Chaos in a three-body self-gravitating cosmological spacetime

Abstract

We investigate the equal-mass 3-body system in general relativistic lineal gravity in the presence of a cosmological constant Λ. The cosmological vacuum energy introduces features that do not have a non-relativistic counterpart, inducing a competing expansion/contraction of spacetime that competes with the gravitational self-attraction of the bodies. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. As for the system with Λ = 0, we find that the structure of the phase space yields a rich variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasi-periodicity while the latter is a region of chaos. However unlike the Λ = 0 case, we find that a negative cosmological constant considerably diminishes the amount of chaos in the system, even beyond that of the Λ = 0 non-relativistic system. By contrast, a positive cosmological constant considerably enhances the amount of chaos, typically leading to KAM breakdown.

Chaotic behavior in a charged three-body self-gravitating system

“…This is consistent with the fact that same-sign charges induce a local positive cosmological constant, which is known to lead to a rapid increase in chaos. 11 The chaotic behavior is enhanced in the upper left pretzel region of the upper graphs and lower pretzel region of the lower graphs of these figures. We see that the upper and right chaotic regions expand in size, filling the pretzel regions and pushing the annulus region in the opposite direction.…”

“…The key difference is the presence of the electric field, which modifies the effect induced by a cosmological constant. 11 Since the electric field appears in the combination ͑E 2 − ⌳ / ͒ in all equations, we set…”

“…Since the potential is invariant for e i → −e i and the charge configurations interchange as the particles cross, we have only six inequivalent configurations: ͑ne ne ne͒, ͑+ ne ne͒, ͑+ + ne͒, ͑+ − ne͒, ͑+ + −͒, and ͑+ + +͒ where ne denotes a neutral particle. The first of these-the fully neutral configuration-has been considered before, 9,10 and the second of these is equivalent to the cosmological case 11 unless ⌳ e = 0, in which case it becomes equivalent to the neutral case. Note that when one particle is neutral the relative magnitudes of the charges are irrelevant-it is only the product of their magnitudes that matters.…”

“…11 In fact, at higher energies, even for relatively small positive ⌳, the lines in the pretzel regions of the Poincaré section started to thicken and very small regions of chaos appear between the groups of ellipses, reminiscent of the preliminary stages of Kolmogorov-Arnold-Moser ͑KAM͒ breakdown. For negative ⌳ the situation was reversedregions of chaos tended to decrease in size as ⌳ became more negative.…”

“…7 Previous work on the three-body system 8 has been carried out for both equal and unequal masses 9,10 with zero cosmological constant and for equal masses with nonzero cosmological constant ⌳. 11 In all cases the system can be transformed into one of a single particle moving in a two-dimensional potential well of hexagonal symmetry. This potential grows linearly as a function of radial distance in the nonrelativistic case but nonlinearly in the relativistic case, yielding a potential in the shape of a hexagonal carafe.…”