Chaos in an exact relativistic three-body self-gravitating system

We consider the problem of three-body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the three-body Hamiltonian, implicitly determined in terms of the four coordinates and momentum degrees of freedom in the system. Nonrelativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal-mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtain orbits in both the hexagonal and three-body representations of the system, and plot the Poincaré sections as a function of the relativistic energy parameter η. We find two broad categories of periodic and quasiperiodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase space between these two types. Despite the high degree of nonlinearity in the relativistic system, we find that the global structure of its phase space remains qualitatively the same as its nonrelativistic counterpart for all values of η that we could study. However, the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing η. For the post-Newtonian system we find that it experiences a chaotic transition in the interval $0.21<\mathrm{}\eta <0.26,$ above which some of the near-integrable regions degenerate into chaos.