{"id":"c354d6aa-4886-478a-9d0c-a798ea893979","arxiv_id":"1908.07823","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a Hooke's law solid, a sound wave packet produces a net backward displacement of matter, equivalent to a mass deficit M = -(9/8) E/c_L^2 per unit area.","lead":"A theoretical analysis shows that a sound wave packet in a solid pushes the material slightly backward, creating a moving mass deficit. The result confirms a recent claim but gives a different, geometry-specific coefficient, and challenges the earlier derivation's method.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (65) conflicts with Ref [1]'s coefficient for the same model, but the direct derivation is internally sound; an independent check would settle it.","rationale":"The reader's weakest assumption was the Hooke's-law form of the potential, which I agree is a scope limitation but not a flaw in the central claim as stated. The more pointed issue is the unresolved discrepancy with Ref [1]'s coefficient, which the paper flags but does not definitively close. However, Soper's own derivation is self-contained and algebraically consistent: the stress expansion (17), the sourced equation (21), the solution (36)-(46), and the mass integral (64) all line up. The coefficient 9/8 is independent of λ and μ, consistent with the plane-wave uniaxial-strain setup which only involves λ+2μ. I therefore cannot identify a load-bearing internal error. The proposed numerical test would be a worthwhile independent check, but I would not change the ACCEPT verdict on the strength of the current concern.","tokens_in":7558,"tokens_out":45421,"duration_ms":523683,"concrete_test":"Numerically integrate the one-dimensional equations derived from Eq. (6) (either the exact finite-strain equations or the second-order system (20)-(21)) for the localized source (29), using at least two distinct ratios λ/μ such as λ=μ and μ=0. Extract the asymptotic permanent displacement q_after and the energy per unit area E, and compare c_L^2 q_after/E with -9/8. If it instead tracks (13λ+14μ)/(6(λ+2μ)), Eq. (65) is wrong.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only substantive risk is the unresolved quantitative disagreement with Ref [1]. For the same Hooke's law potential, Soper's Eq. (51) evaluates Ref [1]'s C as (13λ+14μ)/(6(λ+2μ)), which never equals 9/8 for positive elastic constants. The paper answers by criticizing Ref [1]'s time-averaging procedure, but it does not independently verify that this criticism fully explains the factor. I checked the chain from Eq. (17) through Eq. (46) and the mass integral (64); it is internally consistent, and the coefficient 9/8 is a direct consequence of the retained cubic terms in the Hooke's law Lagrangian. Thus I have no internal-error objection. The Hooke's-law specificity is a scope limit, not a flaw: the abstract and Section 2 state it explicitly.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a direct second-order perturbative analysis of longitudinal sound wave packets in a homogeneous isotropic Hooke's law solid. Using the Lagrangian formulation of continuum mechanics, the author solves the equations of motion to second order in the wave amplitude and finds a net backward displacement of the material after the packet passes, corresponding to a negative effective mass per unit area M = -(9/8) E/c_L^2 carried by the wave. The paper also compares the result with Ref. [1], finds a different coefficient, and attributes the discrepancy to Ref. [1]'s time-averaging procedure.","tokens_in":7732,"tokens_out":8050,"duration_ms":82829,"significance":"This is a clean, self-contained derivation of a surprising effect. The calculation is explicitly parameter-free: the coefficient 9/8 emerges from the Hooke's law Lagrangian rather than being fitted. The paper gives a concrete physical picture (atoms return to rest but are displaced backward) and provides a quantitative estimate for a gel-like material. It serves as an important check on the recent claim that sound waves carry gravitational mass, although the coefficient differs from Ref. [1]. The main limitations, restriction to Hooke's law and to plane-wave geometry, are clearly stated in the abstract and Section 2.","major_comments":[],"minor_comments":[{"comment":"The critique of Ref. [1]'s time-averaging is stated qualitatively ('I am not able to justify this method of attack') but the paper does not identify the precise step in Ref. [1] where a term is incorrectly dropped. Since this section explicitly aims to explain the difference in coefficients, please provide a more detailed diagnosis or state clearly that the source of the discrepancy remains unresolved.","section":"RESULT OF REF. [1]"},{"comment":"In Eq. (29), the notation θ(-R < x+ < R) is ambiguous; it should be written as a product of step functions, e.g., θ(x+ + R) θ(R - x+), to avoid confusion.","section":"SOLUTION OF THE EQUATIONS OF MOTION"},{"comment":"The leap from the nonrelativistic mass deficit to the gravitational coupling is asserted rather than derived. A short explanation of why the nonrelativistic treatment suffices for the gravitational mass claim would help the reader.","section":"CONNECTION TO GRAVITY"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a well-written, careful calculation. The unresolved disagreement with Ref. [1] is a concern, but it does not undermine the internal consistency of the derivation. I believe it is publishable after minor revisions to clarify the comparison with Ref. [1]. The paper is likely to attract attention given the recent PRL."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Soper calculates the mass carried by a longitudinal sound wave packet in a Hooke's law solid and gets 9/8 times the energy over c_L^2, with a minus sign. The PRL (Esposito et al.) had a different coefficient, d log c_L / d log rho. For the same Hooke's law potential, Soper evaluates that expression as (13λ+14μ)/(6(λ+2μ)), which is sensitive to the ratio λ/μ and never 9/8 for positive constants. So there's a real conflict.\n\nWhat's new and good: The derivation is explicit and self-contained. He starts from a standard Lagrangian, writes the displacement as φ + ψ, solves the equations of motion to second order in the source, and shows that ψ asymptotes to a constant proportional to the energy integral. The coefficient 9/8 is a direct consequence of the cubic terms in the Lagrangian, not an input. He also gives a simple physical story: atoms are displaced permanently backward, so the material density in the packet is reduced. The math looks internally consistent; I checked equations (16)–(21) and the mass integral (64) and found no errors. He is upfront that the Hooke's law assumption fixes the coefficient and that a real material with anharmonic terms would give a different answer. That is a scope limitation, not a flaw.\n\nThe soft spot: The unresolved disagreement with Ref [1]. Soper criticizes their time-averaging method, saying it drops terms that are not small, but he doesn't demonstrate exactly which step in their derivation fails. That is a gap. It might be that both are right but describing different observables—his plane-wave setup versus their generic 3D averaging—but he doesn't fully address that either. The paper would be stronger with a back-to-back calculation on the same wave packet using their averaging, to show directly where the factor appears.\n\nWho it's for: people working on effective mass in condensed matter or on nonlinear elasticity. It deserves a serious referee; the derivation is careful and the result challenges a published PRL. I'd recommend accepting it after the authors either pin down the discrepancy or at least state more precisely what in Ref [1] they think is wrong.","headline":"A clean perturbative derivation that yields 9/8 for the sound-wave mass, but the paper never fully explains the discrepancy with the PRL's density-derivative formula.","tokens_in":8203,"tokens_out":3272,"would_cite":true,"duration_ms":34782,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["62.30.+d"],"model":"deepseek-v4-flash","headline":"In a Hooke's-law solid, a sound wave packet permanently shifts material backward and carries negative mass 9/8 E/c_L^2.","keywords":["sound waves","negative mass","mass transport","Hooke's law solid","nonlinear elasticity","continuum mechanics","wave packet","momentum conservation"],"falsifier":"Send a calibrated longitudinal sound pulse down a long rod of known cross-section $A$, density $\\rho$, sound speed $c_L$, and total energy $E_{\\mathrm{tot}}$, and measure the net displacement of the far end after the pulse has left; the $9/8$ claim predicts a permanent backward displacement of $\\frac{9}{8}\\frac{E_{\\mathrm{tot}}}{A\\rho c_L^2}$, so a measured displacement clearly different from this would rule out the claim.","tokens_in":7395,"feed_emoji":"🔊","tokens_out":10420,"duration_ms":104602,"temperature":0.7,"pith_summary":"This paper establishes that a longitudinal sound-wave packet moving through an isotropic Hooke's-law solid leaves the material permanently displaced in the direction opposite to its motion, even though no net momentum is transferred. At second order in the wave amplitude, the packet carries a negative mass per unit area $M = -(9/8)E/c_L^2$, where $E$ is the energy per unit area and $c_L$ is the longitudinal sound speed. The mechanism is ordinary nonlinear elasticity together with momentum conservation, so no relativistic or gravitational input is needed. If correct, the result differs from the coefficient found in a recent Letter that reported gravitational mass carried by sound, replacing $d\\log c_L/d\\log\\rho$ with $9/8$ for a Hooke's solid. The coefficient depends on the assumed elastic energy, so it is not universal.","feed_headline":"Sound wave packets carry negative mass in ordinary solids","feed_subtitle":"Passing sound permanently shifts a Hooke's-law solid backward by a mass deficit of 9/8 E/c_L^2.","key_machinery":"The central machinery is the Lagrangian field theory of continuum mechanics: atoms carry labels $R^a(x)$, the current $J^\\mu$ and strain $s_{ab}$ are built from $\\partial_\\alpha R^a$, and a Hooke's-law potential $U = \\frac{1}{2n}[\\lambda(s_{aa})^2 + 2\\mu s_{ab}s_{ab}]$ fixes the elastic response. Expanding $R^a = x^a + \\phi^a + \\psi^a$ and using momentum conservation $\\partial_0 T^{30} + \\partial_3 T^{33} = f^3$ in the symmetric plane-wave geometry gives linear wave equations for $\\phi^3$ and a driven wave equation for $\\psi^3$. In coordinates $x_\\pm = c_L t \\pm z$, the second-order field evaluates after the packet to $\\psi_3 = \\frac{9}{8}\\int dx^3\\,(\\partial_3\\phi_3)^2$, the identity that carries the argument. This is the mechanism: quadratic terms in the stress produce a secular, time-independent second-order displacement rather than a momentum transfer.","core_discovery":"The paper's central claim is that a plane sound wave packet in a Hooke's-law solid, with amplitude independent of $x$ and $y$ and traveling in the $+z$ direction, produces a small net backward motion of the material. Writing the displacement as $R^a = x^a + \\phi^a + \\psi^a$, where $\\phi$ is first order in the source and $\\psi$ is second order, the author solves the momentum-conservation equation exactly to that order and finds that after the packet has passed, $\\psi_3$ approaches a nonzero constant, $\\psi_3 = \\frac{9}{8 mn c_L^2}\\int dx^3\\,T^{00}$. Consequently, the mass per unit area associated with the packet is $M = -mn\\psi_3 = -(9/8)E/c_L^2$. The paper also computes the corresponding average material velocity inside the packet and notes that this $9/8$ coefficient disagrees with the earlier Letter's $C = d\\log c_L/d\\log\\rho = (13\\lambda+14\\mu)/(6(\\lambda+2\\mu))$ for arbitrary $\\mu/\\lambda$.","pith_inferences":["Because the coefficient $9/8$ comes from the harmonic elastic energy, measuring the backward shift of a rod after a sound pulse could serve as a probe of anharmonic elastic constants, with departures from $9/8$ signalling non-Hooke behavior.","Repeated pulses should make the per-pulse displacement accumulate, so a train of pulses or a resonant cavity could amplify the effect enough for precision displacement measurements.","Since $E/c_L^2$ exceeds $E/c^2$ by roughly $(c/c_L)^2$, the negative mass associated with sound is much larger than the relativistic rest-mass equivalent of its energy; if sound sources gravity, this enhancement may be testable in torsion-balance or optomechanical experiments."],"forward_implications":["A longitudinal sound pulse of energy per unit area $E$ carries a negative mass per unit area $-(9/8)E/c_L^2$, so a sound-bearing solid behaves as if its mass were reduced while the pulse is inside it.","The second-order displacement is permanent: after the packet passes, the material is at rest but shifted in the $-z$ direction, so no net momentum is left behind.","The result is independent of the detailed shape of the packet once its total energy is fixed, at least for source profiles satisfying the paper's zero-net-force condition.","For a 100 W/m$^2$ sound pulse in a water-like gel, the average material velocity inside the packet is about $-5\\times10^{-8}$ m/s, fixing the scale of the effect."],"supporting_citations":[{"why":"The prior result that sound waves carry gravitational mass, which this paper rederives in a restricted plane-wave geometry and with which it compares coefficients.","marker":"[1]"},{"why":"The analogous phonon-in-superfluid result that motivates the same physical question for solids.","marker":"[2]"},{"why":"The Lagrangian continuum-mechanics formulation that supplies the definitions of current, strain, stress tensor, and energy density used throughout.","marker":"[3]"}],"fun_headline_variants":["Sound wave packets shift Hooke-law solids backward","Plane sound waves drag matter opposite wave travel","Sound wave packets carry a 9/8 E/c_L^2 mass deficit","Net backward mass drift from passing sound wave packets","Sound waves shift solid mass opposite to propagation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the Hooke's-law form of the elastic energy, Eq. (6), which fixes the nonlinear coefficients that produce the $9/8$; with any other stress-strain relation the coefficient changes, so the specific number is not universal.","fun_headline_variants_meta":{"raw":{"variants":["Sound wave packets shift Hooke-law solids backward","Plane sound waves drag matter opposite wave travel","Sound wave packets carry a 9/8 E/c_L^2 mass deficit","Net backward mass drift from passing sound wave packets","Sound waves shift solid mass opposite to propagation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000514,"raw_usage":{"total_tokens":2448,"prompt_tokens":847,"completion_tokens":1601,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":463,"completion_tokens_details":{"reasoning_tokens":1524}},"tokens_in":463,"tokens_out":1601,"duration_ms":11949,"temperature":1.0,"reasoning_tokens":1524,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:43:26.044809+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Send a calibrated longitudinal sound pulse down a long rod of known cross-section $A$, density $\\rho$, sound speed $c_L$, and total energy $E_{\\mathrm{tot}}$, and measure the net displacement of the far end after the pulse has left; the $9/8$ claim predicts a permanent backward displacement of $\\frac{9}{8}\\frac{E_{\\mathrm{tot}}}{A\\rho c_L^2}$, so a measured displacement clearly different from this would rule out the claim.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The Lagrangian continuum-mechanics formulation that supplies the definitions of current, strain, stress tensor, and energy density used throughout."}],"review_version":1}