Gyroscopic Gravitational Memory

We study the motion of a gyroscope located far away from an isolated gravitational source in an asymptotically flat space-time. As seen from a local frame `tied to distant stars', the gyroscope precesses when gravitational waves cross its path, resulting in a net `orientation memory' that carries information on the wave profile. The effect is related to asymptotic symmetries: at leading order in the inverse distance to the source, the precession rate coincides with the (covariant) dual mass aspect, providing a celestially local measurement protocol for dual supertranslation charges. Furthermore, the net gyroscopic memory is derived from flux-balance equations for superrotations and contains an extra flux generating electric-magnetic duality. The spin memory effect is reproduced as a special case.


Introduction
The direct observation of gravitational waves from a merger of binary black holes marked the birth of an immensely exciting scientific era: gravitational radiation now provides a new window to observe the universe, complementing the electromagnetic signals traditionally used in astrophysics.Multi-messenger astronomy, however, requires a precise understanding and modelling of gravitational waves, which is made difficult by the nonlinear nature of general relativity.It is therefore of interest to find new, potentially measurable quantities sensitive to gravitational wave profiles.The goal of this paper is to describe one such observable affecting what is, perhaps, the simplest measuring device of astronomers: a spinning gyroscope.
A prominent manifestation of gravitational nonlinearity appears in so-called hereditary effects [1], which depend on the entire past history of the system.In particular, gravitational wave memory [2][3][4][5][6] is a permanent net change of certain metric components.The simplest consequence of this behaviour is displacement memory, i.e. the permanent change of distance between two nearby freely falling test masses after the passage of gravitational waves.Here we shall argue that gyroscopes similarly display an 'orientation memory' sensitive to gravitomagnetic components of the radiating field.
Remarkably, memory is a Newtonian addition to the oscillatory waveforms emitted by bounded gravitational sources [7].As a result, the effect may realistically be quite large for binary systems [8][9][10], but its detection is hampered by the poor sensitivity of gravitational wave detectors at low frequencies.It is nevertheless conceivable that future experiments will be able to detect it: see e.g.[11][12][13].One of the corollaries of this work is thus a novel proposal for the observation of memory-or at least a specific kind thereof.Indeed, there exists by now a plethora of distinct manifestations of gravitational memory, ranging from standard displacement memory [2][3][4] to the kick (velocity) memory effect caused by planar waves [14,15] or compact sources [16,17], spin memory in a Sagnac interferometer [18,19], and novel memory effects in modified theories of gravity [20][21][22][23].The unifying thread [24,25] underlying these seemingly unrelated observables is the existence of asymptotic Bondi-Metzner-Sachs (BMS) symmetries of gravity [26,27] and their various extensions [28][29][30][31]-an active field of research on its own.As we shall explain, gyroscopic memory is related to space-time symmetries as well: the precession rate turns out to coincide with the 'covariant dual mass aspect' [32] that generates dual supertranslations [33][34][35][36][37][38][39][40] in an extended version of the standard BMS group.The resulting net change of orientation-the memory effect as such-stems from flux-balance equations for the angular momentum aspect, and is thus related to superrotation charges in the sense of [30,41,42].Remarkably, an additional nonlinear flux turns out to contribute; we show that it coincides with the local Hamiltonian generator of duality transformations in radiative phase space.As a bonus, the spin memory effect [18] is a special case of our considerations, since a Sagnac interferometer is but a gyroscope made of optical fibres.
To conclude this introduction, a comment is called for regarding the magnitude of precession as a function of the observer's distance to the source.While usual displacement memory is proportional to the inverse distance, gyroscopic memory (like spin memory) is proportional to its square: it is a weaker effect.This is not surprising, however, as even the best known precession effect in gravitational backgrounds with angular momentumnamely Lense-Thirring precession (see e.g.[43, sec. 40.7])-involves the cube of the inverse distance.In that respect, our computation generalizes the Lense-Thirring effect to radiative space-times and 'corrects' it by a dominant, overleading term at large distances.
The paper is organized as follows.Section 2 is devoted to a lightning review of the Bondi formalism for asymptotically flat gravitational fields, and introduces asymptotic geodesics therein.These are then used in section 3 to describe the kinematics of freely falling (or mildly accelerated) gyroscopes with respect to a natural choice of local tetradnamely one that is tied to distant stars and obtained by suitably rotating a 'sourceoriented' frame where one vector points towards the origin of radiation.Finally, section 4 displays the precession rate and ensuing gyroscopic memory for freely falling observers, whereupon the result is related to dual asymptotic symmetries and flux-balance equations of gravitational data.Static observers are also briefly considered, and their gyroscopic memory effects are compared to earlier statements in the literature, such as asymptotic vorticity [44,45] and spin memory [18].

Radiative asymptotically flat metrics
This preliminary section serves to set up our notation, review the Bondi formalism for asymptotically Minkowskian metrics, and derive geodesics near null infinity.These geodesics will eventually be the world lines of freely falling gyroscopes in section 3.
Off-shell metric in Bondi gauge.Consider a Lorentzian space-time manifold endowed with (retarded) Bondi coordinates (u, r, θ a ), where u is retarded time, r is a radius, and θ a (a = 1, 2) are coordinates on a celestial sphere at future null infinity (see fig. 1).The manifold carries a metric ds 2 = g µν dx µ dx ν whose components are taken to satisfy the Bondi gauge conditions Any such metric is commonly written as in terms of a metric γ ab that will eventually include the effects of gravitational radiation, a vector field U a that will eventually carry information on angular momentum, and two functions β, F , the second of which will eventually sense the mass of the source.Off-shell, all these quantities are a priori arbitrary functions on space-time.To ensure that the metric (2) is in fact asymptotically flat, one imposes the extra boundary condition lim r→∞ γ ab (u, r, θ) ≡ q ab (θ), where the right-hand side is the static, radius-independent metric of a unit sphere (with Ricci scalar R[q] = 2) written in coordinates θ a .Eq. ( 3) and the last condition in (1) then imply that a sphere at constant (u, r) has area 4πr 2 , so r measures areal distance.
No further boundary conditions are needed: in vacuum Einstein gravity, eq. ( 3) turns out to fix the on-shell radial dependence of all other variables, as we recall below.
The gauge conditions (1) imply that constant u hypersurfaces are light-like, as g uu = g µν ∂ µ u ∂ ν u = 0. Their normal vector ≡ −g µν ∂ ν u ∂ ν = e −2β ∂ r satisfies • ∇ = 0, thus Figure 1: A Penrose diagram of some asymptotically flat space-time, including Bondi coordinates (u, r, θ a ) in terms of which the (off-shell) metric takes the form (2). Throughout this work, we assume that some source of gravitational radiation is located near the origin (r = 0).The resulting gravitational waves cross the world line of a gyroscope (red) and cause its orientation to change with respect to a tetrad built in section 3.This change of orientation may be seen as a gravitomagnetic memory effect and is studied in detail in section 4.
representing a congruence of outgoing affine null geodesics.Indeed, we are interested in asymptotically flat space-times with a gravitational source near the origin1 -the initial center of mass of the source is assumed to be at r = 0-so is tangent to null rays emitted from the source, and integral curves of eventually reach future null infinity (the region r → ∞ where all other coordinates are kept finite).One should thus think of the large r region as the typical location of detectors of gravitational waves.Indeed, it is the tangent vector that will eventually be used in section 3 to build a source-oriented tetrad carried by observers far away from the source.
On-shell metrics.The Bondi framework naturally suggests solving Einstein's equations perturbatively in 1/r, i.e. as an expansion near null infinity.Vacuum dynamics thus constrains the arbitrary functions of (2) in the form of asymptotic expansions [46] where indices a, b on celestial spheres are raised and lowered with the static metric (3), D denotes the spherical Levi-Civita connection, and each coefficient of 1/r n is some function of (u, θ), as follows.First, the subleading metric correction C ab in ( 6) is the Bondi shear [5], measuring (as its name indicates) the shear of outgoing null geodesic congruences near infinity.It is a symmetric and traceless tensor that depends freely on (u, θ); in fact, it is the only genuinely free data of on-shell metrics, and it will be the key object contributing to precession in sections 3-4.It determines all other subleading coefficients in (4)-( 7) according to where C2 ≡ C ab C ab for brevity.The only quantities that are not directly fixed by shear are the Bondi mass aspect m(u, θ) of eq. ( 4), and the angular momentum aspect L a (u, θ) in ( 7) (respectively measuring densities of energy and angular momentum on celestial spheres): their initial configuration at some time u 0 is arbitrary, but their time evolution is otherwise given by the news tensor N ab ≡ ∂ u C ab according to the balance equations [46] ṁ where the dot denotes partial derivatives with respect to retarded time u, and we have introduced local fluxes of energy and angular momentum: The balance equations (10) will play an important role in section 4, allowing us to relate gyroscopic memory to angular momentum fluxes.We stress once more that the timedependence of news is arbitrary, unless a specific gravitational source has been chosen. 2  To summarize, the on-shell form of the Bondi metric (2) satisfies the asymptotics at large r, where the omitted components rr and ra vanish identically owing to (1).The lead actor in this expression is the asymptotic shear C ab (u, θ), which determines F 2 according to eq. ( 8) and defines the news N ab that crucially affects the balance equations (10) for energy and angular momentum.Subleading corrections to the metric components in ( 12) turn out to be irrelevant in what follows, so we systematically neglect them.
Asymptotic geodesics.We conclude this section by writing down the large r asymptotic solution of the geodesic flow equation u•∇u = 0. (This is a key preliminary for freely falling tetrads and their spin connections built in section 3.) For definiteness, integration constants are fixed by demanding that freely falling observers have an O(r −2 ) radial velocity relative to the source at some reference time u = u 0 .Thus, upon expressing proper velocity as a mildly tedious but straightforward computation yields where m 0 = m(u 0 , θ) is the initial mass aspect and (D • C) 2 ≡ D b C ab D c C ac .Note the following elementary consistency check: in non-radiating backgrounds (say C ab = 0, hence ṁ = 0), the leading radial velocity (15) contains a term −m 0 u/r 2 caused by the usual Newtonian gravitational acceleration vr = −m 0 /r 2 .

Gyroscopes in local frames
This section is devoted to the kinematics of a gyroscope near null infinity in an asymptotically flat gravitational field whose metric takes the Bondi form (2). Accordingly, we start by reviewing the equations of motion of a spin vector relative to any local reference frame in terms of the associated spin connection.We then construct a frame tied to distant stars by first building a source-oriented tetrad, then performing suitable angle-dependent rotations.We end by displaying the spin connections associated with freely falling observers and static observers in Bondi coordinates.Note that most of the discussion for now will be performed off-shell: the metric (2) need not satisfy Einstein's equations.This is all a key prerequisite for section 4, where the spin connection gives rise to gyroscopic precession, hence to memory effects once gravitational dynamics is taken into account.

Gyroscopic kinematics
Consider a (small3 ) gyroscope with proper velocity u = u µ ∂ µ and spin S = S µ ∂ µ such that S • u = 0.When the gyroscope falls freely, its spin obeys the parallel transport equation u • ∇S = 0.More generally, if external forces cause an acceleration a µ = u ν ∇ ν u µ , the spin vector obeys Fermi-Walker transport (u • ∇S) µ = (u µ a ν − u ν a µ )S ν so as to remain orthogonal to velocity [43, chap. 6].Now let an observer carrying the gyroscope measure its orientation with respect to some local tetrad of vectors e μ = e ν μ ∂ ν that are orthonormal in the sense that e μ • e ν = η μν .The spin vector can then be written with hatted components as S = S μe μ, and the Fermi-Walker equation becomes dS where u μ = e μν u ν are the components of proper velocity in the observer's frame, while is the spin connection one-form associated with the tetrad {e μ}.(Hatted indices are raised and lowered with η μν , plain indices with g µν .)Eq. ( 17) states that the precession of gyroscopes is determined by the spin connection, so the latter is a key object.
In what follows, we always restrict attention to frames that are adapted to the observer in the sense that e 0 = u coincides with proper velocity.Then the spin vector S = S îe î is purely spatial (i = 1, 2, 3) and the precession equation (17) becomes where the observer's acceleration no longer contributes.The antisymmetric tensor Ω îĵ thus yields the gyroscope's precession rate.It can be dualized into a vector Ω = Ω îe î ≡ − 1 2 îĵk Ω ĵk , so that eq. ( 19) reads Ṡ = Ω × S in terms of the usual cross product.
Determining the gyroscope's motion thus requires that one chooses a velocity u and a local frame to compute the spin connection (18).Accordingly, we now build two tetrads: the first will be source-oriented (section 3.2), while the second is tied to distant stars (section 3.3).As for the observer's velocity, we shall consider two cases: free fall, and static locations at constant (r, θ) in Bondi coordinates.Both types of world lines are understood to live at large r, so asymptotic expansions near infinity will often be used.

Source-oriented frame
Regardless of whether an observer traces a geodesic or not, let her proper velocity read in Bondi coordinates, where the 'time dilation factor' normalizes u with respect to the off-shell metric (2).The earlier geodesic velocity ( 13) is a special case of (20), while the velocity of a static observer just reads u = γ∂ u .We now build a local tetrad {e 0, e r, e â} in a way that is compatible with the observer so that e 0 = u, and adapted to the source in that e r is aligned with emitted light rays.
To obtain e r, recall from section 2 that = e −2β ∂ r is tangent to outgoing null rays; the radial tetrad vector thus coincides with up to the fact that its component along e 0 is projected out, i.e.
The tetrad is completed by two more vectors e â whose orthogonality to e 0, e r yields for some linearly independent vectors ζ â tangent to the sphere.The remaining orthonormality conditions e â • e b = δ âb then imply that these ζ â's form an orthonormal dyad with respect to the time-dependent, radius-dependent metric γ ab (u, r, θ): The choice of dyad is of course not unique; in fact we do not intend to fix it, so as to preserve general covariance on celestial spheres.A natural choice is nevertheless to pick some 'distant stars' at infinity that happen to be oriented orthogonally to the direction of the source, and take these stars to fix the axes of the dyad.Note that, in any case, the dyad typically depends on time and radius: using the on-shell expressions ( 6)-( 9) and the identity The set of vectors {e 0, e r, e â} is thus a source-oriented tetrad; it is good enough for many applications, including radiative effects at order O(r −1 ), and one may compute the corresponding spin connection (18) to deduce a precession rate (19).There are two issues here, however.The first is that gravitational waves turn out to cause precession at order O(r −2 ), so a more sensitive frame is needed to capture the effects we are seeking.The second is that source-oriented frames rotate continuously, regardless of the presence of radiation, simply to remain directed towards the source (see fig. 2).As we now show, both of these problems are solved by a frame suitably 'tied to distant stars' whose existence is guaranteed by the asymptotically flat nature of space-time.
Figure 2: An observer looks at a source at the origin.The observer's local frame (in black, on the far right) is initially built so that one of its axes coincides with the source's direction, and it also happens to point towards some distant star.As the observer moves, the source-oriented tetrad rotates so as to remain directed towards the source (red frame).Reorienting the tetrad so that its (black) axes point once again towards distant stars requires a (red) compensating rotation (28).

Frame tied to distant stars
There is an obvious drawback in the source-oriented frame.Indeed, one expects free gyroscopes in Minkowski space-time to experience no precession, but the spin connection (18) of the origin-oriented frame in flat space (F = 1, β = 0, U a = 0, γ ab = q ab ) reads where bars denote quantities evaluated in a Minkowskian background.This is manifestly non-zero: a free gyroscope with angular velocity v a precesses at a rate −v a ω a îĵ (θ a ) with respect to the source-oriented frame, purely due to the fact that the tetrad needs to rotate continuously in order to keep pointing towards the origin.It would be much more convenient to devise a tetrad whose spin connection vanishes in flat space.In fact, in pure Minkowski space, the solution simply consists in using the Cartesian frame whose spin connection is indeed zero.However, the proper frame is not as obvious for generic asymptotically flat space-times, where one can at best define an asymptotically inertial structure.One thus needs to build a frame tied to 'distant stars' whose distance from the observer is effectively infinite, contrary to the source, which is at large but finite distance.
In practice, a frame {f μ} tied to distant stars can be obtained from the source-oriented tetrad {e μ} by a celestially local rotation cancelling the spurious time-dependent rotation of the e μ's.Thus we declare that where the rotation matrix R îĵ (θ) only depends on angular coordinates θ a , is independent of the dynamical bulk metric, and may be chosen to reduce to the identity at some reference point θ 0 to be thought of as the initial angular position of the observer.The change of tetrad (26) entails a modification ω → ω of the spin connection (18), expressed in terms of one-forms by the standard transformation law Accordingly, the rotation that yields a frame whose spin connection ω vanishes in Minkowski space is given by a path-ordered exponential where ω is given by (25) and should be thought of as an so(3)-valued one-form.The curve connecting the reference point θ 0 to θ is, in principle, arbitrary, and its choice affects the value of R since the curvature of S 2 does not vanish.The choice is nevertheless ultimately irrelevant, as one merely has to pick some path for every pair (θ 0 , θ) in a suitable open neighbourhood of the observer's initial position. 4We shall assume that one such choice has been made, in such a way that R(θ 0 ) = I is the identity matrix.
The geometric justification of this construction goes as follows (see fig. 2).Suppose the source-oriented frame, centred at θ 0 at some arbitrary initial time u 0 , points towards three stars at infinity, one of them aligned with the source's direction.Then let time flow, whereupon the source-oriented frame rotates in a time-dependent manner so as to keep pointing towards the source.As it does so, the observer's position on a celestial sphere also changes from θ 0 to some θ.Now let the observer perform a rotation (28) to her frame at every step of her motion; doing so reorients the frame so that its vectors are aligned once more with the three stars chosen at time u 0 , regardless of the (possibly dynamical) bulk metric or of other time-dependent quantities.Accordingly, we refer to the tetrad (26) as being 'tied to distant stars' at infinity.
The spin connection of a tetrad tied to distant stars is readily deduced from the transformation law (27).Since the rotation R was chosen so as to annihilate the spin connection of Minkowski space (RωR −1 + RdR −1 = 0), one finds indeed where ω is the spin connection ( 18) of the source-oriented frame built in section 3.2.The actual components of ω and ω depend on one's choice of world line-either a geodesic or a static line at fixed (r, θ), as mentioned at the end of section 3.1.It is therefore time to display these results: using the velocity ( 13)-( 16) of geodesics at large r and the ensuing source-oriented frame ( 22)-( 23), the spin connection (18) follows after a long and painful, but otherwise straightforward calculation.The rotation (29) then cancels the spurious change of orientation due to the leading angular velocity in (16), and one eventually finds (Free fall) up to subleading corrections in 1/r that are implicitly neglected.(Our convention for antisymmetrization is ) This is expressed here in terms of on-shell metric data appearing in eq. ( 12), specifically in terms of the asymptotic shear C ab and the news tensor N ab = ∂ u C ab .(Mass and angular momentum aspects do not contribute to the spin connection at this order in 1/r; in particular, Lense-Thirring precession [43, sec. 40.7] is hidden in subleading terms that are unimportant for us.) The computation is identical for world lines at constant (r, θ), save for the fact that proper velocity (20) simplifies to u = γ∂ u (instead of the more involved expressions ( 13)-( 16)).Using once more the tetrad ( 22)-( 23) and the source-oriented spin connection (18), the rotated spin connection (29) for a static world line at constant (r, θ) reads (Static) again up to neglected subleading corrections.Note the identical âb components of the spin connection (33) and its free fall analogue (31).By contrast, the râ components (32) differ sharply from their geodesic cousin (30): the u component of the former scales as 1/r at large r, while that of the latter goes as 1/r 2 .We return to this in section 4.3.
Eqs. ( 30)-( 33) are crucial results for the remainder of this paper.They determine the precession rate (19) according to Ω îĵ = −u µ ω µ îĵ , where u is either the geodesic velocity ( 13)-( 16) or the static velocity u = γ∂ u , respectively requiring the free fall connection (30)- (31) or its static analogue ( 32)- (33).The next section will indeed be devoted to a detailed study of this precession rate and its implications in terms of gravitational memory.(Note that one can, in fact, use a simplified expression for the precession rate: observers initially at rest in Bondi coordinates have an angular velocity v a = O(r −2 ), so the rotation matrix (28) evaluated along the angular position of an observer satisfies

Gravitational memory from gyroscopic precession
The spin connections ( 30)-(33) determine the angular velocity ( 19) of a spinning gyroscope, located far away from a gravitational source, with respect to a frame whose axes are tied to distant stars.We now investigate this precession in depth, mostly focussing on freely falling observers.Precession then takes place in a plane orthogonal to the source's direction and turns out to be related to a suitable notion of dual mass aspect [32].Bursts of gravitational radiation thus lead to net changes of orientation that may be interpreted as a memory effect.The latter turns out to involve a non-local superrotation charge, a flux term for angular momentum, and a duality generator, all owing to flux-balance equations for gravitational data.We conclude by turning our attention to static, accelerated observers whose leading precession stems from the u component of (32) and yields a memory effect that contains the same information as standard displacement memory [2][3][4].The notion of spin memory [18] is also recovered as a special case.

Precession as a dual mass aspect
Consider a freely falling observer with velocity ( 13)-( 16), carrying a gyroscope whose precession rate (19) involves the spin connection ( 30)- (31).From this one readily finds where the dual of any symmetric tensor X ab on the celestial sphere is defined as [34] (symmetrization may be dropped when X is traceless).This states that, at leading order in 1/r, the gyroscope's axis only rotates in the plane tangent to the celestial sphere at the observer's location.The precession frequency is set by the quantity M, dubbed 'dual covariant mass aspect' [32] in analogy with the usual covariant mass aspect m + 1 8 N ab C ab [41,49]. 5(Indeed, the latter reduces to M when the shear C ab is replaced by its dual C ab and the mass aspect m is replaced by − 1 4 ab D b D c C ac , i.e. by the vorticity of the angular motion of the geodesic flow stemming from eq. ( 16).) Gyroscopic precession near null infinity thus provides an observational protocol for the dual covariant mass aspect, as the latter essentially coincides with the rate (34).
It is convenient, for later use, to Hodge-decompose tensors on the sphere in terms of scalar quantities with definite parity.Accordingly, in what follows we write angular momentum and shear as where superscripts ± refer to parity eigenvalues of the respective functions, while angular brackets denote the symmetric, trace-free projection D a D b ≡ D (a D b) − 1 2 q ab D 2 .Using this, one finds that the linear term in the precession rate (34) has odd parity: Moreover, the balance equation (10) for angular momentum implies the relation DC − = L− + J − , thanks to which the dual mass aspect (36) can finally be recast as A corollary of this rewriting is the absence of precession in non-radiative space-times, where the angular momentum aspect is constant and fluxes vanish since they are proportional to news.We now exploit the decomposition (37) to relate gyroscopic memory to certain symmetries of the gravitational phase space.

Gyroscopic memory and gravitational symmetries
The gyroscope carried by a freely falling observer precesses according to eq. ( 19), with a rate (34) that vanishes in the absence of radiation thanks to the rewriting (37) of the dual mass aspect.It follows that any finite burst of gravitational radiation leaves a permanent imprint-a memory-of its passage on the gyroscope's orientation.To compute this effect, our starting point is the precession equation ( 19), whose solution can formally be written as a time-ordered exponential of Ω acting on the initial spin vector.Since the angular velocity ( 34) is only accurate up to order O(r −2 ), the expansion of the exponential is only reliable up to first order in Ω.We conclude that the net change of orientation that accompanies gravitational waves is where we used eqs.( 34) and (37), while ∆L − ≡ +∞ −∞ du L− is the net change of the parity-odd component of gravitational angular momentum.
Eq. ( 38) is the sought-for memory effect.In order to relate it to gravitational symmetries, note from the Hodge decomposition (35) that L − = D −2 ( ab D b L a ) where D −2 is the inverse of the Laplacian on the sphere, i.e. a suitable Green's function G. Accordingly, the term in square brackets in the memory formula (38) can be written as where Y a (θ) ≡ ab D b G(θ, θ ) is a divergence-free vector field on S 2 whose superrotation charge Q Y and flux F Y are given by 6 while the last term of ( 39) is a seemingly exotic flux contribution: In fact, this quantity is closely related to gravitational electric-magnetic duality: the Hamiltonian generator of local duality transformations δC ab = ε(θ) C ab , δN ab = ε(θ) N ab with respect to the radiative symplectic form [51] is nothing but √ q d 2 θ ε F. One can think of this extra flux F as an analogue of the super-centre of mass flux affecting usual displacement memory for test masses having an initial relative velocity [52]; it just happens to be a duality generator in the present case.
Eqs. (34) and (39) are our main results: they explicitly display the relation between precession velocity and dual mass, and recast the net change of orientation (38) as a sum of three terms related to various symmetries of the gravitational radiative phase space.To our knowledge, eq. ( 39) is actually the first appearance of a duality generator in a gravitational memory effect.The remainder of this work is devoted to a comparison between these considerations and established results in the literature.

Static observers and spin memory
We have so far focussed on freely falling observers whose spin connection is given by ( 30)-( 31), but most of the literature on gravitational memory relies on static observers, whose spin connection takes the form (32)-(33) instead.Accordingly, we now briefly discuss the effects of the two kinds of components of (32)- (33) and compare them to asymptotic vorticity [44,45] and the spin memory effect [18].Incidentally, the latter only involves transverse components (33) which coincide with their freely falling cousin (31), 6 There is a subtlety here: one can redefine the charge and flux (40) by mapping N a → Na = N a + α a and J a → Ĵa = J a − αa for any one-form α a , and using hatted quantities in (40).For the charge to match its Wald-Zoupas version [50], one has to take α a = −uD a m − 1 16 D a C bc C bc − 1 4 C ab D c C bc [46].Fortunately, the sum ∆Q Y + F Y in (39) is unaffected by this ambiguity.so the comparison could equally well be carried out for freely falling observers, but the computation of [18] was carried out for static observers so we stick to that convention.
For static observers (with velocity u = γ∂ u ), the dominant precession takes place in the râ plane and stems from the O(r −1 ) term of eq. ( 32), yielding a precession velocity first found in [44,45] thanks to the vorticity of static world lines in Bondi coordinates.That the effect occurs already at order O(r −1 ) is no surprise, since world lines at constant (r, θ) have a non-zero proper acceleration u ν ∇ ν u µ ∼ ( ṁ r , − ṁ r , 1 2r 2 D b N ab ) at large r.The memory effect induced by ( 43) is proportional to the parity-even variation of shear ∆C + (recall the Hodge decomposition ( 35)), so it contains the same information as standard displacement memory, determined by the balance equation for supertranslations [2][3][4].
We now turn to the spin connection's âb component (33) and show that it reproduces spin memory [18], i.e. the Sagnac effect caused by gravitational waves on an interferometer facing a source of gravitational waves.To see this, first recall the principle of Sagnac interferometers: when a light ray is split in two separate beams moving in opposite directions along some closed path C (say an optical fibre), the apparatus's angular velocity Ω causes a difference ∆T in the time it takes for each beam to orbit the path [53]: Now consider a Sagnac interferometer located far away from a gravitational source, and assume that the closed path C lies in a plane transverse to the direction of wave propagation (i.e. that C is tangent to the celestial sphere at some large radius).The precession rate (34) can then be plugged into (44) and readily reproduces spin memory.Indeed, assuming that the spatial extent of the interferometer is much smaller than the typical gravitational wavelength λ, the second term of the dual mass in (34) is smaller by a factor /λ 1 than the first one, yielding which is an exact two-form.(The factor 1/r 2 of (34) has disappeared in (45) because the precession rate is written as a two-form in coordinate basis dθ a , noting that e â ∼ r ζ âa dθ a at leading order.)It then follows from eq. ( 44) that which coincides as announced with spin memory [18]. 7We momentarily reinstate the speed of light c.