Symplectic Integration of Post-Newtonian Equations of Motion with Spin

TL;DR

This paper introduces a symplectic integrator specifically designed for simulating the post-Newtonian dynamics of spinning black-hole binaries, effectively handling orbital and spin interactions with high accuracy and stability.

Contribution

It develops a novel non-canonical symplectic integration scheme that analytically integrates spin-orbit and spin-spin Hamiltonian terms, improving long-term simulation accuracy.

Findings

Fourth-order integrator exhibits minimal energy and angular momentum drift.

Integrator maintains favorable properties even with weak dissipative forces.

Method achieves excellent error control over long simulations.

Abstract

We present a non-canonically symplectic integration scheme tailored to numerically computing the post-Newtonian motion of a spinning black-hole binary. Using a splitting approach we combine the flows of orbital and spin contributions. In the context of the splitting, it is possible to integrate the individual terms of the spin-orbit and spin-spin Hamiltonians analytically, exploiting the special structure of the underlying equations of motion. The outcome is a symplectic, time-reversible integrator, which can be raised to arbitrary order by composition. A fourth-order version is shown to give excellent behavior concerning error growth and conservation of energy and angular momentum in long-term simulations. Favorable properties of the integrator are retained in the presence of weak dissipative forces due to radiation damping in the full post-Newtonian equations.