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We extend the Ori and Thorne (OT) procedure to compute the transition from anadiabatic inspiral into a geodesic plunge for any spin, with emphasis onnear-extremal ones. Our analysis revisits the validity of the approximationsmade in OT. In particular, we discuss possible effects coming from eccentricityand non-geodesic past-history of the orbital evolution. We find three differentscaling regimes according to whether the mass ratio is much smaller, of thesame order or much larger than the near extremal parameter describing how fastthe primary black hole rotates. Eccentricity and non-geodesic past-historycorrections are always sub-leading, indicating that the quasi-circularapproximation applies throughout the transition regime. However, we show thatthe OT assumption that the energy and angular momentum evolve linearly withproper time must be modified in the near-extremal regime. Using our transitionequations, we describe an algorithm to compute the full worldline in propertime for an extreme mass ratio inspiral (EMRI) and the resultant gravitationalwaveform in the high spin limit.

Transition from Inspiral to Plunge: A Complete Near-Extremal Trajectory and Associated Waveform