Relativistic Mean Motion Resonance

TL;DR

This paper explores how mean motion resonances operate in relativistic systems, specifically involving stellar black holes orbiting a supermassive black hole, and discusses their implications for gravitational wave signals.

Contribution

It develops a relativistic Hamiltonian framework for mean motion resonances and analyzes their behavior in Schwarzschild and post-Newtonian spacetimes.

Findings

Resonance conditions are derived for relativistic black hole systems.

Relativistic effects modify the resonance capture and breaking conditions.

Resonances could influence gravitational wave signals detected by LISA.

Abstract

Mean motion resonances are commonly seen in planetary systems, e.g., in the formation of orbital structure of Jupiter's moons and the gaps in the rings of Saturn. In this work we study their effects in fully relativistic systems. We consider a model problem with two stellar mass black holes orbiting around a supermassive black hole. By adopting a two time-scale expansion technique and averaging over the fast varying orbital variables, we derive the effective Hamiltonian for the slowly varying dynamical variables. The formalism is illustrated with a n'_phi : n'_r : n_phi= 2:1:-2 resonance in Schwarzschild spacetime, which naturally becomes the 3:2 resonance widely studied in the Newtonian limit. We also derive the multi-body Hamiltonian in the post-Newtonian regime, where the radial and azimuthal frequencies are different because of the post-Newtonian precession. The capture and breaking conditions for these relativistic mean motion resonances are also discussed. In particular, pairs of stellar mass black holes surrounding the supermassive black hole could be locked into resonances as they enter the LISA band, and this would affect their gravitational wave waveforms.