Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime

Abstract

This work constitutes the second part of a series of studies that aim to utilise tools from Hamiltonian mechanics to investigate the motion of an extended body in general relativity. The first part of this work [Refs. [1, 2]] constructed a ten-dimensional, covariant Hamiltonian framework encompassing all the linear-in-spin corrections to the geodesic motion in arbitrary spacetime. This framework was proven to be integrable in the Schwarzschild and Kerr spacetimes, specifically. The present work translates this abstract integrability result into tangible applications for linear-in-spin Hamiltonian dynamics of a compact object in a Schwarzschild spacetime. In particular, a canonical system of coordinates is constructed explicitly, which exploits the spherical symmetry of the Schwarzschild spacetime. These coordinates are based on a relativistic generalization of the classical Andoyer variables of Newtonian rigid body motion. This canonical setup allows us to derive ready-to-use formulae for action-angle coordinates and gauge-invariant Hamiltonian frequencies, which automatically include all linear-in-spin effects. No external parameters or ad hoc choices are necessary, and the framework can be used to find complete solutions by quadrature of generic (bound or unbound), linear-in-spin orbits, including orbital inclination, precession and eccentricity, as well as spin precession. The efficacy of the formalism is demonstrated here in the context of circular orbits with arbitrary spin and orbital precession, with the results validated against known results in the literature.

73 pages, 3 figures, 2 tables. V3: an expanded introduction and minor changes in response to referee comments

References in corpus (41)

• Computationally efficient models for the dominant and sub-dominant harmonic modes of precessing binary black holes
• Scattering Amplitudes and the Conservative Hamiltonian for Binary Systems at Third Post-Minkowskian Order
• The TianQin project: current progress on science and technology
• Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory
• Concepts and status of Chinese space gravitational wave detection projects
• Analysis of spin precession in binary black hole systems including quadrupole-monopole interaction
• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• Conservative Dynamics of Binary Systems to Third Post-Minkowskian Order from the Effective Field Theory Approach
• Quasi-Normal Modes and Gravitational Wave Astronomy
• Laying the foundation of the effective-one-body waveform models SEOBNRv5: improved accuracy and efficiency for spinning non-precessing binary black holes
• Science with the TianQin observatory: Preliminary result on extreme-mass-ratio inspirals
• The Overlap of Numerical Relativity, Perturbation Theory and Post-Newtonian Theory in the Binary Black Hole Problem
• Computing inspirals in Kerr in the adiabatic regime. I. The scalar case
• Comparing second-order gravitational self-force, numerical relativity and effective one body waveforms from inspiralling, quasi-circular and nonspinning black hole binaries
• Intermediate mass-ratio black hole binaries: Applicability of small mass-ratio perturbation theory
• Self-Force Calculations with a Spinning Secondary
• Growth of resonances and chaos for a spinning test particle in the Schwarzschild background
• Construction of Explicit Symplectic Integrators in General Relativity. I. Schwarzschild Black Holes
• Construction of explicit symplectic integrators in general relativity. IV. Kerr black holes
• Conservative corrections to the innermost stable circular orbit (ISCO) of a Kerr black hole: a new gauge-invariant post-Newtonian ISCO condition, and the ISCO shift due to test-particle spin and the gravitational self-force
• Self-force effects in post-Minkowskian scattering
• Comparing second-order gravitational self-force and effective one body waveforms from inspiralling, quasi-circular and nonspinning black hole binaries II: the large-mass-ratio case
• Comparison of post-Minkowskian and self-force expansions: Scattering in a scalar charge toy model
• Spinning test body orbiting around a Kerr black hole: Eccentric equatorial orbits and their asymptotic gravitational-wave fluxes
• Extreme mass-ratio inspiral and waveforms for a spinning body into a Kerr black hole via osculating geodesics and near-identity transformations
• Effective-one-body waveforms for extreme-mass-ratio binaries: Consistency with second-order gravitational self-force quasicircular results and extension to nonprecessing spins and eccentricity
• Scattering and bound observables for spinning particles in Kerr spacetime with generic spin orientations
• Analytic solutions for the motion of spinning particles near spherically symmetric black holes and exotic compact objects
• Adiabatic equatorial inspirals of a spinning body into a Kerr black hole
• Spin-geodesic deviations in the Schwarzschild spacetime
• Integrability of eccentric, spinning black hole binaries up to second post-Newtonian order
• Complete set of quasi-conserved quantities for spinning particles around Kerr
• Generalized Carter constant for quadrupolar test bodies in Kerr spacetime
• Black hole scattering near the transition to plunge: Self-force and resummation of post-Minkowskian theory
• Singular Lagrangians, Constrained Hamiltonian Systems and Gauge Invariance: An Example of the Dirac-Bergmann Algorithm
• Action-Angle formalism for extreme mass ratio inspirals in Kerr spacetime
• Extended-body motion in black hole spacetimes: What is possible?
• Post-Newtonian expansions of extreme mass ratio inspirals of spinning bodies into Schwarzschild black holes
• Basic notions of Poisson and symplectic geometry in local coordinates, with applications to Hamiltonian systems
• On the integrability of extended test body dynamics around black holes
• Action-angle variables of a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order

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