the standard linear algebraic analysis. In this paper, we start with the Einstein-Infeld-Hoffman form of equations of motion for $N$-body systems in the uniformly rotating frame. As an extension of the previous work, we consider general perturbations to the equilibrium, i.e. we take account of perturbations orthogonal to the orbital plane, as well as perturbations lying on it. It is found that the orthogonal perturbations depend on each other by the first post-Newtonian (1PN) three-body interactions, though these are independent of the lying ones likewise the Newtonian case. We also show that the orthogonal perturbations do not affect the condition of stability. This is because these always precess with two frequency modes; the same with the orbital frequency and the slightly different one by the 1PN effect. The same condition of stability with the previous one, which is valid even for the general perturbations, is obtained from the lying perturbations.