# Gyroscopic Gravitational Memory from quasi-circular binary systems

Guillaume Faye, Ali Seraj

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Guillaume Faye, Ali Seraj. Gyroscopic Gravitational Memory from quasi-circular binary systems. Classical and Quantum Gravity, 2025, 42 (3), pp.035005. 10.1088/1361-6382/ada339 . hal-04701152

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# Gyroscopic gravitational memory from quasi-circular binary systems

To cite this article: Guillaume Faye and Ali Seraj 2025 *Class. Quantum Grav.* **42** 035005

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# Gyroscopic gravitational memory from quasi-circular binary systems

Guillaume Faye<fnref n="1" /> ID and Ali Seraj<fnref n="2" />,∗ ID

E-mail: ali_seraj@ipm.ir and faye@iap.fr

Received 4 September 2024; revised 10 December 2024

Accepted for publication 24 December 2024

Published 7 January 2025

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CrossMark

## Abstract

Gravitational waves cause freely falling spinning objects to precess, resulting in a net orientation change called gyroscopic memory. In this paper, we will consider isolated gravitational sources in the post-Newtonian (PN) framework and compute the gyroscopic precession and memory at leading PN orders. We compare two competing contributions: the spin memory and the nonlinear helicity flux. At the level of the precession rate, the former is a 2PN oscillatory effect, while the latter is a 4PN adiabatic effect. However, the gyroscopic memory involves a time integration, which enhances subleading adiabatic effects by the fifth power of the velocity of light, leading to a 1.5PN memory effect. We explicitly compute the leading effects for a quasi-circular binary system and obtain the angular dependence of the memory on the celestial sphere.

Keywords: gravitational memory effects, gyroscope, post-Newtonian formalism, null infinity, precession, multipole expansion, gravitational waves

## 1. Introduction

Spin and rotation effects play a crucial role in the dynamics of gravitating systems in general relativity (GR). The mass currents arising in rotating bodies, source a ‘gravitomagnetic’

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Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.

<footnote n="1">¹ *GRéCO*, Institut d'Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98bis boulevard Arago, 75014 Paris, France</footnote>

<footnote n="2">² School of Physics, Institute for Research in Fundamental Sciences (IPM), PO Box 19395-5531, Tehran, Iran</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
potential entering the parametrization of the linearized metric, with no Newtonian counterpart. The primary example of a stationary rotating solution is the Kerr black hole, which has many distinct features from a static black hole of the same mass. Examples of such features include the Lense-Thirring frame dragging [1] and the black hole's ergosphere, whose interaction with the surrounding matter fields leads to interesting physical effects, such as superradiance [2] and the Blandford–Znajek process [3].

In addition to stationary situations, it is necessary to describe the dynamical interaction between gravity and spinning matter. This topic has been extensively studied, beginning with the seminal works of Mathisson and Papapetrou [4, 5], who formulated the motion and precession of test spinning particles in GR. Their results were rephrased by Tulczyjew, and extended by Dixon and others to include the higher multipole structure of test particles [6–10]. The Mathisson, Papapetrou, Dixon (MPD) equations can be derived in full generality from an effective worldline action [11], laying the basis of the effective field theory approach to gravitational waves (GWs). This formalism may be used to describe self-gravitating systems of spinning particles in the post-Newtonian (PN) framework as an important class of GW sources (see [12, 13] for the derivation of the PN equations of motion for compact binaries from MPD equations, or [14] for the calculation of the corresponding Fokker Lagrangian, following theoretical grounds elucidated in [15]).

In this paper, we will be interested in spinning bodies as probes of GWs. In particular, we will investigate DC effects of GW that accumulate over time and lead to persistent observables lasting after the passage of the wave [16]. The primary example of such observables is the displacement effect, consisting of a net change in the distance between nearby freely falling test particles [17–21]. If the test particle is endowed with a spin, as a gyroscope, an additional observable is the precession of the spin caused by the GWs. Different experiments to measure the effect can be thought of. One consists in computing the change in the spin of a rotating object with respect to a parallel-transported frame at finite distance, as suggested in [16]. Another observable is the precession of the spin of a gyroscope with respect to a comoving optical frame tied to distant stars, and the subsequent persistent rotation after the passage of the wave. The latter proposal was examined in [22, 23], where it was found that, restricting to the regime in which the spin vector is approximately parallel transported along a geodesic motion, GWs cause a precession in the local transverse plane of the propagation and is proportional to the inverse square of the distance to the source. As a result, the passage of the wave induces in that case a net rotation in the orientation of the gyroscope, dubbed as the ‘gyroscopic memory’. The dynamics is however more involved when nonlinear spin interactions cannot be neglected so that the full MPD equations must be taken into account. In section 2, we will clarify the limit in which the parallel-transport assumption is valid.

The goal of this paper is to investigate the PN expansion of the gyroscopic memory when the GWs are sourced by binary systems of compact objects. We will actually focus on *quasi-circular* binaries, consisting of two non-spinning compact bodies on a nearly circular orbit whose radius adiabatically shrinks in time. We will address the following questions: (1) How does the test gyroscope’s precession rate depend on its angular position on the celestial sphere? (2) What is the magnitude of the precession in terms of the parameters of the binary system? (3) What are the DC components of the precession and the accumulation effect resulting in the gyroscopic memory?

The paper is organized as follows. In section 2, we introduce the MPD equations to show how, in a certain limit, the spin evolution reduces to the parallel transport used in deriving the gyroscopic memory effect in [22, 23]. This construction and its implications are reviewed
in 2.3. In section 3, we reformulate the precession rate and gyroscopic memory using ‘spin-weighted’ functions on the celestial sphere. The latter approach is advantageous not only formally, due to the simplicity of the resulting expressions, but also for practical computations. Resorting to those tools, it is straightforward to perform the multipole expansion of the precession and to compute memory effects in terms of the radiative moments, which is achieved in section 3. In section 4, we introduce quasi-circular binary systems, discuss their radiation, and compute the precession of a distant freely falling gyroscope as well as the gyroscopic memory in the PN framework. We conclude in section 5, by briefly discussing observational aspects of the gyroscopic memory.

## 2. Brief review of spin dynamics in GR

### 2.1. Effective approach to spinning objects

The motion and precession of a free point-like body with velocity $v^\mu$ (normalized so that $v^\mu v_\mu = -c^2$), momentum $p_\mu$, carrying some spin represented by the antisymmetric tensor $S^{\mu\nu}$ is given by the equations [7]

$$ \frac{Dp_\mu}{D\tau} = -\frac{1}{2}R_{\mu\nu\rho\sigma}v^\nu S^{\rho\sigma} - \frac{c^2}{6}J^{\nu\lambda\rho\sigma}\nabla_\mu R_{\nu\lambda\rho\sigma} + \dots, \quad (1a) $$

$$ \frac{DS^{\mu\nu}}{D\tau} = 2p^{[\mu}v^{\nu]} + \frac{4c^2}{3}R^{[\mu}_{\lambda\rho\sigma}J^{\nu]}\lambda\rho\sigma + \dots, \quad (1b) $$

with $D/D\tau \equiv v^\mu \nabla_\mu$, where the Levi-Civita derivative $\nabla_\mu$, and the corresponding Riemann tensor $R^\mu_{\nu\rho\sigma}$ are derived from the metric $g_{\mu\nu}$, and the square brackets denote the antisymmetrization over the enclosing indices. The tensor $S^{\mu\nu}$ contains nonphysical degrees of freedom related to the choice of the body centroid. To fix them, the above equations must be supplemented by a spin supplementary condition [24–26]. The covariant Tulczyjew condition $p_\nu S^{\mu\nu} = 0$ [6, 7, 27] is particularly convenient, as it implies that in the ‘center-of-mass’ frame—defined as a frame whose temporal basis vector lies along $p^\nu$—, the spin vector is purely spatial. The internal structure of the body manifests itself through the Dixon moments, such as the quadrupole tensor $J^{\mu\nu\rho\sigma}$, as well as higher-order moments, $J^{\nu\lambda\rho\sigma\tau}$, $J^{\nu\lambda\rho\sigma\tau\nu}$, etc which enter the subleading corrections represented by the dots in (1).

From general considerations in statistical physics, we expect that in thermodynamic equilibrium, the macroscopic state of a body—and thus its Dixon multipole moments—should be entirely determined by its mass, energy, linear and angular momenta, equation of state, and stationary external gravitational field<fnref n="3" />. This justifies to resort, when describing the dynamic of point-like particles, to effective Lagrangians that depend only on the velocities and spins of the particles, as well as the metric and (derivatives of) the Weyl tensor evaluated at the locations of the particles [15, 29]. Those Lagrangians can be used to derive effective expressions for

<footnote n="3">³ Such equilibrium can only hold approximately, on timescales that are sufficiently short to prevent any significant exchange of energy with the environment so that, in particular, back-reaction effects remain negligible. See section II.E in [28] for a discussion on macroscopic configurations of non-relativistic non-gravitating isolated systems at thermodynamic equilibrium. In binary systems, because the external field is not stationary, the best we can achieve is an ‘instantaneous’ quasi-equilibrium, provided the relaxation time is much smaller than the orbital period. This practically implies that both companions should behave as fluids.</footnote>
the Dixon moments. If tidal deformations are negligible compared to spin-induced ones, the Dixon quadrupole reads [29, 30]

$$J^{\mu\nu\rho\sigma} = \frac{3\kappa}{mc^4} v^{[\mu} S^{\nu]} \lambda S_{\lambda}^{[\rho} v^{\sigma]}, \quad (2)$$

where the dependence on the equation of state arises through the dimensionless constant $\kappa$. Equations (1) and (2) must be thought of as effective and, as such, they do remain valid for self-gravitating systems. It should be noted that only the first terms in the right-hand sides of each equation in (1) are universal, i.e. independent of the nature of the body, while the others terms containing Dixon moments depend on the system's equation of state. The full MPD equations for test extended bodies, in the zero radius limit, were first derived by Dixon [9], while Bailey and Israel [11] recovered this result starting from an effective Lagrangian (see [29] for a short modern review). This approach was later incorporated in the effective field theory to investigate the dynamics of GW sources [14, 31].

Contracting the momentum with (1b), employing the Leibniz rule, the supplementary condition and (1a) imply that $p_{\nu}DS^{\mu\nu}/D\tau = O(S^2)$. Using this information in (1b) next reveals that $2p_{\nu}p^{[\mu}v^{\nu]} = O(S^2)$. Defining the bare mass as $m \equiv -p^{\mu}v_{\mu}/c^2$, it follows that $p^{\mu}p_{\mu} + m^2c^2 = O(S^2)$ and $p^{\mu} = mv^{\mu} + O(S^2)$. Inserting this back in (1b), we find that

$$\frac{DS^{\mu\nu}}{D\tau} = 0 + O(S^2), \quad (3)$$

i.e. the spin obeys the parallel transport equation to leading order in the spin. Moreover, the body's worldline is geodesic up to linear corrections in the spin, namely, $Dv^{\mu}/D\tau = O(S^{\mu\nu})$. In the rest of this paper, we will neglect higher order spin corrections and will therefore restrict our attention to the parallel transport along a geodesic. In this regime, it is more convenient to define a spin vector $S^{\mu} \equiv -\epsilon_{\mu\nu\rho\sigma}p^{\nu}S^{\rho\sigma}/(2mc)$, so that, at leading order in the spin, the evolution equations reduce to

$$\frac{Dv^{\mu}}{D\tau} = 0, \quad \frac{DS^{\mu}}{D\tau} = 0. \quad (4)$$

## 2.2. Parallel transport in orthonormal frames

The parallel transport equation can alternatively be formulated in a local orthonormal frame $e_a^\mu$, normalized as $e_a^\mu e_b^\nu g_{\mu\nu} = \eta_{ab}$, where $\eta_{ab} = \text{diag}(-1, 1, 1, 1)$. Using $S^\mu = S^a e_a^\mu$ in equation (4), one obtains

$$v^{\mu} \partial_{\mu} S^{a} = -v^{\mu} \omega_{\mu}^{a} S^{b}, \quad \omega_{\mu}^{a} = e^{a}_{\alpha} \nabla_{\mu} e_{b}^{\alpha}, \quad (5)$$

where $\omega_{\mu}^{a}{}_{b}$ is the spin connection associated to the tetrad. The form (5) is not fully satisfactory for our purpose. On the one hand, neither side of (5) is covariant with respect to a general local Lorentz transformations $S^a \to \Lambda^a_b(x^\mu)S^b$. On the other hand, for a generic choice of the tetrad, the equation predicts a precession for the gyroscopes in the flat Minkowski background, since the spin connection is not necessarily vanishing in that case. It is yet possible to rewrite it in a way that resolves both issues and highlights its physical content. This may be achieved by adding $v^{\mu}\bar{\omega}_{\mu}{}^{a}{}_{b}S^b$ on both sides, where $\bar{\omega}_{\mu}{}^{a}{}_{b}$ is a background spin connection such that $\omega_{\mu}{}^{a}{}_{b} \to \bar{\omega}_{\mu}{}^{a}{}_{b}$ as the metric $g_{\mu\nu}$ approaches a background metric $\eta_{\mu\nu}$ (the flat metric in this setup). The resulting equation reads

$$v^{\mu} \bar{D}_{\mu} S^{a} = -v^{\mu} (\omega_{\mu}^{a}{}_{b} - \bar{\omega}_{\mu}^{a}{}_{b}) S^{b}, \quad (6)$$
where we have used that $\overline{D}_{\mu}S^a \equiv \partial_{\mu}S^a + \overline{\omega}_{\mu}{}^a S^b$. Each side of (6) is now covariant under arbitrary local Lorentz transformations, and the right-hand side vanishes over the background spacetime. We can think of (6) as the time evolution of the spin with respect to a frame that is parallel transported with the background structure.

To simplify further, we choose a comoving orthonormal frame $e_a^\mu = (v^\mu, e_i^\mu)$, where the temporal basis vector coincides with the geodesic velocity $v^\mu$ of the gyroscope, while the spatial basis vectors, denoted by $e_i^\mu$ with $\hat{i} \in \{1, 2, 3\}$, are not further specified. In this frame, the spin is purely spatial, $S^\mu = S^i e_i^\mu$, and (6) reduces to

$$ \frac{dS^i}{d\tau} = \Omega_{\hat{j}}^{i\hat{j}} S^{\hat{j}}, \quad \Omega_{\hat{j}}^{i\hat{j}} = -v^{\mu} \left( \omega_{\mu}^{i\hat{j}} - \overline{\omega}_{\mu}^{i\hat{j}} \right) \qquad (7) $$

where $d/d\tau \equiv v^\mu \overline{D}_\mu$ and all frame indices are raised or lowered with the Kronecker delta $\delta_{ij}$. Now, it turns out that the asymptotic flatness, which we impose as a boundary condition, induces a natural background structure $\overline{\omega}_{\mu}^{i\hat{j}}$ uniquely fixed by the boundary metric on the celestial sphere [22, 23]. In physical terms, the corresponding inertial frame is realized by ‘distant stars’ on the celestial sphere. Therefore, (7) is interpreted as the time evolution of the spin with respect to a frame tied to distant stars, and the antisymmetric tensor $\Omega_{\hat{j}}^{i\hat{j}}$ is the precession rate in the $\hat{ij}$ plane.

## 2.3. Gyroscopes in asymptotically flat spacetimes

### 2.3.1. Precession equations for asymptotic gyroscopes.
The metric far from a localized source of GWs is suitably expressed in Bondi coordinates $(u, r, \theta^A)$ with $u$ the null retarded time, $r$ the areal distance from the source localized around $r=0$, and $\theta^A$ a coordinate system on the sphere. As an asymptotic expansion in $r$, it reads

$$ ds^2 = - \left(1 - \frac{2G\mu}{rc^2} + \mathcal{O}\left(\frac{1}{r^2}\right)\right) du^2 - 2 \left(1 + \mathcal{O}\left(\frac{1}{r}\right)\right) du dr + r^2 \left(\gamma_{AB} + \frac{1}{r} C_{AB} + \mathcal{O}\left(\frac{1}{r}\right)\right) d\theta^A d\theta^B + \left(D_B C_A{}^B du d\theta^A + \mathcal{O}\left(\frac{1}{r}\right)\right), \qquad (8) $$

where $G$, $c$ are the Newton constant of gravitation and the speed of light, respectively. The metric $\gamma_{AB}$ on the unit celestial sphere (or its inverse $\gamma^{AB}$) allows raising (lowering) spherical indices. In other occurrences, it is used to define the covariant derivative $D_A$ on the sphere and the Levi-Civita tensor $\epsilon_{AB}$. The Bondi shear $C_{AB}(u, \theta^A)$, which is symmetric and trace-free with respect to $\gamma_{AB}$, encodes the GWs emitted by the source. The metric (8) shows explicit leading-order deviations from the Minkowski metric, while neither the Bondi mass aspect $\mu(u, \theta^A)$ nor the ‘subleading’ corrections [32–34] are required for the purpose of this paper, since the leading gyroscopic effect is entirely determined in terms of the Bondi shear, as we will see below.

Let us now consider a gyroscope at large but finite distance $r$ from the source. We aim at computing its precession with respect to a frame pointing towards distant stars. The details of the construction of the tetrad is given in [22, 23] (see also [35–37]). Decomposing spatial directions into radial and transverse $\hat{i} = \hat{r}, \hat{A}$, it was found in [22, 23] that the passage of GWs
by the gyroscope induces a dominant precession in the transverse plane given by

$$ \Omega_{\hat{A}\hat{B}} = \frac{\epsilon_{\hat{A}\hat{B}}}{r^2} \hat{\Omega} + \mathcal{O}\left(\frac{1}{r^3}\right), \qquad \hat{\Omega}(u, \theta^A) = \underbrace{\frac{c}{4} D_A D_B \tilde{C}^{AB}}_{\hat{\Omega}_{(S)}} - \underbrace{\frac{1}{8} \dot{C}_{AB} \tilde{C}^{AB}}_{\hat{\Omega}_{(H)}}, \qquad (9) $$

where $\tilde{C}_{AB} \equiv \epsilon_{CA}C^C_B$ is the dual shear. The other component $\Omega_{\hat{A}\hat{B}}$ is subleading at large distance, scaling as $\mathcal{O}(r^{-3})$.

### 2.3.2. Gyroscopic memory.
After the passage of the wave, GWs induce a net rotation in the orientation of the gyroscope in transverse plane, whose angle is simply obtained by the time integral of (9),

$$ \Delta\Psi = \frac{\Delta\hat{\Psi}}{r^2} + \mathcal{O}\left(\frac{1}{r^3}\right), \qquad \Delta\hat{\Psi} = \int_{u_0}^{u} dv \left(\hat{\Omega}_{(S)}(v) + \hat{\Omega}_{(H)}(v)\right). \qquad (10) $$

The precession rate (9) consists of a linear term in the shear and a quadratic term. They are denoted by S and H labels respectively, which is justified in two ways. On the one hand, the time integral in (10) kills all but the zero frequency mode in the Fourier expansion of the linear term, and is thus referred to as *soft*, while the quadratic term includes gravitons of all frequencies, and is referred to as *hard*. On the other hand, the first term coincides exactly with the *spin memory effect* [38–41], while the second term measures the total *helicity*, the difference between the number of right-handed and left-handed gravitons at a given point on the celestial sphere [22, 23, 42–46].

Let us end this section by a remark on terminology. In the following, we will call $\hat{\Omega}_{(S)}$ the linear precession, as it is a linear function of the Bondi shear, and $\hat{\Omega}_{(H)}$ the nonlinear precession, as it is quadratic in the shear. However, in the context of gravitational memory effects, there is a splitting of the memory into linear and nonlinear pieces [20, 33, 39, 47], also called the ordinary and null memory effects [48]. In the latter terminology, linearity (nonlinearity) instead refers to first (second) order effects in a perturbative solution of Einstein’s equations. These two terminologies do not coincide in any way, since the Bondi shear itself contains perturbative effects of all orders. To avoid any confusion, we will never use the term linear or nonlinear memory.

## 3. Multipole expansion of the precession rate

### 3.1. Precession in the holomorphic basis

Our main results, equations (9) and (10), involve tensorial expressions constructed out of the symmetric trace-free (STF) Bondi shear and its covariant derivatives with respect to the round metric $\gamma_{AB}$ on the sphere. Remarkably, STF tensors of any rank in two dimensions have only two independent degrees of freedom, combinable into a single complex scalar. This correspondence provides an elegant and practical formulation of the problem in terms of *spin-weighted* functions, which we explain here and further in appendix A. This formalism was developed in the representation theory of the rotation group [49], and independently in the spin-coefficient formalism of Newman and Penrose [50] and is widely used in the context of GW theory.

Given a real orthonormal basis $E_1^A, E_2^A$ on a two-dimensional Riemannian manifold, one can construct a pair of null vectors, consisting of $m^A = (E_1^A + iE_2^A)/\sqrt{2}$ and its complex
conjugate $\overline{m}^A = (E_1^A - iE_2^A)/\sqrt{2}$. By construction $m^A m_A = 0 = \overline{m}^A \overline{m}^A$ and $m^A \overline{m}^A = 1$. In our setup, the relevant manifold is a unit round sphere, whose metric and volume form can be recovered from the real and imaginary part of the product $m_A \overline{m}_B$

$$m_A \overline{m}_B = \frac{1}{2} (\gamma_{AB} - i \epsilon_{AB}). \qquad (11)$$

Moreover, the product of $m_{A_1} \cdots m_{A_s} \equiv m_{A_1} \cdots m_{A_s}$ and $\overline{m}_{A_1} \cdots \overline{m}_{A_s}$ provide a complete basis for STF tensors of rank $s$. In particular, the Bondi shear takes the form

$$C_{AB} = m_{AB} C + \overline{m}_{AB} \overline{C}, \quad \text{where} \quad C = \overline{m}^{AB} C_{AB}, \qquad (12)$$

where $C, \overline{C}$ have spin-weight $-2, +2$ respectively (see appendix A for the definition and further details). Now, let us compute the precession rate (9) in this basis. The linear part $\hat{\Omega}_{(S)}$ involves

$$D_A D_B \tilde{C}^{AB} = \varepsilon^{AC} \gamma^{BD} D_C D_D C_{AB} = i (m^{AB} \overline{m}^{CD} - \overline{m}^{AB} m^{CD}) D_C D_D C_{AB}. \qquad (13)$$

In deriving the last term, we have used (11) to rewrite the metric and the Levi-Civita tensor in terms of the null dyad, and the STF property $m^C \overline{m}^D C_{CD} = 0$ of the shear tensor to simplify the result. Using the eth derivative $\partial$, which acts on $C$ as

$$\partial C = \overline{m}^A \overline{m}^B m^C D_C C_{AB}, \qquad (14)$$

as well as its conjugate $\overline{\partial}$, defined similarly but with $m^C$ replaced by its complex conjugate $\overline{m}^C$ (see appendix A), we thus find

$$\hat{\Omega}_{(S)} = \frac{c}{4i} (\partial^2 C - \overline{\partial}^2 \overline{C}) = \frac{c}{2} \mathrm{Im} (\partial^2 C). \qquad (15)$$

The nonlinear term $\hat{\Omega}_{(H)}$ in (9) is also easily expressed as

$$\hat{\Omega}_{(H)} = -\frac{1}{8} \dot{C}_{AB} \tilde{C}^{AB} = \frac{i}{8} (m_{AB} \dot{C} + \overline{m}_{AB} \dot{\overline{C}}) (m^{AB} C - \overline{m}^{AB} \overline{C}) = \frac{1}{4} \mathrm{Im} (\dot{C} \overline{C}). \qquad (16)$$

Therefore, the total precession rate is given by

$$\hat{\Omega} = \frac{1}{4} \mathrm{Im} (2c \partial^2 C + \dot{C} \overline{C}). \qquad (17)$$

Note in particular that the result is of spin-weight 0, implying that the precession rate is independent of the choice of frame, see equation (45) in appendix A.

## 3.2. Multipole expansion

By construction, the complex shear $C$ is of spin-weight $s = -2$ and thus can be multipole expanded in the basis of spin-weight $-2$ spherical harmonics as

$$C(u, \theta^A) = \sum_{\ell \ge 2, m} C_{\ell m}(u) \partial_\ell Y^m(\theta^A), \qquad (18)$$
where we use the shorthand notation $\sum_{\ell \ge 2, m} = \sum_{\ell=2}^{+\infty} \sum_{m=-\ell}^{\ell}$. The complex conjugate $\bar{C} = \sum_{\ell \ge 2, m} \bar{C}_{\ell m} - 2 \bar{Y}^{\ell m}$ can be expanded, using the property $-s \bar{Y}^{\ell m} = (-1)^{m+s} Y^{\ell-m}$, as

$$ \bar{C} = \sum_{\ell \ge 2, m} C_{\ell m}^* 2 Y^{\ell m}, \quad C_{\ell m}^* = (-1)^m \bar{C}_{\ell-m}. \qquad (19) $$

The Bondi shear $C_{AB}$ can also be expanded in terms of parity-definite real scalars $U(u, \theta^A)$, $V(u, \theta^A)$ with even and odd parity respectively as

$$ C_{AB} = D_{\langle A} D_{B \rangle} U + \varepsilon^C_{\quad A} D_{\langle B} D_{C \rangle} V, \qquad (20) $$

In this relation, angle brackets denote the STF part of the tensor under investigation. An equivalent expression for (20), which is more democratic between even and odd-parity terms is

$$ C_{AB} = D_{\langle A} D_{B \rangle} U + \varepsilon^C_{\quad A} D_{\langle B} D_{C \rangle} V. \qquad (21) $$

Using (21) in the second equation in (12), we obtain

$$ C = \bar{\partial}^2 \bar{Z}, \quad Z = U + iV. \qquad (22) $$

Multipole expansion is an integral part of the PN/multipolar post-Minkowskian (PN/MPM) formalism [47], which facilitates the derivation of the radiation field in terms of the source parameters. In our language, this corresponds to

$$ Z = \sum_{\ell \ge 0} Z_L n^L, \quad Z_L = \frac{4G}{c^{\ell+2}\ell!} \frac{1}{\ell(\ell-1)} \left( U_L - \frac{2\ell i}{c(\ell+1)} V_L \right). \qquad (23) $$

where $U_L, V_L$ are respectively the mass and current radiative multipoles in the basis of STF harmonics $n^L \equiv n^{(i_1 \dots i_\ell)}$. Alternatively, we can multipole expand in the basis of spherical harmonics, by writing $Z = \sum_{\ell \ge 0} Z_L n^L \delta_{\langle L \rangle}^{(L)}$ and using the completeness relationship $\delta_{\langle L' \rangle}^{(L)} = 4\pi \ell! / (2\ell+1)!! \sum_{m=-\ell}^{\ell} Y_L^{\ell m} \bar{Y}_{L'}^{\ell m}$. The numerical coefficients $\bar{Y}_L^{\ell m}$ relate the two bases as $Y^{\ell m}(\theta^A) = \bar{Y}_L^{\ell m} n_L(\theta^A)$ [51]. The result is

$$ Z = \sum_{\ell \ge 0, m} Z_{\ell m} Y^{\ell m}, \quad Z_{\ell m} = \frac{4\pi \ell!}{(2\ell+1)!!} Z_L \bar{Y}_L^{\ell m}. \qquad (24) $$

Using (22) and (24), the complex shear is expanded as

$$ C = \sum_{\ell \ge 2, m} C_{\ell m} - 2 Y^{\ell m} \qquad C_{\ell m} = \frac{G}{\sqrt{2c^{\ell+2}}} \left( U_{\ell m} - i \frac{V_{\ell m}}{c} \right), \qquad (25a) $$

$$ \bar{C} = \sum_{\ell \ge 2, m} C_{\ell m}^* 2 Y^{\ell m} \qquad C_{\ell m}^* = \frac{G}{\sqrt{2c^{\ell+2}}} \left( U_{\ell m} + i \frac{V_{\ell m}}{c} \right), \qquad (25b) $$

where the spherical multipoles $U_{\ell m}, V_{\ell m}$ are related to STF multipoles $U_L, V_L$ as [51]

$$ U_{\ell m} = \frac{16\pi}{(2\ell+1)!!} \sqrt{\frac{(\ell+1)(\ell+2)}{2\ell(\ell-1)}} U_L \bar{Y}_L^{\ell m}, \qquad (26a) $$

$$ V_{\ell m} = \frac{-32\pi \ell}{(\ell+1)(2\ell+1)!!} \sqrt{\frac{(\ell+1)(\ell+2)}{2\ell(\ell-1)}} V_L \bar{Y}_L^{\ell m}. \qquad (26b) $$
### 3.3. Multipole expansion of the precession

#### 3.3.1. Linear precession rate.
Expanding (15) in terms of radiative multipoles (25), and using the property (50) of the $\partial$ operator, we find

$$ \hat{\Omega}_{(S)} = -\frac{G}{4c^2} \sum_{\ell \ge 2, m} \sqrt{\frac{(\ell+2)!}{2(\ell-2)!}} \frac{V_{\ell m}}{c^\ell} Y^{\ell m}, \quad (27) $$

We observe that the linear precession is purely determined in terms of current multipoles.

#### 3.3.2. Nonlinear precession rate.
The nonlinear precession rate is proportional to the imaginary part of $\dot{\overline{CC}} = \sum_{\ell_1 m_1} \sum_{\ell_2 m_2} \dot{C}_{\ell_1 m_1} C^*_{\ell_2 m_2} - 2Y^{\ell_1 m_1}_1 Y^{\ell_2 m_2}$. Being of spin-weight 0, the latter can be expanded in terms of ordinary spherical harmonics, using the orthogonality property of spin-weighted harmonics (51a). One finds

$$ \dot{C} \overline{C} = \sum_{\ell_1 \ge 2, m_1} \sum_{\ell_2 \ge 2, m_2} \sum_{\ell \ge 0, m} G_{\ell m}^{\ell_1 m_1, \ell_2 m_2} \dot{C}_{\ell_1 m_1} C_{\ell_2 m_2}^* Y^{\ell m}, \quad (28) $$

where

$$ G_{\ell m}^{\ell_1 m_1, \ell_2 m_2} = (-1)^m \sqrt{\frac{(2\ell_1 + 1)(2\ell_2 + 1)(2\ell + 1)}{4\pi}} \times \begin{pmatrix} \ell_1 & \ell_2 & \ell \\ m_1 & m_2 & -m \end{pmatrix} \begin{pmatrix} \ell_1 & \ell_2 & \ell \\ 2 & -2 & 0 \end{pmatrix}. \quad (29) $$

Therefore, $\hat{\Omega}_{(H)}$ takes the form $\hat{\Omega}_{(H)} = \sum_{\ell \ge 0, m} \hat{\Omega}_{\ell m}^{(H)} Y^{\ell m}$ with

$$ \hat{\Omega}_{\ell m}^{(H)} = \frac{1}{8i} \sum_{\ell_1 \ge 2, m_1} \sum_{\ell_2 \ge 2, m_2} G_{\ell m}^{\ell_1 m_1, \ell_2 m_2} (\dot{C}_{\ell_1 m_1} C_{\ell_2 m_2}^* - C_{\ell_1 m_1} \dot{C}_{\ell_2 m_2}^*). \quad (30) $$

Note that by the triangular property of the $3j$ symbols, the above is nonvanishing only for $|\ell_1 - \ell_2| \le \ell \le \ell_1 + \ell_2$ and $|m| \le \ell$. Now, using (25) in the above result, we can express the hard precession rate in terms of radiative multipole moments as

$$ \hat{\Omega}_{\ell m}^{(H)}(u) = \frac{G^2}{16ic^4} \sum_{\ell_1, m_1} \sum_{\ell_2, m_2} c^{-(\ell_1+\ell_2)} G_{\ell m}^{\ell_1 m_1, \ell_2 m_2} \times \left[ \begin{aligned} & \left( \dot{U}_{\ell_1 m_1} U_{\ell_2 m_2} + \frac{1}{c^2} \dot{V}_{\ell_1 m_1} V_{\ell_2 m_2} \right) (1-(-1)^{\ell_1+\ell_2+\ell}) \\ & + \frac{i}{c} \left( \dot{U}_{\ell_1 m_1} V_{\ell_2 m_2} - \dot{V}_{\ell_1 m_1} U_{\ell_2 m_2} \right) (1+(-1)^{\ell_1+\ell_2+\ell}) \end{aligned} \right]. \quad (31) $$

In this result, the time dependence is encoded in the radiative multipoles, and the PN order is made explicit by the factors of $1/c$ in the result. The dominant effect is given by the $\dot{U}U$ term with $\ell_1 = 2 = \ell_2$, since there exist odd multipolar orders $\ell = 1, 3$ for which the prefactor $1 - (-1)^{\ell_1+\ell_2+\ell}$ is nonvanishing, and the triangular condition $0 \le \ell \le 4$ is satisfied.

An alternative expression for $\hat{\Omega}_{(H)}$, which makes the PN order more explicit, is obtained by permuting the sums in such a way that those over $\ell$ and $m$ become the outermost ones. The domain of variation of the dummy indices are to be modified accordingly. It is also convenient
<table><caption>**Table 1.** This table shows the smallest power of $1/c$ that appears in each type of term contributing to $\hat{\Omega}_{\ell m}^{(H)}$ in equations (31) and (32), for given multipolarity $\ell$.</caption><thead><tr><th>Type</th><th>ℓ = 0</th><th>ℓ = 1</th><th>ℓ = 2</th><th>ℓ = 3</th><th>ℓ ≥ 4</th></tr></thead><tbody><tr><td>UU</td><td>—</td><td>8</td><td>9</td><td>8</td><td>ℓ + 5</td></tr><tr><td>VV</td><td>—</td><td>10</td><td>11</td><td>10</td><td>ℓ + 7</td></tr><tr><td>UV</td><td>9</td><td>10</td><td>9</td><td>10</td><td>ℓ + 5</td></tr></tbody></table>

<table><caption>**Table 2.** In this table, we display the smallest power of $1/c$ that appears in each type of term contributing to $\hat{\Omega}_{(H)}$, as well as the minimum and maximum multipolarities for each type of term. See equations (31) and (32).</caption><thead><tr><th rowspan="2">Types of terms</th><th colspan="3">n even, ℓ odd</th><th colspan="3">n odd, ℓ even</th></tr><tr><th>UU</th><th>VV</th><th>UV</th><th>UU</th><th>VV</th><th>UV</th></tr></thead><tbody><tr><td>n<sub>min</sub></td><td>8</td><td>10</td><td>10</td><td>9</td><td>11</td><td>9</td></tr><tr><td>ℓ<sub>min</sub></td><td>1</td><td>1</td><td>1</td><td>2</td><td>2</td><td>0</td></tr><tr><td>ℓ<sub>max</sub></td><td>n − 5</td><td>n − 7</td><td>n − 5</td><td>n − 5</td><td>n − 7</td><td>n − 5</td></tr></tbody></table>

to denote $\ell' = \ell_1$ and eliminate $\ell_2$ in terms of the non-negative integer $p$ such $\ell_1 + \ell_2 + \ell = 2p$ if $\ell_1 + \ell_2 + \ell$ is even, and $\ell_1 + \ell_2 + \ell = 2p + 1$ if $\ell_1 + \ell_2 + \ell$ is odd. This yields

$$ \hat{\Omega}_{(H)} = \sum_{\ell \ge 0, m} \hat{\Omega}_{\ell m}^{(H)} Y^{\ell m}, \quad \text{with} $$

$$ \begin{aligned} \hat{\Omega}_{\ell m}^{(H)} = & \frac{G^2}{8i c^7} \sum_{\ell'=2}^{+\infty} \left[ \sum_{p=0}^{\min(\ell, \ell')-1-\delta_{\ell\ell'}} \frac{1}{c^{\ell+2(\ell'-2)-2p}} \sum_{m'=-\ell'+\max(0,m-\ell+2p+1)}^{\ell+\min(0,m+\ell-2p-1)} G_{\ell m}^{\ell'm', \ell+\ell'-2pm-m'} \right. \\ & \left. \times [\dot{U}_{\ell'm'} U_{\ell+\ell'-2p-1m-m'} + \frac{1}{c^2} \dot{V}_{\ell'm'} V_{\ell+\ell'-2p-1m-m'}] \right. \\ & + \sum_{p=0}^{\min(\ell, \ell')-\delta_{|\ell-\ell'|\le 1}} \frac{i}{c^{\ell+2(\ell'-2)-2p+2}} \sum_{m'=-\ell'+\max(0,m-\ell+2p)}^{\ell+\min(0,m+\ell-2p)} G_{\ell m}^{\ell'm', \ell+\ell'-2pm-1} \\ & \left. \times [\dot{U}_{\ell'm'} V_{\ell+\ell'-2pm-m'} - \dot{V}_{\ell'm'} U_{\ell+\ell'-2pm-m'}]. \right] \end{aligned} \quad (32) $$

From those two expressions of $\hat{\Omega}_{(H)}$, it is straightforward to figure out the leading PN order of the various contributions to $\hat{\Omega}_{\ell m}^{(H)}$ for a given value of $\ell$ i.e. including terms proportional to the product of two mass multipole moments denoted as UU, terms made of the product of two current multipole moments, denoted as VV and the mixed UV contributions depending on both types of moments. The various leading orders are summarized in tables 1 and 2.

It is also useful to know the set of multipolar orders that appear at a given PN order. In table 2, we display for each type of terms its leading PN order, as well as the minimum and maximum multipolarities where it arises.
### 3.4. Total helicity flux

We can compute the total helicity flux by integrating (31) over the sphere. We will see that this quantity vanishes for planar orbits. Therefore a net helicity flux is only possible for binary systems of spinning objects.

Integrating (31) over the sphere kills all spherical harmonics except $\ell = 0$. The triangular property of the 3j-symbols then implies that $\ell_1 = \ell_2$. As a result, $\ell_1 + \ell_2 + \ell$ is even, which kills $UU, VV$ terms. Also, $\begin{pmatrix} \ell & \ell & 0 \\ m & -m & 0 \end{pmatrix} = (-1)^{\ell+m}/\sqrt{2\ell+1}$, which implies that $G_{00}^{2m_1, 2m_2} = (-1)^m/\sqrt{4\pi}$. Using these, we are left with

$$ \int_{S^2} d^2\Omega \hat{\Omega}_{(H)} = \frac{G^2}{8c^5} \sum_{l \ge 2, m} \frac{(-1)^m}{c^{2\ell}} (\dot{U}_{lm}V_{l-m} - \dot{V}_{lm}U_{l-m}). \quad (33) $$

For planar orbits, it can be proven in general that [52]

$$ \begin{aligned} U_{lm} &= 0, & \ell + m &= \text{odd}, \\ V_{lm} &= 0, & \ell + m &= \text{even}. \end{aligned} \quad (34) $$

As a result, the total helicity flux vanishes for planar orbits, no matter the orbit is bound or unbound. We conclude that in order to have a total helicity flux from binary systems, we need to have spinning companions.

## 4. PN sources

In this section, we will compute the gyroscopic memory sourced by simple binary systems. To this end, we will use available analytic expressions for $C_{AB}$ [53], which have been derived within the PN/MPM formalism [54]. The following subsection provides some background material on the dynamics of quasi-circular binaries as well as the PN/MPM formalism and its application to those systems. For practical purposes, the reader can skip these discussions and jump to equations (36), displaying the leading radiative multipole moments of the source under consideration. This is essentially what we need for the rest of the paper.

### 4.1. Quasi-circular binary system

#### 4.1.1. PN dynamics of spinless compact binaries.
In Newtonian gravity, the dynamics of two pointlike masses $m_1, m_2$ is a solvable model, integrable in the sense of Liouville. In the center-of-mass frame, the problem reduces to the motion of a body of reduced mass $\mu = m_1m_2/M$ with $M = m_1 + m_2$. Generic solutions are uniquely determined by three constants of motion: energy $E$, angular momentum $J_i$, and the direction of the Laplace–Runge–Lenz vector $A^i$ in the orbital plane.

At higher PN orders, the Lagrangian for the conservative dynamics of point masses is derived by integrating out the gravitational degrees of freedom. A clear distinction must be made between those referring to the near zone (surrounding the matter source but of size much smaller than the wavelength) and those referring to the exterior zone (excluding the matter source). The evaluation of the functional integral whose logarithm essentially gives the reduced action may be achieved either by means of the saddle point method, as in Fokker's approach, or, alternatively à la Feynman, by expanding the integral around a Gaussian kernel
and applying Wick's theorem [55] (see [14, 54, 56] for recent reviews of this type of computation). In the case of spinless objects, the integrability of the resulting Hamiltonian is preserved at higher PN orders, in GR, still neglecting dissipative effects due to gravitational radiation (see e.g. [57] for a Hamiltonian analysis at the 3PN order). Bound spinless binaries, notably, follow planar quasi-elliptic trajectories subject to periastron advance. When taking radiation into account, both the eccentricity and the separation monotonically decrease, so that, in the last stage of its history, the system becomes quasi-circular and shrinks, up to the merger of the two companions.

In the constant plane of the motion, the binary orbits are then quasi-circular and inspiraling. They are most conveniently described in polar coordinates $\rho, \psi = \int dt \omega(t)$, which are not independent, but are related, instead, through a relativistic version of Kepler's law, $GM/\rho^3 = \omega^2[1 + O(1/c^2)]$. The remainder $O(1/c^2)$ entering the right-hand side of this relation, in the PN framework, takes the form of an asymptotic series in the so-called PN parameter $x \equiv (GM\omega/c^3)^{2/3}$. This quantity is dimensionless, of order $1/c^2$, and is linked to the coordinate velocity $v^i$ through $x = v^2/c^2 + O(1/c^4)$. However, unlike $v^2$, the parameter $x$ is an adiabatic invariant [57, 58], defined independently of the particular PN coordinates that are adopted to describe the dynamics.

Now, due to the shrinking of the orbits, the separation $\rho$ is not exactly constant, nor is the orbital frequency $\omega$, but they both depend adiabatically on time. Their dependence may be calculated from the energy flux-balance equation [51, 59], implying that $\dot{\rho} = O(1/c^5)$, and, by virtue of PN Kepler's law, $\dot{x} = O(1/c^5)$. More precisely, the evolution of $x = x(u)$ at leading order is given by [59]

$$ \dot{x} = \frac{64}{5} \frac{c^3}{GM^2} \mu x^5 [1 + O(x)] . \qquad (35) $$

The corrections $O(x)$ are currently known up to the 4PN order [60]. This approximation starts to break down past the innermost stable circular orbit (ISCO), after which, typically, numerical methods are required to describe the binary evolution.

### 4.1.2. Multipolar post-Minkowskian formalism.
At high orders, the waveform and fluxes are computed using the MPM formalism in harmonic coordinates. It involves three main tasks [54]: (i) Constructing perturbatively the general solution to vacuum Einstein's equations outside the matter, using a post-Minkowskian expansion in formal powers of the gravitational constant $G$; the perturbation of each order is expressed as a multipole expansion, depending on six (yet) unspecified sets of multipole moments. (ii) Solving Einstein's equation in the near zone as a PN expansion in formal powers of $1/c$ up to a homogeneous solution at each order. (iii) By means of asymptotic matching techniques, comparing the multipole expansion of the near-zone gravitational field with the near-zone expansion of the exterior field; this matching condition determines for once the full formal solution, including integral expressions for the multipole moments in terms of the near-zone stress-energy pseudo-tensor, and the homogeneous near-zone solutions in terms of functional integrals of the multipole moments.

To read off the waveform, one has to kill terms proportional to $\ln^q r/r$ (with $q$ being any positive integer) by moving to (asymptotically or exact) radiative coordinates [61–63]. The radiative multipoles $U_L$ and $V_L$ are then obtained as functionals of the six MPM moments, whose explicit expressions are known in terms of the source, as explained earlier. For inspiraling binaries of spinless compact objects, they are given in terms of the PN parameter $x(u)$ and the phase $\phi(u)$ (see [54] for details).
### 4.1.3. PN radiative moments of compact binaries.
As this point, we consider a Cartesian-like grid $(t, x^i)$ in harmonic coordinates, with its origin at the center of mass of the binary and its $z$ axis aligned with the angular momentum. In the corresponding spherical coordinates, the observation point is given by $(\tilde{r}, \theta, \phi)$ while the reduced mass is specified by $(\rho(t), \pi/2, \psi(t))$. The shear in Bondi gauge, used in the previous sections, can be deduced with the help of the method of [62, 63] from the result obtained by means of the aforementioned procedure from tasks (i) to (iii). Up to $O(1/r)$ corrections, as well as a free BMS transformation, Bondi angular coordinates coincide with the harmonic spherical coordinates, while $u = t - \tilde{r} - 2GM_{\text{ADM}}/c^2 \log \tilde{r} + o(1)$ and $r = \tilde{r} + GM/c^2 + o(1)$, with $M_{\text{ADM}}$ being the Arnowitt–Deser–Misner (ADM) mass of the system.

On the other hand, to fix the phase of the complex shear $C$ in equation (12), we must choose a specific vector $m^A$, or, equivalently, the polarization basis $E_1^A, E_2^A$ on the unit sphere. We will resort here to the usual coordinate basis of spherical coordinates $E_1 \equiv E_1^A \partial_A = \partial_\theta$ and $E_2 = (\sin\theta)^{-1} \partial_\phi$. With this convention, $U_L$ and $V_L$ introduced in the multipole expansion (23) are the same as in the PN literature [51]. A similar statement holds for the spherical multipole moments $U_{\ell m}$ and $V_{\ell m}$.

The radiative multipoles required for the purpose of this paper then read

$$U_{22} = -8\sqrt{\frac{2\pi}{5}}Mc^2 \nu x e^{-2i\psi}, \qquad U_{20} = \frac{4}{7}\sqrt{\frac{5\pi}{3}}Mc^2 \nu x, \qquad (36a)$$

$$V_{21} = \frac{8}{3}\sqrt{\frac{2\pi}{5}}\delta Mc^3 \nu x^{3/2} e^{-i\psi}, \qquad V_{3,0} = -\frac{32}{5}\sqrt{\frac{3\pi}{35}}Mc^4 \nu^2 x^{7/2}. \qquad (36b)$$

where $\delta = (m_1 - m_2)/M$, $\nu = \mu/M$. A complete list can be inferred from [53]$^4$. We present the results in terms of the phase of the waveform defined as

$$\varphi(u) = \psi(u) - \phi + \frac{\pi}{2} + \mathcal{O}\left(\frac{1}{c^3}\right). \qquad (37)$$

## 4.2. PN expansion of the precession rates

### 4.2.1. Leading effects.
The leading effect in (27) is given by

$$\hat{\Omega}_{(S)} = -\frac{G\sqrt{3}}{2c^4} \left[ \sum_{|m| \le 2} V_{2m} Y^{2m} + \sqrt{5} \sum_{|m| \le 3} V_{3m} Y^{3m} \right] + \mathcal{O}\left(\frac{1}{c^6}\right), \qquad (38)$$

For the nonlinear precession, as clear from (31), the leading effect originates from $\ell_1 = \ell_2 = 2$. This entails a 4PN effect from UU terms, a 4.5PN effect from UV terms, and a 5PN effect from VV terms. Focusing on the leading order contribution, we note that the factor $(1 - (-1)^{\ell_1+\ell_2+\ell})$, implies that $\ell = 1, 3$, and thus

$$\hat{\Omega}_{(H)} = \frac{G^2}{16i c^8} \sum_{|m_1| \le 2} \sum_{|m_2| \le 2} \sum_{\ell=1,3} G^{2m_1, 2m_2} Y^{\ell m} \dot{U}_{2m_1} U_{2m_2} + \mathcal{O}\left(\frac{1}{c^9}\right), \qquad (39)$$

where $G^{2m_1, 2m_2}$ is non zero for $m = m_1 + m_2 \le \ell$ due to properties of 3j-symbols.

<footnote>$^4$Note however that, due to a different choice of polarization vectors, the quantity $r(h_+ - ih_\times)$ in [53] differs from the complex shear $C$ as defined in the present paper by a minus sign (neglecting $O(1/r)$ corrections).</footnote>
### 4.2.2. Axisymmetric mode.
In the previous results, the axisymmetric mode, i.e. $m=0$ is of special status. The reason is that once the result is matched to the source, it takes the form $A(t)\exp(im\phi(t))$, in terms of an adiabatic amplitude $A(t)$ and a fast oscillating phase $m\phi(t)$. The axisymmetric case $m=0$ is special, since it a DC effect, which accumulates over time and builds the leading contribution in the gyroscopic memory (10). In fact, we will show that the time integral of an axisymmetric mode leads to a $c^5$ enhancement with respect to the integrand. Restricting our attention to axisymmetric modes in (39) by setting $m = m_1 + m_2 = 0$, we find

$$ \hat{\Omega}_{(H)}^{m=0} = 7 = \frac{5G^2}{16i c^8} \sum_{\ell=1,3} \sqrt{\frac{2\ell+1}{4\pi}} \begin{pmatrix} 2 & 2 & \ell \\ 2 & -2 & 0 \end{pmatrix} Y^{\ell 0} \sum_{|m_1| \le 2} \begin{pmatrix} 2 & 2 & \ell \\ m_1 & -m_1 & 0 \end{pmatrix} \dot{U}_{2m_1} U_{2-m_1} + \mathcal{O}\left(\frac{1}{c^9}\right). $$

In the sum over $m_1$, the $3j$ symbol involving $m_1 = 0$ vanishes since $\ell$ is odd. The positive and negative $m$ then combine into

$$ \hat{\Omega}_{(H)}^{m=0} = \frac{5G^2}{8c^8} \sum_{\ell=1,3} \frac{2\ell+1}{4\pi} \begin{pmatrix} 2 & 2 & \ell \\ 2 & -2 & 0 \end{pmatrix} P_{\ell}(\cos\theta) \sum_{m_1=1,2} \begin{pmatrix} 2 & 2 & \ell \\ m_1 & -m_1 & 0 \end{pmatrix} \mathrm{Im}(\dot{U}_{2m_1} U_{2-m_1}) + \mathcal{O}\left(\frac{1}{c^9}\right), \quad (40) $$

where $P_\ell(x)$ is the associated Legendre polynomial.

### 4.3. Precession and memory from quasi-circular binary systems

For planar binary systems, the property (34) of radiative multipoles can be used to simplify the precession rates (38) and (39). In the particular case of quasi-circular binary systems, the leading-order soft precession, as an expansion in powers of the PN parameter $x$ is obtained by inserting (36) in (38). The result is

$$ \hat{\Omega}_{(S)} = \frac{2GM\delta\nu}{c} x^{3/2} \sin\varphi \sin(2\theta) + \alpha \sin 2\varphi x^2 + (\beta_1 \sin\varphi + \beta_3 \sin 3\varphi) x^{5/2} + \frac{3}{5} \frac{GM\nu^2}{c} x^{7/2} (5\cos(3\theta) + 3\cos\theta) + \mathcal{O}(x^4), \quad (41) $$

where $\alpha, \beta_1, \beta_3$ are known functions of $\theta$ and $\nu$, which are irrelevant for our leading order analysis, as will become clear below. On the other hand, the leading term in the hard precession is

$$ \hat{\Omega}_{(H)} = \frac{GM\nu^2}{c} x^{7/2} \left[ \frac{5}{28} \cos(2\varphi)(\cos(3\theta) - \cos\theta) - (\cos(3\theta) + 7\cos\theta) \right] + \mathcal{O}(c^{-9}). \quad (42) $$

We observe that the soft precession is dominant over the hard precession: the former is a 2PN effect (~$c^{-4}$), while the latter is a 4PN effect. Moreover, we note that both effects contain fast modes $m \ne 0$, and adiabatic modes $m=0$. The former depends on time through both $x$
<figure><img src="image_5.png" /><figcaption>Figure 1. The leading axisymmetric contributions to the precession rate, given in (43) as a function of the polar angle on the celestial sphere (with respect to the axis of rotation of the binary). The leading memory effects have similar angular dependence, see (44).</figcaption></figure>

and $\phi$, while the latter depends only on the slow variable $x$. Quite interestingly, the adiabatic modes in soft and hard precession rates are of the same order of magnitude, given in terms of $\Omega_0 \equiv GM\nu^2/c$, as

$$ \hat{\Omega}_{(H)}^{m=0} = -\Omega_0 x^{7/2} (\cos(3\theta) + 7\cos\theta) + \mathcal{O}(c^{-9}), \quad (43a) $$

$$ \hat{\Omega}_{(S)}^{m=0} = \frac{3}{5} \Omega_0 x^{7/2} (5\cos(3\theta) + 3\cos\theta) + \mathcal{O}(c^{-9}). \quad (43b) $$

The leading memory effects are obtained by integrating precession rates over time. Such integrals are investigated in appendix B. The main results are equations (55) and (57), indicating that there is a $c^5$ enhancement for axisymmetric (adiabatic) modes, while the time integration does not affect the PN order for non-axisymmetric modes. Therefore, while the leading term in the hard memory is the time integral of the leading term in (42), the leading soft memory originates from a very subleading axisymmetric contribution, namely, the last term in (41). The leading contribution to the memory effects (10) read

$$ \Delta \hat{\Psi}_{(H)} = \Psi_0 (\cos(3\theta) + 7\cos\theta) \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x_0}} \right) + \mathcal{O}(c^{-4}), \quad (44a) $$

$$ \Delta \hat{\Psi}_{(S)} = -\frac{3}{5} \Psi_0 (5 \cos(3\theta) + 3 \cos\theta) \left( \frac{1}{\sqrt{x}} - \frac{1}{\sqrt{x_0}} \right) + \mathcal{O}(c^{-4}). \quad (44b) $$

The order of magnitude of the memory is given by $\Psi_0 \equiv (5G^2M^2\nu)/(32c^4)$, which is 2PN. The leading gyroscopic memory is however enhanced to 1.5PN order due to the accumulation factor $(1/\sqrt{x} - 1/\sqrt{x_0})$. The angular dependence of the celestial sphere is the same as (43), depicted in figure 1.

This equation shows the accumulated memory effect from an initial time $u_0$ at which $x(u_0) = x_0$, to the current time $u$ with $x = x(u)$. Given that the quasi-circular model breaks down beyond the ISCO, one has to restrict to the regime where $x < x_{\text{ISCO}} = 1/6$. On the other hand, this model cannot be extended to the far past. Binaries are formed over time, instead, through mechanisms such as dynamic capture, or the fragmentation of astrophysical clouds. Here, we must think of quasi-circular evolutions as resulting from an initial value problem,
where the binary is circular at initial time $u_0$. Moreover, note that the Blanchet–Damour formalism typically assumes a ‘past-stationarity’ condition, i.e. that gravitational fields are stationary before an initial time $u < u_0$. A more careful analysis of IR divergences may be dealt with in a scattering setup, see e.g. [64, 65] for related works. In this sense, $0 < x_0 \ll 1$ can be thought of as an infrared cutoff.

## 5. Concluding remarks

Let us briefly summarize the main results of this work and point out some future directions, which we plan to explore. It was showed in [23] that GWs from a binary system lead to a small change in the orientation of a distant gyroscope. There, the effect was estimated to be of order $G^2M^2/(r^2c^4)$, using dimensional analysis. We observe from equation (44) that dimensional analysis had correctly captured the leading result, except for the dimensionless accumulation factor $(1/\sqrt{x} - 1/\sqrt{x_0})$ and the dimensionless symmetric mass ratio $\nu$. If the gyroscope is subject to long-lasting GW signal, the accumulation effect can lead to a large enhancement. At the same time, we note that the result is suppressed for extreme mass-ratio inspirals, for which $\nu \ll 1$. In any case, it is still very small overall, due to the $1/r^2$ factor, and thus unlikely to be observed in a realistic experiment, unless the gravitational radiation source turns out to be extremely close to the spinning body.

On the other hand, in this paper, we have focused on the effect of GWs on a single gyroscope. Another scenario involves measuring the effect across a network of gyroscopes distributed on the celestial sphere. For example, we could think of pulsars surrounding a binary system at short distance, although this does not seem to be very realistic. In this case, we could study the correlation among the gyroscopes accordingly, and contrast it with the result (44).

Another interesting issue is to study more carefully the gyroscopic memory when the test gyroscope is highly spinning. In this case, the parallel transport approximation fails, so that higher-order-in-spin effects must be taken into account, as outlined in section 2. These could be of relevance for the most rapidly spinning neutron stars, as probes of GWs.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

AS thanks Adam Pound and Simone Speziale for useful discussions. GF thanks Geoffrey Compère and Université Libre de Bruxelles for supporting a visit related to this project. This research was supported in part by the Université Libre de Bruxelles and the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities.
# Appendix A. Spin weighted functions on sphere

## A.1. Concept of spin weight, notations

In this appendix, we recall how to define spin-weighted quantities associated to a null dyad on a two-dimensional sphere, while making explicit the notations and conventions used throughout the article. Our generic dyad will be denoted as $m^A$, $\bar{m}^A$ and normalized in such a way that $m^A m_A = 0 = \bar{m}^A \bar{m}_A$ and $m^A \bar{m}_A = 1$. Inverting these relations allows finding the expression (11) of $m_A \bar{m}_B$ in terms of the tensors $\gamma_{AB}$, $\epsilon_{AB}$. Under a $U(1)$ rotation in the tangent space of a given point on the sphere, the dyad transforms as

$$m^A \rightarrow e^{i\theta} m^A, \quad \bar{m} \rightarrow e^{-i\theta} \bar{m}. \qquad (45)$$

For any tensor field $T$ of rank $p+q$, we may construct spin-weighted functions by contraction with basis vectors $m^A$ and $\bar{m}^A$, e.g.

$$T_s \equiv T_{A_1 \dots A_p B_1 \dots B_q} m^{A_1} \dots m^{A_p} \bar{m}^{B_1} \dots \bar{m}^{B_q} \qquad (46)$$

with $p$ factors of $m^A$ and $q$ factors of $\bar{m}^A$. The complex function $T_s$ has spin-weight $s = p-q$, as it transforms under dyad rotations as $T_s \rightarrow e^{is\theta} T_s$ [50]. We note that, using (11) in (46), mutual pairs of factors $m^A \bar{m}^B$ may be traded for the metric and the epsilon tensor. Accordingly, the tensor $T$ in (46) is reduced into its irreducible representations under the rotation group. In particular, a STF tensor $\hat{T}$ on the sphere has only two degrees of freedom, which are encoded in the single complex function $T_s = \hat{T}_{A_1 \dots A_s} m^{A_1} \dots m^{A_s}$. (The function $T_s$ of spin weight $-s$ obtained by contracting all indices with factors of $\bar{m}^B$ is not independent of $T_s$, as they are complex conjugates.)

In this language, the covariant derivative acting on tensors on the sphere is replaced by the so-called eth $\partial$ derivative such that [50]

$$\partial\bar{T}_s = m^{A_1} \dots m^{A_p} \bar{m}^{B_1} \dots \bar{m}^{B_q} m^C D_C T_{A_1 \dots A_p B_1 \dots B_q}, \qquad (47)$$

while $\partial\bar{T}_s$ is defined by conjugating the vector that contracts with the derivative, i.e. $m^C \rightarrow \bar{m}^C$ in the above equation. By construction, $\partial(\bar{\partial})$ increases (decreases) the spin-weight by $+1$ ($-1$). The normalization of $\partial$ in equation (47) differs from the one of [50, 66], for which there would be an extra factor $-\sqrt{2}$ in the right-hand side of (47).

## A.2. Spin-weighted harmonics

In this context, spin-weighted harmonics $s Y^{\ell m}$ [50, 66] provide a natural basis for spin-weighted functions on the sphere. They are defined in terms of standard harmonics $Y^{\ell m} \equiv {}_0Y^{\ell m}$. For a positive integer $s$,

$$s Y^{\ell m} \equiv (-\sqrt{2})^s \sqrt{\frac{(\ell - s)!}{(\ell + s)!}} \partial^s Y^{\ell m}, \quad -s Y^{\ell m} \equiv (\sqrt{2})^s \sqrt{\frac{(\ell - s)!}{(\ell + s)!}} \bar{\partial}^s Y^{\ell m}, \qquad (48)$$

and the spin-weighted harmonics vanish when $|s| > \ell$. The Condon-Shortley phase is chosen as in [51], hence a factor $(-1)^m$ with respect to [66]. The definition implies several important properties of spin-weighted harmonics. Under conjugation, it satisfies

$$s \bar{Y}^{\ell m} = (-1)^{m+s} s Y^{\ell-m}. \qquad (49)$$
Moreover, for any non-negative $s \le l$, the operators $\partial$ and $\bar{\partial}$ act as

$$ \partial_s Y^{lm} = -\sqrt{\frac{(l-s)(l+s+1)}{2}} Y^{lm}, \quad \bar{\partial}_s Y^{lm} = \sqrt{\frac{(l+s)(l-s+1)}{2}} Y^{lm}, \quad (50) $$

respectively. Finally, as a complete basis, the spin-weighted harmonics (48) obey orthogonality relations given in terms of the 3j symbols:

$$ \int_{S^2} d^2\Omega_s Y^{l_1 m_1} Y^{l_2 m_2} = \delta_{l_1 l_2} \delta_{m_1 m_2}, \quad (51a) $$

$$ \int_{S^2} d^2\Omega_{s_1} Y^{l_1 m_1} Y^{l_2 m_2} Y^{lm} = \sqrt{\frac{(2l_1+1)(2l_2+1)(2l+1)}{4\pi}} \times \begin{pmatrix} l_1 & l_2 & l \\ m_1 & m_2 & m \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l \\ -s_1 & -s_2 & -s \end{pmatrix}, \quad (51b) $$

where $d^2\Omega$ denotes the element of two-dimensional solid angle.

## Appendix B. Time integration and memory effect

To compute the leading order contribution to the gyroscopic memory, we need to integrate the precession rates (41) and (42) over time. These integrals take the form

$$ I_{\alpha,m}(u) = \int_{-\infty}^{u} dx (t)^{\alpha} e^{im\varphi(t)}. \quad (52) $$

Note that $x(t)$ is an adiabatic variable, while $\phi(t)$ is a fast oscillatory variable. Therefore the case $m=0$ in (52) is a special case which reads

$$ I_{\alpha,0}(u) = \int_{-\infty}^{u} dx x^{\alpha} = \int_{-\infty}^{u} dx \frac{x^{\alpha}}{\dot{x}}. \quad (53) $$

The time evolution of $x$ is obtained by the flux-balance equation for energy [59], implying that

$$ \dot{x} = \frac{64 c^3 \nu}{5 GM} x^5 \left(1 + \mathcal{O}\left(\frac{1}{c^2}\right)\right). \quad (54) $$

This equation enables the computation of axisymmetric mode integrals, when inserted into (53) for $m=0$, which yields

$$ I_{\alpha,0}(u) = \frac{5 GM}{64 c^3 \nu \alpha - 4} x^{\alpha-4} \left(1 + \mathcal{O}\left(\frac{1}{c^2}\right)\right), \quad \alpha \neq 4. \quad (55) $$

On the other hand, the case $m \ne 0$ can be processed by the following replacement in (52)

$$ e^{im\varphi} = \frac{1}{im} \frac{\partial}{\partial\varphi} e^{im\varphi} = \frac{1}{im\omega} \frac{\partial}{\partial t} e^{im\varphi} = \frac{GM}{c^3} \frac{x^{-3/2}}{im} \frac{\partial}{\partial t} e^{im\varphi}, $$

and performing integration by parts to move time derivatives from the fast variable to the slow variable. We thus find

$$ I_{\alpha,m}(u) = \frac{GM}{c^3} \frac{1}{im} \left[ x^{\alpha-3/2} e^{im\varphi} \Big|_{-\infty}^{u} - \int_{-\infty}^{u} dt e^{im\varphi} \frac{d}{dt} \left( x^{\alpha-3/2} \right) \right]. \quad (56) $$
Assuming that $\alpha > 3/2$, the first term vanishes at the lower bound and the integral localizes to an instantaneous expression (this does not hold for $\alpha = 3/2$ which we come back to later). Moreover, using (54), the integrand of the integral in (56) is $d(x^{\alpha-3/2})/dt = (\alpha - 3/2)(64/5)(c^3\nu)/(GM)x^{\alpha+5/2}$, and the integral takes the same form as (52). We can therefore repeat the above algorithm, which implies

$$ I_{\alpha,m}(u) = \frac{GM}{c^3} \frac{e^{im\varphi(u)}}{im} x(u)^{\alpha-3/2} \left[ 1 - (\alpha - 3/2) \frac{64}{5} \frac{\nu}{im} x^{5/2} + \dots \right], \quad (57) \\ \times \alpha > \frac{3}{2}, m \neq 0. $$

We observe that the result is a perturbative PN expansion, where the first correction is 2.5PN subleading with respect to the leading term. Therefore, for the purpose of our work, only the leading term is sufficient. Subleading terms can be found by repeating the above procedure to reach the desired order.

For $\alpha \le 4$ in (55) and $\alpha \le 3/2$ in (57), the integrals are not convergent. However, the quasi-circular orbit cannot be trusted in the far past, as we discussed below (44). Therefore, we regulate the integrals by modifying the lower bound from $-\infty$ to $u_0$, which can be a large but finite negative constant.

## ORCID iDs

Guillaume Faye https://orcid.org/0000-0003-2921-1525

Ali Seraj https://orcid.org/0000-0002-8923-0077

## References

[1] Misner C W, Thorne K S and Wheeler J A 1973 *Gravitation* (W. H. Freeman)

[2] Brito R, Cardoso V and Pani P 2015 *Superradiance: New Frontiers in Black Hole Physics* (Lecture Notes in Physics vol 906) (Springer) pp 1–237

[3] Blandford R D and Znajek R L 1977 Electromagnetic extractions of energy from Kerr black holes *Mon. Not. R. Astron. Soc.* **179** 433–56

[4] Mathisson M 1937 Neue mechanik materieller systemes *Acta Phys. Polon.* **6** 163–200

[5] Papapetrou A 1951 Spinning test-particles in general relativity. I *Proc. R. Soc. A* **209** 248–58

[6] Tulczyjew W 1959 Equations of motion of rotating bodies in general relativity theory *Acta Phys. Polon.* **18** 37

[7] Dixon W G 1970 Dynamics of extended bodies in general relativity. I. Momentum and angular momentum *Proc. R. Soc. A* **314** 499–527

[8] Dixon W G 1970 Dynamics of extended bodies in general relativity. II. Moments of the charge-current vector *Proc. R. Soc. A* **319** 509–47

[9] Dixon W G 1974 Dynamics of extended bodies in general relativity III. Equations of motion *Phil. Trans. R. Soc. A* **277** 59–119

[10] Ehlers J and Rudolph E 1977 Dynamics of extended bodies in general relativity center-of-mass description and quasirigidity *Gen. Relativ. Gravit.* **8** 197–217

[11] Bailey I and Israel W 1975 Lagrangian dynamics of spinning particles and polarized media in general relativity *Commun. Math. Phys.* **42** 65–82

[12] Tagoshi H, Ohashi A and Owen B J 2001 Gravitational field and equations of motion of spinning compact binaries to 2.5 post-Newtonian order *Phys. Rev. D* **63** 044006

[13] Faye G, Blanchet L and Buonanno A 2006 Higher-order spin effects in the dynamics of compact binaries. I. Equations of motion *Phys. Rev. D* **74** 104033

[14] Porto R A 2016 The effective field theorist’s approach to gravitational dynamics *Phys. Rep.* **633** 1–104
[15] Levi M and Steinhoff J 2015 Spinning gravitating objects in the effective field theory in the post-Newtonian scheme *J. High Energy Phys.* JHEP09(2015)219

[16] Flanagan É É, Grant A M, Harte A I and Nichols D A 2019 Persistent gravitational wave observables: general framework *Phys. Rev. D* **99** 084044

[17] Zel'dovich Y B and Polnarev A G 1974 Radiation of gravitational waves by a cluster of superdense stars *Sov. Astron.* **18** 17

[18] Braginsky V B and Grishchuk L P 1985 Kinematic resonance and memory effect in free mass gravitational antennas *Sov. Phys. - JETP* **62** 427-30

[19] Braginsky V B and Thorne K S 1987 Gravitational-wave bursts with memory and experimental prospects *Nature* **327** 123-5

[20] Christodoulou D 1991 Nonlinear nature of gravitation and gravitational wave experiments *Phys. Rev. Lett.* **67** 1486-9

[21] Blanchet L and Damour T 1992 Hereditary effects in gravitational radiation *Phys. Rev. D* **46** 4304-19

[22] Seraj A and Oblak B 2023 Gyroscopic gravitational memory *J. High Energy Phys.* JHEP11(2023)057

[23] Seraj A and Oblak B 2022 Precession caused by gravitational waves *Phys. Rev. Lett.* **129** 061101

[24] Corinaldesi E and Papapetrou A 1951 Spinning test particles in general relativity. 2 *Proc. R. Soc. A* **209** 259-68

[25] Barker B M and Oconnell R F 1979 The gravitational interaction: spin, rotation and quantum effects—a review *Gen. Relativ. Gravit.* **11** 149-75

[26] Costa L F O and Natário J 2015 Center of mass, spin supplementary conditions and the momentum of spinning particles *Equations of Motion in Relativistic Gravity (Fundamental Theories of Physics)* vol 179) (Springer) pp 215-58

[27] Dixon W G 1964 A covariant multipole formalism for extended test bodies in general relativity *Nuovo Cimento* **34** 317-39

[28] Diu B, Guthmann C, Lederer D and Roulet B 1989 *Physique Statistique* (Hermann)

[29] Marsat S 2015 Cubic order spin effects in the dynamics and gravitational wave energy flux of compact object binaries *Class. Quantum Grav.* **32** 085008

[30] Buonanno A, Faye G and Hinderer T 2013 Spin effects on gravitational waves from inspiraling compact binaries at second post-Newtonian order *Phys. Rev. D* **87** 044009

[31] Goldberger W D and Rothstein I Z 2006 An effective field theory of gravity for extended objects *Phys. Rev. D* **73** 104029

[32] Godazgar H, Godazgar M and Pope C N 2019 Subleading BMS charges and fake news near null infinity *J. High Energy Phys.* JHEP01(2019)143

[33] Grant A M and Nichols D A 2022 Persistent gravitational wave observables: curve deviation in asymptotically flat spacetimes *Phys. Rev. D* **105** 024056
Grant A M and Nichols D A 2023 *Phys. Rev. D* **107** 109902 (erratum)

[34] Compère G, Oliveri R and Seraj A 2022 Metric reconstruction from celestial multipoles *J. High Energy Phys.* JHEP11(2022)001

[35] Freidel L, Oliveri R, Pranzetti D and Speziale S 2021 The Weyl BMS group and Einstein's equations *J. High Energy Phys.* JHEP07(2021)170

[36] Godazgar M, Long G and Seraj A 2022 Gravitational memory effects and higher derivative actions *J. High Energy Phys.* JHEP09(2022)150

[37] Seraj A and Neogi T 2023 Memory effects from holonomies *Phys. Rev. D* **107** 104034

[38] Pasterski S, Strominger A and Zhiboedov A 2016 New gravitational memories *J. High Energy Phys.* JHEP12(2016)053

[39] Nichols D A 2017 Spin memory effect for compact binaries in the post-Newtonian approximation *Phys. Rev. D* **95** 084048

[40] Compère G, Oliveri R and Seraj A 2020 The Poincaré and BMS flux-balance laws with application to binary systems *J. High Energy Phys.* JHEP10(2020)116

[41] Mitman K, Moxon J, Scheel M A, Teukolsky S A, Deppe N, Kidder L E and Throwe W 2020 Computation of normal and spin memory in numerical relativity (arXiv:2007.11562)

[42] Maleknejad A 2023 Photon chiral memory effect stored on celestial sphere *J. High Energy Phys.* JHEP06(2023)193

[43] Liu W-B, Long J and Zhou X-H 2024 Electromagnetic helicity flux operators in higher dimensions (arXiv:2407.20077)
[44] Dong J, Long J and Yu R-Z 2024 Gravitational helicity flux density from two-body systems (arXiv:2403.18627)

[45] Kol U 2022 Duality in Einstein's gravity (arXiv:2205.05752)

[46] Hosseinzadeh V, Seraj A and Sheikh-Jabbari M M 2018 Soft charges and electric-magnetic duality
*J. High Energy Phys.* JHEP08(2018)102

[47] Blanchet L 2014 Gravitational radiation from post-Newtonian sources and inspiralling compact binaries *Living Rev. Relativ.* **17** 2

[48] Bieri L and Garfinkle D 2014 Perturbative and gauge invariant treatment of gravitational wave memory *Phys. Rev. D* **89** 084039

[49] Gel'fand I M, Minlos R A and Shapiro Z Y 2018 *Representations of the Rotation and Lorentz Groups and Their Applications* (Courier Dover Publications)

[50] Newman E T and Penrose R 1966 Note on the Bondi-Metzner-Sachs group *J. Math. Phys.* **7** 863–70

[51] Thorne K S 1980 Multipole expansions of gravitational radiation *Rev. Mod. Phys.* **52** 299–339

[52] Faye G, Marsat S, Blanchet L and Iyer B R 2012 The third and a half post-Newtonian gravitational wave quadrupole mode for quasi-circular inspiralling compact binaries *Class. Quantum Grav.* **29** 175004

[53] Blanchet L, Faye G, Iyer B R and Sinha S 2008 The third post-Newtonian gravitational wave polar- isations and associated spherical harmonic modes for inspiralling compact binaries in quasi- circular orbits *Class. Quantum Grav.* **25** 165003
Blanchet L, Faye G, Iyer B R and Sinha S 2012 *Class. Quantum Grav.* **29** 239501 (erratum)

[54] Blanchet L 2024 Post-Newtonian theory for gravitational waves *Living Rev. Relativ.* **27** 4

[55] Damour T and Jaranowski P 2017 Four-loop static contribution to the gravitational interaction potential of two point masses *Phys. Rev. D* **95** 084005

[56] Schäfer G and Jaranowski P 2018 Hamiltonian formulation of general relativity and post-Newtonian dynamics of compact binaries *Living Rev. Relativ.* **21** 7

[57] Damour T, Jaranowski P and Schäfer G 2000 Poincare invariance in the ADM Hamiltonian approach to the general relativistic two-body problem *Phys. Rev. D* **62** 021501
Damour T, Jaranowski P and Schäfer G 2001 *Phys. Rev. D* **63** 029903 (erratum)

[58] Arnold V I 1989 *Mathematical Methods of Classical Mechanics* (Graduate Texts in Mathematics) (Springer)

[59] Landau L and Lifshitz E 1971 *The Classical Theory of Fields* (Pergamon)

[60] Blanchet L, Faye G, Henry Q, Larrouturou F and Trestini D 2023 Gravitational-wave flux and quad- rupole modes from quasicircular nonspinning compact binaries to the fourth post-Newtonian order *Phys. Rev. D* **108** 064041

[61] Blanchet L 1987 Radiative gravitational fields in general relativity. 2. Asymptotic behaviour at future null infinity *Proc. R. Soc. A* **409** 383–99

[62] Blanchet L, Compère G, Faye G, Oliveri R and Seraj A 2021 Multipole expansion of gravitational waves: from harmonic to Bondi coordinates *J. High Energy Phys.* JHEP02(2021)029

[63] Blanchet L, Compère G, Faye G, Oliveri R and Seraj A 2023 Multipole expansion of gravitational waves: memory effects and Bondi aspects *J. High Energy Phys.* JHEP07(2023)123

[64] Sahoo B and Sen A 2019 Classical and quantum results on logarithmic terms in the soft theorem in four dimensions *J. High Energy Phys.* JHEP02(2019)086

[65] Kehrberger L 2023 Mathematical studies on the asymptotic behaviour of gravitational radiation in general relativity *PhD Thesis* Department of Applied Mathematics And Theoretical Physics, University of Cambridge

[66] Goldberg J N, MacFarlane A J, Newman E T, Rohrlich F and Sudarshan E C G 1967 Spin spherical harmonics and edth *J. Math. Phys.* **8** 2155