Transition from adiabatic inspiral to plunge into a spinning black hole

A test particle of mass mu on a bound geodesic of a Kerr black hole of mass M >> mu will slowly inspiral as gravitational radiation extracts energy and angular momentum from its orbit. This inspiral can be considered adiabatic when the orbital period is much shorter than the timescale on which energy is radiated, and quasi-circular when the radial velocity is much less than the azimuthal velocity. Although the inspiral always remains adiabatic provided mu << M, the quasi-circular approximation breaks down as the particle approaches the innermost stable circular orbit (ISCO). In this paper, we relax the quasi-circular approximation and solve the radial equation of motion explicitly near the ISCO. We use the requirement that the test particle's 4-velocity remain properly normalized to calculate a new contribution to the difference between its energy and angular momentum. This difference determines how a black hole's spin changes following a test-particle merger, and can be extrapolated to help predict the mass and spin of the final black hole produced in finite-mass-ratio black-hole mergers. Our new contribution is particularly important for nearly maximally spinning black holes, as it can affect whether a merger produces a naked singularity.