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Tidal perturbations of an extreme mass ratio inspiral around a Kerr black hole

Marta Cocco, Gianluca Grignani, Troels Harmark, Marta Orselli, David Pereñiguez, Maarten van de Meent

TL;DR

We develop an analytic framework to model tidal perturbations of a Kerr black hole by solving the Teukolsky master equation for static modes with $=0$ and quadrupolar order $ =2$ in the Hartle–Hawking tetrad, followed by metric reconstruction in the outgoing radiation gauge. Using the reconstructed metric, we derive a secular Hamiltonian for a test particle and compute tidal shifts of the innermost stable circular orbit (ISCO) and the light ring (LR), with shifts showing strong spin dependence and larger effects for retrograde orbits at high spin. The approach yields explicit spin–tidal coupling expressions and a transparent link between external tidal fields and strong-field orbital dynamics, enhancing gravitational-wave modeling for Extreme Mass Ratio Inspirals (EMRIs) in the LISA band. These results provide a robust analytic tool for studying tidal interactions and tests of General Relativity in the strong-field regime, with potential impact on waveform phasing and parameter estimation.

Abstract

We determine the metric of a Kerr black hole subject to external tidal fields using metric reconstruction techniques. Working within the Newman-Penrose formalism, we solve the Teukolsky master equation for static, quadrupolar modes associated with a slowly varying tidal environment, and reconstruct the corresponding metric perturbation in the outgoing radiation gauge. As an application, we derive the secular Hamiltonian governing the motion of a test particle in the tidally deformed Kerr spacetime and investigate long-term tidal effects relevant to extreme-mass-ratio inspirals. In particular, we compute tidal-induced shifts of the innermost stable circular orbit and the light ring. We find that these tidal corrections are strongly spin dependent, with significantly larger effects for retrograde orbits around rapidly rotating black holes. Our results provide a fully analytic framework for studying tidal interactions and secular dynamics in rotating black-hole spacetimes, with direct applications to gravitational-wave modeling and tests of gravity in the strong-field regime.

Tidal perturbations of an extreme mass ratio inspiral around a Kerr black hole

TL;DR

We develop an analytic framework to model tidal perturbations of a Kerr black hole by solving the Teukolsky master equation for static modes with and quadrupolar order in the Hartle–Hawking tetrad, followed by metric reconstruction in the outgoing radiation gauge. Using the reconstructed metric, we derive a secular Hamiltonian for a test particle and compute tidal shifts of the innermost stable circular orbit (ISCO) and the light ring (LR), with shifts showing strong spin dependence and larger effects for retrograde orbits at high spin. The approach yields explicit spin–tidal coupling expressions and a transparent link between external tidal fields and strong-field orbital dynamics, enhancing gravitational-wave modeling for Extreme Mass Ratio Inspirals (EMRIs) in the LISA band. These results provide a robust analytic tool for studying tidal interactions and tests of General Relativity in the strong-field regime, with potential impact on waveform phasing and parameter estimation.

Abstract

We determine the metric of a Kerr black hole subject to external tidal fields using metric reconstruction techniques. Working within the Newman-Penrose formalism, we solve the Teukolsky master equation for static, quadrupolar modes associated with a slowly varying tidal environment, and reconstruct the corresponding metric perturbation in the outgoing radiation gauge. As an application, we derive the secular Hamiltonian governing the motion of a test particle in the tidally deformed Kerr spacetime and investigate long-term tidal effects relevant to extreme-mass-ratio inspirals. In particular, we compute tidal-induced shifts of the innermost stable circular orbit and the light ring. We find that these tidal corrections are strongly spin dependent, with significantly larger effects for retrograde orbits around rapidly rotating black holes. Our results provide a fully analytic framework for studying tidal interactions and secular dynamics in rotating black-hole spacetimes, with direct applications to gravitational-wave modeling and tests of gravity in the strong-field regime.
Paper Structure (16 sections, 85 equations, 2 figures)

This paper contains 16 sections, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Teukolsky master equation
  3. Static modes
  4. Metric reconstruction
  5. Hertz potential
  6. Perturbation to the Kerr metric: results
  7. m=0 mode
  8. m=±1 modes
  9. m=±2 modes
  10. Fixing the coefficients c2m
  11. Secular dynamics in tidally deformed Kerr spacetime
  12. Secular Hamiltonian
  13. Tidal deformation of ISCO and LR
  14. ISCO shifts
  15. LR shifts
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Secular tidal perturbations to the radius $r$, energy $E$, angular momentum $L$, frequency $\Omega$, and redshift factor $U$ of a massive test particle moving on the ISCO of a tidally deformed Kerr black hole. The shifts are shown as functions of the spin $a$ and are normalized to the Schwarzschild ($a=0$) case. The left panel corresponds to prograde motion, while the right panel corresponds to retrograde motion.
  • Figure 2: Secular tidal perturbations to the radius $r$, impact parameter $b$, and frequency $\Omega$ of a massless test particle moving on the LR of a tidally deformed Kerr black hole. The shifts are shown as functions of the spin $a$ and are normalized to the Schwarzschild ($a=0$) case. The left panel corresponds to prograde motion, while the right panel corresponds to retrograde motion.