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Exploring Traversable Wormholes in $f(Q)$ gravity: Shadows and Quasinormal modes

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Pith's one-line read In f(Q) gravity, negative non-metricity enlarges traversable-wormhole shadows and sets the quasinormal-mode imaginary frequency to zero, a signature that could separate wormholes from black holes.

desk verdict New f(Q) wormhole solutions, but the advertised QNM signature rests on a false V''00=0 and the grey-body argument is broken. read the letter →

arxiv 2504.19282 v1 pith:EVJC7LFZ submitted 2025-04-27 gr-qc

Madhukrishna Chakraborty , Subenoy Chakraborty This is my paper · ORCID
classification gr-qc MSC 83D0583C5783C15 PACS 04.50.Kd04.70.Bw
keywords traversablewormholesf(Q)gravitynon-metricityscalarphotonshadowquasinormalmodesgreybodyfactorsenergyconditionsMorris-Thornemetric
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 constructs two traversable Morris–Thorne wormhole solutions in f(Q) gravity with a power-law model f(Q)=αQ^n, verifies their asymptotic flatness, flaring-out condition, and energy conditions, and then computes their photon shadows and quasinormal-mode spectra. Its central claim is that negative non-metricity Q acts like repulsive gravity: it is what allows a photon shadow to form, it enlarges the shadow radius as Q becomes more negative, and it lowers the real part of the fundamental quasinormal frequency while leaving the imaginary part zero, meaning the modes are undamped and the wormhole is stable. These signatures are presented as a distinct observational fingerprint of non-metricity, one that could in principle distinguish an f(Q) wormhole from a black hole through high-resolution imaging and gravitational-wave ringdown observations. The connection matters because wormhole candidates are otherwise very hard to tell apart from black holes in current telescope images.

What carries the argument

The machinery is the eikonal (large-l) limit of the Schrödinger-like perturbation equation (d²/dr_T²+ω²-V)Ψ=0, with V(r)=$e^{{2Φ}}$[l(l+1)/r² - (rb'-b)/(2r³) + Φ'(r)(1-b/r)/r]. Its l² part is V0(r_T)=$e^{{2Φ(r_T)}}$/r_T², whose maximum fixes the photon sphere via rΦ'(r)=1; the shadow radius is then rsh=rph $e^{{-Φ(rph)}}$. The WKB eikonal formula gives Re(ω0)=l√V00 and Im(ω0)=-(1/2)√(-V0''/(2V00)), and the paper's Im(ω0)=0 follows from asserting V0''(rph)=0. The wormhole solutions are generated from the ansatz f(Q)=αQ^n combined with non-constant redshift functions and the shape functions b(r)=b0/r³ (WH1) and b(r)=-(4μ1/λ)$e^{{-λr}}$(1+3/(λr)+6/(λ²r²)+6/(λ³r³)) (WH2).

What would settle it

Evaluate V0''(r_T) from V0(r_T)=$e^{{2Φ(r_T)}}$/r_T² at the largest positive root of rΦ'(r)=1. For WH1, direct differentiation gives V0''(rph) = -$6e^{{2Φ(rph)}}$/$rph^{4}$ ≠ 0, so the eikonal formula would yield Im(ω0)≠0; checking this one derivative settles whether the no-damping and near-unity grey-body conclusions are correct.

Watch

Extended reading notes

Core claim

The central claim is that in both wormhole solutions the observational quantities respond monotonically to the sign and magnitude of the non-metricity scalar: for Q<0 the effective potential develops a barrier that traps photons, the shadow radius rsh=rph $e^{{-Φ(rph)}}$ increases as Q becomes more negative, and the eikonal quasinormal frequency Re(ω0)=l $e^{{Φ(rph)}}$/rph decreases while Im(ω0)=0. The paper interprets the vanishing imaginary part as the absence of damping and therefore stability, and it reads the combination of a larger shadow, a softer oscillation tone, and near-total grey-body transmission below the threshold frequency as the signature that would separate this wormhole from a black hole. The final remark condenses the claim: 'more repulsive is the gravity, larger is the radius of the shadow and smaller will be the frequency of oscillation.'

Load-bearing premise

The no-damping conclusion rests on the assertion that the eikonal potential has vanishing second derivative at the photon sphere, V0''(rph)=0; if that derivative is nonzero, the imaginary part of the quasinormal frequency does not vanish and the modes would damp.

Editorial extensions

If this is right

  • A wormhole with f(Q)=αQ^n and negative non-metricity would appear to a distant observer with a photon shadow whose radius grows monotonically as Q becomes more negative, giving a concrete prediction for high-resolution imaging campaigns.
  • The real part of the fundamental quasinormal frequency decreases with more negative Q, so any ringdown-like signal from such a wormhole would be lower-pitched than the signal from a comparable black hole.
  • With Im(ω0)=0, the modes do not damp, so instead of the decaying 'ringdown' of a black hole the wormhole would show a long-lived oscillatory response.
  • The grey-body factor Γ_l(Ω) tends to unity for frequencies below the threshold Re(ω0), meaning low-frequency radiation passes through the throat with almost no reflection.
  • Because both the shadow radius and the oscillation frequency are controlled by the same factor e^{Φ(rph)}/rph, measuring one fixes the other within this model.

Reading between the lines

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

  • If the shadow–QNM correspondence is robust, the product rsh × Re(ω0) equals the angular momentum number l, a parameter-free relation that could be tested whenever the same compact object is observed both by imaging and by ringdown analysis.
  • Extending the calculation to rotating wormholes, the spin-induced splitting of the quasinormal spectrum would provide a sharper discriminator between f(Q) wormholes and Kerr black holes than the spherically symmetric case.
  • The no-damping conclusion depends on a single second derivative of the eikonal potential; computing that derivative numerically for the exact perturbation equation would show whether Im(ω0)=0 is an artifact of the WKB shortcut.
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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

5 major / 6 minor

Summary. The manuscript constructs two static spherically symmetric traversable-wormhole solutions in f(Q)=alpha Q^n gravity with nonconstant redshift functions (Eqs. (16)-(19) and (21)-(24)), checks the throat, flaring-out, asymptotic-flatness, and energy conditions in Section 3, computes photon-sphere shadows in Section 4, and then applies WKB eikonal methods to obtain quasinormal-mode frequencies and grey-body factors in Sections 5-6. The advertised result is that negative non-metricity Q enlarges the shadow and lowers Re(omega0), while Im(omega0)=0, so the wormholes are stable and undamped, giving a distinct observational signature of non-metricity. The paper itself flags restrictions: Q is singular at r=sqrt(c1/c2) for WH1, and the power-law f(Q) diverges as Q approaches 0 for n<1, so integer n>=1 is used.

Significance. The paper has useful elements: explicit wormhole solutions with checked flaring-out and asymptotic-flatness conditions, a clear energy-condition analysis, and an explicit admission that for WH1 the non-metricity scalar is singular at r=sqrt(c1/c2) and that the power-law model diverges as Q approaches 0 for n<1. If a fully corrected QNM computation were supplied, the shadow analysis could serve as a starting point for comparing f(Q) wormholes with EHT observations. However, the central result of Section 6 is built on a false curvature condition; after correcting it, Im(omega0) is nonzero and the advertised undamped, stable QNM signature disappears. The circular substitution of integration constants in terms of Re(omega0) means that Figs. 9-10 do not demonstrate a physical effect, and the grey-body interpretation is incorrect. The shadow claim additionally assumes the black-hole photon-sphere formula without a wormhole-specific derivation and uses parameter choices with a photon sphere inside the throat or a singular Q at the throat. The paper's principal advertised conclusion is therefore not established.

major comments (5)
  1. [Section 6, Eqs. (62)-(64)] The statement immediately after Eq. (62) that at the photon sphere V02=V00''=0 is false. For V0(r)=exp(2Phi(r))/r^2, the photon-sphere condition rPhi'(r)=1 makes V0'(rph)=0, but V0''(rph) is not forced to vanish. For WH1 with Phi(r)=ln(c2-c1/r^2), direct differentiation gives V0''(rph)=-6 exp(2Phi(rph))/rph^4, and in the tortoise coordinate the second derivative is multiplied by exp(-2Phi(rph))(1-b(rph)/rph), which is nonzero. For (c1,c2)=(1,2) one obtains rph approximately 1.225 and V_{rT}'' approximately -1.48. Consequently Eqs. (58)-(59) yield Im(omega0)=-1/2 sqrt(-V02/(2V00)) not equal to zero, rather than Eq. (64); the advertised Im(omega0)=0, the stability statement, and the grey-body discussion built on Eq. (61) are not supported.
  2. [Section 6, Eqs. (65)-(67) and Figs. 9-10] The analysis presented as an effect of non-metricity is circular. Eq. (65) is inverted to express c2 in terms of Re(omega0), rph, and c1, and Eq. (66) is inverted to express Phi0 in terms of Re(omega0) and rph; these values are then inserted into Q(r) and plotted against chosen values of Re(omega0). The resulting monotonic relation between Q and Re(omega0) is therefore constructed by definition rather than derived from the dynamics, so Figs. 9-10 provide no independent evidence that non-metricity lowers the quasinormal frequency.
  3. [Section 6, Eq. (61) and following paragraph] The grey-body interpretation misreads the WKB transmission formula. Eq. (61) is derived under the assumption Im(omega0) is nonzero; if Im(omega0)=0 the expression is singular. Taking the limit Im(omega0) approaching 0 from above gives Gamma_l(Omega) approaching 1 for Omega<Re(omega0) and Gamma_l(Omega) approaching 0 for Omega>Re(omega0), not an infinite transmission for frequencies above the threshold. The statement in the paragraph after Eq. (61) that a frequency above Omega0 gives infinite transmission is therefore incorrect, and the claimed threshold behavior does not follow.
  4. [Section 4, Eq. (40)] The shadow radius is taken as rsh=rph exp(-Phi(rph)), which is the standard black-hole photon-sphere result, but no justification is given for applying this formula to a traversable wormhole with two asymptotic regions. In a wormhole, a photon with impact parameter below the critical value can pass through the throat rather than being captured, so the definition of the shadow and the relation between rph and the observed boundary must be derived for the two-sided geometry. Without this, the quantitative claim that negative Q enlarges the shadow is not established.
  5. [Section 4.3, Figs. 4-5] The parameter choices used in the shadow plots are inconsistent with the stated domain r>=r0=1. For (c1,c2)=(0.2,2), the photon-sphere equation rPhi'(r)=1 gives rph=sqrt(3c1/c2)=0.548, which is smaller than r0, so the photon sphere lies inside the throat and Eq. (40) cannot be used. For (c1,c2)=(1,1), the non-metricity scalar in Eq. (20) is singular at the throat r=1, a regime that Section 7 explicitly says is avoided. Only the set (c1,c2)=(1,2) appears to lie in the admissible exterior region.
minor comments (6)
  1. [Section 3.1, Eq. (20)] Eq. (20) appears to be algebraically inconsistent with Eq. (11) and the redshift function in Eq. (16): combining 2Phi'(r)+1/r gives a numerator 3c1+c2r^2, not c1+c2r^2, before division by c2-c1/r^2; please verify the display.
  2. [Section 6, Eq. (65)] The displayed expression for Re(omega0) in Eq. (65) and the substitution c2=Re(omega0)*rph/l+c1/rph^2 are inconsistent with Eq. (63), which contains a square root; the correct inversion is c2=(rph Re(omega0)/l)^2+c1/rph^2.
  3. [Section 6, Remark] The stability condition is written as 'Im(omega0 <= 0)' in the Remark; this should be stated as Im(omega0) <= 0.
  4. [Section 5, Eqs. (55)-(59)] The notation V0 is used both for the l-independent coefficient in the expansion (55) and for the maximum of the full potential in Eq. (51); V00 and V02 are introduced without explicit definitions in terms of V0(rph) and its second tortoise-coordinate derivative, which makes the eikonal formulas unnecessarily hard to follow.
  5. [Section 2.2, Eq. (10)] The discussion of the Raychaudhuri equation is internally inconsistent: it says NEC violation leads to geodesic focusing or convergence, while the following sentences correctly state that R_mu nu k^mu k^nu<0, i.e., NEC violation, is the criterion for defocusing; please reconcile these statements.
  6. [References] Reference [8] is incomplete as printed ('Phys. Dark Univ. 47(8):101793' with no year), and a few references are cited in duplicated or ambiguous forms; please standardize the bibliography.

Circularity Check

2 steps flagged · score 6.0 of 10

The advertised non-metricity QNM signature reduces to a reparameterization: c2 and Φ0 are solved for from Re(ω0) and substituted into Q, so the plotted Q–Re(ω0) trend is an identity, not a prediction.

  1. fitted input called prediction [Section 6, Eq. (65) and following sentence, Fig. 9]
    "Re(ω0) = l ( (c2 − c1/r^2_ph)/rph ), Im(ω0) = 0 (65) Substituting c2 = Re(ω0) × rph/l + c1/r^2_ph in equation (20) we analyze the effect of non-metricity in the QNM spectra for l = 1, 2, 3 ..."

    Equation (65) defines Re(ω0) from the metric constants c1, c2 and rph. The paper then inverts this relation to express c2 in terms of Re(ω0) and inserts that expression into Eq. (20) for Q(r). The plotted claim that more negative Q accompanies smaller Re(ω0) is therefore an algebraic consequence of the chosen parameterization, not a result obtained by solving the wave equation in an f(Q) background with fixed parameters. The 'effect of non-metricity' is read off from curves whose independent variable was already used to set the metric constant; the trend is built in by construction.

  2. fitted input called prediction [Section 6, Eqs. (66)–(67) and following sentence, Fig. 10]
    "Re(ω0) = l ( (1 − Φ0/rph)/rph ) (66) Im(ω0) = 0 (67) so that substituting Φ0 = rph × (1 − rphRe(ω0)/l) in equation (25) and using rsh = rph e^{−Φ(rph)} we can see the effect of non-metricity in QNM spectra for WH2 with radius of shadow as shown in FIG (10)."

    The same reparameterization occurs for WH2: the metric constant Φ0 is solved from the QNM formula (66) and substituted into Eq. (25) for the non-metricity scalar Q. Consequently the displayed relation between Q and Re(ω0) is not an independent prediction of the f(Q) model but a rewriting of the input relation between Re(ω0) and Φ0. Any monotonic trend in the plots is imposed by the substitution, not derived from the perturbation dynamics.

full rationale

The wormhole solutions, energy conditions, and shadow radii in Sections 3 and 4 are direct algebraic consequences of the stated metric ansätze, the f(Q) field equations, and the null-geodesic equations; I find no circularity in those parts. The self-citations appearing in the paper (e.g., refs. [8], [34], [40], [43], [44]) are mostly background or motivational and are not load-bearing for the shadow or QNM formulas, which are standard. The circularity is confined to Section 6. There, Eqs. (65)–(67) are solved for the metric constants c2 and Φ0 in terms of the advertised output Re(ω0), and those solutions are substituted into the expressions for Q. The resulting plots and the Remark that more negative Q gives smaller Re(ω0) therefore express the parameterization itself, not a physical effect. In addition, the paper asserts V02 = V''00 = 0 immediately after Eq. (62); this is an unproved premise, and direct differentiation of V0 = e^{2Φ}/r^2 at rΦ' = 1 does not give zero in general. That affects the validity of Im(ω0)=0 and the grey-body threshold discussion, but it is a mathematical-support issue rather than circularity. Because the central advertised non-metricity QNM signature reduces by construction to a reparameterization of the input constants, while the rest of the manuscript is self-contained, the overall circularity score is 6.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

The central construction uses only the standard Morris-Thorne metric and f(Q) field equations plus a power-law model, so no new particles or forces are invented. However, the analysis carries several free integration constants and model parameters that are chosen by hand for plotting, and the QNM section adds an incorrect technical premise, V''00=0. The constraint ansatze are ad hoc assumptions used to close the system, and the WKB transfer from black holes to wormholes is an unexamined domain assumption.

free parameters (10)
  • alpha = chosen as 4, 5, 6 in plots
    Overall coupling in f(Q)=alpha Q^n; values selected for plotting, not constrained by data.
  • n = chosen as 2, 3, 5 in plots
    Power-law exponent; restricted to n>=1 in Section 7 to avoid divergence of f(Q) as Q goes to 0.
  • c1 = 0.2 or 1 in WH1 plots
    Integration/redshift constant; affects Q, shadows, and the QNM parameterization.
  • c2 = 1 or 2, later re-expressed via Re(omega0)
    Integration/redshift constant; also used in Section 6 to parameterize Q in terms of the QNM frequency.
  • Phi0 = 0.1, 0.2, 0.3 in WH2
    Redshift constant in Phi=ln(1-Phi0/r); affects Q, shadows, and QNM plots.
  • mu1 = -0.0001, -0.0002, -0.0003 in shadows; -0.1, -0.2, -0.3 in energy conditions
    Shape-function amplitude in WH2; values differ between sections and are not fixed by data.
  • lambda = 0.3, 0.4, 0.5 in shadows; 0.1, 0.2, 0.3 in energy conditions
    Exponential decay rate in WH2; used as a free plotting parameter.
  • mu = 1, from asymptotic flatness in Solution I
    In Solution I, m=-1 and mu=1 are forced by the b(r)/r to 0 condition, so this parameter is fixed rather than fitted.
  • b0 = 1, from throat condition r0=1
    Integration constant set by b(r0)=r0 in Solution I.
  • mu0 = 1, from asymptotic flatness in Solution II
    In Solution II, mu0=1 is required for a feasible shape function and asymptotic flatness.
assumptions (6)
  • domain assumption Morris-Thorne line element (9) with no horizon and standard traversability conditions
    The entire paper models wormholes in this metric class, with conditions on b(r), Phi(r), and flaring.
  • domain assumption Field equations (12)-(14) for f(Q) gravity are correct as taken from Refs. [21], [22], [33]
    No independent derivation or consistency check is given; the analysis builds directly on these equations.
  • ad hoc to paper Power-law form f(Q)=alpha Q^n is a viable gravity model
    Chosen for mathematical convenience; n>=1 is imposed to avoid the Q to 0 divergence mentioned in Section 7.
  • ad hoc to paper Constraint ansatze (15), (17), and (22) are valid ways to close the underdetermined system
    These constraints are imposed without physical motivation beyond solvability.
  • domain assumption The WKB/eikonal relation between QNM frequency and photon-sphere potential applies to wormholes as in the black-hole case
    Borrowed from black-hole literature; wormhole boundary conditions differ and this transfer is not justified.
  • ad hoc to paper V''00=0 at the photon sphere
    The paper asserts this after Eq. (62); direct differentiation for WH1 gives a nonzero second derivative, so this axiom is false.

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Pith. "Pith review of Exploring Traversable Wormholes in $f(Q)$ gravity: Shadows and Quasinormal modes." pith.science (2026). https://pith.science/paper/EVJC7LFZ

@misc{pith2026250419282,
  author       = {Pith},
  title        = {Pith review of: Exploring Traversable Wormholes in $f(Q)$ gravity: Shadows and Quasinormal modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVJC7LFZ}},
  note         = {Machine review of arXiv:2504.19282}
}
abstract

The present work deals with some WH solutions in $f(Q)$ gravity theory for non-constant red-shift function subject to a power law model of $f(Q)$, $Q$ being the non-metricity scalar. Important properties of a WH configuration like the asymptotic flatness, traversability, flairing out condition and energy conditions have been checked for the obtained solutions. Shadows for these WHs have been determined and effect of non-metricity has been discussed. Finally, the paper gives an essence of Quasinormal modes and grey body factors for traversable WHs in general and $f(Q)$ wormholes in particular indicating a distinct signature of non-metricity influencing these observational tools.

Figures

Figures reproduced from arXiv: 2504.19282 by the authors.

Figure 1
Figure 1. NEC,WEC,SEC and DEC in case of Solution I for variou [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Flairing out condition in case of Solution II for va [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. NEC,WEC,SEC and DEC in case of Solution II for vario [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Shadows for various model parameters in case of WH1 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Variation of non-metricity Q with r and Vef f with proper radial distance l(r) = ±r for WH1. Remark: In both the WHs, negative non-metricity allows shadow formation. Further, the shadow becomes larger in size (radius of the shadow increases) as non-metricity becomes mo…
Figure 6
Figure 6. Figure 6: Shadows for various model parameters in case of WH2 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Variation of non-metricity Q with r and Vef f with proper radial distance l(r) = ±r for WH2. the non-metricity induced part can be interpreted effective energy momentum contribution. So when, Q < 0 it often leads to repulsive contributions or negative pressure which ar…
Figure 8
Figure 8. Figure 8: V (rT ) vs rT for various choices of the parameters specified in panel that, the QNM gives information about the evolution of the perturbed field as follows: the real part of ω identifies the oscillation of the signal while the imaginary part measures the damping facto…
Figure 9
Figure 9. Figure 9: QNM spectra for WH1 showing effect of non-metricity a [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]
Figure 10
Figure 10. Figure 10: QNM spectra for WH2 showing effect of non-metricity [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]

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