  Extreme mass ratio inspirals (EMRIs) -- systems with a compact object
orbiting a much more massive (e.g., galactic center) black hole -- are of
interest both as a new probe of the environments of galactic nuclei, and their
waveforms are a precision test of the Kerr metric. This work focuses on the
effects of an external perturbation due to a third body around an EMRI system.
This perturbation will affect the orbit most significantly when the inner body
crosses a resonance with the outer body, and result in a change of the
conserved quantities (energy, angular momentum, and Carter constant) or
equivalently of the actions, which results in a subsequent phase shift of the
waveform that builds up over time. We present a general method for calculating
the changes in action during a resonance crossing, valid for generic orbits in
the Kerr spacetime. We show that these changes are related to the gravitational
waveforms emitted by the two bodies (quantified by the amplitudes of the Weyl
scalar $\psi_4$ at the horizon and at $\infty$) at the frequency corresponding
to the resonance. This allows us to compute changes in the action variables for
each body, without directly computing the explicit metric perturbations, and
therefore we can carry out the computation by calling an existing black hole
perturbation theory code. We show that our calculation can probe resonant
interactions in both the static and dynamical limit. We plan to use this
technique for future investigations of third-body effects in EMRIs and their
potential impact on waveforms for LISA.
