Wormholes with low energy density

Peter K.F. Kuhfittig

REVIEW 3 major objections 4 minor 23 references

Wormholes with low energy density

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Low-energy-density wormholes can be macroscopic in f(Q) modified gravity.

desk verdict The paper's new exponential f(Q) ansatz fails because the key near-throat approximation is wrong; the macroscopic-wormhole conclusion rests on an unjustified term drop, so the work adds little to the author's earlier linear case. read the letter →

arxiv 2508.06597 v1 pith:S6M72OGB submitted 2025-08-08 physics.gen-ph

Peter K.F. Kuhfittig This is my paper

classification physics.gen-ph PACS 04.20-q04.20.Jb04.20.Cv

keywords low-energy-densitywormholesnoncommutativegeometryCasimireffectf(Q)gravityMorris-Thornewormholeexoticmatternullenergyconditionmacroscopic

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 tries to establish that wormholes with very low energy density—those based on noncommutative geometry and the Casimir effect—can be macroscopic if gravity is described by f(Q) modified gravity rather than general relativity. In Einstein gravity, such wormholes are tiny because their mass is tied directly to the small energy density. The paper shows that the freedom in f(Q) introduces a free parameter α that can be chosen small enough to make the wormhole's mass and throat radius large, even with the same low energy density. The claim matters because it would remove a central obstacle to treating traversable wormholes as astrophysical objects rather than microscopic curiosities.

What carries the argument

The central mechanism is f(Q) gravity, an extension of general relativity in which the action contains an arbitrary function of the non-metricity scalar Q. With the Morris-Thorne line element and the zero-tidal-force assumption Φ'(r)≡0, the density equation reduces to Eq. (21). The paper chooses f(Q)=$de^{{αQ}}$, so f_Q=$dαe^{{αQ}}$ and f_QQ=$dα²e^{{αQ}}$; near the throat, the term f_Q(2r-b)(b'r-b)/(r-b) is declared dominant, giving Eq. (22). Solving for b'(r) and integrating yields Eq. (24), where the 1/α factor is what makes b(r)—and hence the mass—macroscopic. The same factor appears explicitly in the linear-model result of Eq. (25).

What would settle it

Take a concrete low-energy-density profile, such as the Lorentzian smeared density $\rho(r)=m\sqrt{\gamma}/[\pi^2(r^2+\gamma)^2]$, insert it into the full f(Q) density equation Eq. (21) with $f(Q)=de^{\alpha Q}$ and $\Phi'(r)\equiv 0$, and compute all three bracketed terms near the throat $r=r_0$. If the terms dropped in passing to Eq. (22) are not negligible compared with the retained term for the stated small $\alpha$, then the macroscopic-mass conclusion is unsupported.

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Extended reading notes

Core claim

The paper claims that traversable wormholes sourced by low energy densities—such as those from noncommutative geometry and the Casimir effect—can still be macroscopic if the spacetime is described by f(Q) modified gravity. The key freedom is the choice of f(Q): taking f(Q)=$de^{{αQ}}$, a near-linear form, injects a free parameter α that enters the denominator of the reconstructed shape function. Because the wormhole mass satisfies m(r)=∫ρ 4πr² dr = ½b(r), a sufficiently small positive α inflates b(r) and hence the mass even when the energy density is small. The paper therefore concludes that low energy-density wormholes are not necessarily microscopic.

Load-bearing premise

The argument rides on the unproven near-throat approximation that one term in the f(Q) energy-density equation dominates the others; if the dropped terms are comparable, the formula that turns a small density into a large mass no longer follows.

Editorial extensions

If this is right

If the claim is correct, noncommutative-geometry wormholes need not be microscopic: choosing α sufficiently small makes both the throat radius r0/√γ and the mass macroscopic.
Casimir-effect wormholes can likewise reach macroscopic size, so the laboratory-scale negative energy of the Casimir effect could in principle support an astrophysical wormhole under f(Q) gravity.
The mass formula m(r)=½b(r) means the same low-energy-density profile can yield a much larger mass by tuning α, as long as the near-throat approximation holds.
The null energy condition is still violated at the throat (Eq. 19), so f(Q) gravity does not remove the need for exotic matter; it changes the scale at which such matter can build a wormhole.
The derivation yields an integral equation, Eq. (24), that applies to general low-density profiles, making the proposed mechanism generic rather than tied to one form of matter.

Reading between the lines

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

My inference: the same α-scaling mechanism should apply to any modified-gravity Lagrangian with a free multiplicative parameter in front of a near-linear term, not only to f(Q)=de^{αQ}, since the 1/α factor comes from solving for b'(r) rather than from the exponential's specific shape.
My inference: Eq. (24) suggests a practical tuning procedure—for a given low-energy-density profile and desired throat radius, solve for α; observational upper bounds on α could then be inferred by demanding the wormhole mass not exceed that of known compact objects.
My inference: if the required values of α turn out to be extremely small, the regime may be physically fine-tuned; a quantitative estimate of the allowed α range, computed from a concrete density profile, would settle whether the mechanism is plausible rather than merely formal.

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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

The punchline: this paper doesn't deliver a new valid result. The general-case derivation rests on a claim that one term in Eq. (21) dominates near the throat, but that isn't true; the dropped terms are comparable or larger in every neighborhood of the throat. So Eq. (22) is not an approximation, and the integral equation (24) and the macroscopic-mass conclusion don't follow.

What's actually here: a clear statement of the problem (noncommutative and Casimir wormholes have low energy density and are naturally microscopic) and a sensible idea—if f(Q) gravity has a free parameter that can rescale the shape function, that could make them macroscopic. The linear-case result from Ref. [21] is correctly quoted, and the paper discusses the NEC cleanly in Eq. (19).

The soft spots are serious. With f(Q)=d e^{αQ}, Φ'=0, and b≈r0+kx, let y=x/r0 and η=α/r0². The kept term K scales as −d α e^{αQ} r0²(1-k)/x. The two terms dropped in (22) scale as D₂ ~ −2d α² e^{αQ} r0²(1-k)/x² and D₃ ~ d e^{αQ} r0³. So |D₂/K|≈2η/y and |D₃/K|≈y/η. For y≪η, D₂ dominates; for y≫η, D₃ dominates; at y≈η they're comparable. The kept term is never the leading one. So the approximation is invalid, and everything built on it—the general shape function (23)-(24) and the claim that the general case behaves like the linear case—is unsupported. The linear case itself is fine, but it's the author's earlier work, and the macroscopic scale comes simply from choosing α small. Since α is a free parameter with no constraints discussed, the conclusion "low-energy-density wormholes can be macroscopic" is more a restatement of the input than a derived result. The exponential ansatz doesn't add anything new; the paper explicitly reduces it to the linear form.

Who this is for: someone curious about how a scale parameter in f(Q) gravity could affect wormhole masses. But the new part doesn't hold up. I wouldn't send it to a referee; the error is elementary and the central claim is already in the literature. Desk reject seems right.

Referee Report

3 major / 4 minor

Summary. The paper studies traversable Morris-Thorne wormholes sourced by low energy densities, specifically noncommutative-geometry and Casimir-energy densities, and argues that f(Q) modified gravity can make such wormholes macroscopic. After reviewing the Morris-Thorne conditions and f(Q) gravity, the paper proposes an exponential form f(Q)=d e^{αQ}, derives an approximate shape-function equation near the throat, and obtains an integral expression for b(r). It then claims that because α is a free parameter that can be chosen sufficiently small, the wormhole mass m(r) = b(r)/2 can be macroscopic, so low-energy-density wormholes can be macroscopic.

Significance. If the central derivation were sound, the paper would offer a simple mechanism in f(Q) gravity for promoting microscopic low-density wormhole solutions to macroscopic size. The paper is concise and the algebraic step leading to Eq. (19) is correct; the manuscript also honestly identifies that the NEC is violated only for fQ>0. However, the load-bearing approximation in Eq. (22) is demonstrably invalid, and the subsequent conclusion relies on tuning the free parameter α. The claimed general result is therefore not established. The paper does not provide machine-checked proofs, numerical solutions, or falsifiable predictions, and its significance, as it stands, is limited by the unsubstantiated central approximation.

major comments (3)

[Sec. 5, Eq. (22)] The assertion that, near the throat, the term fQ(2r−b)(b′r−b)/(r−b) dominates all other terms in Eq. (21) is not only unproved but false. Set x=r−r0, k=b′(r0)<1, and η=α/r0². Near the throat b≈r0+kx, Q≈1/(r0x), Q′≈−1/(r0x²). The retained term is K≈−dαe^{αQ}r0²/x, while the dropped terms have magnitudes |D2|≈2dα²e^{αQ}r0/x² (from 2rfQQQ′b) and |D3|≈de^{αQ}r0³ (from fr³). Thus |D2/K|≈2η/(x/r0) and |D3/K|≈(x/r0)/η. For x/r0≪η the D2 term dominates; for x/r0≫η the D3 term dominates; at x/r0≈η both are at least comparable. Hence the retained term is never the leading term in the near-throat region, so Eq. (22), and consequently Eqs. (23)–(24), do not follow from Eq. (21).
[Sec. 5, Eq. (24)] Eq. (24) is presented as a solution for the shape function, but it is actually an implicit integral equation: b(r) appears on the right-hand side both in b(r′)/r′ and inside e^{αQ} through Q(r′,b(r′),b′(r′)). The text states that the general case behaves like the linear case because α is sufficiently small, but no argument is supplied that the implicit equation reduces to Eq. (25), nor is existence or uniqueness of solutions to Eq. (24) discussed. The conclusion that low-energy-density wormholes can be macroscopic is therefore unsupported by the displayed derivation.
[Sec. 5, macroscopic-mass claim] The macroscopic-mass conclusion is driven by the free parameter α. In Eq. (25), b(r) is proportional to 1/α, and the text explicitly says α is chosen sufficiently small. Since no independent physical constraint on α is given, the statement that the mass can be made macroscopic is equivalent to assuming the desired result: for any energy density, one can tune α to make b(r) arbitrarily large. This is a circularity, not a derivation. Moreover, the extrapolation from the linear model f(Q)=αQ+β to the exponential model f(Q)=de^{αQ} is not justified by α small, because the exponential model introduces nonlinear dependence on Q and on b through the kept term in Eq. (22).

minor comments (4)

[Sec. 5, Eqs. (20)–(24)] The paper switches between φ and Φ for the redshift function. Eq. (15) uses φ′, Eq. (16) uses ϕ′, and the text later says Φ′(r)≡0. This notation should be unified.
[Sec. 5, Eq. (20)] For f(Q)=de^{αQ}, the condition fQ>0 used in Eq. (19) requires not only α>0 but also d>0. The paper only states α is positive.
[Sec. 3, Eq. (11)] The rescaling B=b/√γ and the use of r/√γ in the metric is confusing. In particular, the physical throat radius is still r0, not r0/√γ; the latter is the value of the dimensionless coordinate at the throat. The statement that 'r0/√γ is macroscopic' conflates coordinate rescaling with a physical size and should be clarified.
[Sec. 5, Eq. (24)] The integration variable is written as r′, but the notation in the exponent and denominators is ambiguous about whether b in the integrand is b(r′). This should be made explicit.

Circularity Check

2 steps flagged · score 8.0 of 10

Macroscopic-mass conclusion is obtained by declaring the free parameter α sufficiently small, and the general case is inherited from the author's own Ref. [21]; the result is equivalent to its input.

self definitional [Sec. 5, paragraph after Eq. (24)]
"Thanks to the free parameter α from f (Q) gravity, the mass of the wormhole, m(r) = ∫ ρ 4πr² dr = (1/2) b(r) from Eq. (3), can now be macroscopic. The point is that this conclusion is also valid for the general case, Eq. (24), due to the assumption that α is sufficiently small."

The claimed result—macroscopic mass despite low energy density—is obtained solely by declaring the free parameter α 'sufficiently small.' Equation (24) gives b(r) ∼ (1/α)∫ρ..., so any target mass can be reached by taking α small; no independent constraint fixes α. The conclusion 'can be macroscopic' is therefore equivalent to the input 'choose α small enough to make it macroscopic.' This is not a prediction from f(Q) gravity but a restatement of the freedom to tune α.
self citation load bearing [Sec. 5, paragraph 'To see the significance of using a small positive α…' and Eq. (25)]
"It is shown in Ref. [21] that for the noncommutative-geometry case, Eq. (6), the linear form f (Q) = αQ + β, with β = 0, yields b(r) = ... [Eq. (25)] ... Thanks to the free parameter α from f (Q) gravity, the mass ... can now be macroscopic. The point is that this conclusion is also valid for the general case, Eq. (24), due to the assumption that α is sufficiently small."

The macroscopic-mass conclusion for the general exponential case is carried by an appeal to the author's own Ref. [21], which already contains the same 1/α division by a free parameter. Since Ref. [21] is the same author's prior work and the argument there is the same tuning of α, this citation is load-bearing rather than independent support: the general case is said to inherit the linear result because α is small, so the chain of justification loops back to the same free-parameter input.

full rationale

The paper's stated goal is to show that low-energy-density wormholes can be macroscopic. The derivation in Sec. 5 ends with Eq. (24), in which b(r) = (1/α)∫... plus a term; there is no equation fixing α from the wormhole's low energy density. The final step after Eq. (25) explicitly says the mass 'can now be macroscopic' thanks to the free parameter α, and that the general case follows 'due to the assumption that α is sufficiently small.' That is not a derivation from f(Q) gravity: it is a choice of α to produce the desired scale. A statement that one can make an object macroscopic by taking a free parameter small is an existence statement, but the paper presents it as the conclusion of the field equations. In addition, the only bridge from the exponential form to the explicit linear result is Ref. [21], the author's own earlier paper, which uses the same 1/α division. Thus the citation is load-bearing and does not supply independent support. The near-throat dominance claim in Eq. (22) is a separate mathematical-validity concern, not circularity; it does not change this assessment.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

No new entities are introduced; the 'extra degrees of freedom' are the free function f(Q) and its parameter α, both from the cited f(Q) gravity framework.

free parameters (1)

α (parameter in f(Q))
The macroscopic scale of the wormhole is produced by making α small; no constraint, bound, or physical origin is given. Appears in Eqs. (23)-(25) as 1/α, so the final claim depends entirely on this choice.

assumptions (6)

standard math Morris-Thorne wormhole metric and flare-out condition b'(r0)<1
Imported from Ref [1]; used in Eq (19) to obtain the sign of the NEC violation.
domain assumption Noncommutative geometry energy density ρ(r)=m√γ / [π²(r²+γ)²] (Eq. 6)
Taken from Refs [5-9]; assumed to model the matter source in the noncommutative case.
domain assumption f(Q) field equations (16)-(18) with the zero-tidal-force condition Φ'(r)=0
Field equations taken from Ref [16]; the simplification Φ'=0 is stated in Sec. 5 for tractability.
ad hoc to paper Choice f(Q)=d e^{αQ} with α>0 small
Justified only by 'leeway' in f(Q); the entire conclusion depends on this ad hoc choice (Sec. 5).
ad hoc to paper Near-throat dominance of the fQ term in Eq. (22)
The paper states the fractional part of the second term dominates but provides no estimate of the discarded f r³ and fQQQ' terms.
domain assumption Positivity fQ > 0
Assumed after Eq. (19) to make the NEC violation have the required sign.

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Cite this review

Pith. "Pith review of Wormholes with low energy density." pith.science (2026). https://pith.science/paper/S6M72OGB

@misc{pith2026250806597,
  author       = {Pith},
  title        = {Pith review of: Wormholes with low energy density},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S6M72OGB}},
  note         = {Machine review of arXiv:2508.06597}
}

abstract

In spite of their speculative nature, traversable wormholes are a topic of interest that started with the Einstein-Rosen bridge in 1935 and became a major research area with the introduction of the Morris-Thorne wormhole in 1988. It is also become apparent in time that such wormholes are likely to be compact stellar objects, akin to neutron stars. Although widely discussed, wormholes having a low energy density may therefore not be massive enough to exist on a macroscopic scale. Important examples are wormholes based on a noncommutative-geometry background and wormholes supported by the negative energy density sourced by the Casimir effect. The main goal of this paper is to invoke $f(Q)$ modified gravity to provide the extra degrees of freedom to help overcome these obstacles.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [21]

(2023) Macroscopic noncommutative-geometry wormholes as emer- gent phenomena in f (Q) gravity

Kuhfittig, P.K.F. (2023) Macroscopic noncommutative-geometry wormholes as emer- gent phenomena in f (Q) gravity. Letters in High Energy Physics 2023, ID: 399

work page 2023

[2] [1]

and Thorne, K.S

Morris, M.S. and Thorne, K.S. (1988) Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity.American Journal of Physics, 56, 395-412

work page 1988

[3] [2]

(1948) On the attraction between two perfectly conducting plates

Casimir, H.G.B. (1948) On the attraction between two perfectly conducting plates. Proceedings of the Royal Netherlands Academy of Arts and Sciences , 51, 793-795

work page 1948

[4] [3]

(1996) Bound states of strings and p-branes

Witten, E. (1996) Bound states of strings and p-branes. Nuclear Physics B , 460, 335-350. 7

work page 1996

[5] [4]

and Witten, E

Seiberg, N. and Witten, E. (1999) String theory and noncommutative geometry. Journal of High Energy Physics , 1999, ID: 032

work page 1999

[6] [5]

and Spallucci, E

Smailagic, A. and Spallucci, E. (2003) Feynman path integral on the non-community plane. Journal of Physics A 36, L-467-L-471

work page 2003

[7] [6]

(2006) Noncommutative geometry in- spired Schwarzschild black hole

Nicolini, P., Smailagic, A., and Spallucci, E. (2006) Noncommutative geometry in- spired Schwarzschild black hole. Physics Letters B 632, 547-551

work page 2006

[8] [7]

and Spallucci, E

Nicolini, P. and Spallucci, E. (2010) Noncommutative geometry inspired dirty black holes. Classical and Quantum Gravity 27, ID: 015010

work page 2010