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Nonlocal Gravitomagnetism

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Pith's one-line read 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…

desk verdict A solid, algebraically careful extension of nonlocal gravity to the gravitomagnetic sector, whose central bound is robust despite being parameter-dependent. read the letter →

arxiv 1908.05431 v3 pith:64YB7LGV submitted 2019-08-15 gr-qc astro-ph.GAhep-th

Bahram Mashhoon , Friedrich W. Hehl This is my paper · ORCID
classification gr-qcastro-ph.GAhep-th MSC 83C2583D0583C1083C40 PACS 04.20.Cv11.10.Lm95.35.+d
keywords nonlocalgravitygravitoelectromagnetismgravitomagneticfieldteleparallelconstitutivekerneldarkmattersimulationLense-ThirringeffectgravitationalLarmortheorem
topics Dark Matter
open problems Dark Matter
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

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.

What carries the argument

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 $(r/L_N)^2$.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

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.

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.

minor comments (5)
  1. [Eq. (69)] 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.
  2. [III, after Eq. (80)] 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.
  3. [II.B] 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.
  4. [IV] 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.
  5. [III, after Eq. (81)] The phrase "LN /greaterorsimilar1 pc" appears in the text; it should read "L_N ≳ 1 pc" for consistency with the notation used elsewhere.

Circularity Check

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No significant circularity: the nonlocal gravitomagnetic corrections are derived consequences of kernels calibrated to independent galaxy-rotation and solar-system data, not fitted to the gravitomagnetic signals they bound.

full rationale

The paper's central gravitomagnetic results, Eqs. (80) and (82), are obtained by explicitly solving the linearized nonlocal gravity field equations, Eq. (59), with the reciprocal kernels q1 and q2 of Eqs. (50)-(51). The derivation is algebraic and self-contained: the nonlocal potential correction (2−δ)|x|²/L_N² emerges from solving ∇²A = −(8πG/c)(j + j_D) with j_D = ∫ q(x−y)j(y)d³y. The numerical smallness of the correction relative to the GR value depends on the parameters λ0, μ0, and a0, which are fitted to galaxy rotation curves and solar-system orbital data (Refs. [23]-[26]), not to any gravitomagnetic observable. Thus the predicted gravitomagnetic bound is a derived consequence of externally calibrated inputs, not a quantity used in the fit. The self-citations establish the NLG framework and earlier kernel fits, but those fits are against independent astronomical data, so the citations are not load-bearing in a circular sense. No step defines the predicted quantity in terms of itself, and no fitted parameter is renamed as a prediction. The paper's own acknowledged limitations—absence of exact solutions, use of linearized approximation, and phenomenological kernel—are caveats about model uncertainty, not evidence of circular reasoning.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on two empirically fitted constitutive kernels and a set of standard weak-field approximations. No new particles, forces, or conserved quantities are introduced; the effective dark matter density and current are derived composites of matter fields and the kernel, not new entities.

free parameters (3)
  • lambda0 = approximately 3 kpc
    Fundamental nonlocality length scale of NLG; fitted to galaxy rotation curves in Rahvar and Mashhoon 2014 [23]. Appears in kernels q1, q2 and in the nonlocality scale L_N.
  • mu0 = mu0^-1 approximately 17 kpc
    Inverse length scale in kernels q1, q2; fitted to galaxy rotation curves [23]. Determines the exponential falloff of the nonlocal kernel.
  • a0 = greater than or approximately the size of the solar system
    Short-distance nonlocality parameter; bounded below by solar-system data (orbit of Saturn) [24-26]. Used in the small-r expansion (71)-(72) and in the definition of L_N.
assumptions (5)
  • domain assumption The reciprocal kernel R has the causal form (49), with R and K satisfying the reciprocity relation (47) and L1 and L2 integrability conditions.
    This is the constitutive input of NLG, patterned on nonlocal electrodynamics of media; it is not derived from Einstein's equations.
  • domain assumption The Newtonian reciprocal kernel q takes one of the two fitted forms q1 or q2, Eqs. (50)-(51).
    q is determined empirically from galaxy rotation curves [23]; the derivation of B_g depends on these explicit forms.
  • domain assumption The gravitational field is weak and stationary, and the source is compact with |v| much less than c; the matter current is conserved.
    Linearization (36) and the slow-motion energy-momentum expressions in Section II.C rely on this physical regime.
  • standard math The transverse gauge h^mu nu,nu = 0 and phi^mu nu = 0, Eq. (58), is imposed, which is allowed by the gauge freedom of the linearized theory.
    These gauge conditions simplify the torsion pseudovector to zero and lead to Eq. (59), making the nonlocal term proportional to the Einstein tensor.
  • standard math The multipole expansion is truncated to first order in |y|/|x| and to O(r^3/a0^3) in Eq. (72).
    Standard far-field approximation for an isolated source; validity requires |x| much greater than source size and r much less than a0.

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Pith. "Pith review of Nonlocal Gravitomagnetism." pith.science (2026). https://pith.science/paper/64YB7LGV

@misc{pith2026190805431,
  author       = {Pith},
  title        = {Pith review of: Nonlocal Gravitomagnetism},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/64YB7LGV}},
  note         = {Machine review of arXiv:1908.05431}
}
read the original abstract

We briefly review the current status of nonlocal gravity (NLG), which is a classical nonlocal generalization of Einstein's theory of gravitation based on a certain analogy with the nonlocal electrodynamics of media. Nonlocal gravity thus involves integro-differential field equations and a causal constitutive kernel that should ultimately be determined from observational data. We consider the stationary gravitational field of an isolated rotating astronomical source in the linear approximation of nonlocal gravity. In this weak-field and slow-motion approximation of NLG, we describe the gravitomagnetic field associated with the rotating source and compare our results with gravitoelectromagnetism (GEM) of the standard general relativity theory. Moreover, we briefly study the energy-momentum content of the GEM field in nonlocal gravity.

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