Adiabatic Inspiral Transition and Induction to Plunge of a Compact Body in Equatorial Plane Around a Massive Kerr Black Hole

This paper reconstructs the derivation process from the Kerr metric to the adiabatic inspiral, transition, and plunge regimes, aiming to highlight the details and logical connections often overlooked in previous derivations. The first half provides a comprehensive roadmap for readers familiar with advanced general relativity to follow the entire logic of the inspiral-transition-plunge regime from this paper alone. The second half addresses the discontinuity between the adiabatic inspiral and the plunge, including analyses and reinterpretations of the Ori-Thorne and Ori-Thorne-Kesden transition procedures, and proposes two new interpretations: a variant of the Kesden Y-correction and the Adiabatic Inspiral Perturbation-Induced Plunge. The paper also introduces the concept of the Most Stable Circular Orbit (MSCO) and analyzes its properties as a characteristic radius of Kerr black holes.