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

Abstract

We consider the problem of three body motion for a relativistic one-dimensional selfgravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically 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 obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter η. We find two broad categories of periodic and quasi-periodic 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 non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of η that we could study. However 1 email: fburnell@physics.ubc.ca 2 email: jjmaleck@uwaterloo.ca 3 email: mann@avatar.uwaterloo.ca 4 email: t-oo1@ipc.miyakyo-u.ac.jp 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 KAM breakdown for η ≃ 0.26: above which the near integrable regions degenerate into chaos.

Canonical reduction for dilatonic gravity in3+1dimensions

“…It also reduces to the Newtonian N -body gravitational action in the nonrelativistic limit [5][6][7]. The action for the gravitational scalar-tensor formulation in 1 + 1 dimensions [1,3,4,8] coupled to N particles is, in the presence of a cosmological constant Λ …”

The 4-body problem in a (1+1)-dimensional self-gravitating system

“…A time step in the numerical code has a valuet = 1. All the numerical calculations were carried out using the rescaled variables (9). Henceforth we drop all of the "hats" of the dimensionless variables for convenience.…”

“…Consider first the 3-body case. This problem can be mapped to that of a single particle moving in a hexagonal-shaped well, with the bisectors of the hexagon denoting pair-crossings of the particles [9]. In this case we have {σ 1 , σ 2 } as the Braid operators.…”

“…The 3-body OGS is equivalent to a system of two elastically colliding billiard balls in a uniform, gravitational field [6], as well as to a bound state of three quarks to form a "linear baryon" [7]. It can be extended to fully include relativistic gravitational interactions [8], and investigations have been carried out for both the equal mass [9] and unequal mass [10] cases . Furthermore, it is isomorphic to a system in which a billiard elastically collides with a wedge of fixed angle in a uniform, gravitational field [3].…”

“…We begin with an overview of the problem of 4-body motion in Newtonian gravity, and describe a general classification scheme for the motion of the particles that is an extension of that employed in the 3-body case [9]. Two numerical solution methods employed in the paper are described: the first using numerical integration and the equations of motion to obtain smooth particle trajectories.…”