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Possible Formation of Traversable Wormholes and Their Thermodynamic Analysis in $\mathcal{F}(Q,\mathcal{L}_{m},\mathcal{T})$ Gravity

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Constructing traversable wormholes in $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ gravity via the Karmarkar embedding condition, this paper finds that the null and averaged null energy conditions remain violated near the throat, so exotic…

desk verdict The shape function in Eq. (40) is not a Karmarkar-derived solution—the added parameter P breaks the embedding condition—so the paper's central construction is an ansatz, not a derivation. read the letter →

arxiv 2507.09327 v1 pith:EH6QUH3Q submitted 2025-07-12 gr-qc

Rounak Manna , Krishna Pada Das , Ujjal Debnath This is my paper · ORCID
classification gr-qc PACS 04.20.Jb04.50.Kd
keywords wormholeF(QLmT)gravitytraversableenergyconditionsexoticmatterKarmarkarconditionnon-metricitythermodynamicstability
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 establish that static, spherically symmetric traversable wormholes can be constructed in $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ gravity — an extended symmetric-teleparallel theory where the Lagrangian depends on non-metricity $Q$, the matter Lagrangian $\mathcal{L}_m$, and the trace $T$ of the energy–momentum tensor — and that these wormholes still demand exotic matter. Using the Karmarkar embedding condition and the redshift function $\Phi(r)=-2\mu/r$, the authors derive a shape function that satisfies the throat, flaring-out, and asymptotic-flatness requirements. They then show that the null energy condition and the averaged null energy condition are violated near the throat for every parameter choice explored, which they read as evidence that the $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ framework does not remove the need for exotic matter. A thermodynamic analysis reports negative wormhole temperature, positive average pressure and work density, and a narrow radial band of positive specific heat just outside the throat, which the authors interpret as local thermal and equilibrium stability.

What carries the argument

The load-bearing object is the shape function of Eq. (40) with $0<P<r_t$, derived from the Karmarkar (embedding class-1) condition $R_{2323}R_{1414}=R_{1224}R_{1334}+R_{1212}R_{3434}$ under the redshift choice $\Phi(r)=-2\mu/r$. The Karmarkar condition alone gives $b(r)=r-r^5/(r^4+4\mu^2 D e^{-2\mu/r})$, for which the throat condition $b(r_t)=r_t$ has only the trivial solution; the paper adds the free parameter $P$ to make the throat condition hold by construction. All subsequent results — flaring-out plots, embedding diagrams, proper radial distance, energy-condition inequalities, the volume-integral quantifier, and the thermodynamic quantities — are computed from this shape function in the model $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})=Q+mQ^2+\alpha \mathcal{L}_m+\beta T$.

What would settle it

Check the original Karmarkar shape function in Eq. (38) for a non-trivial throat radius: the paper asserts that $b(r_t)=r_t$ gives only the trivial solution, which is why $P$ is inserted. If a non-trivial root exists for any allowed parameter range, then the ad hoc parameter is unnecessary and the constructed solution is not the unique embedding-class-1 wormhole; if no non-trivial root exists, the entire construction rests on the unconstrained parameter $P$.

Watch

Extended reading notes

Core claim

In the authors' own terms, the central discovery is a traversable wormhole solution in $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ gravity whose shape function, $b(r)=P - P r^5/[P r^4 + r_t^4(r_t-P)e^{2\mu(1/r_t-1/r)}] + r$ (with $0<P<r_t$), is obtained by applying the Karmarkar condition to the redshift function $\Phi=-2\mu/r$ and then forcing the throat condition $b(r_t)=r_t$ through an inserted free parameter $P$. This shape function meets the geometric traversability criteria, yet the associated matter fluid violates the null, weak, strong, and dominant energy conditions near the throat; the volume-integral quantifier is negative, so the averaged null energy condition is also violated. The authors take this to show that exotic matter remains indispensable for sustaining traversable wormholes even in this extended gravity theory. They further report that the solution's Hawking and wormhole temperatures are negative, the average pressure and work density are positive, the total energy and energy flux are negative, and the specific heat is positive only in a narrow band immediately outside the throat, which they interpret as locally stable thermal equilibrium supported by exotic matter.

Load-bearing premise

Every conclusion depends on the shape function that is obtained by adding a free parameter $P$ to the Karmarkar-derived formula, with no physical justification or independent constraint on $P$; if that insertion is invalid, the wormhole solution and all of the paper's results collapse.

Editorial extensions

If this is right

  • If the central claim is right, then modifying the gravitational sector to $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ does not bypass the Morris–Thorne conclusion: any traversable wormhole in this theory requires matter that violates the null energy condition near the throat.
  • The volume-integral-quantifier result gives an explicit, parameter-dependent measure of the amount of exotic matter needed, so different choices of $\alpha$, $\beta$, $m$, and $P$ can be ranked by how much NEC-violating fluid they require.
  • The thermodynamic results imply a locally stable configuration only in a narrow band just outside the throat (the specific heat is positive for ranges such as $(1, 1.21]$ for $P=0.1$), with unstable behavior away from the throat and at the throat itself.
  • Negative wormhole and Hawking temperatures, positive work density, and negative total energy are each consistent with a metastable exotic-matter-supported equilibrium, so the model offers a concrete starting point for dynamical perturbation studies.

Reading between the lines

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

  • The ad hoc parameter $P$ is doing the real work: because the throat condition on the Karmarkar-derived shape function is trivial, every conclusion below depends on an unconstrained insertion. A physically motivated derivation of $P$ (for example, from junction conditions or from demanding a specific asymptotic mass) would turn this construction into a genuine prediction rather than a curve-fit.
  • The conclusion that exotic matter is unavoidable may be tied to the particular choices $\Phi=-2\mu/r$ and $f(Q)=Q+mQ^2$; other non-minimal couplings in $\mathcal{F}(Q,\mathcal{L}_m,\mathcal{T})$ could in principle mimic exotic matter at the throat, so a systematic scan over $\mathcal{F}$ forms would be a direct test of how general the claim is.
  • The reported stability is purely thermodynamic and local; it does not address dynamical stability against perturbations, which is the standard requirement for an astrophysically viable wormhole. A linear-perturbation analysis around this background would settle whether the narrow stability band survives.
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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 / 6 minor

Summary. The paper studies static, spherically symmetric traversable wormhole solutions in the extended symmetric teleparallel gravity theory F(Q, L_m, T). It chooses a specific redshift function, derives a shape function from the Karmarkar condition, and then checks the standard traversability requirements (throat condition, flaring-out condition, asymptotic flatness). The authors plot embedding diagrams, compute the proper radial distance, analyze the energy conditions (NEC, WEC, SEC, DEC) and the averaged null energy condition via the volume integral quantifier, and carry out a thermodynamic analysis involving Hawking temperature, wormhole temperature, average pressure, work density, total energy, energy flux, and specific heat. The paper's central conclusion is that exotic matter remains essential for sustaining traversable wormholes even in F(Q, L_m, T) gravity.

Significance. If the construction were valid, the paper would provide a concrete wormhole model in a relatively new modified-gravity framework and would support the general expectation that traversable wormholes require energy-condition violations. The authors give explicit (though lengthy) expressions for the effective density and pressures, and the graphical treatment of the energy conditions is systematic. However, the validity of the central construction is undermined by a load-bearing error: the shape function used in all subsequent computations is not actually a solution of the Karmarkar condition, and the parameter P that fixes the throat condition is introduced ad hoc. The thermodynamic analysis also relies on an unjustified entropy definition. These issues must be resolved before the results can be accepted.

major comments (3)
  1. [Section 4, Eqs. (38)-(40)] The shape function obtained by adding the free parameter P is not a solution of the Karmarkar condition. For the Karmarkar solution, e^γ = (r^4 + 4μ^2 D e^{-2μ/r})/r^4, while inserting Eq. (39) into e^γ = 1/(1 - b/r) gives e^γ = r(r^4 + 4μ^2 D e^{-2μ/r}) / [r^5 - P(r^4 + 4μ^2 D e^{-2μ/r})]. Equality of these two expressions forces P = 0, contradicting the stated range 0 < P < r_t. Thus Eq. (40) does not satisfy the Karmarkar condition, and the claim that the shape function was 'derived through the Karmarkar condition' is unsupported. Because all subsequent results—energy conditions, ANEC violation, and thermodynamics—are computed with Eq. (40), the central construction collapses unless Eq. (40) is re-derived or the Karmarkar claim is dropped and the solution is presented as a phenomenological ansatz. If the latter route is taken, the paper must state clearly that the traversability conditions are imposed by construction rather than derived from the embedding geometry.
  2. [Section 9, Eqs. (56)-(60) and Table 2] The thermodynamic analysis depends on an unjustified entropy definition. The paper states S = 8πr^2 for the wormhole entropy without derivation or discussion. For a spherically symmetric throat of radius r, the standard geometric entropy would be S = A/4 = πr^2; the factor 8 is nonstandard and appears to be chosen ad hoc. The specific heat C_V = T_Hawk dS/dT_Hawk in Eq. (60) and the stability intervals in Table 2 are direct consequences of this choice. The paper's conclusion that the wormhole is thermodynamically stable in a narrow band near the throat is therefore not robust. In addition, the interpretation of negative Hawking temperature as 'thermal stability' is nontrivial and requires support beyond a reference to the exotic-matter literature; the negativity here is driven by the parameter choice μ = -8, so it is not a parameter-independent feature.
  3. [Section 4, Eq. (37) and Section 3, conditions (1)-(5)] The introduction of P is explicitly described as a remedy after the throat condition b(r_t) = r_t 'results in only a trivial solution' for the Karmarkar shape function. This exposes a circularity in the construction: the geometric traversability conditions are not derived from the embedding formalism but are enforced by hand through the free parameter P. The paper provides no physical or geometric justification for P, nor any independent constraint on it. Even if the Karmarkar claim is retracted and Eq. (40) is treated as a phenomenological ansatz, the authors need to demonstrate that the chosen range 0 < P < r_t is natural and that the results are not sensitive to the arbitrary choice of P within that range.
minor comments (6)
  1. [Section 5, Eq. (45)] The integrands in Eqs. (45) and (47) contain garbled radical notation; they should be typeset clearly so that the integration is unambiguous.
  2. [Figure 2 caption] The caption lists P = 0.1 → ♠, P = 0.3 → ♠, P = 0.5 → ♠, P = 0.7 → ♠, P = 0.9 → ♠; all five cases are assigned the same marker, which makes the legend uninformative. Distinct markers should be used.
  3. [Section 9, paragraph before Fig. 10] The text says 'as illustrated in Fig. 1 for various selected values of the parameters β, α, m, and P', but the average pressure is plotted in Fig. 10, not Fig. 1; the cross-reference is incorrect.
  4. [Table 2] In the row for r = 0.8, the entry for P = 0.5 is marked 'Stable' while all other P columns at the same r are marked 'Unstable'. Since the left panel of Fig. 14 shows negative C_V for r < r_t across all P values, this entry appears to be a typo and should be corrected.
  5. [Section 2, Eq. (13)] The field equations are derived for a general F(Q, L_m, T), but the paper later specializes to F = f(Q) + αL_m + βT. It would help the reader if this specialization were announced before the long expressions in Eqs. (24)-(26), rather than only in the text, and if the lengthy reduced field equations were moved to an appendix.
  6. [Concluding remarks] The concluding section states that the shape function b(r) 'derived using the Karmarkar condition Eq. (31)' proves viable, but this repeats the central issue that Eq. (40) does not satisfy the Karmarkar condition. The language should be revised to match the actual derivation, whether or not the solution is re-derived.

Circularity Check

1 steps flagged · score 5.0 of 10

The throat condition is enforced by the ad hoc parameter P and then reported as a verified traversability condition; the energy-condition results are genuine outputs, so the circularity is partial.

  1. fitted input called prediction [Section 4, Eqs. (38)-(40); reaffirmed in the Abstract and Section 10]
    "Although the application of the throat condition results in only a trivial solution. This problem is resolved by adding a free parameter P to Eq. (38), which is written as follows: b(r)=... Next, in order to remove the integration constant D, we once again apply the throat condition, b(rt)=rt, and following a successful removal, Eq. (39) provides b(r)=... with 0<P<rt."

    The Karmarkar-derived b(r) in Eq. (38) makes b(rt)=rt trivial, so P is inserted and b(rt)=rt is then used to eliminate D. Consequently Eq. (40) satisfies the throat condition identically for every allowed P: the later statements that the solution 'adheres to the fundamental geometric requirements... including the presence of a throat' and that 'the analysis confirms the satisfaction of key conditions such as the throat condition' are restatements of this construction, not independent results. The traversability of the wormhole is therefore partly built into the ansatz rather than derived from the F(Q,Lm,T) field equations. In addition, Eq.

full rationale

The central abstract claim—that exotic matter remains necessary in F(Q,Lm,T) gravity—is not circular: ρ, pr, pt are computed from the field equations (24)-(26) with the chosen b(r), Φ(r) and f(Q), and their signs are not imposed by construction. The thermodynamic quantities and ANEC integrals are likewise outputs of those expressions. However, the geometric backbone is not fully derived: the throat condition is made nontrivial by adding a free parameter P and then using b(rt)=rt to fix D, so the paper's confirmation of the throat condition (and any traversability claim resting on it) is a tautological by-product of the ansatz. The self-citations [23,35,85,88,89] are used for standard framework elements (anisotropic fluid, embedding procedure, proper-distance condition) and are not load-bearing uniqueness claims, so they do not add circularity. The score is set at 5 rather than higher because the main energy-condition conclusion has independent computational content; it is set above 2 because a key geometric prediction is enforced by construction rather than derived.

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

The central construction rests on the arbitrary ansatz for the action, the choice of redshift and shape functions, and a set of unconstrained parameters. The most significant free parameter is P, which is introduced specifically to force the throat condition. No new entities beyond the usual anisotropic fluid are introduced.

free parameters (6)
  • P = 0.5 (central); varied 0.1-0.9
    Inserted ad hoc in Eq. (39) to make the throat condition solvable after the original shape function (38) gave a trivial solution. All geometric traversability checks depend on it.
  • mu = -8
    Constant in the redshift function Phi=-2mu/r; chosen by hand. Controls the exponential in the shape function and the magnitude of Hawking temperature.
  • alpha = 1 (varied 0-10)
    Coupling constant in F(Q,Lm,T)=f(Q)+alpha Lm+beta T. No physical constraint is given.
  • beta = 3 (varied 0-10)
    Coupling constant in F(Q,Lm,T). No physical constraint is given.
  • m = 0.1 (varied 0.05-0.5)
    Coefficient of the Q^2 term in f(Q)=Q+mQ^2. No physical constraint is given.
  • rt = 1 (kpc)
    Throat radius. Used as the number 1 in all numerical plots despite being labeled 1 kpc, which creates a unit ambiguity.
assumptions (6)
  • domain assumption The spacetime connection is symmetric teleparallel: curvature-free and torsion-free, with non-metricity Q, and the action is S=integral(F(Q,Lm,T)/(16pi)+Lm) sqrt(-g) d^4x.
    This defines the F(Q,Lm,T) gravity framework; all field equations follow from it.
  • domain assumption The wormhole metric is static, spherically symmetric Morris-Thorne form in Eq. (15).
    The entire analysis restricts to this metric ansatz.
  • domain assumption The matter content is an anisotropic perfect fluid with T_mu_nu given by Eq. (17) and Lm=-P.
    This fluid model is assumed; the specific Lm=-P choice is stated without loss of generality but is a restriction.
  • domain assumption The spacetime is of embedding class-1 and satisfies the Karmarkar condition in Eq. (31).
    This strong geometric condition is imposed to derive the shape function; not all wormhole geometries satisfy it.
  • ad hoc to paper The redshift function is chosen as Phi(r)=-2mu/r, Eq. (37).
    This specific form is selected by hand; the analysis does not explore other redshift functions.
  • domain assumption Energy conditions derived from the Raychaudhuri equation are applied to the matter fluid (rho, pr, pt), not to the effective total fluid.
    The interpretation that NEC violation implies exotic matter assumes the matter fluid is the physical stress-energy.

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Pith. "Pith review of Possible Formation of Traversable Wormholes and Their Thermodynamic Analysis in $\mathcal{F}(Q,\mathcal{L}_{m},\mathcal{T})$ Gravity." pith.science (2026). https://pith.science/paper/EH6QUH3Q

@misc{pith2026250709327,
  author       = {Pith},
  title        = {Pith review of: Possible Formation of Traversable Wormholes and Their Thermodynamic Analysis in $\mathcalF(Q,\mathcalL_m,\mathcalT)$ Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EH6QUH3Q}},
  note         = {Machine review of arXiv:2507.09327}
}
abstract

In this work, we investigate static and spherically symmetric traversable wormhole solutions within the framework of the extended symmetric teleparallel gravity, specifically the $\mathcal{F}(Q,\mathcal{L}_{m},\mathcal{T})$ gravity theory, where $Q$, $\mathcal{L}_{m}$, and $\mathcal{T}$ are the respective representations of the non-metricity scalar, the matter Lagrangian, and the trace of the energy-momentum tensor. By employing a specific redshift function and deriving the shape function through the Karmarkar condition, we examine the fundamental geometric features required for a viable wormhole structure. The analysis confirms the satisfaction of key conditions such as the throat condition, flaring-out condition, and asymptotic flatness. A detailed study of energy conditions for various values of model parameters reveals that the null energy condition and averaged null energy condition are violated near the throat, indicating the presence of exotic matter. Additionally, thermodynamic quantities such as temperature, pressure, specific heat, work density, and energy flux are analyzed, all of which support the thermal and equilibrium stability of the wormhole. Our findings demonstrate that even in extended theories like $\mathcal{F}(Q,\mathcal{L}_{m},\mathcal{T})$ gravity, exotic matter remains essential for sustaining traversable wormholes. This work lays the foundation for further investigations into their stability under dynamical perturbations and potential astrophysical implications.

Figures

Figures reproduced from arXiv: 2507.09327 by the authors.

Figure 1
Figure 1. Plots of the shape function b(r) and related expressions fo [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Embedding two-dimensional diagram (left panel) for various v [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Graphical representation of l(r) as a function of r for various values of the parameter P ∈ [0.1, 0.9] with step size 0.1 with throat radius rt = 1 kpc and µ = −8. A method introduced by Kruskal [100] and Szekeres [101, 102] in 1960 was regarded as the most extensive extension of the Schwarzschild solution, developed in response to the discovery of the Einstein-Rosen bridge. They further developed the Kruskal coordi… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: Energy condition expressions ρ (above left), ρ+pr (above right), ρ+pt (middle left), ρ+pr + 2pt (middle right), ρ − |pr| (below left), ρ − |pt| (below right) plotted against the radial coordinate r for various values of parameter β ∈ [0, 10] with step size 1.2 with thr…
Figure 5
Figure 5. Figure 5: Energy condition expressions ρ (above left), ρ+pr (above right), ρ+pt (middle left), ρ+pr + 2pt (middle right), ρ − |pr| (below left), ρ − |pt| (below right) plotted against the radial coordinate r for various values of parameter α ∈ [0, 10] with step size 1.2 with thr…
Figure 6
Figure 6. Figure 6: Energy condition expressions ρ (above left), ρ + pr (above right), ρ + pt (middle left), ρ + pr + 2pt (middle right), ρ − |pr| (below left), ρ − |pt| (below right) plotted against the radial coordinate r for various values of parameter m ∈ [0.05, 0.5] with step size 0.…
Figure 7
Figure 7. Figure 7: Energy condition expressions ρ (above left), ρ+pr (above right), ρ+pt (middle left), ρ+pr + 2pt (middle right), ρ − |pr| (below left), ρ − |pt| (below right) plotted against the radial coordinate r for various values of parameter P ∈ [0.1, 0.9] with step size 0.1 with …
Figure 8
Figure 8. Figure 8: Volume integral quantifier profiles plotted as functions of th [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Profiles of Hawking temperature and wormhole temperature [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: Average pressure profiles plotted as functions of the rad [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: Work density profiles plotted as functions of the radial coo [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Total energy profiles plotted as functions of the radial co [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Energy flux profiles plotted as functions of the radial coor [PITH_FULL_IMAGE:figures/full_fig_p019_13.png]
Figure 14
Figure 14. Figure 14: Variation of specific heat CV with radial distance r for various values of parameter P ∈ [0.1, 0.9] with step size 0.1, keeping µ = −8 and throat radius rt = 1 kpc fixed. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]

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