{"id":"402bb37e-9829-4707-9621-91d8cfe0c952","arxiv_id":"1908.05431","paper_version":3,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In the linearized regime of nonlocal gravity, the gravitomagnetic field of a rotating mass acquires a tiny nonlocal correction, below one part in 10^10 of the general-relativistic value for Earth and Moon.","lead":"This paper works out the gravitomagnetic (frame-dragging) side of nonlocal gravity, a modified theory of gravity, for a rotating star or planet. It finds the corrections are at most ten orders of magnitude smaller than Einstein's theory predicts, so current experiments cannot see them.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the nonlocal gravitomagnetic bound is algebraically consistent and robust to kernel-parameter uncertainty.","rationale":"The paper's central claim is that the nonlocal gravitomagnetic field of the Earth is at most ten orders of magnitude smaller than the GR value. The derivation of this bound is internally consistent: the linearized field equations, the use of the reciprocal kernels q1 and q2, the multipole expansion for a compact source, and the resulting expressions for A and B_g in Eqs. (80) and (82) all check out. The reader's weakest assumption—that the fitted parameters λ0, μ0, and a0 carry unpropagated uncertainties—is a real caveat, but it does not threaten the central conclusion. Because the ratio scales as r²/LN² and even the most conservative allowed parameter values put LN at the 0.1–1 pc scale, the ratio at Earth-system distances is 10⁻¹³ or smaller, leaving several orders of margin below the 10⁻¹⁰ ceiling. To overturn the bound one would need LN of order 10⁴ km, incompatible with the galactic scales that fix λ0 and μ0 and with the solar-system lower bound on a0. Thus the missing error bars are a presentation issue rather than a load-bearing correctness risk. The paper's larger limitations—no exact solutions, linearized treatment, and a kernel fitted to observations rather than derived from first principles—are explicitly acknowledged in the text and are appropriate for a phenomenological comparison. I therefore find no reason to change the reader's conditional verdict.","tokens_in":17079,"tokens_out":12110,"duration_ms":127213,"concrete_test":"Recompute the nonlocal-to-GR ratio from Eq. (82) at LEO, GPS, and lunar distances using the extreme allowed values of λ0 (0.1–10 kpc), μ0⁻¹ (5–50 kpc), and a0 (1–10³ AU), and check whether r²/LN² ever exceeds 10⁻¹⁰; if it does not, the bound is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"After checking the derivation from Eq. (59) through Eqs. (80)–(82), I find no internal inconsistency or numerical error that would change the central claim. For the q1 kernel, the nonlocal-to-GR gravitomagnetic ratio is r²/LN², where LN = (2λ0a0/(1+ς))^{1/2}; for the q2 kernel the leading nonlocal term vanishes because ∫j d³y = 0. With λ0 ≈ 3 kpc and a0 ≳ the size of the solar system, LN is at least roughly 0.1–1 pc, giving r²/LN² ≈ 10⁻¹⁶ at Earth's surface and ≈10⁻¹³ at the Moon. These are already below the paper's stated 10⁻¹⁰ ceiling by several orders of magnitude, and the claim is an upper bound, not a precision prediction. The reader's concern about unpropagated uncertainties in λ0, μ0, and a0 is a legitimate reporting weakness, but it is not load-bearing: order-of-magnitude changes in these fitted parameters cannot move the ratio above 10⁻¹⁰ without violating the galactic-rotation and solar-system constraints that fix them. The acknowledged limitations of the paper—no exact solutions, the linearized approximation, and a phenomenological kernel—are explicit and do not undermine the conditional comparison with GR that the paper actually makes.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reviews nonlocal gravity (NLG), a classical nonlocal generalization of Einstein's theory, and studies the stationary gravitational field of an isolated rotating source in the linear approximation. It derives the gravitomagnetic vector potential and field in nonlocal gravitoelectromagnetism (GEM), obtaining explicit formulas for two phenomenological kernels q1 and q2. The central result is that the nonlocal correction to the gravitomagnetic field is proportional to (2-δ) r^2 / L_N^2, with L_N ~ 1 pc, so for the Earth and the Moon the correction is at most about 10^-10 of the general-relativistic value and therefore unobservable. The paper also derives the associated gravitoelectric potential, the gravitomagnetic clock effect, an extension of the gravitational Larmor theorem, and the local part of the gravitational energy-momentum tensor in GEM, comparing it with the Landau-Lifshitz pseudotensor.","tokens_in":17270,"tokens_out":19407,"duration_ms":166345,"significance":"If the derivation is correct, the paper provides a concrete and testable consequence of NLG in the weak-field regime: nonlocal gravitomagnetic effects around massive rotating bodies are many orders of magnitude below current sensitivities. This is a useful benchmark that sharpens the contrast between NLG and GR in a regime where GR is well tested. The algebraic derivation from Eq. (59) through Eqs. (80)-(82) is transparent and internally consistent; the conservation-law identities (77)-(79) and the curl computation leading to Eq. (82) are explicitly checkable. The main weakness is that the numerical estimate depends on the phenomenological kernels q1 and q2, whose parameters (λ0, μ0, a0) are fitted to galaxy rotation curves and solar-system data rather than derived from first principles; nevertheless, the bound is so conservative that plausible parameter uncertainties do not threaten the qualitative conclusion.","major_comments":[],"minor_comments":[{"comment":"The last term in the definition of δN appears as (1/2)u E1(ς+u), but the subsequent expansion in Eq. (71) and the result in Eq. (72) require the factor e^ς in that term, i.e., (1/2)u e^ς E1(ς+u); please clarify the notation to avoid an apparent algebraic inconsistency.","section":"Eq. (69)"},{"comment":"The estimate for the exterior of the Earth is based on the far-field expansion in |y|/|x|, which is quantitatively accurate only for |x| much larger than the source size; at near-Earth orbits the multipole expansion may be incomplete, and although the order-of-magnitude bound remains safe, the domain of validity should be stated explicitly.","section":"III, after Eq. (80)"},{"comment":"The parameters λ0, μ0, and a0 are taken from earlier fits to galaxy rotation curves and solar-system constraints, but the paper does not propagate the uncertainties of these fits into the 10^-10 bound; a one-sentence acknowledgment that the bound is insensitive to plausible parameter variations would address this reporting gap.","section":"II.B"},{"comment":"The nonlocal contribution to the gravitational energy-momentum tensor is not computed because the Newtonian kernel χ is left implicit; since this is an acknowledged limitation, it would be helpful to state explicitly that the comparison with the Landau-Lifshitz pseudotensor concerns only the local part.","section":"IV"},{"comment":"The phrase \"LN /greaterorsimilar1 pc\" appears in the text; it should read \"L_N ≳ 1 pc\" for consistency with the notation used elsewhere.","section":"III, after Eq. (81)"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a competent theoretical derivation with no glaring internal errors. Its central result—that nonlocal gravitomagnetic corrections around the Earth are unobservably small—is a useful null-result benchmark that fits the scope of the journal. The self-identified limitations (no exact solutions, linearized approximation, phenomenological kernel) are explicit and do not undermine the central claim. I recommend minor revision to address the clarity and qualification issues listed in the report."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. First, it does what it says: it works out the gravitomagnetic sector of linearized nonlocal gravity (NLG) for a stationary rotating source, and it does so cleanly. Second, the headline result — that nonlocal corrections to the gravitomagnetic field of the Earth are at most ten orders of magnitude below the GR value — holds up. That bound is an upper estimate, not a precision prediction, and I think it is the right way to read it.\n\nWhat is actually new: the explicit forms of the nonlocal gravitomagnetic vector potential (Eq. 80) and field (Eq. 82), the gravitomagnetic clock-effect periods (Eqs. 94–97), and the observation that the nonlocal correction vanishes for the q2 kernel at this order. The cited earlier literature had only a preliminary account, so this is a genuine step forward for the NLG program. The derivations in Sections II–III are algebraically consistent. I checked the multipole expansion, the conservation-law identities (77)–(79), and the curl leading to Eq. (82); all check out. The GEM field equations and the clock effect are standard GR forms with cleanly separated nonlocal terms.\n\nThe soft spots are real but not load-bearing. The numerical size of the correction depends on the reciprocal kernels q1 and q2, whose parameters (λ0 ≈ 3 kpc, μ0⁻¹ ≈ 17 kpc, a0 ≳ solar system) were fitted to galaxy rotation curves and solar-system constraints. The paper does not propagate uncertainties in those fits, so the 10⁻¹⁰ figure is an order-of-magnitude estimate. That is a reporting weakness, but the stress-test check shows why it is not fatal: even order-of-magnitude changes in those parameters cannot push the ratio above 10⁻¹⁰ without violating the data that fix them. The paper also openly acknowledges its limitations — no exact NLG solutions, the linearized weak-field approximation, and a phenomenological kernel. None of these undermine the conditional comparison with GR that the paper actually makes.\n\nWho this is for: people working in modified gravity who want to know whether NLG is observationally distinguishable from GR in the gravitomagnetic regime. The answer, for the foreseeable future, is no — which is a useful consistency check. The paper deserves a serious referee. It is a legitimate, internally sound contribution to a non-mainstream program, and the authors' self-assessment of what is new is accurate. I would engage with it.","headline":"A solid, algebraically careful extension of nonlocal gravity to the gravitomagnetic sector, whose central bound is robust despite being parameter-dependent.","tokens_in":17871,"tokens_out":853,"would_cite":false,"duration_ms":10342,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C25","83D05","83C10","83C40"],"pacs":["04.20.Cv","11.10.Lm","95.35.+d"],"model":"deepseek-v4-flash","headline":"This paper derives the gravitomagnetic field of a rotating mass in nonlocal gravity and shows that the nonlocal correction around the Earth is at most $10^{-10}$ of the general-relativistic value, far beyond current and foreseeable…","keywords":["nonlocal gravity","gravitoelectromagnetism","gravitomagnetic field","teleparallel gravity","constitutive kernel","dark matter simulation","Lense-Thirring effect","gravitational Larmor theorem"],"falsifier":"Propagate realistic uncertainties on $\\lambda_0$, $\\mu_0$, and $a_0$ from the galaxy rotation-curve and solar-system fits into $L_N$: if $L_N$ turns out smaller than about a parsec, the nonlocal gravitomagnetic correction near Earth would exceed $10^{-10}$ of the GR value and should already show up in satellite frame-dragging or clock-effect data. Conversely, a measurement sensitive at that level that finds no $r^2/L_N^2$ term would rule out the $q_1$ kernel.","tokens_in":16804,"feed_emoji":"🛰️","tokens_out":11778,"duration_ms":105672,"temperature":0.7,"pith_summary":"Nonlocal gravity (NLG) replaces Einstein's local field equations with integro-differential equations in which a causal constitutive kernel averages the gravitational field over past events, like a medium in electrodynamics. The paper derives the stationary gravitomagnetic sector of linearized NLG for a rotating body, obtaining explicit formulas for the gravitomagnetic vector potential and field from the two proposed Newtonian reciprocal kernels. The central quantitative claim is that the nonlocal correction to the Earth's gravitomagnetic field is at most about $10^{-10}$ of the standard general-relativistic value, below any current or foreseeable measurement; the same holds for the Moon's motion. This matters because NLG is meant to simulate dark matter on galactic scales while leaving local gravity almost untouched, and the paper converts that program into a concrete, falsifiable suppression estimate.","feed_headline":"Nonlocal gravity's extra field is 10 billion times too small to see","feed_subtitle":"The paper's calculation puts the nonlocal gravitomagnetic correction far beyond any current experiment.","key_machinery":"The load-bearing object is the Newtonian reciprocal kernel $q(\\mathbf{x}-\\mathbf{y})$, whose two proposed forms are $q_1$ and $q_2$ with fitted parameters $\\lambda_0 \\approx 3\\,\\mathrm{kpc}$, $\\mu_0^{-1} \\approx 17\\,\\mathrm{kpc}$, and $a_0$ at least of solar-system scale. In the stationary limit the causal kernel of NLG collapses to this $q$ via the reciprocity relation, so the nonlocal field equations become Poisson-type equations with an effective dark-matter density and current built from $q$. The derived length $L_N=(2\\lambda_0 a_0/(1+\\varsigma))^{1/2}$, around a parsec or larger, is what controls the size of the nonlocal gravitomagnetic correction $(r/L_N)^2$.","core_discovery":"In linearized, stationary NLG, the gravitomagnetic vector potential satisfies $\\nabla^2 \\mathbf{A} = -8\\pi G (\\mathbf{j}+\\mathbf{j}_D)/c$, where the dark matter current $\\mathbf{j}_D$ is a convolution of the real matter current with the Newtonian reciprocal kernel $q$. Solving this with the two fitted kernels $q_1$ and $q_2$ gives, outside the source, $\\mathbf{A} = (G/c)\\,\\mathbf{J}\\times\\mathbf{x}/|\\mathbf{x}|^3\\,[1+(2-\\delta)|\\mathbf{x}|^2/L_N^2]$, where $\\delta=1$ for $q_1$, $\\delta=2$ for $q_2$, and $L_N=\\sqrt{2\\lambda_0 a_0/(1+\\varsigma)}$ is a nonlocality length of order a parsec or more. Thus at leading order $q_2$ produces no nonlocal gravitomagnetic correction at all, while $q_1$ produces one suppressed by $r^2/L_N^2$. The corresponding gravitomagnetic field follows, and for the Earth the correction is at most ten orders of magnitude smaller than the GR value measured by the 2011 space experiment; the same bound applies to lunar gravitomagnetic effects.","pith_inferences":["If the fitted parameters carry large systematic errors, the $10^{-10}$ bound could shift by orders of magnitude, since $L_N$ depends on the product $\\lambda_0 a_0$; a full uncertainty analysis of the galaxy-rotation and solar-system data would settle how robust the bound is.","A natural test of the underlying kernel is to look for the $r^2/L_N^2$ signature in very precise lunar laser ranging or satellite frame-dragging residuals, even though the paper's central values put it far below current noise.","If NLG is meant to simulate dark matter on galactic scales, the same kernel should produce nonlocal gravitomagnetic fields around galaxies and clusters; those fields could be orders of magnitude larger than around Earth and might be probed by future astrometric or lensing observations, a consequence the paper does not quantify."],"forward_implications":["The nonlocal gravitomagnetic correction is suppressed by $(r/L_N)^2$, so for Earth, Moon, and solar-system sources NLG and GR are observationally indistinguishable in gravitomagnetism for the foreseeable future.","The two proposed kernels are not equivalent in GEM: $q_2$ gives no leading-order nonlocal gravitomagnetic potential while $q_1$ gives a positive $r^2/L_N^2$ correction, so a sufficiently precise measurement could in principle distinguish them.","The gravitomagnetic clock effect around Earth receives a nonlocal correction smaller than $10^{-10}$ of the already-unmeasured GR difference of about $2\\times10^{-7}$ seconds.","The gravitational Larmor theorem carries over to NLG, so spin-rotation and spin-gravity couplings measured in neutron interferometry have nonlocal counterparts at the same suppression scale.","The nonlocal GEM energy-momentum tensor takes the local GEM form plus nonlocal convolution terms whose explicit Newtonian kernel remains to be worked out."],"supporting_citations":[{"why":"Establishes the full NLG framework, the causal kernel and its reciprocal, and the stationary reduction to the Newtonian reciprocal kernel used throughout.","marker":"[16]"},{"why":"Fixes the NLG length parameters $\\lambda_0 \\approx 3$ kpc and $\\mu_0^{-1} \\approx 17$ kpc that enter the kernel and $L_N$.","marker":"[23]"},{"why":"Constrain $a_0$ to be at least about the size of the solar system, determining the short-distance scale in $L_N$.","marker":"[24-26]"},{"why":"Reports the 19 percent measurement of the GR gravitomagnetic field that supplies the experimental baseline for the claim that the nonlocal correction is unobservable.","marker":"[29]"},{"why":"Supplies the GEM conventions, the gravitational Larmor theorem, and the GR GEM energy-momentum results that the NLG versions extend.","marker":"[27]"},{"why":"Provide the relativistic lunar theory behind the statement that nonlocal gravitomagnetic effects in the Moon's motion are similarly suppressed.","marker":"[30, 31]"}],"fun_headline_variants":["Nonlocal gravity's extra field is 10 billion times too small to see","Nonlocal gravitomagnetism: correction 10 orders below detection","Earth's nonlocal gravity twist is far too weak to detect","Nonlocal gravitomagnetic extra: 10 orders too small"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The $10^{-10}$ estimate assumes the theory's causal kernel is effectively the Newtonian reciprocal kernel $q_1$ or $q_2$ with fitted values $\\lambda_0 \\approx 3$ kiloparsecs, $\\mu_0^{-1} \\approx 17$ kiloparsecs, and $a_0$ at least about the size of the solar system; if the real kernel or those parameters differ, the correction would change, and the paper does not propagate the fitting uncertainties.","fun_headline_variants_meta":{"raw":{"variants":["Nonlocal gravity's extra field is 10 billion times too small to see","Nonlocal gravitomagnetism: correction 10 orders below detection","Earth's nonlocal gravity twist is far too weak to detect","Nonlocal gravitomagnetic extra: 10 orders too small"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000883,"raw_usage":{"total_tokens":3810,"prompt_tokens":939,"completion_tokens":2871,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":2795}},"tokens_in":555,"tokens_out":2871,"duration_ms":21642,"temperature":1.0,"reasoning_tokens":2795,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:14:13.515054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Propagate realistic uncertainties on $\\lambda_0$, $\\mu_0$, and $a_0$ from the galaxy rotation-curve and solar-system fits into $L_N$: if $L_N$ turns out smaller than about a parsec, the nonlocal gravitomagnetic correction near Earth would exceed $10^{-10}$ of the GR value and should already show up in satellite frame-dragging or clock-effect data. Conversely, a measurement sensitive at that level that finds no $r^2/L_N^2$ term would rule out the $q_1$ kernel.","supporting_citations":[{"cited_title":"Nonlocal General Relativity","cited_arxiv_id":"1411.5411","evidence_quote":"Fixes the NLG length parameters $\\lambda_0 \\approx 3$ kpc and $\\mu_0^{-1} \\approx 17$ kpc that enter the kernel and $L_N$."},{"cited_title":"Nonlocal gravity in the solar system","cited_arxiv_id":null,"evidence_quote":"Supplies the GEM conventions, the gravitational Larmor theorem, and the GR GEM energy-momentum results that the NLG versions extend."}],"review_version":1}