Relativistic mean motion resonance

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 timescale 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__φ:n__r:n_φ=2:1:-2 resonance in Schwarzschild spacetime, which naturally becomes the 3:2 resonance widely studied in the Newtonian limit. We also derive the multibody 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, it is possible that pairs of stellar mass black holes surrounding the supermassive black hole are locked into resonances as they migrate toward the supermassive black hole. As the inner object enter the LISA band, the outer object could reside at a close distance that significantly affects the inner object's gravitational waveform.