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 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.
