FirstPage Symplectic mechanics of relativistic spinning compact bodies I. Covariant foundations and integrability around black holes

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Symplectic mechanics of relativistic spinning compact bodies I.:

Covariant foundations and integrability around black holes

Paul Ramond, 1, ∗

1IMCCE, Observatoire de Paris , Universit´e PSL,

77 Avenue Denfert-Rochereau, FR-75014 Paris

(Dated: February 5, 2024)

Abstract

In general relativity, the motion of an extended body moving in a given spacetime can be de-

scribed by a particle on a (generally non-geodesic) worldline. In ﬁrst approximation, this worldline

is a geodesic of the underlying spacetime, and the resulting dynamics admit a covariant and 4-

dimensional Hamiltonian formulation. In the case of a Kerr background spacetime, the Hamiltonian

was shown to be integrable by B. Carter and the now eponymous constant. At the next level of

approximation, the particle possesses proper rotation (hereafter spin), which couples the curvature

of spacetime and drives the representative worldline away from geodesics. In this article, we lay

the theoretical foundations of a series of works aiming at exploiting the Hamiltonian nature of the

equations governing the motion of a spinning particle, at linear order in spin. Our formalism is

covariant and 10-dimensional. It handles the degeneracies inherent to the local Lorentz invariance

of general relativity with tools from Poisson geometry, and accounts for the center-of-mass/spin-

supplementary-condition using constrained Hamiltonian system theory. As a ﬁrst application, we

consider the linear-in-spin motion in a Kerr background. We show that the resulting Hamiltonian

system admits exactly ﬁve functionally independent integrals of motion related to Killing symme-

tries, thereby proving that the system is integrable. We conclude that linear-in-spin corrections to

the geodesic motion do not break integrability, and that the resulting trajectories are not chaotic.

We explain how this integrability feature can be used to reduce the computational cost of waveform

generation schemes for asymmetric binary systems of compact objects.

∗paul.ramond@obspm.fr

arXiv:2210.03866v4 [gr-qc] 2 Feb 2024

INTRODUCTION

A. The relativistic motion of test bodies

The last decades have seen both the advent of gravitational wave astronomy [1–4] and

the ﬁrst direct observations of a black hole’s strong ﬁeld region [5,6]. These revolutionary

events come as a reward for decades of eﬀort in understanding the family of compact objects,

namely black holes, neutron stars, and possibly exotic stars [7,8]. Of particular interest

are binary systems of compact objects, which have been at the core of many still ongoing

research programs, in view of the future of gravitational wave detectors on Earth [9,10]

or in space [11]. Our ability to predict with great accuracy the motion of such systems

directly impacts the likeliness of detecting them through their gravitational-wave emission.

Thankfully, when it comes to the motion of compact binary systems under their mutual

gravitational attraction, not all of their physical properties inﬂuence their motion the same

way. This “universality” feature, unique to gravitation, is the main reason why multipolar

expansions work: one models a real, extended compact object as a point particle endowed

with a ﬁnite number of multipoles, e.g, mass, linear and angular momentum, deformability

coeﬃcients, etc. The higher the number of multipoles, the ﬁner (and the closer to reality)

the description of the object. The motion of the body is then obtained by solving a coupled

problem: an equation of motion determining the wordline of the particle representing the

body, and some evolution equations for the particle’s multipoles along that worldline. This

very general result holds for any suﬃciently compact object with a conserved stress-energy-

momentum tensor [12,13].

For a small compact object moving in a given, background spacetime, the foundations

of the multipolar schemes were laid in 1937 by Miron Mathisson. Key contributions were

then made by Papapetrou in the forties, Tulzyjew in the ﬁfties and culminated in the six-

ties with a series of articles by Dixon, essentially clarifying previous works and giving the

“big picture.” We refer to Chap. 2 of [14] for a detailed historical account and references

for these pioneering works. The most modern and enlightening framework is undoubtedly

that based on “generalized Killing ﬁelds”, developed by Harte [13,15,16]. This approach

sheds light on many uncovered aspects of previous derivations, makes explicit the diﬀer-

ence between kinematical and dynamical eﬀects, and is able to account for the self-ﬁelds

of the body. The relevant evolution equations, now known as the Mathisson-Papapetrou-

Tulczyjew-Dixon (MPTD) equations, are simple manifestations of the attempt to maintain

Poincar´e invariance as much as possible along the body’s representative worldline [17]. These

equations have been thoroughly studied and solved for a plethora of background spacetimes

both theoretically and numerically, in particular exact and hairy black holes (see references

below) and cosmological metrics [18–20]. The MPTD equations are also recovered in the

context of black hole perturbation theory [21,22], making them not just “eﬀective” equations

replacing the extended body’s true motion, but a real sub-part of Einstein’s ﬁeld equations.

B. Relativistic Hamiltonian formulation

When the multipole description of the orbiting body is truncated at dipolar order, the

MPTD system possesses two equations: one evolves a linear momentum 1-form paand the

other an antisymmetric spin tensor Sab. Since the MPTD equations describes a purely

kinematical evolution [13] of pa, Sab, it is natural to look for a Hamiltonian formulation1of

these equations. In the case of general relativity, two distinct approaches have been proposed

in the past. One of them is based on extending, to curved spacetime, the Lagrangian

formulation of a free spinning particle in ﬂat spacetime [23]. Inspired by this work and

similar development in eﬀective ﬁeld theory [24], the authors of [25] provided a Hamiltonian

formulation of the dipolar MPTD system via a singular Legendre transformation of this

Lagrangian. Extensions to quadrupolar [26] and octupolar orders [27] have been derived,

allowing for the SSC to be understood as a gauge freedom [28]. One drawback of these

Lagrangian formulations, however, is their non-covariance, i.e., they require (i) a particular

3+1 split of the background spacetime and (ii) a ﬁne-tuned choice of spin supplementary

condition with no clear physical interpretation.

The other approach can be traced back to Souriau’s symplectic approach [29] to the prob-

lem, cf. the recent note on Souriau’s pioneering ideas [30]. While our treatment does not

follow directly Souriau’s, it is solely based on symplectic geometry too, and borrow Souriau’s

general philosophy: the Hamiltonian system will be considered the prime ingredient, and

not a mere Legendre transform of some Lagrangian. The general symplectic structure that

we shall use is already present (but somewhat hidden) in Souriau’s [29] and Kunzle’s work

[31], but it seems that its ﬁrst clear appearance is in2[32] (see also references [33]). Natu-

rally, our formulation will share some similarities with other covariant approaches proposed

in [32–35]. However, the works [32,34] only work at the level of the equations of motion for

simple orbits and do not exploit the Hamiltonian nature of the system. Similarly, the Hamil-

tonians proposed in [33,35] have a number of drawbacks that are unsatisfactory for practical

applications, in particular ad-hoc mass parameters too many phase space dimensions, both

issues related to unresolved constraints.Lastly, the necessary choice of spin supplementary

condition for the well-posedness of the MPTD system will lead us, inevitably, to algebraic

constraints. In the Lagrangian formalism, this is usually treated using the Dirac-Bergmann

algorithm [25,36]. Here, in the context of symplectic geometry, this will be handled within

the classical theory of “constrained Hamiltonian systems” following classical textbooks [37]

(or [38] for detailed proofs).

C. The question of integrability around black holes

At monopolar order, the spin tensor Sab is not involved in the description of the body.

The two MPTD equations reduce to (i) pais parallel-transported along the worldline and (ii)

pais tangent to the worldline. Combining the two implies that the worldline is a geodesic of

the background spacetime. The equations of geodesic motion are the most general setup to

describe “free systems”, as was ﬁrst realized by Arnold in [39].Owing to their purely free (i.e.,

purely kinematical) evolution, such geodesic systems are usually simple enough that they

can sometimes be integrable, i.e., possess enough integrals of motion that make the solution

to their equations of motion solvable by quadrature (i.e., using algebraic manipulations

and standard calculus). In general relativity, the most remarkable example of this is the

integrability of geodesics in the Kerr spacetime, ﬁrst demonstrated by Carter [40], ﬁve

years after Kerr’s discovery of the eponymous metric describing rotating black holes [41].

Integrability of Kerr geodesics is far from trivial: there are not enough spacetime symmetries

1By “Hamiltonian formulation”, we mean a system of ordinary diﬀerential equations generated by a scalar

function and some geometric structure on a phase space. We shall be more precise when necessary.

2However, in [32] it is claimed that these brackets are symplectic, although they are not.

to produce enough constants of motion to ensure integrability. Interestingly, Carter’s proof

actually relied on pure symplectic mechanics of the geodesic Hamiltonian, unaware of the

existence of the existence of the Killing-St¨ackel tensor while which only appeared two years

later [42].

While the integrability of the monopolar MPTD system (i.e., geodesics) around black

holes is well understood [40,43], its extension to the dipolar MPTD system (i.e., the motion

of spinning particles) is not so clear. Based on existing numerical simulations exhibiting

artefacts classically associated to chaos, it is reasonable to believe that the integrability

of geodesics is generally broken by the eﬀect of the body’s spin, in the Schwarzschild case

[44–46] as well as in Kerr [47–51]. This seems to be the case for other types of (non-spin)

perturbations, as explored in [52–54]. However, when limiting to spin eﬀect that are linear

in Sab, no clear answer exists in the literature. Our results ﬁll this gap with an analytical

proof of integrability (thus, no chaos arises at linear-in-spin order).

D. Motivation and main results

In this work, our primary motivation lies in the fact that integrability (or lack thereof) for

the motion of a small compact object around a black hole is directly related to the possibility

of using gravitational self-force theory to build gravitational waveforms for extreme mass

ratio inspiral (EMRI) systems [55,56], one of the primary sources of gravitational waves

for Laser Interferometer Space Antenna (LISA) [11]. The Hamiltonian formulation of the

equations governing EMRIs is currently of major interest in this particular ﬁeld because it

can account for most of the non-geodesic eﬀects that require implementing to meet LISA’s

accuracy requirements [57]. This includes the small body’s spin [26,33,58,59], the conser-

vative part of the gravitational self-force [60–62] and possibly its dissipative sector too [63],

while also being adapted to eﬃcient and consistent numerical resolution schemes such as

symplectic integrators [64,65]. In addition, if the Hamiltonian system is integrable, it can

be written in terms of action-angle variables, which are Taylor-made for two-timescale ex-

pansion methods [66], gauge-invariant analysis of resonances [66], an elementary derivation

of the ﬁrst law of mechanics [67,68] as well as ﬂux-balance laws [69,70], all of which are

invaluable tools used in the modeling and understanding of binary system mechanics. Spin-

ning degrees of freedom of the secondary object are not yet fully implemented in state-of-the

art results of EMRI waveform modeling, even though they are under heavy investigation

[71–73]. Our work contributes to this ongoing eﬀort, in order to exploit future gravitational

wave detectors at their full potential [59,74].

Our objective, through this series of work (of which this article is the ﬁrst part), is to

extend all the aforementioned results to the linear-in-spin, dipolar case. As they are strongly

tied to Kerr geodesics being (i) Hamiltonian and (ii) integrable, our ﬁrst aim is to (i) give a

Hamiltonian formulation of the linear-in-spin dynamics and (ii) apply it the Kerr spacetime

and show its integrability.3

As argued above, various proposals have already been made for a Hamiltonian formulation

of the dipolar MPTD system that only satisﬁes some of the criteria below. Our ﬁrst main

result is a formulation of the linear-in-spin MPTD system (1.9) as a Hamiltonian system

that is at once:

3These two items are this article’s content, while aforementioned applications will be published in subse-

quent parts.

•symplectic : it comes with a Poisson structure free of degeneracies

•a ﬁrst integral : it is autonomous and not a pure phase space constraint,

•self-consistent : it does not keep terms of the same order in spin as it omits,

•covariant : it is independent of a choice of spacetime coordinates or tetrad ﬁeld,

•10-dimensional : as a simple counting of the independent degrees of freedom require,

Besides the formalism, a number of secondary formulae that may be useful in other con-

texts are given, and clariﬁcations of various statements made in the literature are discussed.

The proof of integrability in the Schwarzschild and Kerr spacetimes relies on exhibiting

enough functionally independent integrals of motion. Since our Hamiltonian formulation of

the linear-in-spin, dipolar MPTD system is 10-dimensional, integrability follows from the ex-

istence of 5 such integrals, denoted (µ, E, Lz,K,Q). All have a simple physical interpretation

and are associated to some kind of symmetry:

•the mass µof the body is conserved because the particle evolves freely at dipolar order

(no self-induced force or torque from higher-order multipoles), its associated Killing

structure is the metric gab, a particular Killing-St¨ackel tensor,

•the particle’s total energy E, as measured at spatial inﬁnity, is conserved thanks to

the stationarity of the Kerr background, with associated timelike Killing vectors (∂t)a,

•the component Lzof total angular momentum along the primary black hole’s spin

axis, as measured at spatial inﬁnity, is conserved thanks to the axisymmetry of the

Kerr background, with associated spacelike Killing vector (∂ϕ)a,

•two additional integrals K,Q, known as R¨udiger’s invariants, are built from the hid-

den symmetry associated to a Killing-Yano tensor. While Kis the projection of the

particle’s spin 4-vector onto the covariant angular momentum, Qis a linear-in-spin

generalization of the geodesic Carter constant.

We note that we did not discover any of these ﬁve integrals of motion: they have been

known (at least) since R¨udiger’s work [75,76] on linear-in-spin invariants of motion for the

dipolar MPTD system. The novelty rather comes from the rigorous and covariant Hamil-

tonian setup in which they can be understood as functionally independent, ﬁrst integrals,

ensuring integrability.

E. Organization of the paper

This article will serve as the basis for many extensions, starting with [77], and contains

detailed that will be otherwise omitted in further application-oriented publications. Every-

thing is justiﬁed or proved as rigorously as possible. It is organized as follows.

•We start in Sec. Iby reviewing the main aspects of the MPTD system, focusing on the

dipolar order in Sec. I A and the important notion of “spin supplementary condition

(SSC) for spinning particles in Sec. I B. Our choice for the Tulczyjew-Dixon SSC is

motivated in Sec. I C and the linear-in-spin dynamics are covered in Sec. I D. The

equations of interest for the rest of the article, are Eqs. (1.9) and (1.6) (hereafter “the

system”).

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FirstPageSymplecticmechanicsofrelativisticspinningcompactbodiesI.:CovariantfoundationsandintegrabilityaroundblackholesPaulRamond,1,∗1IMCCE,ObservatoiredeParis,Universit´ePSL,77AvenueDenfert-Rochereau,FR-75014Paris(Dated:February5,2024)AbstractIngeneralrelativity,themotionofanextendedbodymovinginagiv...

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