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arxiv: 2605.07224 · v1 · submitted 2026-05-08 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Traversable wormholes in boldsymbol{f(Q)} gravity: Energy conditions, stability and quasinormal modes

Jaydeep Goswami , Rupam Jyoti Borah , Umananda Dev Goswami
Authors on Pith no claims yet

Pith reviewed 2026-05-11 01:19 UTC · model grok-4.3

classification 🌀 gr-qc
keywords traversable wormholesf(Q) gravityenergy conditionsquasinormal modesstabilityshape functionanisotropic matterTOV equation
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The pith

f(Q) gravity permits analytic traversable wormhole solutions that remain dynamically stable under scalar perturbations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates whether modified gravity of the f(Q) type can support traversable wormholes by replacing the need for exotic matter throughout spacetime. Using a power-law form f(Q) equal to gamma times negative Q to the m, together with anisotropic matter and a linear equation of state, the authors construct an exact shape function that meets the flaring-out and asymptotic flatness conditions at the throat. For the interval 0 less than m less than one-half, the null and weak energy conditions are violated only in a narrow region near the throat while positive anisotropy supplies the repulsive stress needed to keep the throat open. Equilibrium is verified through the generalized Tolman-Oppenheimer-Volkoff equation, and linear stability follows from a Schrödinger-like wave equation whose effective potential yields quasinormal frequencies with negative imaginary parts together with confirming time-domain evolutions.

Core claim

By adopting the power-law model f(Q) = gamma (-Q)^m with 0 < m < 1/2 and an equation of state p_r = omega rho, an analytic shape function is obtained that satisfies the geometric requirements for a traversable wormhole. The energy conditions are violated near the throat but the violations remain localized. The anisotropy parameter stays positive, the generalized TOV equation shows force balance with anisotropic effects providing the dominant outward support, and scalar perturbations produce a single-peak effective potential whose sixth-order WKB quasinormal modes have negative imaginary parts, indicating stable damping confirmed by time-domain simulations.

What carries the argument

The analytic shape function derived from the power-law f(Q) = gamma (-Q)^m together with the linear equation of state p_r = omega rho, which adjusts the effective energy-momentum tensor so that the geometric flaring-out condition holds while energy-condition violations stay localized.

If this is right

  • Wormhole solutions exist geometrically for the parameter range 0 < m < 1/2 corresponding to quintessence-like behavior.
  • Null and weak energy conditions are violated only near the throat while the strong energy condition holds outside a small region.
  • Positive anisotropy throughout the spacetime supplies the repulsive stress that sustains the wormhole throat.
  • Both constant and logarithmic redshift functions yield equilibrium configurations via the generalized TOV equation.
  • Quasinormal modes computed by sixth-order WKB with Padé approximants have negative imaginary parts and agree with time-domain stability simulations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The localization of energy-condition violations suggests that f(Q) modifications can confine exotic-matter requirements to small regions, a feature that might appear in other modified-gravity wormhole constructions.
  • The link between the allowed m interval and quintessence-like equations of state offers a concrete bridge between wormhole solutions and dark-energy models without invoking new fields.
  • Agreement between WKB frequencies and time-domain results indicates that the stability conclusion is insensitive to the precise numerical method used for the perturbation analysis.

Load-bearing premise

The solution depends on choosing both a power-law form for f(Q) and a specific linear relation between radial pressure and energy density that together allow an exact analytic shape function within a narrow window of m.

What would settle it

A calculation or simulation in which the effective potential for scalar perturbations produces quasinormal modes with positive imaginary parts, or in which time-domain evolution shows exponential growth of perturbations instead of damping.

Figures

Figures reproduced from arXiv: 2605.07224 by Jaydeep Goswami, Rupam Jyoti Borah, Umananda Dev Goswami.

Figure 1
Figure 1. Figure 1: FIG. 1. 2-D (left) and 3-D (right) embedded plots of the wormhole defined by the shape function ( [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Plots of functions [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. 2-D (left panel) and 3-D (right panel) embedded plots of the wormhole defined by the shape function ( [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Variation of effective energy density [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Plots of [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plot of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Variation of [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Plot of [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Variation of [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Plot of [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Variation of effective potential [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Variation of real (left panel) and imaginary (right panel) parts of QNMs with respect to the model parameter [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Variation of real (left panel) and imaginary (right panel) parts of QNMs with respect to the model parameter [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Plots of effective potential [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Variation of real (left panel) and imaginary (right panel) parts of QNMs with respect to model parameter [PITH_FULL_IMAGE:figures/full_fig_p020_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Variation of real (left panel) and imaginary (right panel) parts of QNMs with respect to the model parameter [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. Time-domain evolution of the scalar perturbation [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Fitting of the time-domain profile of the wormhole to estimate its scalar QNMs for multipole numbers [PITH_FULL_IMAGE:figures/full_fig_p022_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Time-domain evolution of the scalar perturbation [PITH_FULL_IMAGE:figures/full_fig_p023_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: FIG. 21. Fitting of the time-domain profile of scalar perturbation of the wormhole spacetime with logarithmic redshift function to estimate the [PITH_FULL_IMAGE:figures/full_fig_p024_21.png] view at source ↗
read the original abstract

We investigate static and spherically symmetric traversable wormhole solutions in the framework of $f(Q)$ gravity by considering a power-law model of the form $f(Q)=\gamma(-Q)^m$. By adopting an anisotropic matter distribution and imposing an equation of state relating the radial pressure and energy density, we obtain an analytic shape function that satisfies the geometric requirements for a traversable wormhole. The model parameter is constrained to $0<m<1/2$, corresponding to a quintessence-like regime with $-1<\omega<-1/3$. The energy conditions are analyzed in detail, showing that violations of the null and weak energy conditions are unavoidable but remain localized near the wormhole throat. The anisotropy parameter is positive throughout the spacetime, indicating that repulsive anisotropic stresses play a key role in sustaining the wormhole. The equilibrium configuration is examined using the generalized Tolman-Oppenheimer-Volkoff (TOV) equation for both zero and logarithmic redshift functions, where a consistent force balance is achieved with anisotropic effects providing the dominant outward support. Dynamical stability is studied through scalar perturbations, leading to a Schr\"odinger-like wave equation with a single-peak effective potential. The quasinormal modes are computed using the sixth-order WKB method with Pad\'e approximation. The resulting frequencies possess negative imaginary parts, indicating stable damping of perturbations. Time-domain simulations further confirm the stability of the solutions and show good agreement with the WKB results, with small deviations in the damping rates. Thus, these results establish that $f(Q)$ gravity admits traversable wormhole solutions that are geometrically consistent and dynamically stable, with $f(Q)$ gravity effects effectively regulating the required matter content.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript constructs static spherically symmetric traversable wormhole solutions in f(Q) gravity for the power-law model f(Q)=γ(-Q)^m. An anisotropic fluid with a linear equation of state p_r=ωρ is assumed, yielding an analytic shape function b(r) that satisfies the flaring-out condition at the throat for the parameter range 0<m<1/2 (corresponding to quintessence-like ω). Energy conditions are analyzed, showing localized violations of the null and weak energy conditions near the throat. Equilibrium is checked via the generalized Tolman-Oppenheimer-Volkoff equation for both constant and logarithmic redshift functions. Dynamical stability is examined through scalar perturbations, producing a Schrödinger-like equation whose quasinormal frequencies (computed via sixth-order WKB with Padé approximants) have negative imaginary parts; this is corroborated by time-domain evolution.

Significance. If the results hold, the work provides explicit analytic examples of traversable wormholes in f(Q) gravity that are geometrically consistent and linearly stable, with energy-condition violations confined to a finite region around the throat. The agreement between WKB and time-domain methods, together with the closed-form shape function and direct TOV balance, strengthens the evidence that modified gravity can regulate the required matter content without global exotic-matter distributions. This adds a concrete, falsifiable case to the literature on wormholes beyond general relativity.

major comments (2)
  1. [§3] §3 (shape-function derivation): the analytic b(r) is obtained by direct substitution of the chosen EOS p_r=ωρ into the f(Q) field equations; while this produces a closed-form solution satisfying b(r0)=r0 and b'(r0)<1, the restriction 0<m<1/2 is an output of matching the quintessence range -1<ω<-1/3 rather than an independent prediction, so the existence result is tied to this specific EOS choice.
  2. [§5.2] §5.2 (WKB quasinormal modes): the sixth-order WKB frequencies with Padé approximants are reported to have negative imaginary parts for the single-peak effective potential, but the manuscript does not tabulate the explicit frequency values or perform a convergence check against fifth- or seventh-order WKB, which is needed to confirm that the damping rates are robust rather than sensitive to the approximation order.
minor comments (3)
  1. [Abstract and §4] The abstract and §4 claim that 'f(Q) gravity effects effectively regulating the required matter content,' but this phrasing should be qualified to note that the regulation occurs through the interplay of the specific power-law f(Q) and the imposed EOS.
  2. [Figures 2-4] Figure captions for the energy-condition and potential plots should include the specific numerical values of m, γ, and r0 used, to allow direct reproduction of the localized-violation profiles.
  3. [Throughout] Notation: the redshift function is denoted Φ(r) in some sections and Φ in others; consistent use throughout would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments. We address each major comment point by point below, indicating the revisions we plan to implement in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (shape-function derivation): the analytic b(r) is obtained by direct substitution of the chosen EOS p_r=ωρ into the f(Q) field equations; while this produces a closed-form solution satisfying b(r0)=r0 and b'(r0)<1, the restriction 0<m<1/2 is an output of matching the quintessence range -1<ω<-1/3 rather than an independent prediction, so the existence result is tied to this specific EOS choice.

    Authors: We acknowledge that the parameter range 0<m<1/2 is indeed determined by consistency with the chosen linear equation of state and the flaring-out condition. The linear EOS p_r=ωρ was deliberately selected to permit an exact analytic solution for the shape function b(r), a standard strategy in the wormhole literature when seeking closed-form expressions. This choice is physically motivated by the requirement of negative radial pressure to support the throat, corresponding to the quintessence regime. We will add a brief clarifying paragraph in Section 3 to explicitly state the rationale for the EOS and note that the resulting constraints are tied to this assumption rather than being a model-independent prediction of f(Q) gravity. revision: partial

  2. Referee: [§5.2] §5.2 (WKB quasinormal modes): the sixth-order WKB frequencies with Padé approximants are reported to have negative imaginary parts for the single-peak effective potential, but the manuscript does not tabulate the explicit frequency values or perform a convergence check against fifth- or seventh-order WKB, which is needed to confirm that the damping rates are robust rather than sensitive to the approximation order.

    Authors: We agree that tabulating the explicit quasinormal frequencies and performing a convergence test across WKB orders would strengthen the stability analysis. In the revised manuscript we will add a table listing the real and imaginary parts of the fundamental and first overtone frequencies obtained with the sixth-order WKB plus Padé approximants. We will also include a short subsection or appendix comparing these results with fifth- and seventh-order WKB calculations to demonstrate that the negative imaginary parts (damping rates) converge and are not sensitive to the truncation order. revision: yes

Circularity Check

0 steps flagged

No significant circularity: solutions constructed explicitly from stated assumptions

full rationale

The derivation begins with an explicit power-law ansatz f(Q)=γ(-Q)^m and an imposed equation of state p_r=ωρ on the anisotropic fluid. These inputs are used to integrate the field equations for a closed-form shape function b(r) that satisfies the flaring-out and asymptotic-flatness conditions by direct substitution. Subsequent checks (energy-condition violations localized at the throat, positive anisotropy, TOV force balance, and stability via the Schrödinger-like potential with negative Im(ω) from WKB and time-domain evolution) are all downstream verifications performed on the constructed metric. No step renames a fitted quantity as a prediction, invokes a self-citation as a uniqueness theorem, or reduces the central existence claim to a tautology; the allowed interval 0<m<1/2 emerges as the consistency requirement for quintessence-like ω rather than an independent forecast. The paper therefore remains self-contained against its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the power-law ansatz for f(Q), the anisotropic fluid with imposed EOS, and the static spherically symmetric metric; these choices enable analytic progress but are not derived from more fundamental principles.

free parameters (2)
  • m = 0 < m < 1/2
    Exponent in power-law f(Q) restricted to 0 < m < 1/2 to obtain quintessence-like equation of state
  • gamma
    Overall scaling factor in f(Q)
axioms (2)
  • domain assumption Static and spherically symmetric metric ansatz with traversable wormhole geometry
    Standard assumption invoked to reduce the field equations to ordinary differential equations
  • domain assumption Anisotropic matter distribution obeying p_r = omega rho with omega in (-1, -1/3)
    Imposed to close the system and obtain an analytic shape function

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Reference graph

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