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

Boyan Wang

TL;DR

This work analyzes the full sequence from Kerr geodesics to adiabatic inspiral, transition, and plunge for a compact body in the equatorial plane, introducing a Spin-Sign-Converted Dimensionless formalism and the MSCO as an intrinsic Kerr-radius benchmark. It reconciles adiabatic evolution with the plunge via Ori-Thorne and Kesden Y-corrections, extends the transition framework to include radial background forces, and proposes a perturbation-based pathway to a smooth plunge. The paper provides explicit analytic and semi-analytic constructions (including SCCO/ISCO relations, SSCD variables, and Painlevé I-based transition dynamics) together with numerical schemes to solve the governing ODEs near ISCO, and it highlights observational implications for LISA by identifying MSCO behavior and potential signatures in gravitational-wave signals. The Adiabatic Inspiral Perturbation Induced plunge phase offers a novel route to connect slow inspiral with rapid plunge through controlled perturbations, enriching the toolkit for modeling extreme-mass-ratio inspirals (EMRIs) in Kerr spacetimes.

Abstract

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.

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

TL;DR

This work analyzes the full sequence from Kerr geodesics to adiabatic inspiral, transition, and plunge for a compact body in the equatorial plane, introducing a Spin-Sign-Converted Dimensionless formalism and the MSCO as an intrinsic Kerr-radius benchmark. It reconciles adiabatic evolution with the plunge via Ori-Thorne and Kesden Y-corrections, extends the transition framework to include radial background forces, and proposes a perturbation-based pathway to a smooth plunge. The paper provides explicit analytic and semi-analytic constructions (including SCCO/ISCO relations, SSCD variables, and Painlevé I-based transition dynamics) together with numerical schemes to solve the governing ODEs near ISCO, and it highlights observational implications for LISA by identifying MSCO behavior and potential signatures in gravitational-wave signals. The Adiabatic Inspiral Perturbation Induced plunge phase offers a novel route to connect slow inspiral with rapid plunge through controlled perturbations, enriching the toolkit for modeling extreme-mass-ratio inspirals (EMRIs) in Kerr spacetimes.

Abstract

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.

Paper Structure

This paper contains 39 sections, 184 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kerr Metric
  3. Fundamental governing equations
  4. Time-Like Kerr Geodesics
  5. Simplification: equatorial plane
  6. Circular orbits
  7. Stable circular orbit(SCO)
  8. Innermost stable circular orbit(ISCO)
  9. Spin-sign-converted Dimensionless (SSCD) formulas
  10. Spin Sign Convention
  11. Dimensionless
  12. Adiabatic Inspiral and Plunge Regime
  13. Gravitational Wave Emission
  14. Timescale Separation
  15. Adiabatic Inspiral Regime
  16. ...and 24 more sections

Figures (11)

  • Figure 1: Plotting of effective potential DesmosGraph
  • Figure 2: Plotting of Adiabatic Inspiral. DesmosGraph
  • Figure 4: Plotting of Plunge together with Adiabatic Inspiral DesmosGraph
  • Figure 6: Zoom in of plunge and adiabatic plotting DesmosGraph
  • Figure 7: Ori-Thorne Transition ProcedureDesmosGraph The blue dotted line is the ISCO radius, green dotted line is ISCO minus $\delta$ where $\delta$ is the 'inward push'. The purple line is Ori-Thorne transition plot on the previous plots of Adiabatic and Plunge. The plot is the dimensionless proper time $\tilde{\tau}$ as function of dimensionless radius $\tilde{r}$. As $\tilde{r}$ getting close to $\tilde{r}_{isco}$, Ori-Thorne transition function takes over from adiabatic to plunge, jumping through the radius where the other two fails.
  • ...and 6 more figures