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Analogy of space-time as an elastic medium, estimation creep coefficient of space from MOND theory, gravitational lensing and via time data from the GPS effect, discussion of the results for dark matter and Einstein's field equation

T0 review · 5 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a single 'creep coefficient' of space, derived from galaxy rotation and cluster lensing, can replace dark matter in general relativity's field equations.

desk verdict The creep coefficient is a renamed MOND ratio, not a derived property of spacetime; the paper re-labels empirical ratios and offers no falsifiable distinction from dark matter or MOND. read the letter →

arxiv 2412.13277 v1 pith:3CE222JU submitted 2024-12-17 gr-qc

David Izabel This is my paper · ORCID
classification gr-qc PACS 04.50.Kd46.90.+s
keywords darkmattercreepofspacespace-timeelasticmediumMONDgravitationallensingmodifiedgravitygranularvacuumGPStimedilation
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

This paper tries to show that the extra gravity attributed to dark matter is not caused by invisible particles but by a slow, time-dependent deformation of the space-time fabric—what mechanics calls creep. Treating space-time as an elastic medium whose flexibility is set by the coupling constant $\kappa = 8\pi G/c^4$, the author reads the MOND velocity anomaly of galaxy rotation as a change in the effective gravitational constant, $G \to G(1+\varphi)$, and extracts a creep coefficient $\varphi_{\mathrm{space}} = a_0/a - 1$, refined by a local-to-mean density ratio, with values between about 0.2 and 9. The same exercise using the Bullet Cluster's gravitational-lensing mass split gives $\varphi_{\mathrm{space}} = (1-p_v)/p_v$, between 0.66 and 4, while GPS clock data give a near-zero $\varphi_{\mathrm{time}}$ near Earth. If the identification holds, the dark matter problem would be a mechanical property of the vacuum rather than a missing particle, and the field equation of general relativity would be modified by a creep factor $(1+\varphi)$ on the stress-energy side.

What carries the argument

The load-bearing object is the creep coefficient $\varphi$, borrowed from engineering viscoelasticity, where a material's long-term modulus falls as $E_{\mathrm{lt}} = E_{\mathrm{st}}/(1+\varphi)$. The paper maps this onto gravity by writing $G_{\mathrm{lt}} = G(1+\varphi)$ and $\kappa_{\mathrm{lt}} = 8\pi G(1+\varphi)/c^4$, so creep increases the flexibility of space-time. The two empirical inputs are (i) the MOND acceleration scale $a_0$ entering $r_{\mathrm{MOND}} = r_{\mathrm{classical}}\sqrt{a_0/a}$, and (ii) the visible-mass fraction $p_v$ of the Bullet Cluster entering $\alpha_v(1+\varphi)=1$. The machinery turns those ratios into a single mechanical parameter and then into a modified field equation with a creep-amplified source term.

What would settle it

Measure the creep coefficient for the same galaxy twice—once from its rotation curve and once from its weak-lensing shear. If the two values differ by more than the combined measurement errors, the single-fabric-creep identification fails. A second check: if creep is real, the apparent dark-matter fraction of an isolated galaxy should grow with its age at fixed mass and radius, which is testable by comparing young and old galaxies of equal mass.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that a single mechanical parameter—the creep coefficient $\varphi$ of the space-time fabric—can reproduce three independent dark-matter signatures. For galaxy rotation, MOND's replacement of the classical relation $v=\sqrt{GM/r}$ by $v=\sqrt[4]{GMa_0}$ is re-expressed as a growth of the galaxy's equivalent radius, $r_{\mathrm{MOND}} = r_{\mathrm{classical}}\sqrt{a_0/a}$, and that ratio is interpreted as a long-term increase of the gravitational constant, $G_{\mathrm{lt}} = G(1+\varphi)$, giving $\varphi_{\mathrm{space}} = a_0/a -1$ (later multiplied by $\rho_{\mathrm{local}}/\rho_{\mathrm{mean}}$). For gravitational lensing by the Bullet Cluster, the observed $1-p_v$ dark-mass fraction is rewritten as an amplification of the visible deflection angle, $\alpha_v(1+\varphi)=1$, giving $\varphi_{\mathrm{space}}=(1-p_v)/p_v$. For time, the GPS gravitational frequency shift gives $\varphi_{\mathrm{time}}$ of order $6\times 10^{-3}$, consistent with negligible dark matter in the Solar System. The author concludes that the field equation should carry a creep-modified source $(1+\varphi)T_{\mu\nu}$ (or an additional stress-energy tensor of the fabric itself), so that what looks like dark matter is accumulated deformation of space under constant load.

Load-bearing premise

Everything rests on whether the empirical ratio measured in galaxy rotation and the dark-matter mass fraction measured in cluster lensing are the same physical thing as the slow deformation of a material under constant load, rather than new names for the old discrepancy.

Editorial extensions

If this is right

  • The dark matter fraction in any gravitating system becomes a measurable creep coefficient: compute $\varphi$ from rotation curves or lensing and compare with the $0.2$–$9$ range.
  • The field equation of general relativity acquires a creep factor $(1+\varphi)$ multiplying the stress-energy side, so at cosmological scales the effective gravitational coupling grows with loading time.
  • The near-zero $\varphi_{\mathrm{time}}$ from GPS means the Solar System remains essentially classical; dark matter effects would appear only on galactic and cluster scales where loads have acted for billions of years.
  • If creep is real, dark matter abundance should not be constant: its apparent strength should grow with the age and loading history of the structure, a prediction testable by comparing young and old galaxies.
  • The approach predicts that lensing and rotation-curve measurements in the same galaxy should yield the same $\varphi$, offering a direct cross-check independent of dark matter assumptions.

Reading between the lines

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

  • A testable extension: stack weak-lensing profiles of isolated spiral galaxies and compare the inferred $\varphi_{\mathrm{lens}}$ with $\varphi_{\mathrm{MOND}}$ from the same rotation data; agreement would support a single fabric property, disagreement would push toward separate dark matter.
  • The author leaves implicit that if creep of the vacuum is physical, the acceleration scale $a_0$ should itself be derivable from a material property of space, such as a characteristic creep rate or relaxation time.
  • One could look for slow, non-classical relaxation of the metric after a disturbance, for example in the tail of gravitational-wave ringdowns, as a direct signature of viscoelastic space.
  • The GPS result suggests any laboratory or Solar System test should see no anomaly, so the strongest evidence would come from redshift-dependent rotation curves, where older, creep-loaded galaxies should show larger $\varphi$ at fixed mass and radius.
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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

5 major / 5 minor

Summary. The paper develops an analogy between spacetime and an elastic medium, proposing that dark-matter-like gravitational anomalies in galactic rotation curves, galaxy-cluster lensing, and GPS time dilation can be interpreted as a 'creep' of the space fabric. The author defines a creep coefficient phi_space from MOND as a0/a - 1, later modified by a local-to-mean density ratio; from Bullet Cluster lensing as (1 - p_v)/p_v; and from GPS gravitational time dilation as a small residual phi_time about 0.006. The paper then argues that these coefficients suggest a granular or crystalline vacuum and proposes replacing G by G(1 + phi) or adding a space-fabric stress tensor to Einstein's equation. The stated goal is to provide a dark-matter-free approach based on the creep of the texture of space.

Significance. The paper's ambition is significant: if a creep coefficient of spacetime could be derived from first principles and predict rotation curves, lensing, and time dilation with a single value, it would be a substantive alternative to dark matter. It is a strength that the author gives explicit numerical tables and a concrete GPS estimate, and that the analogy with materials creep is clearly stated. However, as presented, the central quantity phi is defined by the observations rather than derived from a creep constitutive law, and the three determinations do not agree where they can be compared. The manuscript is therefore a speculative reinterpretation rather than an established model.

major comments (5)
  1. [§7.1, Eq. (44)] Eq. (44) is a postulate, not a derivation: the text introduces it with 'we can therefore postulate', and the equation simply sets 1 + phi = a0/a. Consequently phi_space = a0/a - 1 in Eq. (45a) is an algebraic restatement of the MOND interpolating function, not an independent mechanical quantity. In the deep-MOND regime a = sqrt(G M a0)/r, so phi(r) = r sqrt(a0/(G M)) - 1; Table 1 exhibits this radial dependence inside a single galaxy (Milky Way: -0.55 at 50 kly, +2.83 at 520 kly). A creep coefficient is a constitutive property of a medium and should not be a function of the position of a test particle within one galaxy. This circular step is load-bearing for the paper's central claim.
  2. [§8, Eqs. (48b)-(53)] The lensing determination is likewise definitional. Eq. (48b) postulates alpha_v(1 + phi) = 1, and with alpha_v identified with p_v, Eq. (53) gives phi = (1 - p_v)/p_v = p_DM/p_v. This is the dark-matter-to-visible-mass ratio written in new variables, not an independent estimate. The values 0.66 and 4 merely reflect the assumed 60% and 20% visible-mass fractions at two cluster radii. Because phi again changes with radius, it cannot serve as a single material coefficient, and the claimed agreement with the MOND values is agreement by construction.
  3. [§7.2, Tables 1 and 2] The paper claims that Table 2, taken from Ref. [21], verifies the magnitude of the Table 1 results, but the two tables disagree for nearly every entry. For the Milky Way at 50 x 10^3 light years, Table 1 gives phi = -0.55 while Table 2 lists R/s = 0.45, which corresponds to phi approximately 1.22; LMC gives -0.37 vs +0.59; SMC -0.62 vs +1.63; and most other rows differ in sign. If Table 2 is meant to quantify a different, geometric effect (as the text suggests), the comparison should be stated explicitly, and the claim that the results are 'consistent' is not supported by the displayed numbers.
  4. [§7.2, second occurrence of Eq. (45a)] The density-calibrated relation phi_space = (a0/a)(rho_local/rho_mean) - 1 is introduced ad hoc, with the author stating that 'it would be necessary to calibrate the creep coefficient as a function of the mass density'. No operational definition or data source for rho_local/rho_mean is given, and the abstract's headline range 0.2-9 depends on this unspecified ratio. Adding such a position-dependent calibration function makes the already definitional relation unfalsifiable.
  5. [§9.2, Eqs. (67)-(69)] The GPS time-dilation estimate does not provide an independent test. Eq. (67) assumes the creep hypothesis by writing G -> G(1 + phi) inside the time-dilation formula, and phi_time is then extracted from the small residual between the measured gravitational frequency shift and the standard G expression. The residual 0.006 is consistent with the known inaccuracies of a simplified two-point GPS calculation, which neglects Earth's quadrupole, higher-order relativistic terms, and satellite orbital details. In addition, phi_time about 0.006 is not compared quantitatively with the claimed phi_space range 0.2-9; the conclusion that the effect is small near Earth is expected regardless of the model.
minor comments (5)
  1. [Title, abstract, §8] The Bullet Cluster is consistently rendered as 'ball cluster' in the title, abstract, and Section 8; please correct this typo throughout.
  2. [Eq. (70)] Eq. (70) writes R_mu nu + 1/2 g_mu nu R, whereas Eq. (4) and the standard Einstein equation use R_mu nu - 1/2 g_mu nu R; please correct the sign or justify the different convention.
  3. [Eq. (69)] Eq. (69) is ambiguous: the fraction should be written as (delta t / delta t_0) / (G M / (r c^2)) - 1 or with clear parentheses to avoid confusion.
  4. [Section 7] Two different equations are labeled (45a): the basic MOND result and the density-calibrated version; please renumber them to avoid ambiguity.
  5. [Opening note] The opening note stating that the paper has been accepted by International Journal of Modern Physics D is not appropriate for a preprint under review and should be removed.

Circularity Check

3 steps flagged · score 9.0 of 10

The creep coefficient phi is not derived but defined: Eq. (44) sets phi = a0/a - 1, Eq. (53) sets phi = (1 - p_v)/p_v, and Sec. 9.2 sets phi_time to the measured-over-predicted GPS ratio minus one.

  1. self definitional [Section 7.1, Eqs. (41)-(45a)]
    "We therefore postulate that the MOND’s theory effect on the equivalent radii of the galaxy is due to a variation of G by the creep effect of space (since G is associated with the Young's modulus Y in the elastic medium analogy): 𝐺(1 + 𝜑) = (𝐺 𝑎0/𝑎 ) (43) Let us consider the following relation which allows us to evaluate the creep coefficient of space considered by analogy as an equivalent elastic medium: 𝑎0/𝑎 = (1 + 𝜑) (44)"

    This is a definition, not a derivation. No creep constitutive law, no time-dependent strain history, and no material test enters; the MOND ratio a0/a is simply relabeled as 1+phi. With deep-MOND kinematics a = sqrt(G M a0)/r, Eq. (45a) becomes phi(r) = r sqrt(a0/(G M)) - 1, so the supposed creep coefficient varies with radius inside one galaxy. Table 1 shows this (Milky Way: phi ~ -0.55 at 50 kly, +2.83 at 520 kly). A material creep coefficient cannot depend on the test particle's position; this is the MOND rotation curve rewritten. The later correction phi = (a0/a)(rho_local/rho_mean) - 1 is an added fitted density factor, again not derived from creep mechanics.

  2. self definitional [Section 8, Eqs. (48b)-(53)]
    "The creep can be seen as an increase of the angle due to the visible matter 𝛼𝑣 by a factor 𝜑: 𝛼𝑣 + 𝜑𝛼𝑣 = 1 (48b) ... 𝛼𝑣(1 + 𝜑)=1 (48c) ... 𝜑𝑠𝑝𝑎𝑐𝑒 = 1/𝛼𝑣 − 1 = 1−𝛼𝑣/𝛼𝑣 = 1−𝑝𝑣/𝑝𝑣 (53)"

    Equation (48b) assumes, rather than derives, that the total deflection is the visible-matter deflection multiplied by (1+phi). Since the deflection angle is taken proportional to mass (Eq. 49), alpha_v is identified with the visible mass fraction p_v, so Eq. (53) is just the dark-matter fraction (1-p_v) divided by p_v. The Bullet Cluster numbers phi=0.66 and phi=4 are therefore the published visible/dark mass ratios in new notation; no independent lensing measurement of a creep coefficient is performed.

1 more flagged steps
  1. fitted input called prediction [Section 9.2, Eqs. (68)-(69) and numerical application]
    "We can extract 𝜑 from (67): − (𝑑𝜏−𝑑𝑡/𝑑𝑡 ) 𝑟𝑐2/𝐺𝑀 − 1 = 𝜑 (68) Or equivalently: (𝑑𝑡−𝑑𝜏/𝑑𝑡 ) 𝑟𝑐2/𝐺𝑀 − 1 = 𝛿𝑡/𝛿𝑡0 𝑟𝑐2/𝐺𝑀 − 1 = (𝛿𝑡/𝛿𝑡0 / 𝐺𝑀/𝑟𝑐2 − 1) = 𝜑 (69) ... 𝛿𝑡/𝛿𝑡0 = 45.9 × 10−6/(24 × 60 × 60) = 5.3125 × 10−10 ... So, the strain variation between the satellite level and the Earth surface is: 6.96077 × 10−10 − 1.681166 × 10−10 = 5.279604 × 10−10 ... 𝜑𝑡𝑖𝑚𝑒 = (5.3125 × 10−10/5.27 × 10−10) − 1 = 0.00632568"

    The time creep coefficient is constructed as the ratio of the observed GPS gravitational time offset to the standard general-relativistic prediction, minus one. That is exactly the empirical residual between measurement and theory; it is a fit, not a prediction. If GR had matched GPS perfectly, phi_time would be zero by construction regardless of whether any creep exists. Thus this section adds no independent support for the creep hypothesis; it merely re-expresses the known GPS result as a coefficient.

full rationale

The paper's central quantity, the creep coefficient phi, is never obtained from a creep law, stress history, or material response function. In the MOND section, Eqs. (43)-(44) set 1+phi equal to a0/a, so phi_space = a0/a - 1 is the MOND interpolating function renamed; the radius dependence of this 'coefficient' inside a galaxy confirms it is not a material property. In the lensing section, Eq. (48b) postulates alpha_v(1+phi)=1, and with alpha_v=p_v, Eq. (53) gives phi = (1-p_v)/p_v, i.e., the dark-matter fraction relabeled. In the GPS section, phi_time is the measured-over-predicted time dilation ratio minus one, i.e., the observational residual. All three 'estimates' reduce to their inputs by construction. The paper also contains self-citations ([6], [7], [8], [16], [41]) for the elastic-space analogy, but the circularity does not depend on those citations; it is definitional. Since the conclusion that dark-matter effects can be replaced by a creep of space rests entirely on these redefined residuals, the derivation is essentially equivalent to its input: MOND, the Bullet Cluster mass fraction, and GPS timing data are not used to test a creep model, they are repackaged as creep coefficients.

Assumptions & free parameters 3 free parameters · 5 assumptions · 1 invented entities

The central claim depends on the elastic-space analogy, on the postulate equating MOND's acceleration ratio with 1+phi, on the Bullet Cluster visible fraction, and on an ad hoc density correction. No free parameter is predicted from first principles.

free parameters (3)
  • Milgrom acceleration a0 = 1.2e-10 m/s^2
    MOND constant fit to galaxy rotation curves, entered in Eq. (25) and used to define phi in Eq. (44).
  • rho_local/rho_mean ratio
    Introduced ad hoc in Eq. (45a) to fix negative creep coefficients; no values or measurement procedure are given.
  • Visible mass fraction p_v (Bullet Cluster) = 0.6 near center to 0.2 at periphery
    Data from [35] used to compute phi_space = (1-pv)/pv in Eq. (53); the coefficient is thus set by the assumed dark matter fraction.
assumptions (5)
  • domain assumption Spacetime can be described as an elastic medium with Young's modulus and Poisson ratio, so creep laws apply.
    Sections 3-4 and Eq. (10)-(17) assume the elastic analogy carries over to creep; no derivation.
  • domain assumption The vacuum has a granular or crystalline structure that permits creep.
    Section 3 and 4 use [17], [32], [39] to infer granularity; this is a heuristic analogy, not established.
  • ad hoc to paper MOND's acceleration ratio a0/a equals 1+phi.
    Eq. (43)-(44) state this equality by postulate; it is the step that turns MOND into a creep coefficient.
  • ad hoc to paper The Einstein equation can be modified by adding a space-fabric stress tensor multiplied by (1+phi).
    Eq. (70)-(73) introduce t_e,mn from [37],[38] and attach the creep factor without a derivation.
  • standard math Weak-field linearized Einstein equations are standard.
    Eq. (5)-(6) rely on the standard linearized gravity framework; this is unproblematic.
invented entities (1)
  • Granular or crystalline space fabric undergoing creep
    purpose: To produce the excess gravitational effects currently attributed to dark matter.
    The paper postulates this from the elastic analogy and quantum-gravity granularity arguments but provides no direct falsifiable handle, such as a predicted signature that distinguishes creep from dark matter particles.

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Cite this review

Pith. "Pith review of Analogy of space-time as an elastic medium, estimation creep coefficient of space from MOND theory, gravitational lensing and via time data from the GPS effect, discussion of the results for dark matter and Einstein's field equation." pith.science (2026). https://pith.science/paper/3CE222JU

@misc{pith2026241213277,
  author       = {Pith},
  title        = {Pith review of: Analogy of space-time as an elastic medium, estimation creep coefficient of space from MOND theory, gravitational lensing and via time data from the GPS effect, discussion of the results for dark matter and Einstein's field equation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3CE222JU}},
  note         = {Machine review of arXiv:2412.13277}
}
read the original abstract

After recalling the principles that allow space-time to be considered by analogy as an elastic medium, we show how the modified gravity according to the MOND theory concerning the anomaly of the velocities of stars at the periphery of galaxies can be seen as a creep of space acting on the radius of galaxies that gives a creep coefficient of Phi(space) = ((a0/a) x (Ro local/ Ro mean) -1). The values vary between 0.2 and 9 depending on the type of galaxy and density distribution. Considering the gravitational lensing effect of the ball cluster we obtain a creep coefficient Phi (space) = (1-pv)/pv. With pv the percentage of visible matter and pDM the percentage of Dark matter from the global mass (pv + pDM =1). The values vary between 0.66 and 4 for this cluster. This paper therefore raises the question, via these creep coefficients, of the possible granular nature of the vacuum and therefore of space fabric on the one hand and proposes another dark matter-free approach based on the creep of the texture of space to explain gravitational anomalies on the other hand.

Figures

Figures reproduced from arXiv: 2412.13277 by the authors.

Figure 1
Figure 1. Velocity of stars in galaxies of radius R according to Newton's theory (2) and observed (1) But for more than fifty years, the mysterious dark matter particles have been untraceable, regardless of the detectors and the sensitivity of the detectors. The ontological approach therefore fails for the moment. This leaves the legislative approach, which consists of changing the law of gravitation, at least in part, in cer… view at source ↗
Figure 2
Figure 2. Effect of the curvature of space on Earth [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

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