Wave-Spin Interaction in an Angular Momentum Lens

1 Wave-Spin Interaction in an Angular Momentum Lens A. Chawla REAL Institute and IIT Delhi Abstract—This paper develops a self-contained theoretical framework for the interaction of localized quantum spins with weak gravitational-wave (GW) perturbations, formulated in an absolute-space / universal-time (3+1) setting. Building on Mathisson–Papapetrou–Dixon dynamics, Thorne–Macdonald 3+1 bookkeeping, and Caves’ distributed-in-time measurement formalism, we derive an explicit worldline spin–gravity interaction action, identify the leading curvature and gradient couplings, and construct the path-integral/influence-functional representation describing the gravitational field as an intrinsic meter. We then integrate these components into a time-distributed measurement model and provide concrete expressions suitable for numerical evaluation (e.g., an N-pulse sequence). The analysis clarifies how angular-momentum exchange between the spin subsystem and the GW field preserves total angular momentum and demonstrates how information about the spin trajectory can, in principle, be carried by outgoing gravitational radiation. Practical scaling laws and limitations are discussed. Index Terms—Gravitational waves, spin precession, MPD equations, distributed-in-time measurement, influence functional, 3+1 formalism, ZAMO, Tulczyjew SSC. I. INTRODUCTION The interaction of localized quantum degrees of freedom with dynamical spacetime — even in the weak-field, linearized regime of gravitational waves (GWs) in General Relativity — provides a probe into rich conceptual territory where quantum measurement, momentum conservation laws, and gauge choices interplay. This work assembles and extends into new territory three strands of formalism that are individually well-established: (i) the Mathisson–Papapetrou–Dixon (MPD) description of a spinning test particle in curved spacetime, (ii) the 3+1 three coordinates of absolute-space / one of universal-time book-keeping developed in Thorne–Macdonald and related treatments which exposits on the relation between proper-time evolution and global observables, and (iii) the distributedin-time path-integral measurement framework (due to Caves et al.) which prescribes how to treat timeextended quantum measurements and meters in a path-integral language. Our central goal is to present a practically usable action and path-integral representation for a localized spin interacting with linearized GWs, to extract the leading terms of interaction kernels (stress–energy source and resulting influence functional), and to show how sequences of impinging GW pulses (in an N-pulse model) produce cumulative spin changes that are constrained by total angular-momentum conservation. The formulation is explicitly done in a zero angular momentum observer (ZAMO)/global-time 3+1 language so that numerical simulation and comparison with earlier Eulerian or Fermi-normal coordinate derivations is straightforward. The remainder of the paper is organized as follows. Section II introduces the physical setup and notation in the linearized gravity framework. Section III develops the Mathisson–Papapetrou–Dixon (MPD) pole-dipole reduction and derives the worldline spin source. Section IV presents the nonrelativistic Hamiltonian forms and maps spin-connection terms to curvature-driven torques. Sections V–VII formulate the distributed-in-time measurement path integral, integrate out the gravitational-wave field to obtain the influence functional, and analyze backaction freedom together with angular-momentum exchange. Section VIII extends the analysis to two localized spins in curved spacetime, deriving the retarded spin–spin coupling kernels and clarifying collective versus relative rotational dynamics. Section IX explores structural correspondences and angular-tensor decompositions that govern the causal spin–spin coupling. Section X connects the formalism to the N-pulse model and sketches numerical-ready expressions. Section XI discusses angular-momentum accounting, scaling laws, and 2 limitations. The paper concludes with a summary and suggested directions for future work. The appendices provide extended derivations, detailed tensor contractions, and supplementary discussions that support and expand upon the results in the main text. TABLE OF NOTATION The adjacent table lists all the symbols used in the paper and their meanings. II. PHYSICAL SETUP AND NOTATION We work in linearized gravity about Minkowski spacetime, gµν =ηµν +hµν,|hµν| ≪ 1,(1) and use Greek indices µ, ν, . . . = 0,1,2,3, spatial Latin indices i, j, k = 1,2,3, and signature (−,+,+,+). Global time tdenotes the Thorne–Macdonald universal time coordinate; proper (ZAMO) time τis related to tby the lapse avia dτ =a, dt locally. We use units with c= 1 except where convenient factors are shown. The localized quantum system is a spin (for clarity we mostly consider spin-1 2but retain tensor notation for generality). The spin tensor is Sµν and the spatial spin 3-vector is Si=1 2εijkSjk,(2) with εijk the Levi–Civita symbol. We adopt the Tulczyjew spin supplementary condition (SSC) Sµνpν= 0 for the derivations that follow (Sec. III); explicit comments appear where other SSCs would alter O(1) prefactors. We label the gravitational-wave field in TT gauge by spatial components hij(t, x)and adopt standard polarization decomposition for plane waves. The central worldline (location of the localized spin) is taken to be at x=0for simplicity; finite-size corrections are discussed qualitatively. III. MPD POLE–DIPOLE STRESS–ENERGY AND THE WORLDLINE SPIN SOURCE The pole–dipole truncation of the multipole expansion for a compact object gives the distributional stress–energy tensor (Dixon form) Tµν(x) = Zdτhp(µuν)δ(4)(x−z(τ))−∇αSα(µuν)δ(4)(x−z(τ))i, (3) where zµ(τ)is the representative worldline, uµ= dzµ/dτ and pµis the momentum monopole. Inserting (3) into the linearized interaction action Sint =1 2Zd4x, hµν(x)Tµν(x)(4) and integrating by parts produces a spin-dipole coupling term S(spin) int =1 2Zdτ, ∂αhµν(z(τ)), Sα(µuν)(τ).(5) This identity is exact at the pole–dipole level, up to surface terms. It shows that the dipole (spin) couples naturally to derivatives of the metric perturbation rather than to hitself; equivalently, the convenient effective source that pairs with hij is distributional and involves spatial derivatives: τij(x, t) = −∂kSk(iuj)δ(3)(x−z(t)).(6) Equation (6) is most simply interpreted in the worldline pairing 1 2Rd3x, hij(t, x)τij(t, x): integrating the derivative onto hij reproduces (5). A. Nonrelativistic rest frame and Tulczyjew SSC For a nearly-rest worldline (ui≈0,u0≈1) and Tulczyjew SSC S0i≈0, the leading coupling to spatial metric values hij(t, 0)vanishes; the leading observable couplings arise from spatial gradients ∂khij or from the Riemann tensor components R0i0j∼ −1 2∂2 thij. This observation explains why a naive local hijSij torque is absent for pure TT plane waves impinging on a pointlike spin at rest — the physical torque is curvature/gradient driven. B. Converting to global time (3+1 lapse factors) Thorne–Macdonald’s 3+1 language clarifies the difference between worldline proper-time evolution and the global-time evolution used in practical bookkeeping. Because dτ =a, dt, any proper-time rate must be converted: d dt =1 a d dτ .(7) Consequently, curvature-driven torques expressed per unit proper time become rescaled by 1/a when expressed per unit global time; this factor is included consistently in the Hamiltonian and influence-functional constructions below. 3 TABLE I TABLE OF NOTATION Symbol Meaning gµν Background Minkowski metric hµν Linearized gravitational-wave perturbation (|hµν |<1) tGlobal (Thorne–Macdonald) universal time coordinate TProper (ZAMO) time αLapse function relating dT =α dt Sµν Spin tensor of localized system Si=ϵijkSjk Spatial spin 3-vector ϵijk Levi-Civita symbol Tulczyjew SSC: Sµν pν= 0 Spin supplementary condition used hij (t, x)TT-gauge spatial GW components Tµν (x)Stress-energy tensor (Dixon pole-dipole form) zµ(T)Representative worldline of the spin uµ=dzµ/dT Worldline four-velocity pµMomentum monopole Sspin Spin-dipole coupling action Tij (x, t)Effective worldline source coupling to hij ωab(t)Linearized spin connection Sab Spin generator H(t) = H0+Hint(t)Nonrelativistic Hamiltonian decomposition Hint(t) = −ωab(t)Sab Spin-connection interaction term Cssc SSC-dependent prefactor in spin-curvature coupling dSi dt Spin evolution equation under curvature torque Seff(t)Effective precession vector from curvature Stot[S, h]Total action (spin + GW + interaction) A(yq, hout)Joint amplitude for distributed measurement outcomes F[S, S′]Influence functional coupling forward/backward spin histories Dij,kl(t, t′)GW propagator kernel Js(t)GW-mode projection of worldline source K(t, t′)Measurement kernel (noise + signal) ∆SCumulative spin change from GW pulses h(n) ij (t)n-th GW pulse strain tensor e(n) +,×Polarization tensors of n-th pulse ∆Jspin Change in spin angular momentum ∆JGW Change in GW angular momentum flux Jtot =S+JGW +Jother Total angular momentum (conserved) R(ˆn, ϕ)Local SU(2) rotation operator (wobble) θkSmall rotation angle of k-th wobble ˆnkRotation axis of k-th wobble ϕkAzimuthal angle of wobble axis ∆ϕkIncremental change in azimuthal angle between bursts ΦNet cumulative rotation about z-axis Hspin, Hfield Enlarged Hilbert space (spin + GW field) PzEffective polarization/precession shift along z-axis LMagnitude of classical rotor angular momentum Gret ij,kl(t, x;t′, x′)Retarded GW Green’s function Ya[S, h]Distributed measurement functional (Caves formalism) Wa(t)Window function for distributed measurement IV. HAMILTONIAN AND NONRELATIVISTIC REDUCTION Start from the curved-space Dirac action or from the MPD equations and reduce to a nonrelativistic Pauli-type Hamiltonian for a localized spin degree of freedom. The schematic structure — valid to first order in hand in the nonrelativistic limit — is H(t) = H0+Hint(t),(8) with H0the internal spin Hamiltonian (zero for an isolated spin except for possible Zeeman terms) and the interaction term obtained from the spinconnection pieces in the Dirac or MPD reduction. Following the standard expansion one finds an operator-form coupling Hint(t) = −1 2, ω0ab(t), Sab,(9) 4 where ω0ab is the linearized spin connection evaluated in the chosen frame and Sab the spin generator. In TT gauge with h0µ= 0 the spatial spinconnection components simplify to ω0ij ≃1 2∂thij; naively substituting yields Hint(t)≃ −1 4∂thij(t)Sij.(10) However, expressing Sij =εijkSkand contracting εkij∂thij gives zero for any symmetric hij, explaining the previously-mentioned vanishing of a firstorder local torque. A. Curvature coupling (MPD result) The leading nonvanishing spin evolution arises from spin–curvature coupling present in the MPD formalism. After a careful reduction and choice of SSC one obtains, in the nonrelativistic rest frame and expressed per unit global time, dSi dt =−CSSC 2a,¨ hij(t), Sj+O(S2, vS),(11) where CSSC =O(1) is an SSCand conventiondependent prefactor (e.g. C ≈ 1/2or 1depending on definitions) and ais the lapse. For plane TT waves ¨ hij is the linearized Riemann component −2R0i0j, so (11) is consistent with the Riemann-driven torque intuition. B. Effective Hamiltonian for the nonrelativistic spin Equation (11) is equivalent (for unitary spin evolution neglecting dissipation) to a time-dependent Hamiltonian Heff(t) = −Ωeff (t)·Swith (Ωeff)∗i(t) = C ∗ SSC 2a, ϵijk,¨ hjk(t), , (12) where the antisymmetric combination maps the Riemann-driven tensor into an effective precession vector. In many practical coordinates ϵijk¨ hjk = 0 for a pure TT plane wave at a single point, and thus the MPD tensor form (component contraction) is the safer canonical object to use numerically. V. DISTRIBUTED-IN-TIME MEASUREMENT — PATH-INTEGRAL FORMULATION We now adapt Caves’ distributed-in-time measurement formalism to the spin+GW system in global time. The objective is to write the joint amplitude for a sequence of measured outcomes yq Q q=1 corresponding to time-distributed readouts of functionals of the spin and/or metric. A. Total action and measurement kernels The full action (in global time) is Stot[S, h] = Sspin[S] + SGW[h] + Sint[S, h],(13) where Sspin encodes the intrinsic quantum rotation action (e.g. Berry-term for spin path integrals plus any H0), SGW is the quadratic linearized-gravity action in TT gauge, and Sint is given by (5) (converted to tvia dτ =a, dt). A time-distributed measurement occurring over a causal window around time tqis modeled by a resolution amplitude (kernel) Yq(yq−Yq[S, h]), where Yqis the functional (Caves’ functional) mapping the histories into the readout variable. B. Joint amplitude and integrating out the GW field The joint amplitude for outcomes yqand final GW configuration hout at time tfis A(yq, hout) = ZD[S]D[h], eiStot[S,h]Y q Yqyq−Yq[S, h], δ(h(tf)−hout). (14) Because SGW[h]is quadratic and Sint is linear in hfor the pole–dipole truncation, the h-path integral is Gaussian and can be performed exactly, yielding an influence functional F[S,S′]coupling forward and backward spin histories. The reduced amplitude for spin histories conditioned on measurement outcomes is then Ared(yq) = ZD[S]D[S′], ei(Sspin[S]−Sspin[S′]),F[S,S′],Y qe Yq(· · · ), (15) where e Yqare effective kernels obtained after integrating out hand possibly conditioning on outgoing GW observables. C. Form of the influence functional The influence functional is (schematically) F[S,S′] = exp−i 2Zdt dt′τij(t)−τ′ij(t)Dij,kl(t, t′)τkl(t′) + τ′kl(t′) 2. (16) where τij(t)is the worldline source (distributional) obtained above and Dij,kl(t, t′)is the appropriate GW propagator (combination of retarded/advanced/Feynman kernels depending on the measurement condition; for unconditional reduced dynamics use the closed-time-path / in-in propagator). Importantly, because τij contains spatial 5 derivatives (Eq. (6)), the kernel effectively involves derivatives ∂k∂k′Devaluated at coincident spatial points, which manifest as time-nonlocal curvaturetype couplings in the reduced spin dynamics. VI. GW AS METER:INFORMATION TRANSFER AND BACK-ACTION Treating the GW field as the measurement apparatus, the outgoing GW modes after interaction with the spin carry information about the integrated worldline source τij[S]. Expanding the influence functional to second order in the weak spin–GW coupling gives a Gaussian conditional probability for measured GW-mode amplitudes ξof the form P(ξ|S)∝exp"−1 2Zdt dt′JS(t)−ξ(t)K(t, t′)JS(t′)−ξ(t′)#. (17) where JS(t)is the GW-mode projection of τij[S](t) and Kencodes the mode kernel (including quantum and classical noise). The same kernel’s imaginary part produces back-action on the spin-state reduced dynamics. The effective signal-to-noise and the ability to infer features of S(t)from ξare determined by the ratio of |JS|to the kernel width K−1; for physically realistic GW amplitudes and microscopic spins this ratio is extremely small, but the formal structure stands. VII. BACK–ACTION FREEDOM AND FORCE-LIKE GRAVITATIONAL SPIN DRIVING The interaction between the localized spin and the weak, impinging gravitational waves (GWs) may be viewed through two fully compatible interpretations: (i) as a curvature-driven torque on the spin through the MPD reduction, and (ii) as a time-distributed measurement in the sense of Caves, where the GW field acts as the intrinsic meter coupled to the spin’s history. In this section we link these two perspectives by recalling the concept of back-action-free force detection developed for a simple harmonic oscillator (SHO) monitored by time-shifted meters. We show that the same structural ideas illuminate how the spin’s local curvature coupling can be probed by distributed GWs and that angular-momentum exchange with the field naturally plays the role of “meter back-action” in a rotational sector. A. From Impulsive Meter Couplings to GW Pulses In standard distributed measurements, the detector is modeled as a sequence of meters coupling to the system in non-overlapping time windows. If the coupling functions Ki(t)satisfy specific orthogonality constraints, the collected meter readouts reveal the applied force while canceling the system’s initial conditions and back-action noise injected by earlier meters. Thus, information flows unidirectionally from the unknown force into the meters. Our spin-GW scenario admits a direct analogy: each short gravitational pulse functions as an external meter which (i) couples locally to the spin’s angular momentum and (ii) carries away angular momentum after the interaction. Because the GW field is dynamical, the “meters” are physically real fields, not auxiliary ancillas. The distributed coupling kernels are instead the time profiles fn(t) of the arriving pulses. These kernels are not under perfect experimental control in astrophysical sources, but the theoretical structure is identical: the mapping from pulse history to final spin state may be expressed as a linear functional to first order in the GW amplitude. Denoting by Si(t)the Heisenberg spin operator in global time, and by h(n) ij (t)the n-th TT-wave pulse arriving in a finite window (tn, tn+Tp), the cumulative first-order spin change is ∆Si≈ −CSSC 2a N X n=1 Ztn+Tp tn dt ¨ h(n) ij (t)Sj(t),(18) which is the rotational analogue of the forcedetermination expression in oscillator detection. Equation (18) is the dynamical counterpart of extracting information from the integrals RKi(t)x(t)dt in the oscillator case. B. Freedom from Initial Spin Direction: A Rotational Orthogonality Condition Where the SHO analysis demands that the Fourier components ˜ Ki(ω0)vanish at the natural oscillator frequency to suppress sensitivity to initial conditions, an analogous rotational condition appears here. If successive pulses have propagation directions and polarization tensors chosen such that their effective curvature-torque vectors span a plane containing the initial spin direction, then the leading-order 6 torque produces only in-plane “wobbles” which, when summed, have vanishing projection along the initial spin axis: N X n=1 Ztn+Tp tn dt ¨ h(n) ij (t)Sj 0≈0.(19) In this configuration, the reduced dynamics become insensitive to the unknown initial spin orientation along that axis, just as carefully designed meter pulses suppress sensitivity to initial SHO position and momentum. Rotational geometry replaces Fourier zeros, but the logic of isolation from initial conditions is identical. The same geometry that removes initial-state dependence also eliminates first-order back-action on the conjugate rotation variable; to detect curvature information without disturbing the protected component of angular momentum is a genuine backaction-free channel. Crucially, this does not violate angular-momentum conservation: the orthogonal component of the spin remains a reservoir exchangeable with GWs. C. Non-Commutativity and Constructive Holonomy Back-action freedom in the SHO example merely prevents detector noise from obscuring the signal; it does not generate any intrinsic dynamics. In contrast, for the spin system the non-commutativity of SU(2) produces a quintessentially quantum secondorder effect: even when Eq. (19) holds, the small rotations generated by Eq. (18) accumulate through the Baker–Campbell–Hausdorff (BCH) series to yield a net rotation orthogonal to the wobble plane. This is a geometrically robust signature of curvature holonomy on the Bloch sphere. Therefore, the same distributed-measurement structure that suppresses initial-condition backaction also enhances sensitivity to cumulative curvature twisting: the protected component of spin effectively becomes a curvature probe. D. GW Field as Both Signal and Back-Action Channel From the Caves–influence-functional perspective, one may solve for the outgoing GW modes conditioned on the spin history. The resulting conditional probability is Gaussian in the extracted field amplitudes and contains two pieces: 1) A signal term: the outgoing GW radiation carries a linear imprint of the spin evolution through the dipole worldline source; 2) A back-action term: the imaginary part of the influence kernel captures the torque noise the GW field feeds back into the spin’s off-axis components. In the carefully arranged geometry matching Eq. (19), these two terms separate cleanly: the protected spin component experiences no decohering back-action yet remains inferable from the outgoing GW field. This is the rotational analogue of meternoise evasion in SHO force detection. E. Conservation Laws: No Violation of Rotational Symmetry It is essential to emphasize that angularmomentum conservation is never threatened. Whenever the reduced spin gains a longitudinal component by the effective BCH rotation, an equal and opposite correction appears in the GW sector as outgoing helicity flux. The Wigner rotation group remains a symmetry of the full system; any apparent violation arises solely from neglecting the meter’s angular-momentum channel. F. Summary of Structural Correspondences The adjacent structural map clarifies that our GW–spin system is a faithful back-action-free rotational-force detector in principle, with the gravitational field playing both the signal carrier and the conjugate-back-action channel. VIII. TWO SPINS IN CURVED SPACETIME In the preceding sections we examined the dynamics of a single classical spin interacting with a sequence of incident gravitational waves. That analysis built upon the pole–dipole truncation of the Mathisson–Papapetrou–Dixon (MPD) equations and employed the in-in/influence-functional formalism to account for the causal back-action of the gravitational field. Here we extend the formulation to incorporate two localized spinning particles whose worldlines are influenced not only by the externally incident gravitational pulses but also by a “reference” gravitational wave that modulates the spatial separation between them. This allows us to explore whether the pair rotates as a collective rigid object or instead exhibits richer internal relative motion. 7 Oscillator Force Detection Spin–GW Interaction Force F(t)drives x(t)Curvature ¨ hij (t)drives Si(t) Meters Ki(t)couple to x(t)GW pulses couple via MPD dipole source Back-action noise in pAngular-momentum flow in GW helicity Fourier zero removes ICs Geometric alignment removes ICs Final meter state ⇒FOutgoing GW ⇒curvature Noise-evading quadrature Protected spin component TABLE II STRUCTURAL MAP A. Worldline Sources and Influence Functional Let the two particles be labeled by α∈ {A, B} with worldlines zα(t)and spin 3-vectors Sα i(t). For each, the pole–dipole stress-energy density may be written, as before, τα ij(x, t) = −∂kSα k(i(t)uα j)(t)δ(3) x−zα(t), (20) where Sα ij =εijkSα kand uα iis the spatial velocity of particle α. The total source experienced by the gravitational field is then τij(x, t) = τA ij (x, t) + τB ij (x, t).(21) Following the same integration procedure used in the single-spin case, the linearized gravitational field hij is Gaussian and can be integrated out to yield the influence functional F[S, S′]. Because τij is quadratic in the exponent of the CTP integral, the result contains three classes of terms: (i) selfinteraction of A, (ii) self-interaction of B, and importantly, (iii) cross-interaction terms that encode the causal exchange of gravitational radiation and tidal influence between the two spins. The latter take the form IAB[S, S′] = −i 2Zdt dt′hτA ij (t)Dij,kl(t, t′)τB kl (t′)+τB ij (t)Dij,kl(t, t′)τA kl(t′)i+(terms with primes). (22) Here Dij,kl is the relevant CTP propagator for the transverse-traceless graviton. This single expression compactly contains the entire physical content of gravitational mediation between the two spins. B. Two-Spin Equations of Motion As in the one-body case, the effective equations of motion follow from varying the real part of the influence action, leading to torques on each spin from two distinct sources: (i) local curvature at the particle’s own position, arising from the externally supplied incident pulses and the “reference” wave that drives separation oscillations; (ii) retarded curvature sourced by the other spin— the physical back-action of one particle’s spindipole GW field upon the other. To leading order in the weak-field, nonrelativistic limit, dSi A(t) dt =−C 2a¨ hi jt, zA(t)Sj A(t) + Ci AB(t),(23) with a symmetric expression for Si B(t). The coefficient C/a is the same conversion factor identified previously between curvature and spin-precession rate. The new term is the retarded torque Ci AB(t) = −Zdt′Ki jt, t′;zA, zBSj B(t′).(24) This memory integral reflects the causal propagation time required for gravitational radiation from particle Bto reach particle A. C. Structure of the Retarded Kernel The kernel is obtained by inserting the explicit expressions for τα ij into Eq. (22) and extracting the part proportional to SB(for the torque on A). After reducing the antisymmetric spin tensors to 3-vectors using Levi-Civita contractions, one arrives at the compact representation Ki j(t, t′) = 1 2εiab εcd j∂a∂c′Gret bd,pqt, zA(t); t′, zB(t′)upuq, (25) where Gret is the retarded Green’s function for a massless spin-2 field in TT gauge. The derivatives act with respect to the spatial coordinates of Aand Brespectively, and the upuqprojection ensures we are extracting the dynamical curvature components (¨ hij in the particle’s rest frame). Equation (25) is general and exact at linearized order. However, analytic understanding is greatly clarified by special cases. Two regimes of particular physical relevance emerge: 8 (i) Near-zone / uniform-field regime.: If the instantaneous separation R=zB(t′)−zA(t)is much smaller than the gravitational wavelength λGW of the incident pulses and the binding between the particles ensures quasi-rigid motion, then the curvature experienced at the two sites is nearly identical. In this limit the propagator derivatives collapse to local functions of tand the kernel reduces to Ki j(t, t′)≈C 2aδ(t−t′)¨ hi j(t),(26) thereby reproducing the single-spin MPD torque law and confirming that the pair rotates as a single effective object. (ii) Finite-separation / retarded-coupling regime.: When the oscillating separation is comparable to (or larger than) λGW, the full causal propagation structure matters. Using the spherical-wave form of the retarded spin-2 Green function and applying the spatial derivatives yields Ki j(t, t′)≈εiabεcd jPbd,pq(ˆn)"A(0) ac (ˆn) R3δ(∆t−R) + A(1) ac (ˆn) R2δ′(∆t−R) + A(2) ac (ˆn) Rδ′′(∆t−R)#, (27) where ∆t=t−t′,ˆn= (zB−zA)/R, and Pis the usual TT projector. The coefficients A(n) ac (ˆn)encode directional structure. Physically, these three contributions correspond to a quasi-static tidal term (1/R3), an induction term (1/R2), and a radiation-dominated term (1/R). Substituting this into Eq. (24) yields the compact retarded evaluation Ci AB(t)≃ −"Fij(ˆn) R¨ Sj B(t−R)+Gij(ˆn) R2˙ Sj B(t−R)+Hij(ˆn) R3Sj B(t−R)#. (28) The three tensors F,G, and Hare obtained by explicit contraction of the Levi–Civita products with the TT projector and will be detailed momentarily. In summary, the fate of the two-spin system— whether it behaves as a unified rotor or develops nontrivial internal rotational dynamics—depends sensitively on the instantaneous ratio R/λGW, the degree of internal rigidity, and the relative phases of the incident pulses. The stage is now set to evaluate the explicit angular tensors F,G, and Happearing in Eq. (28), which govern the causal spin–spin coupling through the gravitational field. D. Explicit Angular Tensors for Retarded Spin–Spin Coupling The kernel expansion in Eq. (28) requires the angular tensors F,G, and Hthat result from contracting Levi–Civita products with the transversetraceless projector of the graviton propagator. We now compute these objects explicitly. Let ˆn= (zB−zA)/R denote the unit separation vector pointing from particle Ato particle B. In the TT gauge, the projector takes the form Pij,kl(ˆn) = (δik −ˆniˆnk) (δjl −ˆnjˆnl)−1 2(δij −ˆniˆnj) (δkl −ˆnkˆnl), (29) which enforces symmetry, transversality, and tracelessness. The double Levi–Civita contraction in Eq. (25) effectively converts the antisymmetric spin tensors into the physical spin 3-vectors and projects orthogonally to ˆn. After straightforward (but lengthy) algebra we may express all terms in Eq. (28) using two orthogonal projectors on spin space: P⊥ ij =δij −ˆniˆnj, Qij =εikℓ ˆnkP⊥ ℓj .(30) The operator P⊥projects any vector into the plane transverse to the line of separation, while Qrotates that transverse projection about ˆnby π/2. It is then advantageous to parametrize the full causal kernel in the basis {P⊥, Q}: Fi j(ˆn) = f1P⊥ij+f2Qij,Gi j(ˆn) = g1P⊥ij+g2Qij,Hi j(ˆn) = h1P⊥ij+h2Qij. (31) For nonrelativistic motion (u0≃1, ui≃0), the coefficients are f1=3 2, f2=−3 2,(32) g1=−3, g2= 3,(33) h1=15 2, h2=−15 2.(34) Thus we have the compact and explicit forms Fi j(ˆn) = 3 2P⊥ij−Qij(35) Gi j(ˆn) = −3P⊥ij−Qij(36) Hi j(ˆn) = 15 2P⊥ij−Qij(37) Several key structural facts follow immediately: 9 •Each tensor annihilates the component of the spin along the line of separation: Fi jˆnj=Gi jˆnj=Hi jˆnj= 0. Thus the gravitational torque is constrained to the transverse plane. •The operator Qgenerates a phase-shifted rotation in the transverse plane, meaning the retarded influence of Bon Agenerically promotes beat-like and precessional dynamical patterns. •The hierarchy H≫G≫Fas R→0 confirms the dominance of near-field tidal coupling for small separations. Substituting Eqs. (32)–(34) into Eq. (28) yields a fully explicit evolution law for the two-spin system in curved spacetime that automatically encodes retarded causality, directionality, and gravitational radiation reaction. These ingredients allow us to explore whether an oscillating separation driven by a reference gravitational wave enforces rigidbody co-rotation, or instead excites internal differential modes with accumulated noncommutative spin holonomy. We now proceed to analyze the dynamical consequences of this coupling structure in representative pulse-driven scenarios. But before we do that, a segway to look at the macroscopic case. E. Black Holes as Coupled Spinning Objects: Classical vs. Microscopic Spins Up to this point our analysis has treated each spin as a microscopic, polarizable dipole source carried by a compact particle. This is the correct description for fundamental quantum spins or small composite bodies. However, to emphasize the contrast between quantum and classical rotational degrees of freedom, it is illuminating to replace each spinning particle by a Kerr black hole, whose angular momentum is macroscopic and fundamentally a geometric property of the gravitational field itself. The MPD pole–dipole truncation used throughout the previous sections remains formally applicable in this setting: a spinning black hole moving in an external gravitational field is well approximated by a worldline with a mass monopole and spin dipole, provided the curvature gradients across the horizon are small. In this sense, Eqs. (23) and (28) still describe the precession of the black hole spin vectors due to tidal gravitational fields and gravitational radiation exchange. Yet, several key conceptual differences emerge: (i) Classical spin is geometry.: For a black hole, the “spin vector” Siis not a property of internal matter, but a classical charge associated with horizon geometry (the Killing–York quasilocal angular momentum). Its magnitude obeys the Kerr inequality |S| ≤ M2. Therefore, the SU(2) spin space appearing in the microscopic case becomes a purely classical phase space lacking any intrinsic quantization. (ii) Gravitational waves extract horizon angular momentum.: In the microscopic spin case, radiated angular momentum is supplied entirely by the external field. For black holes, radiation reaction physically modifies the horizon geometry: Si(t)decreases when outgoing waves carry positive angular momentum flux in that direction. The influencefunctional derivation makes this explicit, since the imaginary part of the influence action is proportional to the GW energy–momentum flux. (iii) Strong-field tidal response dominates.: The near-zone tensor Hij(ˆn)in Eq. (28) is proportional to 1/R3and represents the leading tidal spin–spin coupling. For black holes, this response is enhanced by the large horizon radius: R∼ O(M), so nearby spinning holes experience differential torques that far exceed the microscopic case. This is consistent with the well-known strong spin–orbit and spin–spin couplings in post-Newtonian binary dynamics. (iv) No inherently quantum holonomy.: Although noncommutative accumulation of rotations still occurs for classically spinning bodies, a Kerr black hole lacks the SU(2) quantization angles and Berry-phase-like geometric phases associated with internal quantum spin. Thus any net rotation accumulated from a sequence of pulses directly reflects classical curvature holonomy rather than microscopic quantum structure. (v) Internal rigidity.: Unlike two microscopic spins, two black holes separated by a substantial distance do not constitute a rigid body. There is no material linkage enforcing co-rotation; only the 16 time-nonlocality because of retarded propagation. For a plane-wave pulse in the near-zone the propagator reduces effectively to local-in-space kernels and the time integrals simplify to convolution with ¨ h∗ij(t), reproducing the MPD form. The noise kernel (Hadamard function) determines the symmetric part of the influence functional and encodes the GW quantum noise experienced by the spin. D: DISTRIBUTED MEASUREMENT KERNELS (CAVES ADAPTATION) We adopt Caves’ resolution-amplitude formalism for time-distributed measurements. For measurement label qcentered at tqwith causal window of width bqdefine the measured functional Yq[S, h] = Ztq tq−bq !dt;wq(t), F[S(t), h(t)], with window function wq(t)normalized appropriately and Fa linear or nonlinear functional of the instantaneous fields (e.g., a particular GWmode projection or a local spin observable averaged against a meter field). The resolution amplitude Yq(yq−Yq)models imprecision; for Gaussian meters Yq(∆) = exp[−∆2/(2σ2 q)]. Inserting these kernels into (14) and integrating out the GW degrees of freedom produces effective measurement kernels e Yqthat depend both on the choice of mode-channel being monitored and on the accessible outgoing GW observables. When the GW itself serves as the meter (i.e., we detect the outgoing mode amplitude), e Yqis directly obtained by conditioning the Gaussian path integral to the observed mode amplitude and has the Gaussian form (17) with the integrand determined by the influence-functional kernel. E: N=5 EXAMPLE AND NUMERICAL RECIPE We specify a concrete numerical recipe that implements the MPD/influence-functional informed evolution used previously, but now with the exact derivative-based source. a) Geometry and polarization matrices: Let the lab axes be (x, y, z)with the spin at the origin and pointing along zinitially. Let N= 5 waves each propagate in the y–zplane at angles ϕnto the zaxis; define propagation vectors ˆnn= (0,sin ϕn,cos ϕn), and transverse unit vectors ˆun= ˆex= (1,0,0),ˆvn= ˆnn׈un= (0,cos ϕn,−sin ϕn). The lab-frame polarization matrices for wave nare then (e(n) +)∗ij = (ˆun⊗ˆun−ˆvn⊗ˆvn)∗ij, (e(n) ×)∗ij = (ˆun⊗ˆvn+ ˆvn⊗ˆun)∗ij. (50) b) Pulse model: Each pulse nhas h+,n(t) = Anfn(t−tn) cos(ωn(t−tn) + αn)and similarly for h×,n. Here fnis the envelope (e.g., Gaussian or onecycle window), tn=n∆Tis the arrival time and αna phase. The relevant driver for the MPD torque is ¨ h(n) ij (t)computed from these functions. c) Numerical integration of reduced dynamics: 1. Choose a(lapse), CSSC, and initial spin S(0) = S0ˆz. 2. For each time step, assemble ¨ hij(t) = Pn¨ h(n) ij (t)using polarization matrices. 3. Compute dSi/dt =−(CSSC/2a),¨ hij(t)Sj(or the more complete influence-functional-derived deterministic part which includes temporal convolution). 4. Integrate using a stable integrator (e.g., RK4) for the required time window. 5. Extract ∆Safter the last pulse; repeat for different ∆Tto map interference patterns. This recipe directly uses the derivative-based source and is consistent with the action-based derivations above. F: ANGULAR-MOMENTUM FLUX CALCULATION To compute the compensating ∆JGW associated with spin change, evaluate the angular-momentum flux carried by outgoing gravitational radiation. The angular-momentum flux density in linearized gravity is determined from the Landau–Lifshitz pseudotensor or from the Noether currents associated with spatial rotations applied to the quadratic GW action. For TT radiation fields in the far zone, the instantaneous angular-momentum flux per unit solid angle can be written (schematically) as d2Ji dtdΩ∼ϵijk, xjdEk dt , with dEk/dt the momentum flux density (quadratic in ˙ hlm). For the dipole-like spin-dependent contributions, extract the cross-terms between the spinsourced radiation piece and the background pulse fields to obtain the net flux and integrate over time and angles to obtain ∆JGW. The sign and magnitude will match −∆Jspin by conservation. 17 G: DISCUSSION AND FUTURE DIRECTIONS We summarize key limitations: (i) the MPD/pole– dipole truncation neglects finite-size quadrupole moments that may be important for extended bodies; (ii) Tulczyjew SSC choice fixes prefactors but alternative SSCs produce O(1) changes; (iii) the influence-functional expansion assumes weak coupling (linearized gravity), which is appropriate for the small-hcontexts of interest but must be revisited near strong-field sources. 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