On the Dynamics of Extended Bodies and the GravoThermo Memory Effect

Ali Shojai; Raihaneh Moti

REVIEW 3 major objections 4 minor 29 references

On the Dynamics of Extended Bodies and the GravoThermo Memory Effect

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper claims that a gravitational wave pulse permanently shifts the azimuthal orientation of an extended body's spin and writes its polarization into the body's thermodynamic state.

desk verdict The claimed permanent spin memory vanishes on the paper's own equations: G(v) in Eq. (36) is odd, so Δφ and the thermodynamic results don't follow as printed. read the letter →

arxiv 2508.19231 v2 pith:EQNYYHR5 submitted 2025-08-26 gr-qc hep-th

Raihaneh Moti , Ali Shojai This is my paper · ORCID

classification gr-qchep-th

keywords gravitationalwavememoryextendedbodiespole-dipoleequationsspinpartitionfunctionthermodynamicpolarizationnonlinear

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that gravitational wave memory acts on extended rotating bodies, not just on point particles: after a Gaussian gravitational wave pulse passes, the angle between a body's spin and the wave direction returns to its original value, but the spin's azimuthal orientation in the plane perpendicular to the wave does not, leaving a permanent small shift. It further claims that when many such bodies with random orientations form an ensemble, the pulse causes a permanent redistribution of spin orientations, and that redistribution shows up as changes in the partition function, entropy, and internal energy that depend on the wave's polarization ratio. The paper also claims the body re-radiates a small gravitational wave, so the original wave acquires a nonlinear memory component. A sympathetic reader would care because the result suggests that thermodynamic systems can record the past passage of a gravitational wave and encode information about its source.

What carries the argument

The load-bearing object is the spin orientation in the plane transverse to the wave, specifically the azimuthal angle $\varphi$ of the spin vector. The argument is carried by an order-by-order perturbation of the pole-dipole equations of motion for an extended body around a flat background, with zeroth-order momentum fixed to $P^\mu_{(0)}=(1,0,0,0)$ and the zeroth-order spin confined to the $XZ$ plane; the gravitational wave enters as a small metric perturbation with plus and cross polarizations, chosen as Gaussian pulses $h_+(t,z) = H_+ e^{-(z-ct)^2/2\sigma^2} e^{ik(z-ct)}$ and similarly for $h_\times$. The pulse determines forcing functions $F(v)$ and $G(v)$, and integrating the perturbed spin equations $dS_x/dt = g_2$ and $dS_y/dt = g_1$ plus the position equations numerically produces the permanent $\varphi$ shift. The same perturbed trajectory feeds the quadrupole formula for the re-radiated field, yielding the nonlinear wave memory.

What would settle it

Repeat the first-order calculation with quadrupole-moment terms included (the next order in the multipole expansion) for the same Gaussian pulse: if the permanent $\Delta\varphi$ vanishes or becomes initial-velocity dependent instead of initial-spin dependent, the reported memory is an artifact of pole-dipole truncation. A second check is to measure the final azimuth of a spinning test mass after a calibrated pulse in a tabletop experiment, where the predicted shifts for the plotted parameters with $\epsilon=0.1$ are around $10^{-6}$ degrees; a null result at that level would rule the claim out.

Watch

Extended reading notes

Core claim

On the paper's terms, the central discovery is that the dynamics of an extended rotating body at pole-dipole (spin-only) order contains a gravitational memory: integrating the first-order perturbed equations of motion in the field of a Gaussian gravitational wave pulse yields a net change $\Delta\varphi$ in the azimuthal spin orientation while the polar angle $\theta$ returns to its initial value, and the sign and magnitude of $\Delta\varphi$ oscillate with the initial spin angle $\alpha$ and grow with the polarization ratio $\varsigma = H_+/H_\times$. The same integration gives a permanent longitudinal jump and a spiral transverse motion of the body's reference point, so the memory appears in both rotational and linear degrees of freedom. For an ensemble of randomly oriented bodies in a magnetic field, the paper argues that the wave-induced reorientation changes the partition function, producing a polarization-dependent entropy change and, for strongly plus-polarized waves, a resonance in internal energy near the characteristic temperature; this is presented as a gravo-thermo memory that encodes information about the wave and its source.

Load-bearing premise

The result assumes that during the pulse each body is adequately described as a spinning object with no quadrupole or higher internal structure, with its momentum locked to the rest frame $P^\mu_{(0)}=(1,0,0,0)$, and that in the ensemble intermolecular forces exactly cancel the wave-induced linear motion; if any of these fail, the permanent azimuthal shift and thermodynamic memory could be artifacts of the truncation.

Editorial extensions

If this is right

Every extended rotating body that encounters a gravitational wave pulse should retain a small, permanent azimuthal spin shift while its polar alignment is restored; the shift oscillates with initial orientation and increases with the plus-to-cross polarization ratio.
An ensemble of such bodies will have a permanent redistribution of spin orientations, so thermodynamic state variables after the pulse differ from their initial values even after the wave has gone.
The entropy change grows as the wave becomes more plus-polarized, and for strongly plus-polarized waves the internal energy exhibits a temperature resonance near the system's characteristic temperature.
A body on the wave's path re-radiates a small transverse-traceless field, so the wave itself acquires nonlinear memory; for the Earth and a 10 kHz merger signal the extra amplitude is about $2.7 \times 10^{-3}/r$.
Although a single body's $\Delta\varphi$ is a rapidly oscillating function of initial orientation, averaging over an ensemble gives smooth thermodynamic curves, making the effect statistically observable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the pole-dipole truncation is reliable, the same mechanism should appear in any geometric theory with a nonlinear tensor field that couples to spin; the memory is a fingerprint of nonlinearity rather than of a specific metric theory.
One testable extension the paper does not pursue: replace the assumption of exactly balanced intermolecular forces with a dilute gas or suspension so translational and rotational memory compete; the resulting partition-function shift could be larger or partially randomized, a difference that can be checked by simulation.
The oscillatory $\Delta\varphi(\alpha)$ curve suggests that a sufficiently large ensemble could act as a polarimeter: the statistics of final spin orientations encode the polarization ratio $\varsigma$, so thermodynamic measurements might infer source properties of a past gravitational wave.
The paper's point-particle comparison implies that taking the body size to zero should recover the earlier memory result, so comparing the extended-body and point-particle curves would isolate the role of tidal coupling in the memory.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on 2508.19231. The main result, as written, is not supported by the equations. In Section III the spin evolution is dS_x/dt = g2, dS_y/dt = g1, with g1,g2 proportional to G(v) from Eq. (36). For the Gaussian pulse, G(v) = exp(-v^2/2σ^2)(σ^2 sin v + v cos v) is an odd function of v, so its integral over the pulse vanishes. That means ΔS_x = ΔS_y = 0 and there is no permanent change in the transverse spin orientation at first order. Figures 2, 6, and everything downstream (the partition function, entropy, energy) are not reproducible from the printed equations. This looks like a typo — the real part of (σ^2 - i v)e^{iv} = σ^2 cos v + v sin v has the right parity and would give a nonzero memory — but as submitted the central claim doesn't follow.

The paper has real strengths. The extension of Mathisson-Papapetrou to first order in the GW perturbation is a sensible setup, the progression from point particles [19] to extended bodies and then to thermodynamics is natural, and the authors are appropriately restrained about detectability (0.27% per unit distance for Earth is honestly tiny). The thermodynamic imprint idea is genuinely worth thinking about.

The soft spots beyond the parity error: Section IV is mislabeled. The scattered radiation computed there is transient — it has the same Gaussian support as the incoming pulse — so calling it 'nonlinear memory' is wrong. The equations (17)-(21) are asserted as 'straightforward' without derivation; that's minor if the rest is sound, but with the central issue it matters. The numerics have no convergence checks or error bars. And the ensemble analysis assumes intermolecular forces exactly cancel the wave-induced linear motion; that's a strong assumption, stated in a footnote-like sentence, and it carries the thermodynamic calculation.

Conclusion: this deserves a serious referee, because the idea is interesting and the flaw is likely fixable. Send it to review with a request to check the parity of Eq. (36) and recompute the numerics. I wouldn't cite it in its current form, but after a corrected version it could be a reasonable contribution. The reader's 'conditional' verdict is fair — I'd push it closer to 'major revision, possibly reject if the typo isn't the issue.'

Referee Report

3 major / 4 minor

Summary. The paper studies the motion of an extended spinning body described by the Mathisson-Papapetrou equations in a linearly polarized gravitational-wave pulse, working to first order in the wave amplitude. It claims that after the pulse the body retains a permanent change in the azimuthal orientation of its spin, that this change is imprinted on the gravitational wave through reradiation, and that an ensemble of such bodies acquires a permanent change in its partition function, entropy, and internal energy. The central quantitative results are a purported finite spin memory, shown in Figures 2 and 6, and the thermodynamic memory derived from it in Section V.

Significance. If the claims were correct, the paper would extend gravitational memory from point particles to extended bodies and would connect memory to thermodynamics of an ensemble, which is an interesting and potentially useful direction. The paper is self-contained in the sense that no free parameter is fitted to the thermodynamic output, and the mathematical framework is standard. However, the central spin-memory result does not follow from the equations as written: the forcing function G(v) in Eq. (36) is odd in v, so its integral over the pulse vanishes. Consequently, the claimed permanent azimuthal spin shift and all thermodynamic consequences derived from it are not supported by the presented calculation. The paper also relies on a set of first-order equations, Eqs. (17)-(21), that are asserted without derivation, and the numerical plots are presented without error estimates or convergence checks. The underlying idea is salvageable, but the manuscript in its current form does not establish its main claim.

major comments (3)

[Section III, Eqs. (36) and (38)] The permanent spin-memory result is contradicted by the written equations. Equations (38) give dS_x/dt = g_2 and dS_y/dt = g_1, with g_1 and g_2 proportional to G(v) from Eq. (36). Since G(v) = exp(-v^2/2σ^2)(σ^2 sin v + v cos v) is an odd function of v, the integral over the pulse at fixed z vanishes: ∫_{-∞}^{∞} G(v) dv = 0. The S_12 component in Eq. (27) is zero, so S_z is unchanged at first order as well. Therefore the spin returns to its initial orientation, and the permanent Δφ shown in Figures 2 and 6 does not follow from the printed equations. Please correct the expression for G(v) (or the real-part convention used to define h_+ and h_×) and derive the resulting Δφ analytically, or explain explicitly why the first-order spin shift is nonzero despite the vanishing integral.
[Section III, Eqs. (17)-(21)] The first-order equations that drive the entire analysis are stated as 'straightforward to show' but are not derived. These equations determine P_μ^{(1)}, S_{μν}^{(1)}, and U_μ^{(1)}, and any sign or factor error in them propagates into the forcing functions in Eqs. (31)-(36). For example, Eq. (27) sets S_{31}^{(1)} = -ϵ g_1, while Eq. (38) gives dS_y/dt = g_1; if S_y is identified with S_{31}, the sign is inconsistent, and if S_y is instead identified with S_{13}, the identification should be stated. Please provide the derivation of Eqs. (17)-(21) or a supplementary calculation so that the forcing terms can be checked.
[Section V, Eqs. (53)-(59)] The thermodynamic memory is built entirely on the numerical values of Δφ_i(θ_i). Since the first-order calculation gives Δφ = 0 when Eq. (36) is used, the entropy and energy changes plotted in Figures 11-13 have no basis unless the spin-memory calculation is corrected. Additionally, the paper does not state how Δφ_i(θ_i) was extracted from the numerical solutions or how the averaging over θ_i was implemented; please specify the discretization, the number of samples, and the numerical accuracy. Without this information, the claimed smooth thermodynamic behavior cannot be reproduced.

minor comments (4)

[Section IV, Eq. (50)] The estimate H_{rad}/H_{in} ~ 2.7 × 10^{-3}/r is dimensionally ambiguous. The paper states 'per unit distance from earth' but r is a length; please specify the units of r and the values of ς, ν, and the Earth mass and size used to obtain this number.
[Section IV, Eq. (46)] The term 'nonlinear memory' for the reradiated field is misleading: the body motion is linear in the incoming wave amplitude, and the quadrupole formula used here is also linear in the source's second moment. Calling this 'nonlinear memory' conflates it with the standard Christodoulou memory, which is a nonlinear effect in the gravitational field itself. Please clarify the terminology.
[Figures 2, 6, 9] The numerical plots are presented without error bars, convergence checks, or a description of the numerical integrator. Given that the central claim depends on tiny residual values (about 10^{-6} degrees), the absence of such checks makes it impossible to assess whether the plotted memory is numerical noise.
[Section III, Eq. (27)] The notation for the spin components is not defined consistently. Please state explicitly which components of S^{μν} correspond to S_x, S_y, and S_z in both Eqs. (24) and (27).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the spin and thermodynamic memory results are derived from the Mathisson-Papapetrou equations with no fitted parameters; the authors' prior work appears only as motivation and comparison.

full rationale

The claimed chain is self-contained. Section III expands the Mathisson-Papapetrou equations to first order in the wave amplitude and solves Eqs. (37)-(39) with explicitly written pulse functions; the unperturbed momentum and spin are chosen as initial data, not as outputs. The permanent spin change is obtained by integrating the displayed differential equations, and the Section V partition function is constructed by inserting the numerically obtained Delta-phi_i(Theta_i) into Eq. (56), so the thermodynamic quantities are consequences of the preceding dynamics rather than inputs to it. No parameter is fitted to the entropy or energy curves. References [19] and [20] are the authors' previous papers, but they are cited as a toy-model motivation and as a comparison, not as a premise of the extended-body calculation; the load-bearing equations (17)-(21) and (31)-(38) are derived here. The explicit Section V limitation that inter-molecular forces balance the wave-induced linear motion is a modeling assumption, not a circular step. A separate numerical-consistency concern about the zero integral of the G(v) in Eq. (36) is a correctness issue rather than circularity and is therefore not counted in this score.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new particles, fields, or conserved quantities are introduced. The memory parameter is a derived quantity, not a new entity. The free parameters listed are numerical choices for the plots rather than constants fitted to data, but the quantitative thermodynamic curves depend on them.

free parameters (4)

perturbation amplitude epsilon = 0.1
Chosen for the numerical plots in Section III and never varied; the linearized expansion assumes it is small.
normalized pulse width k sigma = 10
Chosen for all plots in Section III; the memory magnitude and its oscillatory dependence on initial orientation can depend on pulse width.
initial spin angle alpha = 35 degrees
Used in the illustrative dynamics plots; Section V later treats the spin angle as a distributed initial condition in the ensemble.
polarization ratio ς = H+/H× = Values 0.3, 0.5, 1, 2, 3, 10
Varied to explore plus-polarized versus cross-polarized waves; no observational value is derived.

assumptions (5)

domain assumption The Mathisson-Papapetrou-Dixon pole-dipole truncation, with the spin supplementary condition PμSμν=0 and conserved charges m² and s², governs the extended body.
Invoked in Section II to replace the full energy-momentum conservation; if higher multipoles contribute during the pulse, the derived equations miss them.
domain assumption The gravitational wave is a weak linearized perturbation hμν of Minkowski spacetime with Gaussian pulse profiles h+(t,z) and h×(t,z).
Introduced in Section III, Eq. (10) and Eqs. (29)-(30); all results are first order in epsilon.
ad hoc to paper The reference point has initial four-momentum P(0)μ=(1,0,0,0) and spin confined to the XZ-plane.
Chosen in Section III, Eqs. (22)-(24), as initial conditions; the claimed memory may depend on this frame choice.
ad hoc to paper For the ensemble, intermolecular interactions balance the wave-induced linear motion so that only spin reorientation contributes to the thermodynamic changes.
Stated in Section V before Eq. (52); this isolates rotation but is a strong modeling assumption.
standard math The standard quadrupole formula (40) estimates the radiation emitted by the body.
Used in Section IV to estimate hrad; this is a standard weak-field result from Ref. [29].

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Cite this review

Pith. "Pith review of On the Dynamics of Extended Bodies and the GravoThermo Memory Effect." pith.science (2026). https://pith.science/paper/EQNYYHR5

@misc{pith2026250819231,
  author       = {Pith},
  title        = {Pith review of: On the Dynamics of Extended Bodies and the GravoThermo Memory Effect},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQNYYHR5}},
  note         = {Machine review of arXiv:2508.19231}
}

read the original abstract

The non-linearity of the theory of gravity induces a hysteresis effect in both the systems interacting with gravity and in the gravitational field. The effect is usually referred to as the memory effect. In this paper, we explore this phenomenon in the context of the dynamics of extended rotating objects in the presence of a gravitational wave pulse. Then, we consider an ensemble of such objects and show that a redistribution of spin orientations takes place as a gravitational wave passes through. We examine how this redistribution, changes the partition function and the thermodynamic quantities including entropy and energy of the system. The result shows that some information about the source of the gravitation wave is encoded in the thermodynamics of the system.

Figures

Figures reproduced from arXiv: 2508.19231 by the authors.

**Figure 1.** Change in θ orientation for ς = 10. -100 -50 50 100 kt -2 2 4 6 ϕ in degrees Spin orientation -100 -95 -90 -85 -80 -4 × 10-13 -3 × 10-13 -2 × 10-13 -1 × 10-13 0 Starting orientation 80 85 90 95 100 -1.41 × 10-6 -1.41 × 10-6 -1.41 × 10-6 -1.41 × 10-6 -1.41 × 10-6 Ending orientation [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

**Figure 2.** Change in ϕ orientation for ς = 10. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Transverse motion of the reference point for [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Longitudal motion of the reference point for [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: Change in θ orientation for ς = 0.3. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

**Figure 6.** Figure 6: Change in ϕ orientation for ς = 0.3 [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: Transverse motion of the reference point for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Longitudal motion of the reference point for [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]

**Figure 9.** Figure 9: Memory of ϕ-angle of spin for ς = 3. In section V, we will use an ensemble of extended bodies and study their thermodynamics. For such randomly oriented bodies, this highly oscillatory nature is integrated out leading to a smooth contribution to the thermodynamic quant…

**Figure 10.** Figure 10: An ensemble of rotating objects with initial random orientations in constant mag [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]

**Figure 11.** Figure 11: Footprint of the gravitational wave passage in the entropy. [PITH_FULL_IMAGE:figures/full_fig_p022_11.png]

**Figure 12.** Figure 12: Change in the internal energy for small values of [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]

**Figure 13.** Figure 13: Change in the internal energy for large values of [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [19]

On the Gravitational Precession Memory Effect for an Ensemble of Gyroscopes

R. Moti and A. Shojai, On the gravitational precession memory effect for an ensemble of gyroscopes, Class. Quant. Grav. 41 (2024) 025011 [arXiv:2307.04151]

work page Pith review arXiv 2024

[2] [1]

Zel’dovich and A.G

Y.B. Zel’dovich and A.G. Polnarev, Radiation of gravitational waves by a cluster of super- dense stars, Sov. Astron. 18 (1974) 17

work page 1974

[3] [2]

Braginsky and K.S

V.B. Braginsky and K.S. Thorne, Gravitational-wave bursts with memory and experimental prospects, Nature 327 (1987) 123

1987

[4] [3]

Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments , Phys

D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments , Phys. Rev. Lett. 67 (1991) 1486

1991

[5] [4]

Blanchet and T

L. Blanchet and T. Damour, Hereditary effects in gravitational radiation, Phys. Rev. D 46 (1992) 4304. 25

work page 1992

[6] [5]

K. S. Thorne, Gravitational-wave bursts with memory: The Christodoulou effect , Phys. Rev. D 45 (1992) 520

work page 1992

[7] [6]

The First-Order Velocity Memory Effect from Compact Binary Coalescing Sources

A.K. Divakarla and B.F. Whiting, First-order velocity memory effect from compact binary coalescing sources, Phys. Rev. D 104 (2021) 064001 [arXiv:2106.05163]

work page Pith review arXiv 2021

[8] [7]

Grishchuk and A.G

L.P. Grishchuk and A.G. Polnarev, Gravitational wave pulses with ‘velocity coded memory’, Sov. Phys. JETP 69 (1989) 653

work page 1989