Partial suppression of chaos in relativistic three-body problems

¹ National Council of Research – Institute of Complex Systems, Via Madonna del piano 10, 50019 Sesto Fiorentino, Italy
² National Institute of Nuclear Physics (INFN) – Florence unit, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
³ National Institute of Astrophysics – Arcetri Astrophysical Observatory (INAF-OAA), Piazzale E. Fermi 5, 50125 Firenze, Italy
⁴ Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

^★ Corresponding authors; pierfrancesco.dicintio@cnr.it, aatrani@gmail.com

Received: 20 October 2024
Accepted: 27 November 2024

Abstract

Context. Recent numerical results seem to suggest that, in certain regimes of typical particle velocities, when the post-Newtonian (PN) force terms are included, the gravitational N-body problem (for 3 ≤ N ≲ 10³) is intrinsically less chaotic than its classical counterpart, which exhibits a slightly larger maximal Lyapunov exponent Λ_max.

Aims. In this work, we explore the dynamics of wildly chaotic, regular and nearly regular configurations of the three-body problem with and without the PN corrective terms, with the aim being to shed light on the behaviour of the Lyapunov spectra under the effect of the PN corrections.

Methods. Because the interaction of the tangent-space dynamics in gravitating systems – which is needed to evaluate the Lyapunov exponents – becomes rapidly computationally heavy due to the complexity of the higher-order force derivatives involving multiple powers of v/c, we introduce a technique to compute a proxy of the Lyapunov spectrum based on the time-dependent diagonalization of the inertia tensor of a cluster of trajectories in phase-space. In addition, we also compare the dynamical entropy of the classical and relativistic cases.

Results. We find that, for a broad range of orbital configurations, the relativistic three-body problem has a smaller Λ_max than its classical counterpart starting with the exact same initial conditions. However, the other (positive) Lyapunov exponents can be either lower or larger than the corresponding classical ones, thus suggesting that the relativistic precession effectively reduces chaos only along one (or a few) directions in phase-space. As a general trend, the dynamical entropy of the relativistic simulations as a function of the rescaled speed of light falls below the classical value over a broad range of values.

Conclusions. We observe that analyses based solely on Λ_max could lead to misleading conclusions regarding the chaoticity of systems with small (and possibly large) N.