arXiv subjects

Paul Ramond

Publications and source records attributed to Paul Ramond.

11 recordsLinked to original sources

arXiv2026

Quadrupolar tidal effects destroy the integrability of black hole geodesics: analytic proof and numerical evidence of chaos

In general relativity, the motion of a test mass around a rotating black hole is described by Kerr geodesics. Owing to the symmetries of the Kerr spacetime, these geodesics possess four constants of motion, rendering the associated Hamiltonian system integrable. This integrability underlies much of the analytical framework used to model asymmetric-mass-ratio inspirals, key sources for future gravitational-wave detectors. Real compact bodies, however, are not test masses: their internal structure couples to the background curvature. In this work, we show that a non-spinning body endowed with a tidally induced quadrupole admits no deformation of the geodesic Carter constant that remains conserved, for generic tidal couplings and generic Kerr spin. Consequently, the leading-order tidal dynamics is generically non-integrable. The proof is analytic and relies on two key ingredients: a covariant Hamiltonian formulation of tidal dynamics on the same phase space as the geodesic problem, valid in arbitrary background spacetimes, and a novel relation between curvature tidal scalars and the geodesic Carter constant derived from the algebraic and Killing symmetries of Kerr spacetime. We complement this result with numerical diagnostics of the tidally perturbed dynamics, including Poincar\'e sections, Lyapunov exponents, and escape-time maps. These reveal chaotic structures in phase space, such as stochastic layers, sensitivity to initial conditions, and fractal basin boundaries, consistently with the analytic non-integrability result.

gr-qc↗

arXiv2026

Octupole moments and the non-universality of free-fall in general relativity

Extended bodies in general relativity do not necessarily fall along geodesics, but can be accelerated. These accelerations depend on an object's angular momentum, as well as on its quadrupole, octupole, and higher-order moments. However, these multipole moments can evolve differently from one body to another. Different bodies can thus fall differently, even with identical initial data. This paper examines how octupole moments contribute to the non-universality of free-fall in general relativity. We begin by showing that in arbitrary vacuum spacetimes, only the trace-free component of an octupole moment can affect an object's motion. It follows that at least 16 out of 40 octupole components decouple from the laws of motion. Then, we obtain two decompositions for trace-free octupole moments, one in terms of a timelike frame vector and the other in terms of a null tetrad. These decompositions are applied to motion both in generic Newtonian spacetimes and in fully-relativistic vacuum spacetimes that are of Petrov type D. In Newtonian spacetimes, the mass moments are shown to have their ordinary Newtonian effects, while the momentum moments determine a body's hidden momentum---the misalignment between its momentum and its velocity. In Petrov type D spacetimes (such as Kerr), we show that some torques that are impossible with quadrupole moments are possible with octupole moments. Octupole moments can thus have qualitatively-different effects from quadrupole moments.

gr-qc↗

arXiv2026

Symplectic mechanics of relativistic spinning compact bodies. III. quadratic-in-spin integrability in Type-D Einstein spacetimes: persistence and breakdown

We investigate the integrability of spinning compact body dynamics at quadratic order in spin in four-dimensional Einstein spacetimes admitting a non-degenerate Killing--Yano tensor. Working within the Mathisson--Papapetrou--Tulczyjew--Dixon framework under the Tulczyjew--Dixon spin supplementary condition, we model the spin-induced quadrupole with a deformability parameter $\kappa$, where $\kappa=1$ corresponds to black holes. The dynamics is formulated as a Hamiltonian system on a 10-dimensional physical phase space obtained by Dirac--Bergmann reduction. For $\kappa=1$, we establish Liouville--Arnold integrability at quadratic order in spin by constructing five independent, Poisson-commuting first integrals, including a generalization of the Carter constant and the R\"udiger constant to quadratic-in-spin order in Einstein spacetimes beyond Kerr. For $\kappa \neq 1$, the R\"udiger and Carter constants are no longer conserved; integrability does not persist at this order. All our results are carried out in a covariant manner and numerically verified, and Kerr is recovered as a special case. These results show that integrability can extend beyond Kerr and beyond the linear-in-spin regime, while its breakdown for $\kappa \neq 1$ points to the spin-induced quadrupole as a decisive probe of compact body structure.

gr-qc↗

arXiv2024

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

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.

gr-qc↗

arXiv2024

On the integrability of extended test body dynamics around black holes

In general relativity, the motion of an extended test body is influenced by its proper rotation, or spin. We present a covariant and physically self-consistent Hamiltonian framework to study this motion, up to quadratic order in the body's spin, including a spin-induced quadrupole, and in an arbitrary background spacetime. The choice of spin supplementary condition and degeneracies associated with local Lorentz invariance are treated rigorously with adapted tools from Hamiltonian mechanics. Applying the formalism to a background space-time described by the Kerr metric, we prove that the motion of any test compact object around a rotating black hole defines an integrable Hamiltonian system to linear order in the body's spin. Moreover, this integrability still holds at quadratic order in spin when the compact object has the deformability expected for an isolated black hole. Our analytical results shed light on longstanding numerical conjectures regarding spin-induced chaos in the motion of asymmetric compact binaries, and provides a powerful framework to improve current gravitational waveform modelling to account for spin-induced extended body effects.

gr-qc↗

arXiv2022

Symplectic mechanics of relativistic spinning compact bodies. I. linear-in-spin integrability under Killing-Yano symmetry

We study the Hamiltonian dynamics of a neutral, massive spinning test body at linear order in spin, as governed by the Mathisson--Papapetrou--Tulczyjew--Dixon equations in a 4-dimensional spacetime admitting a non-degenerate Killing--Yano tensor. The Tulczyjew--Dixon spin supplementary condition is imposed as a set of algebraic constraints, and the resulting constraint surface is equipped with a Poisson--Dirac bracket, yielding a non-degenerate, 10-dimensional physical phase space. On this reduced space we identify five functionally independent first integrals in involution: the autonomous Hamiltonian, two constants of motion associated with two commuting Killing vectors, a generalised Carter constant, and the R\"udiger constant; all four descending from the Killing--Yano tensor. All calculations are covariant, and the result is a purely geometric statement: it requires no background field equations and holds for any metric admitting a non-degenerate Killing--Yano tensor, beyond Kerr, beyond vacuum and beyond general relativity. Our results identify Killing--Yano symmetry as the sole geometric source of linear-in-spin integrability. Extensions to quadratic-in-spin dynamics, including the spin-induced quadrupole, and to tidally-induced quadrupolar effects for non-spinning bodies, are treated in companion papers.

gr-qc↗

arXiv2022

First Law of Mechanics for Spinning Compact Binaries: Dipolar Order

Building upon the Noether charge formalism of Iyer and Wald, we derive a variational formula for spacetimes admitting a Killing vector field, for a generic energy-momentum distribution with compact support. Applying this general result to the particular case of a binary system of spinning compact objects moving along an exactly circular orbit, modelled using the multipolar gravitational skeleton formalism, we derive a first law of compact binary mechanics at dipolar order. We prove the equivalence of this new result with the canonical Hamiltonian first law previously derived for binary systems of spinning compact objects, for spins colinear with the orbital angular momentum. This paper paves the way to an extension of the first law of binary mechanics to the next quadrupolar order, thereby accounting for the spin-induced and tidally-induced deformability of the compact bodies.

gr-qc↗

arXiv2021

New Methods of Isochrone Mechanics

Isochrone potentials, as defined by Michel Hénon in the fifties, are spherically symmetric potentials within which a particle orbits with a radial period that is independent of its angular momentum. Isochrone potentials encompass the Kepler and harmonic potential, along with many other. In this article, we revisit the classical problem of motion in isochrone potentials, from the point of view of Hamiltonian mechanics. First, we use a particularly well-suited set of action-angle coordinates to solve the dynamics, showing that the well-known Kepler equation and eccentric anomaly parametrisation are valid for any isochrone orbit (and not just Keplerian ellipses). Second, by using the powerful machinery of Birkhoff normal forms, we provide a self-consistent proof of the isochrone theorem, that relates isochrone potentials to parabolae in the plane, which is the basis of all literature on the subject. Along the way, we show how some fundamental results of celestial mechanics such as the Bertrand theorem and Kepler's third law are naturally encoded in the formalism.

math-ph↗

arXiv2021

Multipolar Particles in Helically Symmetric Spacetimes

We consider a binary system of spinning compact objects with internal structure, moving along an exactly circular orbit, and modelled within the multipolar gravitational skeleton formalism, up to quadrupolar order. We prove that the worldline of each multipolar particle is an integral curve of the helical Killing vector field, and that the 4-velocity, 4-momentum, spin tensor and quadrupole tensor of each particle are Lie-dragged along those worldlines. The geometrical framework developed in this paper paves the way to an extension of the first law of compact-object binary mechanics up to quadrupolar order.

gr-qc↗

arXiv2021

The Geometry of Isochrone Orbits: from Archimedes' parabolae to Kepler's third law

In classical mechanics, the Kepler potential and the Harmonic potential share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and Harmonic potentials are called \it{isochrone}. In this paper, we solve the following general problem: are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Michel Hénon in 1959, in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometric generalization of Kepler's third law, and give a similar law for the apsidal angle, for any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations, and use this to derive an analytical parameterization of any isochrone orbit. Along the way we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem, and argue that our geometrical methods can be exported to more generic orbits in potential theory.

physics.class-ph↗

arXiv2020

Abel-Ruffini's Theorem: Complex but Not Complicated!

In this article, using only elementary knowledge of complex numbers, we sketch a proof of the celebrated Abel--Ruffini theorem, which states that the general solution to an algebraic equation of degree five or more cannot be written using radicals, that is, using its coefficients and arithmetic operations $+,-,\times,÷,$ and $\sqrt{\ }$. The present article is written purposely with concise and pedagogical terms and dedicated to students and researchers not familiar with Galois theory, or even group theory in general, which are the usual tools used to prove this remarkable theorem. In particular, the proof is self-contained and gives some insight as to why formulae exist for equations of degree four or less (and how they are constructed), and why they do not for degree five or more.

math.HO↗