Celestial mechanics in Kerr spacetime

Abstract

The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy $E$, the axial component of the angular momentum, $L_z$, and Carter's constant $Q$. These parameters are obtained by solving the Hamilton-Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies ${\omega }_r$, ${\omega }_{\theta }$ and ${\omega }_{\varphi }$ associated with the radial, polar and azimuthal components of orbital motion. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this article, it is shown that the fundamental frequencies are geometric invariants and explicit formulas in terms of quadratures are derived. The numerical evaluation of these formulas in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters such as the semi-latus rectum $p$, the eccentricity $e$ or the inclination parameter ${\theta }_{-}$ are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulas.

25 pages (LaTeX), 5 figures, submitted to Class. Quantum Grav

Cited by in corpus (178)

• Self-force and radiation reaction in general relativity
• Zoom & Whirl: Eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction
• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• Mapping spacetimes with LISA: inspiral of a test-body in a `quasi-Kerr' field
• Using LISA EMRI sources to test off-Kerr deviations in the geometry of massive black holes
• Gravitational wave snapshots of generic extreme mass ratio inspirals
• Analytical solutions of bound timelike geodesic orbits in Kerr spacetime
• A Periodic Table for Black Hole Orbits
• Extreme- and Intermediate-Mass Ratio Inspirals in Dynamical Chern-Simons Modified Gravity
• Confusion Noise from LISA Capture Sources
• The Fast and the Fiducial: Augmented kludge waveforms for detecting extreme-mass-ratio inspirals
• Gravitational radiation reaction and inspiral waveforms in the adiabatic limit
• Rotating black hole orbit functionals in the frequency domain
• Adiabatic waveforms for extreme mass-ratio inspirals via multivoice decomposition in time and frequency
• Spacetime and orbits of bumpy black holes
• Towards adiabatic waveforms for inspiral into Kerr black holes: I. A new model of the source for the time domain perturbation equation
• The influence of the hydrodynamic drag from an accretion torus on extreme mass-ratio inspirals
• Foundations of an effective-one-body model for coalescing binaries on eccentric orbits
• An observable signature of a background deviating from Kerr
• Testing Chern-Simons Modified Gravity with Gravitational-Wave Detections of Extreme-Mass-Ratio Binaries
• The Science of the Einstein Telescope
• Stellar transits across a magnetized accretion torus as a mechanism for plasmoid ejection
• Black hole perturbation theory and gravitational self-force
• Homoclinic Orbits around Spinning Black Holes I: Exact Solution for the Kerr Separatrix
• Metric perturbations produced by eccentric equatorial orbits around a Kerr black hole
• Level sets of the lapse function in static GR
• The location of the last stable orbit in Kerr spacetime
• Charged particle motion in Kerr-Newmann space-times
• Extreme Mass Ratio Inspirals: LISA's unique probe of black hole gravity
• Gravitational Waves from a Particle in Circular Orbits around a Rotating Black Hole to the 11th Post-Newtonian Order
• Adiabatic waveforms from extreme-mass-ratio inspirals: an analytical approach
• Gravitational waves from Extreme Mass Ratio Inspirals in non-pure Kerr spacetimes
• Binary black hole merger gravitational waves and recoil in the large mass ratio limit
• Gravitational self-force on eccentric equatorial orbits around a Kerr black hole
• Periastron advance in spinning black hole binaries: comparing effective-one-body and Numerical Relativity
• First Law of Mechanics for Compact Binaries on Eccentric Orbits
• Periastron Advance in Spinning Black Hole Binaries: Gravitational Self-Force from Numerical Relativity
• Importance of transient resonances in extreme-mass-ratio inspirals
• Growth of resonances and chaos for a spinning test particle in the Schwarzschild background
• Improved analytic extreme-mass-ratio inspiral model for scoping out eLISA data analysis
• Gravitational waves from bodies orbiting the Galactic Center black hole and their detectability by LISA
• Analytic solutions for parallel transport along generic bound geodesics in Kerr spacetime
• Hamilton-Jacobi equation for spinning particles near black holes
• Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole
• The role of the supermassive black hole spin in the estimation of the EMRI event rate
• Resonantly enhanced and diminished strong-field gravitational-wave fluxes
• Comparison Between Self-Force and Post-Newtonian Dynamics: Beyond Circular Orbits
• Exciting black hole modes via misaligned coalescences: I. Inspiral, transition, and plunge trajectories using a generalized Ori-Thorne procedure
• Conditions for Sustained Orbital Resonances in Extreme Mass Ratio Inspirals
• Gravitational waves from merging compact binaries
• The harmonic structure of generic Kerr orbits
• The Astrophysics of Resonant Orbits in the Kerr Metric
• Assessing the impact of transient orbital resonances
• Hamiltonian Formulation of the Conservative Self-Force Dynamics in the Kerr Geometry
• Extreme mass ratio inspirals on the equatorial plane in the adiabatic order
• Homoclinic Orbits around Spinning Black Holes II: The Phase Space Portrait
• Eccentric binary black holes: Comparing numerical relativity and small mass-ratio perturbation theory
• Astrophysically relevant bound trajectories around a Kerr black hole
• Spinning test body orbiting around a Kerr black hole: Eccentric equatorial orbits and their asymptotic gravitational-wave fluxes
• Gyroscopes orbiting black holes: A frequency-domain approach to precession and spin-curvature coupling for spinning bodies on generic Kerr orbits
• A census of transient orbital resonances encountered during binary inspiral
• Fast and Fourier: Extreme Mass Ratio Inspiral Waveforms in the Frequency Domain
• Periodic orbits around Kerr Sen black holes
• Orbital resonances around Black holes
• Isofrequency pairing of geodesic orbits in Kerr geometry
• Non-local parameter degeneracy in the intrinsic space of gravitational-wave signals from extreme-mass-ratio inspirals
• Resonantly enhanced kicks from equatorial small mass-ratio inspirals
• "Flux-balance formulae" for extreme mass-ratio inspirals
• Accurate time-domain gravitational waveforms for extreme-mass-ratio binaries
• Analytic Kludge Waveforms for Extreme Mass Ratio Inspirals of Charged Object around Kerr-Newman Black Hole
• Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations
• Importance of tidal resonances in extreme-mass-ratio inspirals
• New Kludge Scheme for the Construction of Approximate Waveforms for Extreme-Mass-Ratio Inspirals
• Modeling transient resonances in extreme-mass-ratio inspirals
• Spacetime Encodings I- A Spacetime Reconstruction Problem
• Transition from Inspiral to Plunge: A Complete Near-Extremal Trajectory and Associated Waveform
• Probing orbits of stellar mass objects deep in galactic nuclei with quasi-periodic eruptions -- II: population analysis
• Relativistic Effects in Extreme Mass Ratio Gravitational Wave Bursts
• Boundary to bound dictionary for generic Kerr orbits
• Gravitational waves from extreme mass ratio inspirals around bumpy black holes
• Evolution of the Carter constant for inspirals into a black hole: effect of the black hole quadrupole
• Observables for bound orbital motion in axially symmetric space-times
• Scattering and bound observables for spinning particles in Kerr spacetime with generic spin orientations
• Self force on a scalar charge in Kerr spacetime: inclined circular orbits
• A Case for a Binary Black Hole System Revealed via Quasi-Periodic Outflows
• Towards a framework for testing general relativity with extreme-mass-ratio-inspiral observations
• Tidal deformations of a binary system induced by an external Kerr black hole
• Kerr Geodesics in Terms of Weierstrass Elliptic Functions
• Resonant self-force effects in extreme-mass-ratio binaries: A scalar model
• Fundamental frequencies and resonances from eccentric and precessing binary black hole inspirals
• Computational Efficiency of Frequency-- and Time--Domain Calculations of Extreme Mass--Ratio Binaries: Equatorial Orbits
• Repeated faint quasinormal bursts in extreme-mass-ratio inspiral waveforms: Evidence from frequency-domain scalar self-force calculations on generic Kerr orbits
• Bound orbits of a slowly evolving black hole
• Geodesic motion in the (rotating) black string spacetime
• Impact of relativistic waveforms in LISA's science objectives with extreme-mass-ratio inspirals
• Gravitational wave emission under general parametrized metric from extreme mass ratio inspirals
• Phase space mixing in the equatorial plane of a Kerr black hole
• Spinning particles near Kerr black holes: Orbits and gravitational-wave fluxes through the Hamilton-Jacobi formalism
• Extreme mass ratio inspirals into black holes surrounded by matter
• Self-forced inspirals with spin-orbit precession
• Deformation of extremal black holes from stringy interactions
• The gravitational capture of compact objects by massive black holes
• Circular and non-circular nearly horizon-skimming orbits in Kerr spacetimes
• Fast inspirals and the treatment of orbital resonances
• Repeating Nuclear Transients as Candidate Electromagnetic Counterparts of LISA Extreme Mass Ratio Inspirals
• Strong-field tidal distortions of rotating black holes: II. Horizon dynamics from eccentric and inclined orbits
• Ultralight scalars and resonances in black-hole physics
• Extreme mass ratio inspirals: perspectives for their detection
• Thermodynamics of a Black Hole with Moon
• Near-horizon geodesics of high-spin black holes
• Generalized gravitomagnetic clock effect
• A Newtonian analogue of Kerr black hole
• Probing fundamental physics with Extreme Mass Ratio Inspirals: a full Bayesian inference for scalar charge
• Action-Angle formalism for extreme mass ratio inspirals in Kerr spacetime
• Particle motion under the conservative piece of the self-force is Hamiltonian
• Gyroscope precession along general timelike geodesics in a Kerr black hole spacetime
• Is motion under the conservative self-force in black hole spacetimes an integrable Hamiltonian system?
• Orbital spin dynamics of a millisecond pulsar around a massive black hole with an general mass quadrupole
• A Note on Celestial Mechanics in Kerr Spacetime
• Lindblad resonance torques in relativistic discs: II. Computation of resonance strengths
• Geometrized effective-one-body formalism for extreme-mass-ratio limits: Generic orbits
• Transition of EMRIs through resonance: corrections to higher order in the on-resonance flux modification
• Post-adiabatic waveform-generation framework for asymmetric precessing binaries
• A geometric origin for quasi-periodic oscillations in black hole X-ray binaries
• Actions of spinning compact binaries: Spinning particle in Kerr matched to dynamics at 1.5 post-Newtonian order
• On the integrability of extended test body dynamics around black holes
• Verifying black hole orbits with gravitational spectroscopy
• Autoparallel orbits in Kerr Brans-Dicke spacetimes
• Axisymmetric, stationary collisionless gas configurations surrounding black holes
• Strong-gravity precession resonances for binary systems orbiting a Schwarzschild black hole
• Black hole photon ring beyond General Relativity: an integrable parametrization
• Secondary spins of extreme mass ratio inspirals: A probe to the formation channels
• Action-angle variables of a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order
• Photon Boomerang in a Nearly Extreme Kerr Metric
• Augmented kludge waveforms and Gaussian process regression for EMRI data analysis
• Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime
• Resonant excitation of eccentricity in spherical extreme-mass-ratio inspirals
• Observables of a test-mass along an inclined orbit in a post-Newtonian approximated Kerr spacetime to leading-order-quadratic-in-spin
• Gravitational Burst Radiation from Pulsars in the Galactic centre and stellar clusters
• Eccentricity distribution of extreme mass ratio inspirals
• Motion of small bodies in general relativity: foundations and implementations of the self-force
• Radio timing in a millisecond pulsar-extreme/intermediate mass ratio binary system
• Gravitational wave fluxes on generic orbits in near-extreme Kerr spacetime: higher spin and large eccentricity
• Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space-time
• Post-Newtonian expansion of gravitational energy and angular momentum fluxes: inclined spherical orbits about a Kerr black hole
• On the Integrability of the Geodesic Flow on a Friedmann-Robertson-Walker Spacetime
• A Study of Elliptical Last Stable Orbits About a Massive Kerr Black Hole
• Dynamical Measurement of Supermassive Black Hole Masses: QPE Timing Method
• Building a bridge between comparable and extreme mass ratio black hole binaries: a single spin precessing model for the final state
• Tidally-induced nonlinear resonances in EMRIs with an analogue model
• Cumulative effect of orbital resonances in extreme-mass-ratio inspirals
• Stealth Chaos due to Frame Dragging
• The first law of binary black hole scattering
• Gravitational wave templates from Extreme Mass Ratio Inspirals
• Spherical inspirals of spinning bodies into Kerr black holes
• Testing Effective Quantum Gravity with Gravitational Waves from Extreme-Mass-Ratio Inspirals
• A note on the conversion of orbital angles for extreme mass ratio inspirals
• Impact of mass transfer on the orbital evolution of a white dwarf close to an intermediate-mass black hole
• Post-Newtonian expansion of fluxes from a scalar charge on an inclined-spherical orbit about a Kerr black hole
• Captured are circularized: A relativistic treatment of extreme mass ratio inspirals crossing accretion disks
• The Carter Constant for Inclined Orbits About a Massive Kerr Black Hole: I. circular orbits
• Constants of motion in gravitational self-force theory
• KerrGeoPy: A Python Package for Computing Timelike Geodesics in Kerr Spacetime
• Observable signature of magnetic tidal coupling in hierarchical triple systems
• Extended mass and spheroidal deformation effects on epicyclic frequencies and periapsis shift in quasi-circular orbits
• Testing Modified Gravity with Gravitational Wave Astronomy
• Classification of radial Kerr geodesic motion
• Estimating High-Order Time Derivatives of Kerr Orbital Functionals
• Unexpected Symmetries of Kerr Black Hole Scattering
• Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits
• Precisely computing bound orbits of spinning bodies around black holes II: Generic orbits
• KAM and geodesic dynamics of blackholes
• Particles with precessing spin in Kerr spacetime: analytic solutions for eccentric orbits and homoclinic motion near the equatorial plane
• Adiabatic evolution due to the conservative scalar self-force during orbital resonances
• Tidal perturbations of an extreme mass ratio inspiral around a Kerr black hole
• Analytical solutions of bound timelike geodesic orbits in effective-one-body frame
• Gravitational Wave Signatures of Quasi-Periodic Eruptions: LISA Detection Prospects for RX J1301.9+2747
• Modified Gravity Corrections in Fundamental Orbital Frequencies in Kerr Spacetime