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REVIEW 4 major objections 4 minor 34 references

Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that general relativity's nonlinear self-interaction makes baryonic matter alone sufficient to explain the tight relation between galaxies' baryonic mass and observed acceleration, and that the MOND acceleration scale…

desk verdict A bold, honest attempt to explain the radial acceleration relation with GR self-interaction, but the load-bearing disk force law is assumed rather than derived, and the claimed a0 emergence inherits that assumption. read the letter →

arxiv 1909.00095 v3 pith:QDUMAVB6 submitted 2019-08-31 astro-ph.GA gr-qc

Alexandre Deur , Corey Sargent , Balša Terzić This is my paper · ORCID
classification astro-ph.GAgr-qc
keywords generalrelativitydarkmatterspiralgalaxiesgalacticdynamicsgravitationalself-interactionmass-accelerationrelationMONDaccelerationscalerotationcurves
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 argues that the puzzling correlation between a galaxy's baryonic mass and its observed acceleration—commonly read as evidence for dark matter—follows from a neglected property of general relativity: gravitational fields interact with themselves. In a flattened disk, that self-interaction confines the gravitational field to the plane, so the force between the center and a point in the disk falls as $1/r$ rather than Newton's $1/r^2$. Using this force law in parameter-free models of disk galaxies, the authors reproduce the observed mass–acceleration relation without dark matter or modified gravity. The same calculation yields a characteristic acceleration at the bulge–disk transition, about $1.2\times 10^{-10}\,\mathrm{m/s^2}$, matching the acceleration scale $a_0$ of the modified-dynamics theory MOND. If the paper is right, dark matter is not needed for disk galaxy dynamics and $a_0$ is a consequence of galaxy structure, not a new fundamental constant.

What carries the argument

The machinery is gravitational field self-interaction, the $n>0$ terms in the expanded Einstein–Hilbert Lagrangian $\mathcal{L}=\sum_n (16\pi G M)^{n/2}[\phi^n(\partial\phi\partial\phi-(16\pi G M)^{1/2}\phi T)]$. By analogy with QCD flux tubes, self-interaction makes the gravitational field lines collapse into the disk plane for a flattened mass distribution, giving a logarithmic potential $\Phi_d(r)=G' M_d^{\mathrm{enc}}(r)\ln r$ and a $1/r$ force in the disk, while a spherical bulge restores the Newtonian $1/r^2$ force inside the transition radius $r_t$. The two-dimensional coupling $G'$ is fixed to $G/r_t$ by continuity at $r_t$. Two complementary galaxy models sample observed parameter correlations among bulge effective radius, Sersic index, disk scale length, and bulge/disk masses, so the resulting $g_{\rm SI}$–$g_{\rm N}$ correlation is parameter-free.

What would settle it

Take a sample of disk galaxies with independent baryonic mass maps and compute $g_{\rm SI}(r)=G M_b^{\mathrm{enc}}(r)/r^2+(G/r_t)M_d^{\mathrm{enc}}(r)/r$ with $r_t$ determined from the bulge–disk force balance, then compare with observed accelerations from rotation curves; if the residuals exceed the observational uncertainties or correlate with galaxy parameters, the central claim fails.

Watch

Extended reading notes

Core claim

The central claim is that the nonlinear terms in the Einstein–Hilbert Lagrangian, which make the gravitational field self-interact, grow important at galactic scales and change the effective force law in a disk galaxy from $1/r^2$ to $1/r$ in the disk-dominated region. The transition happens at a radius $r_t$ where the bulge and disk forces balance, and matching the effective two-dimensional coupling $G'=G/r_t$ leaves the models without adjustable parameters. Computed with this rule, the acceleration including self-interaction, $g_{\rm SI}$, plotted against the Newtonian baryonic acceleration $g_{\rm N}$, reproduces the empirical relation between observed and baryonic accelerations; the direct lattice calculation already does so for bulge-less galaxies, and the two sampling models extend this across the observed range of disk morphologies. The acceleration at the transition radius peaks at $1.25\pm 0.06\times 10^{-10}\,\mathrm{m/s^2}$, consistent with the MOND scale $a_0\approx 1.2\times 10^{-10}\,\mathrm{m/s^2}$. Thus, in the paper's own terms, baryonic matter alone suffices to explain disk galaxy dynamics once general relativity's nonlinearity is included.

Load-bearing premise

The load-bearing premise is that gravitational field lines, once distorted by the dense central mass, remain confined to the disk plane in real disk galaxies, so the disk force truly follows $1/r$ with $G'=G/r_t$; the paper explicitly flags this as the assumption that extending the one-dimensional trapping result to a two-dimensional disk does not compromise the trapping of the field.

Editorial extensions

If this is right

  • The observed tight relation between baryonic mass and acceleration in disk galaxies stops being evidence for dark matter; it becomes a direct consequence of general relativity's nonlinearity.
  • The MOND acceleration scale $a_0$ is not a new constant of nature but the dynamically selected acceleration at the bulge–disk transition, so its value can vary with galaxy morphology and mass distribution.
  • Rotation curves should flatten naturally in disk-dominated regions because the GR self-interaction force $1/r$ exceeds the Newtonian $1/r^2$ expectation increasingly with radius.
  • Dark-matter-free baryonic models with the $1/r$ force law and $G'=G/r_t$ predict the small intrinsic scatter of the empirical relation, since the residual depends only weakly on galaxy parameters.

Reading between the lines

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

  • If the mechanism is right, gravitational lensing reconstructions of disk galaxies should show the extra apparent mass following the $1/r$ potential rather than a spherical dark halo, a signature testable with current lensing data.
  • Because $G'=G/r_t$ depends on galaxy structure, the transition acceleration should shift systematically with bulge-to-disk ratio; MOND predicts a strictly universal $a_0$, so this shift is a distinguishing observation.
  • The same field self-interaction should act in galaxy clusters, but their less flattened geometry predicts a weaker or differently shaped correction; cluster dynamics are therefore a natural place to look for the effect's limits.
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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

4 major / 4 minor

Summary. The paper proposes that nonlinear self-interaction of the gravitational field in General Relativity can explain the baryonic mass--acceleration relation (RAR) of disk galaxies reported by McGaugh et al. 2016, without dark matter or modified gravity. The central mechanism is that, for an axisymmetric disk, gravitational field self-interaction traps the field lines in two dimensions, producing a logarithmic potential and a 1/r force outside a transition radius rt, while a spherical bulge retains the Newtonian 1/r^2 force. The paper uses direct lattice calculations for a few galaxies, then constructs two phenomenological disk models (Model 1 with uniform sampling, Model 2 with random sampling) using a Sersic bulge plus exponential disk, with G' = G/rt fixed by continuity at rt. It reports that the resulting g_SI versus g_N relation reproduces the MLS2016 correlation and that the acceleration at rt peaks at about 1.2e-10 m/s^2, which it identifies with the MOND scale a0.

Significance. If the central premise were established, the paper would be highly significant: it would offer a baryon-only, dark-matter-free and MOND-free explanation of the RAR, and would provide a dynamical origin for the MOND acceleration scale. The manuscript includes useful cross-checks: lattice calculations that recover known free-field potentials and the Cornell potential, a mean-field disk calculation, and quantitative fits to the MLS2016 data with comparison of residuals. However, the significance is entirely contingent on the assumed 1/r gravitational force law in disks and on the transition prescription. Since that force law is not derived from the Einstein-Hilbert Lagrangian for a realistic galaxy and is explicitly acknowledged in the text to be an assumption, the paper does not currently demonstrate its central claim.

major comments (4)
  1. [Sec. 4; Sec. 5; Appendix C] The load-bearing input of the models is the assumed force law: Newtonian 1/r^2 inside rt and 1/r outside rt, with effective coupling G' = G/rt (Eqs. 9, 12--14). This force law is not derived from Eq. (1) for a disk mass distribution. The manuscript itself concedes in Sec. 4 that "Extending the one-dimensional result to the two-dimensional disk case assumes that the spread of the mass within the disk area does not compromise the trapping of the field in two dimensions," and Appendix C states that rt (equivalently G') "cannot presently be analytically calculated from first principles" and is "assessed phenomenologically." Since every downstream result, including the reproduction of the MLS2016 relation and the claimed emergent a0, is an output of this imposed force law, the paper does not independently support its central claim that GR nonlinearities make baryonic matter alone sufficient.
  2. [Sec. 6.2; Eq. (12)] The claimed "dynamically emerging" acceleration scale is largely built into the model definition. In Model 2, rt is defined as the radius where the bulge and disk accelerations are equal, and with G' = G/rt this reduces to M_b^enc(rt) = M_d^enc(rt) (Eq. 13). The acceleration at rt, whose distribution is shown in Fig. 4, is therefore determined by the chosen transition condition, the mass profiles, and the empirical scaling relations Eqs. (3)--(5). It is not an independent prediction from the GR calculation. The statement that a0 can be explained as the acceleration where "the disk mass overtakes the bulge mass" overstates what the model actually derives.
  3. [Sec. 5; Eq. (10)] Despite the abstract's claim of "parameter-free galactic models," the construction in Eqs. (6)--(14) contains several free or adjustable choices: the transition radius rt (2Re in Model 1; force equality in Model 2), the effective coupling G', and the width of the Fermi-Dirac smoothing function D(r). G' is fixed by continuity, G' = G/rt, rather than predicted from the Lagrangian. The statement in Sec. 5 that "there are no adjustable parameters" is therefore misleading: the parameters are not fitted to the MLS2016 data, but they are also not derived from first principles, as Appendix C acknowledges.
  4. [Sec. 4; Fig. 1 (top)] The only calculation based directly on Eq. (1) agrees with the observed RAR for bulge-less Hubble types 5 and 6, while types 3 and 4 overestimate g_SI and "lie on the edge of the observed distribution." Because many observed disk galaxies contain bulges, the direct GR-based evidence covers only a subset of the relevant population. The agreement for the full population therefore rests entirely on Models 1 and 2, which impose the aforementioned force law and transition prescription rather than deriving them.
minor comments (4)
  1. [Eq. (1)] The notation in Eq. (1) is compressed: the coefficients an, the sign conventions, and the meaning of the bracket shorthand are not fully specified, which makes the central equation difficult to audit.
  2. [Fig. 5] The figure caption lists the Kendall coefficient as ck = 0.207, while the text of Sec. 6.3 reports ck = -0.207. The sign inconsistency should be corrected.
  3. [Sec. 2] The statement that the ratio GM/L "becomes 10^-2 for galactic systems" is given without explicit unit conventions or a sample calculation. In natural units the numerical value is not transparent, and a reader cannot verify the claimed threshold without additional definitions.
  4. [Sec. 6.1] The fit of the Model 2 simulation to the MLS2016 functional form, Eq. (2), shows a compatible g†, but the paper should clarify whether the fit is treated as a model comparison or as a phenomenological diagnostic, given that the same functional form is used for both the data and the simulation.

Circularity Check

3 steps flagged · score 7.0 of 10

The claimed 'predictions' are built into the model: the 1/r disk force is assumed (Sec. 4), G' = G/rt is imposed by continuity (Sec. 5.1), and the 'emergent' a0 is the acceleration at the defined transition radius (Sec. 6.2).

  1. ansatz smuggled in via citation [Section 4, direct calculations (paragraph after two-body lattice result)]
    "Extending the one-dimensional result to the two-dimensional disk case of galaxies assumes that the spread of the mass within the disk area does not compromise the trapping of the field in two dimensions."

    The 1/r disk force is the load-bearing input: it produces the outer-region gSI proportional to sqrt(gN) behavior that matches the MLS2016 relation. The paper does not derive this force law from the Einstein-Hilbert action for a disk; it imports it from prior same-author work (Deur 2009, 2017; Deur 2020) built on a QCD flux-tube analogy and point-source lattice calculations. The agreement with the observed radial acceleration relation is therefore a consistency check of the assumed force law, not a derivation of it from first principles.

  2. self definitional [Sec. 5.1 (before Eq. 10); Sec. 6.2]
    "G′ is determined by requiring the accelerations to match at rt: gSI,r<rt(rt) = gSI,r>rt(rt). Thus, G′ = G/rt, by construction."

    G' sets the normalization of the 1/r force; fixing it by continuity at the modeler-chosen transition radius means the transition scale is an input, not an output. In Model 2, rt is itself defined by the force-equality condition (Eq. 12), and Sec. 6.2 identifies the 'dynamically emerging' acceleration a(rt) with MOND's a0. Thus a0 is the acceleration at a radius whose definition and continuity condition were chosen to produce a single characteristic scale; its peak value inherits the empirical scaling relations (3)-(5), so it is not an independent prediction.

1 more flagged steps
  1. fitted input called prediction [Appendix C]
    "Therefore, even for infinitely thin disks, rt—or equivalently G′—is non-universal and cannot presently be analytically calculated from first principles. It can be obtained from numerical calculations such as those in Refs. (Deur 2009, 2017), or assessed phenomenologically as done in this article."

    The key scale rt/G'—which controls the 1/r-force normalization and hence both the RAR reproduction and the value of a0—is 'assessed phenomenologically' rather than derived. The numerical references are same-author works and are for pointlike/two-body sources. The abstract's 'parameter-free galactic models' and 'direct calculations based on General Relativity's Lagrangian' are therefore overstated: the central scale is supplied from phenomenology, and the claimed emergence of a0 is downstream of that phenomenological input.

full rationale

The paper contains genuine calculations: lattice evaluation of the GR Lagrangian for point sources, and a comparison of the resulting gSI-gN curves for bulge-less galaxies to MLS2016. Those direct results for Hubble types 5 and 6 agree with the data. However, the generalization to disk galaxies—the step that produces the full radial acceleration relation—is an explicit assumption (Sec. 4), and the transition scale/coupling is imposed by continuity or 'assessed phenomenologically' (Sec. 5.1, App. C). The emergent a0 is identified with the acceleration at the defined transition radius (Sec. 6.2), so that prediction reduces by construction to the model's chosen transition condition and to the empirical scaling relations used to populate the galaxy sample. Because these load-bearing elements are assumed or imported from prior self-citations rather than independently derived, the central claims are partially circular. The paper's own text candidly states the key limitation, which supports this reading. The score is 7 rather than higher because there is independent content in the direct lattice calculations and the galaxy scaling relations are external to MLS2016, but the central 'parameter-free' predictions are substantially shaped by the model's own definitions.

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

The central claim rests on the assumed 2D confinement of the gravitational field in disks, the Newtonian treatment of bulges, and empirical galaxy scaling relations. The effective 2D coupling G' and the transition radius rt are modeling inputs rather than derived predictions, which is why they are listed as free parameters even though the prose calls the models parameter-free.

free parameters (3)
  • G' (effective 2D gravitational coupling) = G/rt per galaxy
    Set by continuity at the transition radius in Eq. (9). Appendix C states that G' is non-universal, depends on geometry and mass distribution, and cannot be analytically calculated from first principles, so it is an adjustable modeling input rather than a derived prediction.
  • Transition radius rt = Model 1: rt = 2Re; Model 2: solution of M_b(rt) = M_d(rt)
    The radius at which the force law changes from 1/r^2 to 1/r is chosen by definition or fixed as twice the half-light radius. It is not derived from the Einstein-Hilbert Lagrangian and it directly controls the shape of the radial acceleration relation.
  • Transition width of the Fermi-Dirac smoothing function = rt/2 (Model 1)
    Used in Eq. (10) to connect the bulge and disk regimes smoothly. The authors state that the choice has little influence on the results, so it is a minor freedom.
assumptions (5)
  • standard math Einstein-Hilbert Lagrangian can be expanded as in Eq. (1), with the Fierz-Pauli term as the n=0 part.
    This is the standard weak-field expansion of GR used in the paper, introduced in Section 2.
  • domain assumption GR self-interaction becomes important when GM reaches a fraction of the system size L, around 10^-3 to 10^-2.
    Invoked in Section 2 to justify nonlinear effects in galaxies; supported only by self-cited lattice studies, mainly Deur 2017.
  • ad hoc to paper Field-line trapping confines gravity to two dimensions in a disk, yielding a logarithmic potential and a 1/r force.
    Assumed in Sections 2 and 4 and used in Eq. (9). It is not derived from Eq. (1) for a disk, only argued by analogy with QCD and a two-point lattice result.
  • domain assumption Spherical symmetry cancels self-interaction, so the bulge remains Newtonian with a 1/r^2 force.
    Used throughout Section 5 and relied on for Eq. (6). No derivation is given for the cancellation in a realistic spheroid.
  • domain assumption The empirical scaling relations in Eqs. (3)-(5) define a representative galaxy parameter space.
    Used to generate model galaxies in Sections 5.1 and 5.2. The relations are taken from the literature, with scatter handled by cuts, and they shape the resulting correlation.

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Cite this review

Pith. "Pith review of Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies." pith.science (2026). https://pith.science/paper/QDUMAVB6

@misc{pith2026190900095,
  author       = {Pith},
  title        = {Pith review of: Significance of Gravitational Nonlinearities on the Dynamics of Disk Galaxies},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QDUMAVB6}},
  note         = {Machine review of arXiv:1909.00095}
}
read the original abstract

The discrepancy between the visible mass in galaxies or galaxy clusters, and that inferred from their dynamics is well known. The prevailing solution to this problem is dark matter. Here we show that a different approach, one that conforms to both the current Standard Model of Particle Physics and General Relativity, explains the recently observed tight correlation between the galactic baryonic mass and its observed acceleration. Using direct calculations based on General Relativity's Lagrangian, and parameter-free galactic models, we show that the nonlinear effects of General Relativity make baryonic matter alone sufficient to explain this observation.

Figures

Figures reproduced from arXiv: 1909.00095 by the authors.

Figure 1
Figure 1. Correlation between the acceleration accounting for GR’s self-interaction, gSI, and the acceleration computed with Newtonian gravity, gN, plotted along with the correlation observed in MLS2016 (grey circles). Top: Lagrangian-based calculations for various Hubble type galaxies. The galaxies are approximated as pure (bulge-less) disks. The calculations agree well with observation when this approximation is justified (… view at source ↗
Figure 2
Figure 2. Re vs. Md. Observed values are shown as blue circles (Sofue 2015). The best χ 2 fit to the observed data is denoted with a solid line, and one dex in dashed lines. Orange circles denote generated galaxies which passed the first cut only, while the red circles denote those who passed both cuts. The galaxies represented by the red circles are disk galaxies, while those corresponding to the orange circles are too bulge… view at source ↗
Figure 3
Figure 3. The acceleration accounting for GR’s self-interaction, gSI, versus that computed with Newtonian gravity, gN. The yellow line shows the best fit to our data simulated with Model 2 (red color density plot) using the form in Eq. (2). The black x’s are the average hlog(gSI)i. The yellow line and x’s can be compared to the MLS2016 fit, shown by the green line. The insert displays the residual between our simulated data a… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Acceleration at the transition radius rt for the set of galaxies generated in Model 2. The vertical line denotes a0 = 1.2 × 10−10 m/s 2 from MOND. 6.2. Emerging characteristic acceleration scale The transition scale between the two regimes in Model 2 is defined such as…
Figure 5
Figure 5. Figure 5: The bulge radius Re versus the residual between our calculated acceleration gSI and the MLS2016 relation, shown for log(gN) ≈ −11. The Pearson’s (cp = −0.325), Spearman’s (cs = −0.307) and Kendall’s (ck = 0.207) correlation parameters, all with negligible p-values, ind…
Figure 6
Figure 6. Figure 6: Potential around two massive bodies, with the 1/x (free–field, Newtonian case) contribution subtracted. The straight lines demonstrate the approximate linear behavior of the potential away from the mid-distance between the bodies (x = 28). There, by symmetry, the poten…
Figure 7
Figure 7. Figure 7: Distance dependence of the force obtained using a mean-field approximation to compute the self-interaction effects in a disk galaxy (solid red line) (Deur 2020). The total galaxy baryonic mass is Mtot = 5 × 1011M and has an exponentially decreasing density profile char…
Figure 8
Figure 8. Figure 8: Left: potential between two massive bodies, calculated in the static limit with Eq. (1) for n ≤ 2 (black stars), n ≤ 1 (red triangles), and n = 0 (Newtonian case, blue squares). The two sources are located on the lattice x−axis at ±5 lattice spacings u from the lattice…
Figure 9
Figure 9. Figure 9: Dependence of G 0 on the transition scale rt . Field lines emerge radially for a source (here, for clarity, only those emerging from the galaxy center are shown). A coupling constant, here G 0 , determines the density of the field lines emerging from the source (or, in…

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    WEŎXQ (E Aѹ wr[ЅV 6 ͝L䗐ϙ3g(

    eqnarray [ - 16 GM T ]= 1 2 ^ _ _ ^ - 1 2 ^ _ ^ _ _ ^ - ^ _ _ ^ + ^ _ ^ ^ _ - (16 GM) ^ T_ . eq:Fierz-Pauli Lagrangian eqnarray While Eq. ( eq:EH ) is often used to study quantum gravity---with questions raised regarding its applicability in that context, see e.g. Padmanabhan:...

    2020

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