Three-body equations of motion in successive post-Newtonian approximations

Abstract

There are periodic solutions to the equal-mass three-body (and N-body) problem in Newtonian gravity. The figure-eight solution is one of them. In this paper, we discuss its solution in the first and second post-Newtonian approximations to General Relativity. To do so we derive the canonical equations of motion in the ADM gauge from the three-body Hamiltonian. We then integrate those equations numerically, showing that quantities such as the energy, linear and angular momenta are conserved down to numerical error. We also study the scaling of the initial parameters with the physical size of the triple system. In this way we can assess when general relativistic results are important and we determine that this occur for distances of the order of 100M, with M the total mass of the system. For distances much closer than those, presumably the system would completely collapse due to gravitational radiation. This sets up a natural cut-off to Newtonian N-body simulations. The method can also be used to dynamically provide initial parameters for subsequent full nonlinear numerical simulations.

8 pages, 9 figures

References in corpus (7)

• Relativistic Closed-Form Hamiltonian for Many-Body Gravitating Systems in the Post-Minkowskian Approximation
• Foundations of multiple black hole evolutions
• Close encounters of three black holes
• Relativistic hydrodynamics in the presence of puncture black holes
• Post-Newtonian N-body simulations
• Choreographic solution to the general relativistic three-body problem
• Can N-body systems generate periodic gravitational waves?

Cited by in corpus (15)

• Interactions between multiple supermassive black holes in galactic nuclei: a solution to the final parsec problem
• Numerical evolution of multiple black holes with accurate initial data
• Collinear solution to the general relativistic three-body problem
• Gravitational waveforms for 2- and 3-body gravitating systems
• Chaos in self-gravitating many-body systems: Lyapunov time dependence of $N$ and the influence of general relativity
• Tidal deformations of a binary system induced by an external Kerr black hole
• Next-to-leading order spin-orbit and spin(a)-spin(b) Hamiltonians for n gravitating spinning compact objects
• Three-Body Effective Potential in General Relativity at Second Post-Minkowskian Order and Resulting Post-Newtonian Contributions
• Post-Newtonian effects on Lagrange's equilateral triangular solution for the three-body problem
• Effective two-body approach to the hierarchical three-body problem
• Uniqueness of collinear solutions for the relativistic three-body problem
• Effective two-body approach to the hierarchical three-body problem: quadrupole to 1PN
• Regular second order perturbations of binary black holes: The extreme mass ratio regime
• Exploring the Dynamics of General Relativistic Binary-Single and Binary-Binary Encounters of Black Holes
• The hierarchical three-body problem at $\mathcal{O}(G^2)$