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REVIEW 4 major objections 5 minor 47 references

Study of Wormhole in $f(Q)$ gravity with some dark energy models

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Exact wormhole shape functions can be produced in linear f(Q) gravity from three dark energy equations of state, and each solution satisfies the Morris-Thorne viability conditions while violating the null energy condition at the throat.

desk verdict Repackaged f(Q) wormhole constructions, one of which fails asymptotic flatness; not worth refereeing. read the letter →

arxiv 2507.03939 v1 pith:EQUMR27D submitted 2025-07-05 gr-qc

Tanmoy Chowdhury , Prabir Rudra , Tandrima Chowdhury , Farook Rahaman This is my paper · ORCID
classification gr-qc
keywords wormholef(Q)gravityChaplygingasdarkenergyshapefunctionconditionsTOVequationnon-metricity
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 traversable wormholes can be built inside linear f(Q) gravity—a theory of gravity based on non-metricity rather than curvature—by letting the wormhole's radial pressure follow three dark energy equations of state: a varying Chaplygin gas, a generalized Chaplygin gas, and a varying barotropic fluid. For each case the authors derive an exact shape function $b(r)$, check the Morris-Thorne throat, flaring-out, and asymptotic-flatness conditions, and plot the energy conditions. In all three models the null energy condition fails at the throat, which is the standard signal of the exotic matter needed for a traversable wormhole. They also use the Tolman-Oppenheimer-Volkoff balance to argue that the configurations are stable under hydrostatic and anisotropic forces. If true, the result would give a concrete recipe: choose a dark energy equation of state, solve for $b(r)$, and get a wormhole geometry in this modified gravity.

What carries the argument

The central object is the Morris-Thorne shape function $b(r)$ in the static wormhole line element. Together with a constant redshift function and the linear $f(Q)=\alpha Q$ Lagrangian, it turns the f(Q) field equations into simple relations: $-\alpha b'/r^2=\rho$, $\alpha b/r^3=p_r$, and $(\alpha/2)(b'/r^2-b/r^3)=p_t$. The argument works by choosing the radial pressure to obey a dark energy equation of state, solving the resulting differential equation for $b(r)$, and then checking the geometric and energy conditions.

What would settle it

Solve $b(r_0)=r_0$ for each shape function—(5.2), (5.7), and (6.2)—using the paper's stated parameters ($a=0.05$, $u=-4$, $B=1$, $c=0.09$ for WH1; $a=-1.2$, $B=1$, $c_1=10$ for WH2; $\omega=-2$, $c_1=1$ for WH3) and check for a positive real root $r_0$ with $b'(r_0)<1$. If any model has no positive real throat radius or fails the flaring condition at that radius, the claimed traversable wormhole does not exist; separately, evaluating $\rho+p_r$ at $r_0$ from (5.3)-(5.4), (5.8)-(5.9), and (6.3)-(6.4) would settle the NEC-violation claim.

Watch

Extended reading notes

Core claim

Within a linear model $f(Q)=\alpha Q$ with constant redshift function, the paper claims that the field equations reduce to three algebraic relations linking the shape function to energy density and pressures. Substituting the dark energy equations of state $p_r=-B b(r)^u/\rho^a$, $p_r=-B/\rho^a$, and $p_r=-\omega\rho b(r)^u$ yields closed-form shape functions $b(r)$, and with negative $\alpha$ these are claimed to meet the throat condition $b(r_0)=r_0$, the flaring condition $b'(r_0)<1$, and asymptotic flatness $b(r)/r\to 0$. The same solutions are then shown, for chosen parameter values, to give positive energy density but negative radial pressure, so that $\rho+p_r<0$ at the throat—NEC violation—while the tangential-pressure NEC can still hold in some models. The TOV force balance is plotted and found to vanish, establishing equilibrium.

Load-bearing premise

The construction assumes the wormhole's radial pressure is exactly given by one of the three dark energy equations of state with the stated constants, and that $\alpha$ must be negative for asymptotic flatness; if a different equation of state or a positive coupling were used, the derived shape functions and conclusions would no longer follow.

Editorial extensions

If this is right

  • Each derived shape function is an explicit candidate geometry that can be embedded in three-dimensional space, so the models give visualizable wormhole spacetimes rather than abstract existence statements.
  • The same construction can be retried with any equation of state: substituting $p_r$ into the relation $\alpha b/r^3=p_r$ gives a first-order ODE for $b(r)$, so the method is a template for generating new wormhole solutions.
  • Because the TOV balance is satisfied with zero gravitational force (constant redshift), the stability argument depends only on hydrostatic and anisotropic forces canceling; models satisfying that balance are in equilibrium.
  • NEC violation in all three models implies the wormhole throats must contain exotic matter, so the paper does not evade the usual energy-condition cost of traversability.
  • The requirement $\alpha<0$ for asymptotic flatness ties the gravitational coupling sign to the existence of the solutions.

Reading between the lines

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

  • One extension not pursued here is to replace the linear $f(Q)=\alpha Q$ ansatz with nonlinear f(Q) and ask whether the same equations of state still yield asymptotically flat shape functions, or whether a positive coupling could work.
  • The hand-picked constants ($B$, $u$, $a$, $\omega$, $c$) have no independent observational anchor; testing these models would require linking the parameters to cosmological data or to lensing and echo observations of a wormhole candidate.
  • The throat condition $b(r_0)=r_0$ is checked graphically for the plotted parameters; a numerical root-finding scan over parameter space could reveal whether the models remain traversable away from those values or whether viability is confined to narrow intervals.
  • Since the redshift is constant, the gravitational redshift signature is absent; a nonzero $\Phi(r)$ could alter the TOV force balance and the NEC status, so the robustness of the construction under that relaxation is an open question.
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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 / 5 minor

Summary. The paper constructs static, spherically symmetric Morris-Thorne wormhole solutions in linear f(Q) gravity with a constant redshift function. Three dark-energy equations of state are imposed on the radial pressure: varying Chaplygin gas (Eq. 5.1), generalized Chaplygin gas (Eq. 5.6), and varying barotropic fluid (Eq. 6.1). Exact shape functions b(r) are presented for each case, followed by graphical checks of the throat, flaring-out, and asymptotic flatness conditions; energy-condition plots; embedding diagrams; and a TOV-equation force balance. The abstract and conclusion claim that all three models satisfy the Morris-Thorne conditions, violate the NEC, and are therefore traversable wormholes.

Significance. The paper is an example-based study in an active area: exact wormhole solutions in modified gravity with dark-energy fluids. The field equations for the linear f(Q) model are standard, and the strategy of imposing an equation of state on p_r is a common and legitimate way to close the underdetermined system. If the solutions were correct, they would provide a set of explicit shape functions and energy-condition profiles worth comparing with earlier f(Q) wormhole literature. However, the generalized Chaplygin model (WH2) fails the asymptotic flatness condition algebraically and in its own figures, the TOV 'stability' test is an identity rather than a stability analysis, and the parameter values for WH3 are internally inconsistent. The main traversable-wormhole claim is therefore not established. No machine-checked proofs or reproducible code are provided; all checks are graphical.

major comments (4)
  1. [Sec. 5.2, Eq. (5.7), Figs. 4-5] The asymptotic flatness claim for WH2 is false. With the stated parameters a=-1.2, B=1, c1=10, one has a/(a+1)=6 and r^{3/a+3}=r^{1/2}, so b(r) as given in Eq. (5.7) behaves as b(r) ~ D^6 r^3 for large r; consequently b(r)/r ~ r^2, which diverges and violates the Morris-Thorne condition b(r)/r -> 0 stated in Section 3. The text in Section 5.2 that 'we retained the asymptotic flatness condition' is contradicted by the formula itself, and Fig. 4 shows b/r increasing to values of order 10 while Fig. 5 shows b-r > 0 and b' ~ 10^10. Moreover, substituting a power-law ansatz b ~ r^k into Eqs. (4.6)-(4.8) with EoS (5.6) forces k=3 for every a ≠ -1, so an asymptotically flat solution of this type does not exist for the generalized Chaplygin gas in this linear f(Q) model. Since WH2 is one of the paper's three central constructions, the main claim that all three models satisfy the conditions for a viable wormhole is invalid.
  2. [Sec. 8, Eqs. (8.1)-(8.2), Fig. 15] The TOV test does not establish stability. Because the redshift function is constant, phi'=0, and Eq. (8.1) reduces to d p_r/dr + (2/r)(p_r - p_t)=0, which is simply the conservation equation for the anisotropic stress-energy tensor and is identically satisfied by the field equations. The plots of F_A and F_H in Fig. 15 therefore show an algebraic identity, not the reaction of the configuration to perturbations or even a nontrivial equilibrium condition. The abstract and Section 9 use this calculation to support the word 'stability'; that support should be removed or replaced by a genuine stability analysis.
  3. [Secs. 5.1-5.2 and 6] The energy-condition discussion is self-contradictory. Section 5.1 states that the NEC is violated (rho+p_r <= 0, p_r <= 0) and then, in the same paragraph, says 'the solution satisfies all energy conditions.' Section 5.2 says the NEC 'in terms of both pressures remains true even in the locations where the NEC in terms of P_r is broken,' which is not meaningful as written and conflicts with the abstract's claim that every model violates the NEC. Since the NEC violation is the stated physical output of the paper, these statements must be made mutually consistent before the conclusions can be assessed.
  4. [Sec. 6, Eq. (6.2), Figs. 8-10, 14] The parameter values for WH3 are not reproducible. The text specifies 'omega = 2; c1 = 1' for the solution, but the captions of Figs. 8 and 9 state 'omega = -2 and c1 = 1', and Fig. 14 uses 'omega = -1'. These choices give different functional forms for b(r) in Eq. (6.2), and with omega = -2 and non-integer u, the base of the fractional power is negative for small r, leaving the real-valued shape function undefined unless a branch choice is specified. The paper should state one consistent parameter set and the branch convention used to generate all WH3 figures.
minor comments (5)
  1. [Abstract and Sec. 9] The sentence 'Since each wormhole model is shown to violate the NEC, it can be understood that these wormholes are traversable' is logically incomplete: NEC violation is a necessary condition for traversable wormholes in GR-like settings, but it is not sufficient. The paper does not demonstrate the additional traversability requirements (bounded tidal forces, finite crossing time, no horizon).
  2. [Sec. 5.1, Eqs. (5.2)-(5.5)] The constants c and c1 are used in the shape function and in the expressions for rho, p_r, and p_t, but their definitions are not given before they appear; the text only later sets both to 0.09 for the plots.
  3. [Secs. 4-7] Several presentation issues: Eq. (4.4) contains a spurious dot on f_Q, the text around Eq. (4.2) has an apparent typo in the bracketed term, and the embedding-diagram text refers to Figs. 11-12 while the figures are labeled Figs. 13-14 with inconsistent parameter captions.
  4. [Sec. 5.2 and Figs. 6-7] The solutions contain fractional powers of negative quantities such as (-1)^{1/a} and alpha^{-1/a-1} for alpha < 0 and non-integer a. The paper does not specify the principal branch or the convention used to make these expressions real in the figures; this should be stated for the exact formulas to be well defined.
  5. [Introduction, Refs. [10]-[12]] The paper should state explicitly what is new relative to Refs. [10]-[12], which already treat wormholes in f(Q) gravity with varying Chaplygin gas, generalized Chaplygin gas, and barotropic fluids; the current manuscript does not contain a novelty statement.

Circularity Check

2 steps flagged · score 2.0 of 10

No load-bearing circularity: shape functions are obtained by direct ODE integration from assumed EoS; however the NEC-violation conclusion is forced by the assumed negative-pressure EoS and the TOV check is an identity, so the physical 'predictions' add little independent content.

  1. fitted input called prediction [Abstract; Section 5.1 Eq. (5.1); Section 9 conclusion]
    "Since each wormhole model is shown to violate the null energy condition (NEC), it can be understood that these wormholes are traversable. [...] Null energy condition is violated as the model varies with the energy density ρ ≥ 0 and the radial pressure is negative all over the space."

    All three radial-pressure EoS are assumed with an explicit negative sign: pr = −B b(r)^u/ρ^a (5.1), pr = −B/ρ^a (5.6), and pr = −ωρ b(r)^u (6.1). With α<0 the solved energy density is positive, so the paper's own conclusion reduces the NEC violation to the sign of the input EoS. The 'traversable wormhole' punchline is therefore not an independent prediction of f(Q) gravity but a restatement of the assumed dark-energy EoS after integration. The shape-function integration itself is still a genuine derivation, so this is partial, not total, circularity.

  2. other [Section 8, Eqs. (8.1)-(8.2), Figure 15]
    "For our choice of the redshift φ as a constant, we have FG = − φ′/2 (ρ + pr), which vanishes, i.e., the gravitational force does not impact the stability of our model. This simplifies our condition as, FH + FA = 0."

    With Φ=const, the field equations (4.6)-(4.8) imply dpr/dr + (2/r)(pr−pt) = 0 identically for any b(r): this is the conservation content of the solved system. Thus the plotted balance FH + FA = 0 is an identity, not a test. The paper labels this 'stability analysis' and concludes stable wormholes, but the equilibrium check carries no information beyond the field equations already used to construct the solution.

full rationale

The central construction is self-contained: the authors assume a linear f(Q) model (4.5), set Φ=const, and solve the ordinary differential equation obtained by equating the radial pressure (4.7) to a chosen dark-energy EoS; the resulting shape functions are exact integrals of that ODE, not imported from the conclusions. No load-bearing self-citation chain is present. The two concerns above are real but limited: the NEC violation is an assumption-driven consequence of the negative-pressure EoS, and the TOV check is a conservation identity. A separate and more serious correctness problem, not a circularity, is that the generalized-Chaplygin shape function (5.7) at a=−1.2 is not asymptotically flat: the exponent a/(a+1)=6 and the inside grows like r^{1/2}, so b(r) ~ r^3 and b(r)/r → ∞, contradicting the paper's claim in Section 5.2 and its Fig. 4. Because the derivation does not use its conclusions as premises, the circularity score is low.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central construction rests on a handful of hand-picked constants and the assumption that the exotic fluid exactly satisfies the chosen equations of state. No new physical entities are proposed.

free parameters (6)
  • alpha = -0.5, -0.75, -1, -1.25, -1.5
    Coupling constant of linear f(Q); sign chosen negative to satisfy asymptotic flatness, magnitude swept in plots.
  • B = 1
    Chaplygin gas constant; set to unity by hand without justification.
  • a = 0.05 (WH1), -1.2 (WH2)
    Chaplygin index; chosen ad hoc to obtain wormhole solutions.
  • u = -4 (WH1), 1 (WH3)
    Varying index in the EoS; chosen by hand.
  • omega = 2 (text) or -2 (figures)
    Barotropic index; inconsistent between text and figure captions.
  • c, c1 = 0.09 (WH1), 10 (WH2), 1 (WH3)
    Integration constants from solving the shape function ODE; chosen to satisfy throat conditions.
assumptions (5)
  • domain assumption Morris-Thorne static spherically symmetric wormhole metric (3.1) with finite redshift function
    The metric is assumed from the start; all solutions are built on it.
  • domain assumption Linear f(Q) gravity with f(Q)=αQ
    The paper uses the simplest form and asserts non-linear forms would not change the physics, but does not demonstrate this.
  • ad hoc to paper The radial pressure follows the chosen dark energy EoS exactly
    These EoS are assumed without derivation and are the basis for the shape function solutions.
  • standard math Energy conditions from Raychaudhuri equation are used as standard criteria
    Standard textbook relations used to classify matter.
  • standard math The f(Q) field equations (2.7) are taken from the literature (ref [40])
    Background framework.

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Cite this review

Pith. "Pith review of Study of Wormhole in $f(Q)$ gravity with some dark energy models." pith.science (2026). https://pith.science/paper/EQUMR27D

@misc{pith2026250703939,
  author       = {Pith},
  title        = {Pith review of: Study of Wormhole in $f(Q)$ gravity with some dark energy models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EQUMR27D}},
  note         = {Machine review of arXiv:2507.03939}
}
abstract

This study discusses the development of some particular static wormhole models in the background of an extended $f(Q)$ gravity theory. Wormhole solutions are derived by considering the radial pressure to admit an equation of state corresponding to Chaplygin gas. The Chaplygin gas equation of state is taken into consideration in two different forms: $p_{r}=-\frac{Bb(r)^{u}}{\rho^{a}}$, $p_{r}=-\frac{B}{\rho^{a}}$. Wormhole models are also generated assuming that a variable barotropic fluid may explain the radial pressure given by $p_{r} =-\omega\rho b(r)^u$. For every model, the shape function $b(r)$ is the function that can be derived from the wormhole metric in any scenario. The stability analysis of the wormhole solutions and the shape function viability for each situation are then investigated.Since each wormhole model is shown to violate the null energy condition (NEC), it can be understood that these wormholes are traversable. More generally, we investigate whether the model is stable under the hydrostatic equilibrium state condition using the TOV equation.The physical characteristics of these models are shown under the same energy circumstances. The typical characteristic is the radial pressure $p_{r}$ near the wormhole throat, which violates the NEC $(\rho+P_{r} \geq0)$. In some models, it is possible to meet the NEC at the neck and yet violate the DEC $(\rho - P_{r}\geq0)$. In summary, precise wormhole models may be generated, provided that $(\rho\geq0)$, and there may be a potential breach of the NEC at the wormhole's throat.

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