Transient resonances in the inspirals of point particles into black holes

Abstract

We show that transient resonances occur in the two body problem in general relativity, in the highly relativistic, extreme mass-ratio regime for spinning black holes. These resonances occur when the ratio of polar and radial orbital frequencies, which is slowly evolving under the influence of gravitational radiation reaction, passes through a low order rational number. At such points, the adiabatic approximation to the orbital evolution breaks down, and there is a brief but order unity correction to the inspiral rate. Corrections to the gravitational wave signal's phase due to resonance effects scale as the square root of the inverse of mass of the small body, and thus become large in the extreme-mass-ratio limit, dominating over all other post-adiabatic effects. The resonances make orbits more sensitive to changes in initial data (though not quite chaotic), and are genuine non-perturbative effects that are not seen at any order in a standard post-Newtonian expansion. Our results apply to an important potential source of gravitational waves, the gradual inspiral of white dwarfs, neutron stars, or black holes into much more massive black holes. It is hoped to exploit observations of these sources to map the spacetime geometry of black holes. However, such mapping will require accurate models of binary dynamics, which is a computational challenge whose difficulty is significantly increased by resonance effects. We estimate that the resonance phase shifts will be of order a few tens of cycles for mass ratios $\sim 10^{-6}$, by numerically evolving fully relativistic orbital dynamics supplemented with an approximate, post-Newtonian self-force.