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

The last decades have seen both the advent of gravitational wave astronomy B. P. Abbott et al. (2016) (LIGO Scientific Collaboration and Virgo Collaboration); B. P. Abbott et al. (2019) (LIGO Scientific Collaboration and Virgo Collaboration); B. P. Abbott et al. (2020) (LIGO Scientific Collaboration and Virgo Collaboration); Abbott et al. (2021) and the first direct observations of a black hole’s strong field region Akiyama et al. (2019, 2022). 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 Cardoso et al. (2016); Toubiana et al. (2021). 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 Unnikrishnan (2013); Punturo et al. (2010) or in space Amaro-Seoane et al. (2017). 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 Dixon (1974); Harte (2015).

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 sixties with a series of articles by Dixon, essentially clarifying previous works and giving the “big picture.” We refer to Chap. 2 of Ramond (2021) 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 Harte (2012, 2015, 2020). This approach sheds light on many uncovered aspects of previous derivations, makes explicit the difference 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 Poincaré invariance as much as possible along the body’s representative worldline Harte (2008). 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 Harte (2007); Obukhov and Puetzfeld (2011); Zalaquett et al. (2014). The MPTD equations are also recovered in the context of black hole perturbation theory Pound (2015); Miller and Pound (2021), making them not just “effective” equations replacing the extended body’s true motion, but a real sub-part of Einstein’s field equations.

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 $p_{a}$ and the other an antisymmetric spin tensor $S^{ab}$. Since the MPTD equations describes a purely kinematical evolution Harte (2015) of $p_{a},S^{ab}$, it is natural to look for a Hamiltonian formulation¹¹1By “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. of 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 Hanson and Regge (1974). Inspired by this work and similar development in effective field theory Porto (2006), the authors of Barausse et al. (2009) provided a Hamiltonian formulation of the dipolar MPTD system via a singular Legendre transformation of this Lagrangian. Extensions to quadrupolar Vines et al. (2016) and octupolar orders Marsat (2015) have been derived, allowing for the SSC to be understood as a gauge freedom Steinhoff (2011). 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 Souriau (1970) to the problem, cf. the recent note on Souriau’s pioneering ideas Damour (2024). 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 Souriau (1970) and Kunzle’s work Künzle (1972), but it seems that its first clear appearance is in²²2However, in d’Ambrosi et al. (2015) it is claimed that these brackets are symplectic, although they are not. d’Ambrosi et al. (2015) (see also references Witzany et al. (2019)). Naturally, our formulation will share some similarities with other covariant approaches proposed in d’Ambrosi et al. (2015, 2016); Witzany et al. (2019); Witzany (2019). However, the works d’Ambrosi et al. (2015, 2016) 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 Hamiltonians proposed in Witzany et al. (2019); Witzany (2019) 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 Barausse et al. (2009); Brown (2022). Here, in the context of symplectic geometry, this will be handled within the classical theory of “constrained Hamiltonian systems” following classical textbooks Arnold et al. (2006) (or Deriglazov (2022) for detailed proofs).

At monopolar order, the spin tensor $S^{ab}$ is not involved in the description of the body. The two MPTD equations reduce to (i) $p^{a}$ is parallel-transported along the worldline and (ii) $p^{a}$ is 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 Arnold (1966).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 Carter (1968), five years after Kerr’s discovery of the eponymous metric describing rotating black holes Kerr (1963). Integrability of Kerr geodesics is far from trivial: there are not enough spacetime symmetries 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-Stäckel tensor while which only appeared two years later Walker and Penrose (1970).

While the integrability of the monopolar MPTD system (i.e., geodesics) around black holes is well understood Carter (1968); Frolov et al. (2017), 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 Bombelli and Calzetta (1992); Verhaaren and Hirschmann (2010); Zelenka et al. (2020) as well as in Kerr Suzuki and Maeda (1997); Semerák (1999); Kyrian and Semerák (2007); Han (2008); Compère and Druart (2022). This seems to be the case for other types of (non-spin) perturbations, as explored in Witzany et al. (2015); Polcar et al. (2022); Mosna et al. (2022). However, when limiting to spin effect that are linear in $S^{ab}$, 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).

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 Barack and Pound (2018); Pound and Wardell (2021), one of the primary sources of gravitational waves for Laser Interferometer Space Antenna (LISA) Amaro-Seoane et al. (2017). 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 Afshordi et al. (2023). This includes the small body’s spin Boyle et al. (2008); Vines et al. (2016); Witzany et al. (2019); Lukes-Gerakopoulos and Witzany (2021), the conservative part of the gravitational self-force Vines and Flanagan (2015a); Fujita et al. (2017); Blanco and Flanagan (2023a) and possibly its dissipative sector too Galley (2013), while also being adapted to efficient and consistent numerical resolution schemes such as symplectic integrators Haier et al. (2006); Wang et al. (2021). 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 expansion methods Hinderer and Flanagan (2008), gauge-invariant analysis of resonances Hinderer and Flanagan (2008), an elementary derivation of the first law of mechanics Isoyama et al. (2014); Le Tiec (2015) as well as flux-balance laws Isoyama et al. (2018); Grant and Moxon (2023a), all of which are invaluable tools used in the modeling and understanding of binary system mechanics. Spinning 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 Piovano et al. (2020); Mathews et al. (2022); Rahman and Bhattacharyya (2023). Our work contributes to this ongoing effort, in order to exploit future gravitational wave detectors at their full potential Barausse et al. (2020); Lukes-Gerakopoulos and Witzany (2021).

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.³³3These two items are this article’s content, while aforementioned applications will be published in subsequent parts.

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 (9) as a Hamiltonian system that is at once:

• •

  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 contexts 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 existence of 5 such integrals, denoted $(\mu,E,L_{z},\mathfrak{K},\mathfrak{Q})$. All have a simple physical interpretation and are associated to some kind of symmetry:

• •

  the mass $\mu$ of 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 $g_{ab}$, a particular Killing-Stäckel 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 $(\partial_{t})^{a}$,

• •

  the component $L_{z}$ of 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 $(\partial_{\phi})^{a}$,

• •

  two additional integrals $\mathfrak{K},\mathfrak{Q}$, known as Rüdiger’s invariants, are built from the hidden symmetry associated to a Killing-Yano tensor. While $\mathfrak{K}$ is the projection of the particle’s spin 4-vector onto the covariant angular momentum, $\mathfrak{Q}$ is 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 Rüdiger’s work Rüdiger (1981, 1983) on linear-in-spin invariants of motion for the dipolar MPTD system. The novelty rather comes from the rigorous and covariant Hamiltonian setup in which they can be understood as functionally independent, first integrals, ensuring integrability.

This article will serve as the basis for many extensions, starting with Ramond and Isoyama (2024), and contains detailed that will be otherwise omitted in further application-oriented publications. Everything is justified or proved as rigorously as possible. It is organized as follows.

• •

  We start in Sec. I by reviewing the main aspects of the MPTD system, focusing on the dipolar order in Sec. I.1 and the important notion of “spin supplementary condition (SSC) for spinning particles in Sec. I.2. Our choice for the Tulczyjew-Dixon SSC is motivated in Sec. I.3 and the linear-in-spin dynamics are covered in Sec. I.4. The equations of interest for the rest of the article, are Eqs. (9) and (6) (hereafter “the system”).

• •

  A first Hamiltonian formulation of the system is provided in Sec. II, where it is built on a 14-dimensional phase space, denoted $\mathcal{M}$, in Sec. II.1. Sections II.2 and II.3 contain the resulting equations of motion and an important change of coordinates, respectively. The phase space $\mathcal{M}$ is then reduced to a 12-dimensional one, denoted $\mathcal{N}$, by lifting the degeneracies on $\mathcal{M}$ associated to the local Lorentz invariance of general relativity, as explained in Sec. II.4.

• •

  Section III is aimed at covering $\mathcal{N}$ with canonical coordinates, and discussing their physical meaning. The derivation, though technical and relegated to App. B, is based on a method that will be relevant for applications, namely transforming non-canonical coordinates into canonical ones. For an explicit example of its use, see Ramond and Isoyama (2024). A summary of the 12D formulation on $\mathcal{N}$ is then provided in Sec. III.3,

• •

  Section IV is divided in two parts: first in Sec. IV.1 we rigorously incorporate the Tulczyjew-Dixon spin supplementary condition into the framework, thereby reducing the phase space once again, from $\mathcal{N}$ to a final 10-dimensional phase space $\mathcal{P}$ where all physical trajectories lie. The final formulation is summarized in Sec. IV.2 and is the first main result of this paper. Second, in Sec. IV.3, the state-of-the-art results for invariants of the MPTD system are discussed in the context of our Hamiltonian framework.

• •

  Lastly, in Sec. V with apply the 10-dimensional formulation to the Schwarzschild and Kerr spacetime, in Sec. V.2 and Sec. V.3, respectively. All explicit formulae are given, and the proof of integrability follows from straightforward calculations of Poisson brackets on the 10-dimensional phase space $\mathcal{P}$. This section contains many explicit formulae on which subsequent works in this series will be based.

A concluding section Conclusions and prospects contains a summary of our results in Sec. Concluding summary and a number of possible applications and prospects in Sec. Future applications, most of which are already ongoing. All our notations and conventions for Lorentzian and symplectic geometry are described in App. A, and a Mathematica Companion Notebook, accessible by simple request, contains an explicit verification of all calculations made in Sec. V related to integrability in Kerr.

In the present work, we will truncate the MPTD system (1) at dipolar order, so that all the physical information about the compact object is encoded into a four-momentum form $p_{a}$ (not tangent to $\mathscr{L}$ in general) and an antisymmetric spin tensor $S^{ab}$. The dipolar MPTD system is obtained by turning off all beyond-dipolar-order multipoles Eqs. (1), which is equivalent to neglecting the force $F_{a}$ and torque $N^{ab}$ Harte (2015). One is left with the dipolar MPTD equations:

	$\displaystyle(\nabla_{u}p)_{a}$	$\displaystyle=R_{abcd}S^{bc}u^{d}\,,$		(2a)
	$\displaystyle(\nabla_{u}S)^{ab}$	$\displaystyle=2p^{[a}u^{b]}\,.$		(2b)

The physical interpretation of (2) is clear: the coupling between $S^{ab}$ and the background curvature $R_{abcd}$ drives the evolution of $p^{a}$, while the misalignment of $p^{a}:=g^{ab}p_{b}$ of $u^{a}$ drives the evolution of $S^{ab}$. Three scalar fields along $\mathscr{L}$ can be defined from the multipoles of the particle:

$$\mu^{2}:=-p_{a}p^{a}\,,\qquad S_{\circ}^{2}:=\frac{1}{2}S_{ab}S^{ab}\,,\quad\text{and}\quad S_{\star}^{2}:=\frac{1}{8}\varepsilon_{abcd}S^{ab}S^{cd}\,.$$

(3)

By assumption, $p^{a}$ is time like while $S^{ab}$ is spacelike, resulting in $\mu\geq 0$, $S_{\circ}\geq 0$ and $S_{\star}\geq 0$. Among these norms, $\mu$ and $S_{\circ}$ will be referred to as the mass, and $S_{\circ}$ as the spin norm. Note that they are intrinsically defined from the particle’s multipole, and do not depend on any observer. These quantities are, in general, not conserved under the dipolar MPTD system (2).

A remark of importance has to be made about Eqs. (2). It is known that the spin of any compact object will induce a quadratic-in-spin quadrupole Marsat (2015) (see also Chap. 6 in Ramond (2021)), which contributes to both force $F_{a}$ and torque $N^{ab}$ in the full MPTD system (1). Therefore, when truncating (1) at dipolar order to obtain (9), we have effectively neglected all quadratic-in-$S^{ab}$ contributions from the right-hand side of (1). Consequently, to remain self-consistent, Eqs. (9) would also need to be linearized⁵⁵5Technically, one could argue that Eq. (9) are already linear in $S^{ab}$. However, they are not well-posed mathematically, and it will become apparent that they are not linear-in-$S^{ab}$ when completed with an additional condition ensuring well-posedness, cf. Sec. I.2. in $S^{ab}$. If not, one would include quadratic-in-spin corrections from the dipolar sector but omits those of the quadrupolar sector, leading to inconsistency. Therefore, it should be kept in mind that the dipolar MPTD equations (2) are only self-consistent at linear order in spin. This linearization, rather ambiguous if the “spin” means the tensor $S^{ab}$, can be done properly with a choice of spin supplementary condition and the introduction of a spin-related, small parameter, as explained in Sec. I.4.

The number of independent unknowns in $(v^{\alpha},p_{\alpha},S^{\alpha\beta})$ is $4+4+6=14$. Yet, there are only $10$ differential equations in (2). This ambiguity comes from an arbitrariness in the choice of parameter $\lambda$ associated to the tangent vector $v^{\alpha}$, and in the definition of the spin tensor, which contains angular momentum to be defined with respect to some reference (as in classical mechanics). A common way of fixing these issues (and thus having a well-posed differential system), is to 1) take $v^{a}=u^{a}$, the four-velocity along $\mathscr{L}$ so that $\lambda$ is the proper time $\tau$ and the condition $u^{a}u_{a}=-1$ reduces the number of unknowns to 13, and 2) impose a so-called spin supplementary condition (SSC), reducing this number to 10. An SSC takes the form of 3 algebraic constraints of the form

$$f_{a}S^{ab}=0\,,$$

(4)

where $f^{a}$ is some timelike vector defined along $\mathscr{L}$ (only three because this equation projected onto $f_{b}$ gives $0=0$ by antisymmetry of $S^{ab}$). To elucidate the meaning of such SSC, perform a Hodge decomposition of $S^{ab}$ with respect to the timelike vector $f^{a}$, effectively splitting $S^{ab}$ into a spin vector $S^{a}$ and mass dipole vector $D^{a}$, according to⁶⁶6Equation (5) assumes that $f_{a}f^{a}=-1$. If not, extra factors of $f_{a}f^{a}$ must be included.

$$S^{ab}=\varepsilon^{abcd}f_{c}S_{d}+2D^{[a}f^{b]}\quad\Leftrightarrow\quad\begin{cases}\,S^{b}=\frac{1}{2}\varepsilon^{abcd}f_{a}S_{cd}\,,\\ D^{b}=f_{a}S^{ab}\,.\end{cases}$$

(5)

The six degrees of freedom contained in the tensor $S^{ab}$ are encoded in the spacelike vectors $S^{a}$ and $D^{a}$, which have only three degrees of freedom each, as both verify the constraint of being orthogonal to $f^{a}$. Clearly, the SSC (4) means, physically, that the mass dipole $D^{a}$ measured in the spacelike frame of timelike direction $f^{a}$ vanishes. In that frame, $S^{ab}$ possesses three degrees of freedom and can be entirely described in terms of $S^{a}$. This decomposition is entirely analogous to the decomposition of the Faraday tensor $F^{ab}$ of electromagnetism into magnetic $B^{a}$ and electric $E^{a}$ fields measured by a given observer Gourgoulhon (2013), with the dictionary $(F^{ab},B^{a},E^{a})\leftrightarrow(S^{ab},S^{a},D^{a})$.

The natural question is: which frame should be chosen?, i.e., which vector $f^{a}$ in (4) ?. This question goes beyond the scope of the present work, and we refer to the thorough review Witzany et al. (2019) and references therein for details on the matter of SSCs (see also Lukes-Gerakopoulos et al. (2014); Costa and Natário (2015); Timogiannis et al. (2021)). While different choices of SSC will lead to more or less simple evolution equations, physical meaning and mathematical well-posedness, one generally expects that physical observables should all remain the same regardless of the chosen SSC, at least at dipolar order Steinhoff (2011, 2015b); Lukes-Gerakopoulos (2017).

In this article and subsequent works, we shall work with the so-called Tulczyjew-Dixon spin supplementary condition (TD SSC) Tulczyjew (1959); Dixon (1964)⁷⁷7Tulczyjew was the first to acknowledge the fact the other SSCs did not necessarily lead to unique center-of-mass trajectories in special relativity Dixon (2015), and adopted it also in GR in Tulczyjew (1959). :

$$C^{b}:=p_{a}S^{ab}=0\,.$$

(6)

Combining the TD SSC (6) with the dipolar MPTD equations (2) leads to a number of well-known results, recalled here for convenience. First, it follows from Eq. (2b) that the spin norm $S_{\circ}$ is constant along $\mathscr{L}$. Second, from (5), we find $S_{\star}=0$ (as enforced by any SSC Witzany et al. (2019)). Third, as shown first in Ehlers and Rudolph (1977) (see also Obukhov and Puetzfeld (2011) for a covariant proof), the TD SSC implies the following exact expression for the tangent vector $v^{a}$ in terms of $p_{a},S^{ab}$ and the background geometry

$$u^{a}=\frac{m}{\mu^{2}}\left(p^{a}+\frac{2S^{ab}R_{bcde}p^{c}S^{de}}{4\mu^{2}+R_{abcd}S^{ab}S^{cd}}\right)\,,$$

(7)

where $m:=-p_{a}u^{a}$ can be expressed in terms of solely ($p_{a},S^{ab},R_{abcd}$) by contracting Eq. (7) with $v_{a}$ and re-inserting the result. Equation (7) readily implies that all the scalars (3) are conserved.

The momentum-velocity relation (7) is a consequence of the dipolar MPTD system (2) and the SSC (4). One could ask if the converse result holds true, i.e., whether the momentum-velocity relation (7) and the dipolar MPTD together imply $p_{a}S^{ab}=0$. This is almost the case, for Eqs. (7) and (2) actually imply that $\nabla_{v}(p_{a}S^{ab})=0$, i.e., that the four-vector $p_{a}S^{ab}$ is parallel-transported along $\mathscr{L}$. This feature that has no major implication for solving the equations themselves⁸⁸8From the point of view of a Cauchy problem, i.e., a set of ODE’s with some initial conditions, it suffices to require $p_{a}S^{ab}=0$ at some initial time for the TD SSC to hold at all times (by linearity of parallel-transport).. Yet, it must be handled with care in a Hamiltonian formulation of those same equations. This is discussed in Sec. IV.1. Our choice of SSC (6) is not random, but motivated by a number of reasons, detailed below:

• (i)

  to our knowledge, it is the only one for which a unique center-of-mass-like worldline can be unambiguously defined for the actual extended body, and for which mathematical proofs of unicity exist Beiglböck (1967); Madore (1969); Schattner (1979a, b); Harte (2015). Other SSCs tend to correspond to a family of worldlines rather than a unique one Costa et al. (2012).

• (ii)

  it leads to a momentum-velocity relation that can be generalized to any multipolar order and can also account for the presence of electromagnetic fields Gralla et al. (2010). To our knowledge, this has never been shown for any other SSC.

• (iii)

  it naturally emerges when looking at the MPTD equations from the point of view of generalized Killing fields, the formalism developed by A. Harte Harte (2012, 2015) that goes beyond the test body assumption and takes into account self-field effects.

• (iv)

  several works, including Rüdiger (1981, 1983); Compère and Druart (2022); Compère et al. (2023), aimed at finding invariants of motion for the MPTD system beyond the geodesic limit. These works proved successful when relying on the TD SSC, and methods do not seem to generalize to other SSCs.

• (v)

  in spacetimes admitting a Killing-Yano tensor, it leads to a covariant notion of angular momentum Dietz and Rüdiger (1981); Dietz and Rudiger (1982); Compère and Druart (2022), with well-behaved geodesic and Newtonian limits for black hole spacetimes Ramond and Isoyama (2024).

• (vi)

  it is compatible with a covariant Hamiltonian formulation. As pointed out in Witzany et al. (2019), the reason is that the TD SSC is “co-moving” Witzany et al. (2019), i.e., it relies on only the intrinsic properties of the body ($p_{a},S^{ab}$) and no background fields

• (vii)

  for such covariant Hamiltonian formulations, the TD SSC uniquely reduces the number of degrees of freedom by the largest amount Witzany et al. (2019). In a canonical setup, this corresponds to two phase space dimensions.

• (viii)

  as seen in Sec. IV, the TD SSC defines a phase-space sub-manifold $\mathcal{T}$ that is invariant under the flow of the free particle Hamiltonian (12). This is a key aspect of our covariant formulation.

• (ix)

  in a future publication, we shall demonstrate how the linear-in-spin, Hamiltonian formalism presented in this paper can be extended covariantly to the quadratic-in-spin case. We have only been able to do this with the use of the TD SSC.

For convenience, from now on, we take the tangent vector $v^{a}$ to be the four-velocity $u^{a}$ of the particle. The associated parameter along $\mathscr{L}$ is the proper time, denoted $\tau$. Inserting (7), into (2) then leads to a closed system of 10 equations for the ten independent components of the momenta $(p_{a},S^{ab})$. This system of equations, however, has a singular limit (observe the denominator in (7)). To make it explicit, we introduce the (only) dimensionless physical parameter at play: If $1/L^{2}$ is the typical curvature scale of the background spacetime, then Eq. (7) is well-behaved only in the limit where

$$\epsilon:=\frac{S_{\circ}}{\mu L}\lesssim 1\,.$$

(8)

For a generic extended body, the spin angular momentum $S_{\circ}$ scales as $\mu\ell$ where $\ell$ is the typical size of the body, leading to $\epsilon=\ell/L$. We recover the small parameter tied to multipolar expansions that lead to the MPTD equations Dixon (1974); Harte (2012); Ramond and Le Tiec (2021). In the particular case of a compact object (defined by $\ell\simeq\mu$) orbiting around black hole of mass $M$ (for which $L\simeq M$) one readily obtains $\epsilon=\mu/M$. Therefore, the MPTD equations are well-suited to capture the leading effect of the spin in the context of binary systems with asymmetric mass ratio (see the discussions in Witzany et al. (2019); Miller and Pound (2021)).

With the small parameter (8), we can now rigorously expand the dipolar MPTD system to linear order in spin, by (i) setting $S^{ab}=S_{\circ}\bar{S}^{ab}$, $p^{a}=\mu\bar{p}^{a}$ and $R_{abcd}=\bar{R}_{abcd}/L^{2}$, with barred quantities being dimensionless, (ii) inserting this into Eqs. (2) and (7) and (ii) neglecting any non-linearity in $\epsilon$. This leads to

	$\displaystyle p^{a}$	$\displaystyle=\mu u^{a}\,,$		(9a)
	$\displaystyle(\nabla_{u}p)_{a}$	$\displaystyle=R_{abcd}S^{bc}u^{d}\,,$		(9b)
	$\displaystyle(\nabla_{u}S)^{ab}$	$\displaystyle=0\,.$		(9c)

It is worth noting at this order, Eq. (9a) implies that $p^{a}$ is tangent to $\mathscr{L}$, just as in the non-spinning (geodesic) case. Moreover, Eq. (9c) implies that $S^{ab}$ is parallel-transported along $\mathscr{L}$, just as for a test spin. However, Eq. (9b) implies that $\mathscr{L}$ is not a geodesic of $(\mathscr{E},g_{ab})$ because of the spin-curvature coupling. It is tempting to interpret the right-hand side of Eq. (9b) as a “force”, as it drives the evolution of the momentum. But we stress that there is no dynamical evolution in these equations. In fact, the MPTD system at dipolar order (2) is (i) pure kinematics: body-dependent multipoles only enter at quadrupolar order, with non-zero $F^{a},N^{ab}$ Harte (2015); Harte and Dwyer (2023); and (ii) completely universal: all spinning bodies with the same initial conditions will follow the same trajectories.

System (9) along with the TD SSC (6) is the system that we will to study in this paper. More precisely, what we will investigate in subsequent sections is the Hamiltonian formulation of the differential system given by the components of Eq. (9) onto the natural basis $(\partial_{\alpha})^{a}$ associated to some coordinate system $x^{\alpha}$ on $(\mathscr{E})$. This is the object of Sec. II. We also have to keep in mind that the well-posedness of Eqs. (9) relies on the TD SSC (6), which has to holds along $\mathscr{L}$⁹⁹9We emphasize again that the system (9) only implies $\nabla_{u}(p_{a}S^{ab})=0$, and not $p_{a}S^{ab}=0$.. We will ensure that the latter is correctly incorporated into the Hamiltonian formulation in Sec. (IV). Lastly, we emphasize that all calculations, from now on, will be performed consistently at linear order in spin, and what we is meant by that, is that in any equation we can restore the dimensionless, small parameter $\epsilon$ (define in (8)) and neglect any non-linearity in $\epsilon$.

It is well-known that the linearized MPTD + TD SSC system (9) can be written as a Poisson system d’Ambrosi et al. (2015); Witzany et al. (2019). As reviewed in App. D, this type of system is constructed from three ingredients: a phase space $\mathcal{M}$, a Poisson structure $\Lambda$ on $\mathcal{M}$ (or, equivalently, a set of Poisson brackets $\{,\}$), and a Hamiltonian $H$.

The computation of the equations of motion is made with respect to some arbitrary parameter $\lambda$ along the worldline (the “time” conjugated to the Hamiltonian), using the (generalized) Hamilton equations:

$$\frac{\mathrm{d}F}{\mathrm{d}\lambda}=\{F,H\}=\{F,x^{\alpha}\}\,\frac{\partial H}{\partial x^{\alpha}}+\{F,p_{\alpha}\}\,\frac{\partial H}{\partial p_{\alpha}}+\{F,S^{\alpha\beta}\}\,\frac{\partial H}{\partial S^{\alpha\beta}}\,,$$

(13)

with $F$ any function of the phase space variables. To be precise, in Eq. (13) the first equality is a definition (that of the evolution of $F:\mathcal{M}\rightarrow\mathbb{R}$ under the flow of $H$ for the Poisson system $(\mathcal{M},\{,\},H)$), and the second equality makes use of the Leibniz rule satisfied by the Poisson bracket. The evolution parameter $\lambda$ is that uniquely conjugated to the Hamiltonian (such that $\{\lambda,H\}=1$). It can be physically interpreted only once we compare the Hamilton equations to the system (9).

The partial derivatives of the Hamiltonian (12) are easily found to be

$$\frac{\partial H}{\partial x^{\alpha}}=-\Gamma^{\gamma}_{\alpha\beta}p_{\gamma}p^{\beta}\,,\quad\frac{\partial H}{\partial p_{\alpha}}=g^{\alpha\beta}p_{\beta}\quad\text{and}\quad\frac{\partial H}{\partial S^{\alpha\beta}}=0\,,$$

(14)

where we projected the metric compatibility condition $\nabla_{a}g_{bc}=0$ onto the natural basis to get the first equation. We then replace $F$ in Eq. (13) by the phase space coordinates $x^{\alpha},p_{\alpha},S^{\alpha\beta}$ and combine Eqs. (11), (13) and (14) to obtain the evolution for each coordinate:

	$\displaystyle w^{\alpha}:=\frac{\mathrm{d}x^{\alpha}}{\mathrm{d}\lambda}$	$\displaystyle=g^{\alpha\beta}p_{\beta}\,,$		(15a)
	$\displaystyle\frac{\mathrm{d}p_{\alpha}}{\mathrm{d}\lambda}$	$\displaystyle=\Gamma^{\nu}_{\alpha\lambda}p_{\nu}w^{\lambda}-\frac{1}{2}R_{\alpha\beta\mu\nu}S^{\mu\nu}w^{\beta}\,,$		(15b)
	$\displaystyle\frac{\mathrm{d}S^{\alpha\beta}}{\mathrm{d}\lambda}$	$\displaystyle=-(\Gamma^{\alpha}_{\lambda\nu}S^{\lambda\beta}+\Gamma^{\beta}_{\lambda\nu}S^{\alpha\lambda})w^{\nu}\,,$		(15c)

where $w^{\alpha}$ is just a notation for $\mathrm{d}x^{\alpha}/\mathrm{d}\lambda$. Regarding Eqs. (15b) and (15c), the first term on their right-hand side can be absorbed into their left-hand side by introducing covariant derivatives $\nabla_{w}=w^{\alpha}\nabla_{\alpha}$. We note that Eqs. (15b) and (15c) are exactly the linearized MPTD equations at dipolar order (9), written with respect to tangent vector $w^{\alpha}=p^{\alpha}$. Therefore, the parameter $\lambda$ uniquely associated to the Hamiltonian (12) corresponds physically to the worldline parameter associated to $p^{a}$ as a tangent vector to $\mathscr{L}$. By virtue of Eq. (9a), we find $\lambda:=\bar{\tau}=\tau/\mu$, and we shall use $\bar{\tau}$ from now on. We note that this is the same result as in the non-spinning, geodesic case. Since $\mu$ is conserved along $\mathscr{L}$, physically speaking, $\bar{\tau}$ is an affine parameter.

The phase space coordinates (10) are useful to prove that the Hamiltonian system (12)-(11) is equivalent to the original system (9). It is also relevant for interpretation purposes. However, when it comes to reducing the system and to doing Hamiltonian mechanics per-se, they are not practical at all. Here, we introduce a new system of coordinates on $\mathcal{M}$ which still retains some physical intelligibility while also admitting drastically simpler Poisson brackets.

First, we require some geometrical construction within spacetime $(\mathscr{E},g_{ab})$. Consider an orthonormal tetrad field $\{(e_{A})^{a}\}_{A\in\{0,\ldots,3\}}$, whose components in the natural basis are $(e_{A})^{\alpha}$ ($A$ labels the four different vectors, and $\alpha$ the four components of the $A$-th vector). Let $\omega_{aBC}:=g_{bc}(e_{B})^{b}\nabla_{a}(e_{C})^{c}$ be the connection 1-forms associated to the tetrad. Since $\omega_{aBC}=-\omega_{aCB}$, there are six independent such 1-forms. Their components in the natural basis, $\omega_{\alpha BC}$, are the connection coefficients and those in the tetrad, $\omega_{ABC}$, are the so-called Ricci rotation coefficients (we follow Sec. 3.4b of Wald (1984)). As before, all these objects are defined throughout spacetime, and become simple functions of the phase space coordinates $x^{\alpha}$ in the phase space picture. New coordinates on $\mathcal{M}$ are then obtained by (i) keeping the coordinates $x^{\alpha}$, (ii) replacing the linear momentum variables $p_{\alpha}$ by $\pi_{\alpha}$, a combination of linear and angular momentum, and (iii) replacing $S^{\alpha\beta}$ by $S^{AB}$, the tetrad components of the spin tensor. Explicitly, we define these new coordinates $(x^{\alpha},\pi_{\alpha},S^{AB})$ by setting¹²¹²12I thank J. Féjoz for noticing that, at fixed $x^{\alpha}$, the change of coordinates (16) is linear in the momenta $p_{\alpha},S^{\alpha\beta}$, thus making connecting it to the notion of fiber bundles.

	$\displaystyle p_{\alpha}$	$\displaystyle=\pi_{\alpha}+\frac{1}{2}\omega_{\alpha BC}S^{BC}\,.$		(16a)
	$\displaystyle S^{\alpha\beta}$	$\displaystyle=S^{AB}(e_{A})^{\alpha}(e_{B})^{\beta}\,.$		(16b)

Solving these equations for $\pi_{\alpha}$ and $S^{AB}$ shows that this coordinate change is indeed invertible, and takes the form $(x^{\alpha},p_{\alpha},S^{\alpha\beta})\mapsto(x^{\alpha},\pi_{\alpha}(z,p,S),S^{\alpha\beta}(z,S))$ where the right-hand side emphasizes the new coordinates’ dependence on the old set (10), denoted schematically by $(z,p,S)$. The Poisson brackets of the new coordinates can then be computed from the rule (103) and the old brackets (11). This calculation does not appear in the literature, to our knowledge. Therefore, we have detailed it in App. C. It is lengthy but straightforward, using not so well-known formulae involving the connection 1-forms (in particular Eq. (3.4.20) of Wald (1984)). The non-vanishing Poisson brackets are found to be:

	$\displaystyle\{x^{\alpha},\pi_{\beta}\}$	$\displaystyle=\delta^{\alpha}_{\beta}\,,$		(17a)
	$\displaystyle\{S^{AB},S^{CD}\}$	$\displaystyle=2\eta^{A[D}S^{C]B}+2\eta^{B[C}S^{D]A}\,.$		(17b)

Note that, even though the six $S^{AB}$ do not verify the canonical Poisson brackets, the four pairs $(x^{\alpha},\pi_{\alpha})$ do, and have vanishing Poisson brackets with $S^{AB}$. The spin variables $S^{AB}$ have Poisson brackets akin to the commutators of the so(1,3) algebra (see, e.g., Chap. 7 in Gourgoulhon (2013)). This is reminiscent of the definition of $S^{AB}$ as the components in an orthonormal tetrad field, the latter forming a set left invariant by the SO(1,3) group of Lorentz transformations. This property is even more clear when we use, instead of the six independent $S^{AB}$, the $3+3$ variables $(S^{1},S^{2},S^{3},D^{1},D^{2},D^{3})$ defined by

$$S^{I}=\frac{1}{2}\varepsilon^{I}_{\phantom{I}JK}S^{JK}\quad\text{and}\quad D^{I}=S^{0I}\,.$$

(18)

where $\varepsilon^{IJ}_{\phantom{IJ}K}$ is the 3D Levi-Civita symbol, equal to $1$ (resp. $-1$) when $IJK$ is an even (resp. odd) permutation of $123$, and $0$ otherwise. Although they are but notations in phase space, $S^{I}$ and $D^{I}$ have a clear physical meaning: they are the tetrad components of the Euclidean spin and mass dipole vectors as defined in Eq. (5) (i.e., with $(e_{0})^{a}$ replacing $f^{a}$ there). The brackets (17b), with these notations, take their simplest form

	$\displaystyle\{x^{\alpha},\pi_{\beta}\}$	$\displaystyle=\delta^{\alpha}_{\beta}\,,$		(19a)
	$\displaystyle\{S^{I},S^{J}\}$	$\displaystyle=\varepsilon^{IJ}_{\phantom{IJ}K}S^{K}\,,$		(19b)
	$\displaystyle\{D^{I},D^{J}\}$	$\displaystyle=-\varepsilon^{IJ}_{\phantom{IJ}K}S^{K}\,,$		(19c)
	$\displaystyle\{D^{I},S^{J}\}$	$\displaystyle=\varepsilon^{IJ}_{\phantom{IJ}K}D^{K}\,.$		(19d)

Now the brackets (19) are identical to the commutators of the generators for the Lie algebra so(1,3). We also stress that brackets (19) are equivalent to, it is more a renaming of the spin coordinates (through Eq. (18)) than a coordinate transformation.

We can now express the original Hamiltonian (12) in terms of the new coordinates $(x^{\alpha},\pi_{\beta},S^{I},D^{J})$ on $\mathcal{M}$, by inserting Eqs. (16) into Eq. (12). We obtain

$$H:(x^{\alpha},\pi_{\beta},S^{I},D^{J})\mapsto\frac{1}{2}g^{\alpha\beta}\pi_{\alpha}\pi_{\beta}+\frac{1}{2}g^{\alpha\beta}\pi_{\alpha}\omega_{\beta CD}S^{CD}\,.$$

(20)

Even though (20) calls for a rather natural identification into a “non-spinning” term $\tfrac{1}{2}g^{\alpha\beta}\pi_{\alpha}\pi_{\beta}$ corrected by a “linear-in-spin term” $\tfrac{1}{2}g^{\alpha\beta}\pi_{\alpha}\omega_{\beta CD}S^{CD}$. This split, however, should be interpreted with care, as there are spin-related quantities hidden in $\pi_{\alpha}$, cf. Eq. (16). We will always avoid any such interpretation, and rather treat the whole Hamiltonian as one describing an object with both linear and angular momentum (orbital and spin) intertwined.

We now focus on the Poisson brackets (19) between the new coordinates $(x^{\alpha},\pi_{\alpha},S^{I},D^{I})$. Following the notations of App. D.2, we are in the case $N=14$ and we construct the Poisson matrix $\Lambda^{ij}(y)=\{y^{i},y^{j}\}$, where $(y^{1},\ldots,y^{14})$ denote the coordinates $(x^{\alpha},\pi_{\beta},S^{I},D^{I})$, in that order.

Since the 12D leaves $\mathcal{N}$ are symplectic (non-degenerate), it is tempting to try and construct 12 canonical coordinates, i.e., 6 pairs $(q^{i},p_{i})_{i\in\{1,\ldots,6\}}$ for which the brackets on $\mathcal{N}$ satisfy $\{q^{i},p_{j}\}=\delta^{i}_{j}$. To derive these coordinates, we note that the Casimirs (24) are only functions of $(S^{I},D^{I})$ and not $(x^{\alpha},\pi_{\alpha})$. A natural method to construct canonical coordinates on $\mathcal{N}$ emerges naturally: (i) keep the 4 pairs $(x^{\alpha},\pi_{\alpha})$ of $\mathcal{M}$, as they will be automatically canonical on $\mathcal{N}$, from Eq. (19a) (recall that $\{x^{\alpha},\pi_{\beta}\}^{\mathcal{N}}=\{x^{\alpha},\pi_{\beta}\}$), and (ii) construct 2 other canonical pairs covering the spin sector $(\vec{S},\vec{D})$, say $(\sigma,\pi_{\sigma}),(\zeta,\pi_{\zeta})$. The latter can be found by solving the nine spin-dependent brackets (17), now seen as canonical brackets on $\mathcal{N}$ of the 6 $(\sigma,\pi_{\sigma},\zeta,\pi_{\zeta})$-dependent functions $(S^{I},D^{I})$. For example, the bracket $\{S^{1},S^{2}\}=S^{3}$ on $\mathcal{M}$ now becomes a partial differential equation (PDE)

$$\frac{\partial S^{1}}{\partial\sigma}\frac{\partial S^{2}}{\partial{\pi_{\sigma}}}-\frac{\partial S^{2}}{\partial\sigma}\frac{\partial S^{1}}{\partial{\pi_{\sigma}}}+\frac{\partial S^{1}}{\partial\zeta}\frac{\partial S^{2}}{\partial{\pi_{\zeta}}}-\frac{\partial S^{2}}{\partial\zeta}\frac{\partial S^{1}}{\partial{\pi_{\zeta}}}=S^{3}\,,$$

(28)

in which $S^{1},S^{2},S^{3}$ are viewed as functions on $\mathcal{N}$ of the coordinates $(\sigma,\pi_{\sigma}),(\zeta,\pi_{\zeta})$. This calculation is rather non-trivial because the nine Poisson brackets turn into a system of nine nonlinear coupled PDEs, but educated guesses from the so(1,3) symmetries make things manageable.