Galactic Spacetime Solutions with a Varying Newton's Coupling

Rounak Chakraborty; Sandipan Sengupta

REVIEW 3 major objections 5 minor 20 references

Galactic Spacetime Solutions with a Varying Newton's Coupling

T0 review · 3 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper derives exact vacuum solutions in a theory with a spatially varying Newton constant whose rotation curves flatten at large radii and whose gravitational bending of light is slightly reduced.

desk verdict New torsional vacuum solutions with a solid rotation-curve result, but the headline lensing claim is undone by an inverted prefactor in §V. read the letter →

arxiv 2505.04060 v1 pith:AYCZ6FAD submitted 2025-05-07 gr-qc astro-ph.GA

Rounak Chakraborty , Sandipan Sengupta This is my paper · ORCID

classification gr-qcastro-ph.GA PACS 04.50.Kd95.35.+d98.62.Gq

keywords varyingNewton'sconstantgalacticrotationcurvestorsionexactvacuumsolutionsgravitationallensingdeflectionnon-baryonicmassscalar-tensorgravitydarkmatteralternatives

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

The paper derives a new family of exact, static, spherically symmetric vacuum solutions in a four-dimensional gravity theory whose Newton's coupling varies mildly in space. In these solutions the product $f(r)g(r)$ takes the form $(r/R)^{2\alpha}$ with $\alpha\ll 1$, so a test particle's circular velocity approaches $\sqrt{\alpha}$ at large radii, reproducing flat galactic rotation curves without any matter source. The effective mass producing the curve is purely geometric and includes a negative non-baryonic contribution, which the authors emphasize is not negative matter. For light, the spacetime diminishes the Einstein deflection angle by a correction suppressed by $\alpha$, opposite to the enhanced bending typical of dark-matter or modified-gravity models. If correct, the construction offers a geometric alternative to dark matter at galactic scales that lensing observations could in principle distinguish.

What carries the argument

The load-bearing device is the ansatz $f(r)g(r)=(r/R)^{2\alpha}$, a two-parameter family of non-Newtonian static spherical metrics in which $\alpha\ll 1$ is the asymptotic circular-velocity squared. Inserting this ansatz into the integrated torsion equation gives $\xi(r)=1/[C_1(r/R)^\alpha-C_2(r/R)]^2$; setting $C_2=0$ keeps the coupling slowly varying and leads to the exact metric (14). The same ansatz directly makes $v^2\to\alpha$ at large radii, so the flat rotation curve is encoded in the assumed product form rather than emerging independently from the field equations.

What would settle it

Compute the exact deflection integral in the paper for the metric (14) without the weak-field expansion; if the correction to $\delta-4m_B/r_0$ changes sign or grows with the impact parameter anywhere inside the halo, the infinite-halo result fails. Observationally, stacked galaxy-galaxy lensing around galaxies with flat outer rotation curves should show a small deficit in tangential shear relative to a singular isothermal halo of the same rotation velocity, and a measured enhancement would rule the model out.

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Extended reading notes

Core claim

The central discovery is a set of exact vacuum metrics in the torsional phase of a first-order, curvature-linear gravity action with a varying Newton's coupling. For static spherical symmetry the field equations reduce to $R_{\alpha\beta}(\bar{\omega}+K)=0$, with contortion $K^{IJ}{}_\mu=\frac{1}{2\xi}e^{\sigma[J}e^{I]}{}_\mu\partial_\sigma\xi$. Assuming the non-Newtonian parametrization $f(r)g(r)=(r/R)^{2\alpha}$ with $\alpha\ll 1$, the authors obtain the metric $f(r)=(r/R)^{2\alpha}$ and $g^{-1}(r)=(r/R)^{2\alpha}[(1-\alpha)^{-2}-C(r/R)^{-1+3\alpha}]$. The circular velocity then satisfies $v^2(r)\to\alpha$ as $r\to\infty$, so $\alpha$ is the asymptotic rotation velocity squared. The effective density and pressures generated by the torsion give an enclosed mass $m(r)=-\frac{\alpha(2-\alpha)}{2(1-\alpha)^2}r+\frac{CR}{2}(r/R)^\alpha$, whose leading geometric term is linear in $r$ and negative. For light passing through an infinite halo, the deflection angle is $\delta\approx 4m_B/r_0-8\alpha m_B/r_0$, a reduction relative to Einsteinian bending rather than the enhancement found in typical dark-matter models.

Load-bearing premise

The load-bearing premise is the starting assumption that $f(r)g(r)=(r/R)^{2\alpha}$ with $\alpha\ll 1$, a form that already forces the rotation velocity to become constant at large radii, together with the discarding of the $C_2\neq 0$ branch; flat rotation curves are therefore built into the ansatz rather than derived from the field equations.

Editorial extensions

If this is right

Asymptotically flat rotation curves are obtained in vacuum, so the model explains the observed flatness without invoking a dark-matter component.
The effective mass enclosed by a radius $r$ contains a geometric term linear in $r$ with a negative sign; the authors stress this is a torsion-induced effective source, not a genuine negative mass.
The deflection of light is predicted to be $\delta\approx 4m_B/r_0-8\alpha m_B/r_0$, so the model gives less bending than Einstein gravity with the same baryonic mass, opposite to the enhanced bending of dark-matter halos.
At $\alpha=0$ the solution reduces to the Schwarzschild exterior, and in the weak-field limit the circular velocity recovers the Keplerian $m_B/r$ term, so the construction contains general relativity plus small corrections in the appropriate limits.

Reading between the lines

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

Because $\alpha$ is a free parameter of the ansatz rather than a derived quantity, the model as it stands does not predict a specific rotation velocity; its testable content is the relation between the observed $v^2\approx\alpha$ and the fractional suppression $8\alpha$ of the lensing angle.
The finite-halo matching condition in the appendix determines the halo boundary through the baryonic mass and $\alpha$; a precise measurement of where the rotation curve flattens and where it joins the exterior geometry could test this relation.
If the negative geometric mass term is read as an effective density it would violate usual energy conditions; the authors avoid that conclusion by treating it as a torsion artifact, but a fully covariant stress-energy analysis would clarify whether any physical energy condition is actually violated.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The torsional vacuum solutions are genuinely new and the field-equation work in Secs. III-IV is internally consistent as far as I can check. The α=0 limit recovers Schwarzschild, and the circular-velocity formula plus the negative geometric mass term follow cleanly from the metric. If you care about exact solutions in variable-Newton-coupling gravity, this is a real addition.

But the paper's headline—diminished light bending that distinguishes it from CDM—is very likely the product of an algebra error. In §V the identity for the null integrand carries a prefactor 1/(1−α)², but the correct left-hand side for C=0 is (1−α)²[(r/r0)^{2−2α}−1]. That puts the printed identity off by a factor (1−α)^{-4}. With the correct prefactor, the pure-halo integral gives Δφ≈π/2+πα, so δ≈2πα, a positive correction—same sign as the singular isothermal sphere's +2πα. The printed −8αm_B/r_0 in Eq. (19) omits the halo's own contribution; it only picks up the baryonic piece. Since m_B/r_0 is itself of order α for typical galaxies, the dropped term is not negligible. The claimed 'diminished rather than enhanced' bending does not hold.

Two smaller wobbles, both less severe. First, the flat rotation curve is put in by hand through the ansatz fg=(r/R)^{2α}; α is not derived but read off as v². That is not a vice in exact-solution work, but the abstract's 'predicts flat rotation curves' is too strong. Second, the C2=0 branch in Eq. (13) is dismissed as 'practically relevant' with no real argument; a referee should ask what happens for C2≠0 and whether those solutions are also static and stable. The finite-halo appendix inherits the §V prefactor issue when C=0.

The paper is worth a serious referee. The exact solutions and the negative-mass insight are interesting to the torsional/modified-gravity community, and the algebra is checkable. But the deflection section needs recalculation, and the abstract and conclusions should be redrawn if the sign flips as I think it does. A referee should be asked to verify the prefactor carefully; if I'm right, this is a major-revision situation, not a desk reject, because the core geometry still stands.

Referee Report

3 major / 5 minor

Summary. The paper studies static, spherically symmetric vacuum solutions in a four-dimensional first-order gravity theory with a spacetime-varying Newton-type coupling ξ (Sec. II). In the torsional phase with ξ≠0, the authors assume the product form f(r)g(r)=(r/R)^{2α} with α≪1 (Sec. IV, Eq. (12)), obtain the scalar profile (13), discard the C₂≠0 branch, and derive the exact metric (14). They show that the circular velocity satisfies v²(r)→α as r→∞, so α is identified with the asymptotic rotation speed, and they compute an effective energy-momentum tensor with negative energy density at large radii. In Sec. V they derive the deflection of light in the infinite-halo limit and report δ≈4m_B/r_0−8α m_B/r_0, i.e., a diminished bending relative to Einsteinian gravity. A finite-halo matching to Schwarzschild is treated in the Appendix.

Significance. If the calculations were correct, the paper would provide an exact, matter-free realization of asymptotically flat rotation curves in a torsional extension of GR, together with a lensing signature that distinguishes the model from dark-matter halos. The paper is commendably explicit: closed-form metrics, the scalar profile, and the geodesic integrals are all displayed, which makes the results checkable. However, the headline lensing claim is invalidated by an algebraic prefactor error whose correction reverses the sign of the leading halo contribution, and the flat rotation curve is an input assumption rather than an emergent prediction. The exact solutions remain a useful contribution if the presentation is revised to be accurate about what is derived and what is assumed.

major comments (3)

[Sec. V, Eq. (18) and the displayed identity after it] The identity used to evaluate the null geodesic integral has an inverted prefactor. Setting C=0 in the metric (14) gives fg=(r/R)^{2α} and g^{-1}=(1−α)^2, so the left-hand side of the identity is (1−α)^2[(r/r0)^{2−2α}−1], not (1−α)^{−2}[(r/r0)^{2−2α}−1]. The base integral therefore evaluates to Δφ=π/[2(1−α)^2]≈π/2+πα, yielding δ≈2πα for the pure-halo (C=0) case. The positive halo contribution is missing from the printed expansion Δφ≈π/2+2(1−2α)m_B/r_0, and the sign of the leading α correction in Eq. (19) is wrong: the halo enhances the bending in the same direction as CDM, rather than diminishing it. Since the abstract and Section V advertise the diminished deflection as the key observational distinction, this error is load-bearing.
[Sec. IV, Eqs. (12)–(15)] The 'prediction' of flat rotation curves is an input. The ansatz fg=(r/R)^{2α}, together with the standard formula v²=rf'/(2f), gives v²→α at large r irrespective of the field equations; the dynamics fixes the metric only after the ansatz is imposed. The abstract and conclusion describe the solutions as 'corresponding to' and 'leading to' flat rotation curves, which overstates the status of α as a free parameter. In addition, the C₂=0 restriction in Eq. (13) is justified only by 'practically relevant' slow variation of ξ; no quantitative or physical criterion is given, so the claim that the displayed family represents the galactic solutions of the theory is not established.
[Sec. IV, Eq. (14) vs. Eq. (15)] The two displayed forms of the solution are inconsistent in the power of (r/R) multiplying the constant C: Eq. (14) contains (r/R)^{−1+3α}, while Eq. (15) and the subsequent weak-field expansion use (r/R)^{−1+α}. This discrepancy affects the circular-velocity formula and feeds into the lensing integral; the authors should identify the correct exponent, recalculate Eqs. (15)–(19), and check the matching condition in the Appendix with the corrected metric.

minor comments (5)

[Abstract] There is a typo in the abstract: 'observationally dintinguishable' should be 'observationally distinguishable'.
[Sec. IV and Appendix] The name 'Schwarzchild' appears in several places and should be 'Schwarzschild'.
[Sec. IV, Eq. (17)] The statement that the negative linear term in the effective mass m(r) is responsible for the flat velocity profile should be reconciled with the fact that m(r)/r is negative at large r while v² is positive; the relation between the effective mass and the circular velocity in this torsional theory should be stated explicitly.
[References] Reference [12] is incomplete as printed (the article title and publication details are missing).
[Appendix] The finite-halo results in Eqs. (A.4)–(A.5) should be revisited in light of the corrected infinite-halo bending; as written, the arctangent terms in (A.5) cancel in the R→∞ limit and do not reproduce a finite halo contribution.

Circularity Check

1 steps flagged · score 6.0 of 10

Flat rotation curves are put in by the fg=(r/R)^{2α} ansatz and then read out as v²→α; the exact solution follows from the assumed form, so the central 'prediction' reduces to the input parametrization.

self definitional [Sec. IV, Eqs. (12)–(15)]
"we may choose a convenient parametrization [11, 14] of the behaviour above as : f(r)g(r) = (r/R)^{2α} ... The circular velocity ... is given by: v^2(r) = rf'/2f [8]. ... This leads to a constant circular velocity at large radii from the galactic centre: v^2(r) → α as r → ∞. Thus, the small parameter α acquires a physical interpretation in this large distance limit."

Equation (12) fixes the leading large-r scaling f ∼ (r/R)^{2α} (the field equations only determine subleading terms and ξ(r)), and the standard circular-velocity formula v² = rf′/2f then gives v² → α by direct differentiation. Therefore the asymptotically flat rotation curve and the numerical value of α are inputs of the chosen ansatz, not outputs of the theory. The paper itself says α 'acquires a physical interpretation' as the asymptotic velocity, i.e. the input parameter is renamed as the result. The exact metric and negative effective mass are genuine consequences of the assumed form, but the headline claim of finding flat rotation curves is the assumed product form restated.

full rationale

The central circularity is in Sec. IV: the family of 'galactic spacetimes' is generated by imposing fg = (r/R)^{2α}, and the field equations are then solved for ξ and the subleading metric terms. The flat rotation curve v² → α follows immediately from the geodesic formula v² = rf′/2f once f has this power-law scaling, so it is not a prediction of the varying-G theory but a property of the ansatz, with α later identified with the observed circular velocity. This is construction rather than derivation, so the 'prediction' reduces by construction and warrants partial circularity (score 6). No load-bearing self-citation chain was found: refs. [14] and [17] are prior work by the same author defining the action, but the ansatz is stated explicitly and the algebra is self-contained; there is no imported uniqueness theorem and no machine-checked or externally falsifiable result being replaced by self-citation. Separately, the deflection identity in Sec. V appears to have a prefactor inconsistency that would change Eq. (19), but that is a technical correctness concern rather than a circularity and is not counted in the score.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No new particle, force, or dimension is introduced. The scalar ξ and the torsion are part of the already proposed Lagrangian; the negative non-baryonic mass is an interpretation of an effective energy-momentum tensor, not a new entity. The central construction rests on the ansatz that directly encodes flat rotation curves, and on the ad hoc C2=0 branch choice.

free parameters (3)

α = ~10^-6 for spiral galaxies, (v_flat/c)^2
Small parameter in the ansatz fg=(r/R)^(2α); the solution gives v²∞=α, and the paper identifies α with the observed flat rotation velocity, effectively setting it by the data it aims to explain.
R = not specified, matched at halo boundary
Length scale introduced in Eq. (12) to keep f and g dimensionless; fixed by matching to a Newtonian or Schwarzschild metric at some radius. It is a free scale of the solution family.
C, or baryonic mass mB=CR/2 = determined by baryonic mass
Integration constant in the metric; for α=0 it reduces to the Schwarzschild mass parameter. It encodes the baryonic matter contribution but is a free constant of the solution.

assumptions (4)

domain assumption The first-order gravity action (1) with independent ξ, e, ω fields and torsion solution (3) describes a viable theory of gravity at galactic scales.
The whole derivation starts from this Lagrangian from refs. 14 and 17; no experimental or observational support for the framework is provided in this paper.
ad hoc to paper The metric product form fg=(r/R)^(2α) with α much less than 1 represents galactic spacetimes.
This ansatz, Eq. (12), is chosen because it leads to flat circular velocities; it is not derived from the field equations and directly contains the result.
ad hoc to paper The branch C2=0 in Eq. (13) is the only physically relevant one.
The paper discards the C2-not-equal-to-zero branch by requiring ξ to vary slowly, without deriving this condition from the theory or from observations.
domain assumption Weak-field and small-α expansions apply in the deflection computation, with C much less than 1, M/r* much less than 1, and r0 comparable to R.
These assumptions are needed for the closed-form deflection angles in Sec. V and the appendix; the leading correction is only valid in this regime.

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Cite this review

Pith. "Pith review of Galactic Spacetime Solutions with a Varying Newton's Coupling." pith.science (2026). https://pith.science/paper/AYCZ6FAD

@misc{pith2026250504060,
  author       = {Pith},
  title        = {Pith review of: Galactic Spacetime Solutions with a Varying Newton's Coupling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AYCZ6FAD}},
  note         = {Machine review of arXiv:2505.04060}
}

read the original abstract

We find a new family of galactic metrics corresponding to flat rotation curves at the outer radii. These are vacuum solutions to a gravity theory where the Newton's coupling varies mildly in space. The effective `mass', whose origin is purely geometric, receives a negative non-baryonic contribution. The angle of deflection of a light ray propagating in this geometry is found to be diminished rather than enhanced compared to the Einsteinian bending, the effect being highly suppressed though. Hence, these spacetimes are observationally dintinguishable from other geometric or `dark matter' based alternatives invoked to explain mass discrepancies in galaxies.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Zwicky, Die Rotverschiebung von extragalaktischen N ebeln, Helv

F. Zwicky, Die Rotverschiebung von extragalaktischen N ebeln, Helv. Phys. Acta 6, 110 (1933)

work page 1933

[2] [2]

V. C. Rubin and W. K. Ford, Jr., Rotation of the Andromeda N ebula from a Spectro- scopic Survey of Emission Regions, Astrophys. J. 159, 379 (1970); V. C. Rubin, N. Thon- nard, and W. K. Ford, Jr., Rotational properties of 21 SC gala xies with a large range of luminosities and radii, from NGC 4605 /R = 4kpc/ to UGC 2885 /R = 122 kpc/, Astrophys. J. 238, 4...

work page 1970

[3] [3]

M. S. Roberts and R. N. Whitehurst, The rotation curve and geometry of M31 at large galactocentric distances., Astrophys. J. 201, 327 (1975)

work page 1975

[4] [4]

Milgrom, A modiﬁcation of the Newtonian dynamics as a p ossible alternative to the hidden mass hypothesis., Astrophys

M. Milgrom, A modiﬁcation of the Newtonian dynamics as a p ossible alternative to the hidden mass hypothesis., Astrophys. J. 270, 365 (1983); A modiﬁcation of the Newtonian dynamics - Implications for galaxies., Astrophys. J. 270, 371 (1983)

work page 1983

[5] [5]

Bekenstein and M

J. Bekenstein and M. Milgrom, Does the missing mass probl em signal the breakdown of Newtonian gravity?, Astrophys. J. 286, 7 (1984)

work page 1984

[6] [6]

P. D. Mannheim and D. Kazanas, Exact Vacuum Solution to Co nformal Weyl Grav- ity and Galactic Rotation Curves, Astrophys. J. 342, 635 (1989); P. D. Mannheim, 10 Linear Potentials and Galactic Rotation Curves, Astrophys . J. 419, 150 (1993), arXiv:hep-ph/9212304 [hep-ph]

arXiv 1989

[7] [7]

R. H. Sanders, A Stratiﬁed Framework for Scalar-Tensor T heories of Modiﬁed Dynamics, Astrophys. J. 480, 492 (1997), arXiv:astro-ph/9612099 [astro-ph]

arXiv 1997

[8] [8]

Nucamendi, M

U. Nucamendi, M. Salgado, and D. Sudarsky, Alternative a pproach to the galactic dark matter problem, Phys. Rev. D 63, 125016 (2001)

work page 2001