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REVIEW 3 major objections 4 minor 47 references

Dark Matter and Energy-Momentum Squared Gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Energy-momentum squared gravity can flatten galaxy rotation curves without dark matter, the paper argues, while predicting testable lensing and radar-delay effects.

desk verdict The flat rotation curve is imposed by the power-law ansatz, not produced by EMSG—despite the abstract's claim, and the paper's own Sec. V says as much. read the letter →

arxiv 2501.19141 v1 pith:2S6TGZGE submitted 2025-01-31 gr-qc

H. R. Fazlollahi , H. Velten , H. Shojaie This is my paper · ORCID
classification gr-qc PACS 04.50.Kd95.35.+d
keywords energy-momentumsquaredgravitydarkmattergalacticrotationcurvesflatcurvemodifiedlightdeflectionradarechodelaysphericallysymmetricspacetime
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

The paper aims to establish that energy-momentum squared gravity (EMSG), an extension of general relativity obtained by adding the scalar $T_{\mu\nu}T^{\mu\nu}$ to the Lagrangian, can account for the flat rotation curves of galaxies without invoking particle dark matter. Restricting attention to static, spherically symmetric, pressureless dust and to metrics close to general relativity, the authors derive a circular orbital speed whose square is approximately $s/2$, independent of the radial coordinate, with the small parameter $s$ entering through $e^{a+b}=(r/\sigma)^s$. Identifying $s/2$ with the observed $v^2\sim10^{-6}$ for halo speeds of roughly $200$–$500$ km/s gives a parameter-light modified-gravity route to the dark-matter phenomenology of galaxy rotation, and the same spacetime predicts modified light deflection and radar echo delays that reduce to the general-relativistic values when the halo contribution is switched off.

What carries the argument

The load-bearing object is the ansatz $e^{a+b}=(r/\sigma)^s$, equivalently $a'+b'=s/r$ with $s\ll 1$, imposed on the combination of metric functions $a(r)$ and $b(r)$ in the static spherical line element. Its work is to convert the constant parameter $s$ into a constant orbital speed through the weak-field formula $v^2=ra'/2$, while the EMSG source term $T_{\mu\nu}T^{\mu\nu}$ supplies an effective $\rho^2$ pressure that makes the halo density fall as $r^{-2}$; the coupling $\alpha$ enters the velocity only at order $s^2$, so the flat-curve prediction is carried by the ansatz rather than by the strength of the squared-matter term.

What would settle it

Take a galaxy with both a measured rotation curve and measured lensing deflection: fit $s$ from $v^2\simeq s/2$, then check whether the predicted deflection (Eq. 40) and the $\rho\propto r^{-2}$ density profile match the data; a substantial mismatch would refute the claim. A second decisive check is to solve the EMSG field equations for dust without imposing Eq. (14) and see whether the resulting rotation curve is still flat.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that a dust-filled static spherically symmetric spacetime in EMSG, taken in the vicinity of general relativity, has a halo rotation velocity that is essentially independent of radius: $v^2\approx s/2$ up to subleading terms involving the coupling $\alpha$, so the tangential speed is flat and the energy-momentum squared term effectively plays the role of dark matter. The supporting density profile is $\rho\approx s/(\kappa r^2)$, and the same metric yields deflection angles and radar echo delays that grow with the halo scales of the model, returning to the general-relativistic $4GM/r_0$ deflection and the standard radar echo delay when the dark-matter region vanishes.

Load-bearing premise

The whole result rests on the assumed form $e^{a+b}=(r/\sigma)^s$ with a constant small $s$; if that form is not independently justified, the flat rotation curve is placed in by hand rather than predicted by the theory.

Editorial extensions

If this is right

  • Galactic rotation curves are flat in the halo with $v^2\simeq s/2$, so fitting observed speeds of $200$–$500$ km/s fixes $s\sim10^{-6}$, independent of $\alpha$ at leading order.
  • The halo density follows $\rho\propto r^{-2}$, a definite mass-profile prediction inside the dark-matter region.
  • The light deflection angle grows with $s$, so a galaxy's fitted rotation speed gives a predicted gravitational-lensing signature from the same parameter.
  • The radar echo delay increases with the halo radius $r_d$, with the general-relativistic result recovered as $r_d\to r_0$.
  • The analysis is restricted to weak-field, pressureless matter, so the model's claim of dark-matter mimicry is so far limited to that regime.

Reading between the lines

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

  • The flat curve is in effect installed by the assumed $a'+b'=s/r$; a decisive test would be to derive this relation from a microphysical Lagrangian or to solve the EMSG field equations numerically without the ansatz and see whether flatness survives.
  • Because the leading velocity depends only on $s$ and not on $\alpha$, the mimicry may be generic to any modified gravity that enforces a slowly varying $a+b$; comparing theories on this point would show what is unique to EMSG.
  • A falsifiable cross-check is to use one galaxy's fitted $s$ to predict both its lensing deflection and its radar echo delay, since all three observables are tied to the same parameter.
  • The $\rho\propto r^{-2}$ halo implies a total mass that keeps growing with radius until the halo edge, a consequence that puts additional constraints on galaxy masses and satellite dynamics beyond the rotation curve itself.
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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

3 major / 4 minor

Summary. The paper studies energy-momentum squared gravity (EMSG) as a possible explanation of dark matter on galactic scales. After deriving the field equations for a static, spherically symmetric dust spacetime, the authors impose the metric ansatz e^{a+b}=(r/\sigma)^s (Eq. 14) with s\ll 1. They solve for the density and metric functions, obtain an approximate rotation velocity v^2\approx s/2 (Eq. 36), identify s/2 with the observed squared circular speed, and claim that the model 'effectively illustrates the presence of dark matter.' The paper then uses the same fitted metric to compute light deflection angles and radar echo delays, with figures for four galaxies.

Significance. If the central claim were correct, the paper would provide a concrete modified-gravity mechanism for flat galactic rotation curves without particle dark matter. The manuscript is clearly written in structure and includes explicit field equations, a transparent parameter count (s, \sigma, \alpha), and a candid closing admission about the role of \alpha. However, the derivation does not supply what it claims: the flat rotation curve is effectively inserted by the power-law ansatz, and the EMSG coupling \alpha only appears at second order. Because the paper's own Section V concedes this, the result, as stated, does not establish a dark-matter effect of EMSG. The lensing and radar-delay sections inherit the same fitted parameter s and do not provide an independent test of the model.

major comments (3)
  1. [Sec. III, Eqs. (14)-(16), (35)-(36)] The flat rotation curve is not derived from the EMSG field equations; it is imposed by the ansatz e^{a+b}=(r/\sigma)^s. This ansatz immediately gives a'+b'=s/r (Eq. 16). Combined with the weak-field formula v^2=ra'/2 (Eq. 35) and the solution (26) for e^a, the result v^2\simeq s/2 follows at leading order. The accompanying density profile (25), rho\simeq s/(\kappa r^2), is the isothermal-sphere profile that is already known to give flat rotation curves in Newtonian gravity. Thus the flat curve is a consequence of choosing a logarithmic time-time metric component, not a prediction of the energy-momentum squared term.
  2. [Sec. V and Sec. III after Eq. (36)] The central claim stated after Eq. (36) is that the model, 'through the role of the energy-momentum squared term in the field equations for pressureless matter,' illustrates dark matter. This is directly contradicted by the paper's own closing remark in Sec. V: 'the modified gravity features appears due to the adoption of relation (14) since the parameter \alpha enters into the dynamics as a second order effect as seen in (36).' Since the leading-order flat rotation velocity is independent of \alpha, the dark-matter-like signature is not attributable to the energy-momentum squared correction.
  3. [Sec. III, Eqs. (23)-(29)] The 'near general relativity' expansion is not a controlled perturbative limit. The exact density solutions (23) and (24) contain 1/\alpha terms, so the \alpha\to 0 limit is singular, while the later expansions in equations (26)-(29) treat \alpha s^2 as a small correction without defining the dimensionless small parameter. This makes the claim that the model stays 'in the vicinity of general relativity' (Eq. 15 and surrounding text) insufficiently supported. A proper perturbative derivation in which the EMSG coupling is demonstrably small and subdominant is needed before the flat-curve result can be attributed to the modified-gravity sector.
minor comments (4)
  1. [Eq. (32)] The conserved angular momentum should be J=r^2 d\phi/d\tau, not J=r^2(d\phi/d\tau)^2 as written; the subsequent use in Eq. (33) is only consistent with the former definition.
  2. [Eq. (43)] The radar echo delay expression appears dimensionally inconsistent: the first term contains a factor s/\sigma^2 multiplied by r_d^{s-2}, which does not have dimensions of length in geometric units. Please check the derivation and the displayed formula.
  3. [Sec. IV A, Eq. (39)] The split of the integration domain at r_d introduces a sharp transition between the 'dark matter' region and the exterior Schwarzschild region, but no matching or continuity condition for the metric or its derivative is stated. This weakens the quantitative status of the deflection-angle result.
  4. [Sec. III, Eqs. (19)-(22)] The text says Eq. (22) is obtained by eliminating \rho^2 from Eq. (19), but Eq. (22) also involves \rho'' and \rho'^2 and appears to require a separate combination of equations. The derivation should be made explicit.

Circularity Check

3 steps flagged · score 8.0 of 10

Flat rotation curve is imposed by the power-law ansatz (14), with s fitted to the observed velocity; the paper's own Sec. V admits α is only a second-order effect.

  1. self definitional [Section III, Eqs. (14), (16), (35), (36)]
    "As a plausible form, we can assume l(r) = (r/σ)^s, where s is a dimensionless parameter and σ is the length scale of the system. ... one obtains a′ + b′ = s/r ... υ2 = ra′/2 ... υ2 ≈ s/2 − 6α/(κ^4 r^4)(κ^2 r^2 − 6α)s^2."

    The ansatz e^{a+b} = (r/σ)^s with constant s directly fixes a′+b′ = s/r. Using the resulting solution for a(r), the weak-field circular speed formula gives v^2 ≈ s/2, an r-independent constant. Thus the flat rotation curve is written into the assumed metric combination before the EMSG field equations are solved. The EMSG coupling α appears only in the O(s^2) correction and does not generate the flatness, so the advertised 'emergence' of a flat curve is an algebraic consequence of the input ansatz.

  2. fitted input called prediction [Abstract; Section III after Eq. (36)]
    "By fixing the components using the rotational velocities of galaxies, the model demonstrates the emergence of a flat rotation curve in the galactic halo. ... Consequently, by neglecting higher-order terms of s, equation (36) reveals that s/2 is approximately equal to the square of the tangential velocity, i.e., s/2 ≈ υ²."

    The parameter s is not predicted by the theory; it is set equal to 2υ² using the observed (and assumed constant) halo velocity. The same relation v^2 ≈ s/2 is then presented as a demonstration of flatness. Since a constant input s produces a constant output v^2 by construction, the rotation-curve 'prediction' reduces to the fit. The flatness of the observed curve is already assumed when a single representative value (200–500 km/s) is used to fix s.

1 more flagged steps
  1. other [Section V, Remarks]
    "Actually, the modified gravity features appears due to the adoption of relation (14) since the parameter α enters into the dynamics as a second order effect as seen in (36)."

    This sentence concedes the central mechanism: the dark-matter-like feature is attributed to 'relation (14)', i.e., the ad hoc power-law ansatz, not to the energy-momentum squared term. It directly contradicts the claim after Eq. (36) that the model 'through the role of the energy-momentum squared term' produces the dark-matter effect. The central result is therefore self-definitional relative to the assumed metric form, with α contributing only a small correction.

full rationale

The load-bearing circularity here is not a self-citation chain; the cited background, including the EMSG action and the l(r) parametrization of Sobouti, is not used to forbid alternatives. Instead, the central claim reduces to the metric ansatz and to a fitted parameter. The authors posit e^{a+b} = (r/σ)^s with constant s (Eq. 14), which fixes a′+b′ = s/r (Eq. 16). Solving the field equations and using the weak-field formula v^2 = r a′/2 (Eq. 35) gives v^2 ≈ s/2 with α-dependent corrections only at O(s^2) (Eq. 36). Thus the flatness of the rotation curve is a direct consequence of taking s constant in the ansatz, not of the T^2 term. The paper then fixes s through the observed halo velocity, writing 's/2 ≈ υ²', so the predicted flat curve is also a fitted input. The density profile ρ ≈ s/(κ r^2) (Eq. 25) is the classic isothermal-sphere profile long known to produce flat rotation curves, again inserted through the ansatz. The lensing and radar-delay sections inherit the same fitted s and therefore provide no independent confirmation. The authors themselves admit in Sec. V that the modified-gravity features appear 'due to the adoption of relation (14)' and that α is a second-order effect. At leading order, the advertised dark-matter prediction is equivalent to the assumed power-law metric combination plus the identification s = 2v²; this is a definitional/fitted-input circularity, warranting a score of 8 rather than a milder self-citation finding.

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

The central claim rests on the power-law ansatz (14), which imposes flat rotation, the dust-fluid modeling choice, the unconstrained coupling alpha, and the Schwarzschild exterior matching. The only external anchors are the standard geodesic formulas and the EMSG field equations from [35]. The fitted parameter s is the key free input.

free parameters (3)
  • s = approximately 2 v^2, about 10^-6 for typical galaxies
    Dimensionless exponent in the metric ansatz (14). It directly sets the rotation velocity via v^2 = r a'/2 approximately s/2, and the paper sets s from observed galaxy velocities ('s/2 approximately v^2'). Not predicted independently.
  • sigma = not fitted; a galactic length scale
    Length scale in the ansatz e^(a+b) = (r/sigma)^s. Appears in metric components; the paper claims it is unimportant when sigma is much less than r_d, but no physical origin or independent value is given.
  • alpha (EMSG coupling) = unconstrained in this paper
    Coupling of the T_munu T^munu term in the action. Enters the density profile and correction terms; the paper gives no observational bound and only states that alpha is 'critical for constructing a valid model.'
assumptions (5)
  • domain assumption The EMSG field equations from Roshan and Shojai (Eq. 4) are correct, and for dust one may set Lm = p so that the derivative terms vanish.
    The entire derivation rests on Eqs. (7)-(11). The paper chooses Lm = p solely for simplicity and does not justify dropping the 4 T_alpha_beta d^2 Lm / d g d g term.
  • ad hoc to paper The metric ansatz e^(a+b) = (r/sigma)^s with constant s much less than 1 (Eq. 14), taken from Sobouti [42], is a valid near-GR solution.
    This is the central modeling choice. It directly fixes a'(r) approximately s/r and hence the flat rotation curve. No physical mechanism or independent constraint motivates it beyond being 'a plausible form.'
  • domain assumption A galactic halo can be modeled as a static, spherically symmetric, pressureless perfect fluid.
    Used in Eqs. (9)-(11); excludes velocity dispersion, anisotropy, baryonic disk structure, and relativistic fluid effects.
  • domain assumption The exterior region beyond r_d is exactly Schwarzschild, matched at r_d without junction conditions.
    Used to split the integrals in Eqs. (39) and (42). No Israel junction conditions or continuity requirements are imposed at the boundary of the dark-matter region.
  • standard math Stable circular orbit and weak-field circular speed formula v^2 = r a'/2 (Eqs. 33-35).
    Standard geodesic result for static spherically symmetric metrics; used to convert the metric function a(r) into rotation velocity.

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Cite this review

Pith. "Pith review of Dark Matter and Energy-Momentum Squared Gravity." pith.science (2026). https://pith.science/paper/2S6TGZGE

@misc{pith2026250119141,
  author       = {Pith},
  title        = {Pith review of: Dark Matter and Energy-Momentum Squared Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2S6TGZGE}},
  note         = {Machine review of arXiv:2501.19141}
}
abstract

In this study, we examine the energy-momentum squared modified theory of gravity, where the squared term $T_{\mu\nu}T^{\mu\nu}$ is incorporated into the conventional gravitational Lagrangian. This modification aims to account for dark matter effects on galactic scales. Specifically, we analyze the model near general relativity solutions for spherically symmetric and static metrics. By fixing the components using the rotational velocities of galaxies, the model demonstrates the emergence of a flat rotation curve in the galactic halo. Additionally, we investigate the proposed model's predictions for the light deflection angle and radar echo delay.

Figures

Figures reproduced from arXiv: 2501.19141 by the authors.

Figure 1
Figure 1. FIG. 1. The light-deflection angle for different galaxies, given [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The radar echo delay as a function of [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.