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REVIEW 3 major objections 3 minor 84 references

Nonradial perturbations of static charged wormholes

T0 review · 3 major / 3 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Static charged wormholes carry a previously unknown nonradial instability: the fundamental quadrupolar polar mode grows once the mass is large enough, in addition to the known radial instability.

desk verdict Plausible new l=2 polar instability in charged EB wormholes, well-derived but needs an independent numerical check before I'd call it secure. read the letter →

arxiv 2607.18399 v1 pith:NIPZT4UH submitted 2026-07-20 gr-qc

Jose Luis Blázquez-Salcedo , Luis Manuel González-Romero , Fech Scen Khoo , Jutta Kunz , Pablo Navarro Moreno This is my paper · ORCID
classification gr-qc
keywords chargedEllis-BronnikovwormholesquasinormalmodesnonradialinstabilitypolarperturbationsphantomscalarfieldextremalReissner-NordströmlimitEinstein-Maxwelltheoryspectralmethods
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 computes the quasinormal-mode spectrum of static, spherically symmetric charged Ellis-Bronnikov wormholes — solutions of Einstein-Maxwell theory with a phantom scalar field — for both axial and polar perturbations, including gravitational, electromagnetic, and scalar degrees of freedom. Its central result is the discovery of a nonradial instability: for uncharged Ellis-Bronnikov wormholes, the fundamental polar branch labeled b2 at multipole l=2 crosses from damped to growing once the mass-to-throat ratio exceeds about 0.3, and a similar instability appears for subcritical charged wormholes with sufficiently large parameter Λ. This instability is distinct from the well-known radial (l=0) instability and, if real, means that the nonradial sector imposes additional constraints on the dynamical viability of these wormholes. The paper also finds that electric charge generally lengthens damping times, with mode frequencies and damping rates approaching the extremal Reissner-Nordström limit in the most charged configurations.

What carries the argument

The load-bearing object is the b2 polar branch: the second family of polar quasinormal modes that appears because gravitational and phantom-scalar perturbations are coupled, so that each multipole carries two polar branches (b1 and b2) rather than one. The paper follows this branch with a Chebyshev spectral method, a numerical expansion of the perturbation functions in Chebyshev polynomials on a compactified radial grid: the perturbation equations are reduced, via spherical-harmonic decomposition, to a quadratic eigenvalue problem in the complex frequency ω, and the sign of Im(ω) decides stability (negative = damped, positive = growing). The same machinery produces the axial, electromagnetic

What would settle it

Run an independent time-domain integration of the linearized polar perturbation equations for an uncharged Ellis-Bronnikov wormhole with mass-to-throat ratio about 0.4: if the quadrupolar polar perturbation rings down instead of growing exponentially, the reported instability is a numerical artifact.

Watch

Extended reading notes

Core claim

The authors derive the full first-order perturbation equations for charged Ellis-Bronnikov wormholes, decomposing the metric, electromagnetic, and phantom-scalar perturbations into axial and polar sectors. In the uncharged limit they recover the known Ellis-Bronnikov spectrum and confirm the expected isospectrality of electromagnetic modes between axial and polar sectors. Their principal claim is that the fundamental polar branch at multipole l=2 — the branch they call b2, in which gravitational and phantom-scalar perturbations are strongly coupled — becomes unstable for sufficiently massive wormholes. For uncharged Ellis-Bronnikov wormholes its imaginary frequency crosses zero at a mass-to-

Load-bearing premise

The result stands or falls on whether the numerical solver's tracking of one particular oscillation branch — the second polar branch of the quadrupolar mode — remains trustworthy in the large-mass regime, where the authors report degraded accuracy, so that its crossing from damped to growing is physical rather than a numerical artifact.

Editorial extensions

If this is right

  • If the l=2 polar instability is real, stability analyses of Ellis-Bronnikov wormholes that focus only on radial modes are incomplete: sufficiently massive configurations are dynamically unstable in the nonradial sector as well.
  • Electric charge can substantially reduce damping rates: as the extremal Reissner-Nordström limit is approached, modes become long-lived and all branches accumulate near zero imaginary frequency, so charged wormholes can ring almost like the extremal black hole.
  • Subcritical charged wormholes inherit the l=2 instability for large Λ, but its growth rate is suppressed by charge; the critical and supercritical families show no l=2 instability in the computed range, with the caveat that accuracy degrades at large parameters.
  • The radial instability relaxes near the extremal limit (ω_I scales roughly as (1 − M/r_T)^3.1), giving arbitrarily long instability timescales, while supercritical wormholes instead show two unstable radial branches that merge and acquire real frequencies with opposite signs.

Reading between the lines

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

  • Editorial inference: The most direct confirmation would be an independent time-domain evolution of the linearized polar equations for an uncharged Ellis-Bronnikov wormhole at M/r_T ≈ 0.4; if the quadrupolar channel does not grow, the reported crossing is a numerical artifact.
  • Editorial inference: If the instability survives nonlinear evolution, it could set the effective lifetime of massive wormhole black-hole mimickers, possibly making them shorter-lived than the radial instability alone would suggest.
  • Editorial inference: The authors' analogy between charge and rotation suggests a concrete test: compute the l=2 polar modes of rapidly rotating Ellis-Bronnikov wormholes to see whether rotation similarly suppresses the nonradial instability; the paper only conjectures the radial analogue.
  • Editorial inference: The finding that charge suppresses the l=2 growth rate raises the possibility of a critical charge-to-mass ratio above which the nonradial instability disappears entirely — a threshold the paper does not map, but a natural next calculation.
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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 / 3 minor

Summary. The paper derives the axial, radial, and polar perturbation equations for static charged Ellis–Bronnikov wormholes in Einstein–Maxwell theory with a phantom scalar, and computes quasinormal-mode spectra with a Chebyshev spectral method. The uncharged limit is benchmarked against known Ellis–Bronnikov results and the expected axial–polar electromagnetic isospectrality is recovered. For charged configurations, the paper tracks axial and polar branches across critical, subcritical, and supercritical families, finding that charge generally increases damping times near the extremal Reissner–Nordstrøm limit. The headline new claim is a previously unknown polar instability: the fundamental l=2 b2 branch crosses ωI=0 at M/rT≈0.3 for uncharged EB wormholes (Sec. 5.1.2, Fig. 3, Table 2) and is also unstable for subcritical wormholes at sufficiently large Λ (Sec. 5.3, Table 9). The authors also note that critical wormholes do not show this instability in the computed range, while supercritical solutions appear stable.

Significance. If the polar l=2 b2 instability is real, it constitutes a genuinely new dynamical-instability channel for charged and uncharged EB wormholes, distinct from the known radial l=0 instability, and it would strengthen the constraints on the viability of these wormhole models. The paper's strengths are its explicit perturbation systems, the closed-form backgrounds, the reproduction of the known uncharged spectrum, and the demonstration of EM isospectrality in the uncharged limit. There are no fitted parameters in the reported spectra. The main result, however, depends on a single spectral implementation in the parameter regime that the authors themselves flag as least accurate, with no independent numerical confirmation. The catalog of stable modes may well be reliable, but the central instability claim requires additional verification before it can be considered secure.

major comments (3)
  1. [Sec. 5.1.2 / Table 2, with Sec. 4] The headline instability rests on one spectral method without independent validation. Sec. 4 states that accuracy 'remains better than 10^-3' but degrades for higher excited modes and larger Λ and γ1, and no per-mode error estimates, Np-convergence data, or second numerical method are provided. The crossing in Table 2 occurs between ωI r0 = -0.001 (Λ=0.7) and +0.013 (Λ=0.8), and Table 9 reports ωI ≈ 0.067 for Λ=0.75 in the same high-parameter regime. While these values exceed 10^-3 in absolute terms, the absence of convergence documentation for exactly this branch and regime leaves the central instability claim under-supported. I request a convergence table at the crossing and at the unstable points, plus at least one independent check (e.g., time-domain integration, a different discretization, or a separate spectral implementation).
  2. [Sec. 5.1.2 / Figs. 3 and 5] The b1/b2 labels are introduced because the polar modes are mixed gravitational–scalar, and the branches exhibit crossings and repulsions in the complex-frequency plane. The claim that the unstable mode is 'the fundamental l=2 b2 branch' requires unambiguous branch tracking through these features, especially near M/rT ≈ 0.23 where the imaginary parts of the b1 and b2 branches cross and the real parts repel. The paper does not provide eigenfunction overlaps or any explicit branch-tracking criterion. If the labels switch at an avoided crossing, the instability could belong to a different physical mode, or the continuity of the branch used in Tables 2 and 9 could be broken. Please provide evidence of branch continuity, for instance overlap integrals of eigenvectors between adjacent parameter points.
  3. [Sec. 6 / critical and subcritical conclusions] The conclusion that critical wormholes do not exhibit the polar l=2 instability is stated to hold only within the computed range, and the text concedes that 'it is possible that such an instability appears for large values of the parameters, for which our method loses accuracy.' This is appropriately cautious, but it means the claimed dichotomy (stable critical, unstable subcritical) is not a proven boundary but an upper-bound statement. A quantitative statement of the maximum accessible parameter value and the corresponding accuracy would make the limitation precise.
minor comments (3)
  1. [Tables 1–12 vs. Fig. 3] The appendix tables report frequencies with r0=1, while the main figures scale by the throat radius rT. Since rT ≠ r0 for charged wormholes, the reader cannot directly compare, e.g., the Λ=0.8 crossing in Table 2 with the M/rT ≈ 0.3 statement in Sec. 5.1.2. Please state the relation between rT, M, Qe, and r0, and give the key crossing also in rT units.
  2. [Sec. 5.4.2] The phrase 'breaking of isospectrality between the b1 and b2 branches' is imprecise: isospectrality was introduced for axial and polar EM modes. The b1/b2 splitting is a degeneracy-breaking of the scalar–gravitational polar branches. Please rephrase to avoid confusion.
  3. [Eq. (44)] The expansion coefficient i0 is easily misread as the imaginary unit multiplied by zero. Please use a distinct symbol such as \iota_0 or i_0.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QNM spectra and the reported polar instability are computed from the stated field equations and closed-form backgrounds, not obtained by inverting the output or by a self-citation chain.

full rationale

The derivation chain is self-contained. The background solutions are given in closed form (Eqs. 5-8), with the free parameters (r0, gamma1, Lambda or mu) fixed before any mode is computed. The axial, radial and polar perturbation equations are derived from the action (Eqs. 17-19, 27-29, 36-42), and the quasinormal frequencies are eigenvalues of the resulting quadratic eigenvalue problem (Eq. 57) solved with a Chebyshev spectral method. No parameter is fitted to the reported spectra, and the l=2 polar instability is not used to determine any constant; it is presented as a numerical output that crosses omega_I = 0 in Table 2 and reappears in Table 9. The self-citations ([37], [43], [59], [63], [64]-[67]) are used for benchmarking the uncharged spectrum, for the radial-sector companion result, and for the numerical method; these are independent of the new polar-instability claim and none of them supplants the actual computation performed here. The paper even reports that the instability is absent in the critical and supercritical sectors, which is not what one would expect if the claim were forced by an input assumption. Numerical accuracy degradation in the large-mass/large-parameter regime is a correctness and robustness concern, not a circularity concern, and the stated accuracy 'better than 10^{-3}' plus the non-marginal crossing magnitudes provide no evidence that the output was constructed from the inputs by definition.

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

There are no fitted free parameters: r0 is a scaling set to 1, γ1 and Λ/µ parametrize known closed-form backgrounds, and no constant is tuned to reproduce the reported QNM frequencies. The analysis assumes the Einstein–Maxwell–phantom action is the relevant theory, the Regge–Wheeler gauge captures all physical perturbations, outgoing-wave boundary conditions define the QNMs, and the Chebyshev eigensolver converges. No new entities are introduced.

assumptions (5)
  • domain assumption The Einstein–Maxwell–phantom action (1) with a minimally coupled phantom scalar is the theory whose wormhole solutions are physically relevant.
    The entire analysis lives in this model; the phantom field has negative kinetic energy and is not derived from a more fundamental theory.
  • standard math Regge–Wheeler gauge and spherical-harmonic decomposition capture all first-order metric, gauge-field, and scalar perturbations without missing gauge or constraint modes.
    Reduces the PDE system to ODEs (Sec. 3); standard for spherically symmetric backgrounds but an assumption about completeness.
  • domain assumption Quasinormal modes are defined by purely outgoing waves at both asymptotic infinities through the tortoise coordinate (20); no incoming radiation at either end.
    This selects discrete complex frequencies; different boundary conditions would change the spectrum.
  • domain assumption The Chebyshev spectral discretization with Np collocation points converges, and the quadratic eigenvalue solver returns physical modes, not spurious numerical eigenvalues.
    No convergence proof or independent cross-check is supplied; accuracy is asserted via comparison to prior results and internal estimates (Sec. 4).
  • domain assumption The prior uncharged EB QNM results of [37,43] used as benchmark are correct.
    Used to validate the method in the Qe=0 limit; if those results were wrong, the calibration fails.

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Pith. "Pith review of Nonradial perturbations of static charged wormholes." pith.science (2026). https://pith.science/paper/NIPZT4UH

@misc{pith2026260718399,
  author       = {Pith},
  title        = {Pith review of: Nonradial perturbations of static charged wormholes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIPZT4UH}},
  note         = {Machine review of arXiv:2607.18399}
}
abstract

We investigate the nonradial quasinormal-mode spectrum of static charged Ellis--Bronnikov wormholes in Einstein--Maxwell theory minimally coupled to a phantom scalar field. The background solutions are known in closed form and comprise three classes: subcritical, critical and supercritical, which all approach the extremal Reissner--Nordstr\"om geometry at the boundary of their domain of existence. We derive the linear perturbation equations for axial and polar sectors, including the coupled gravitational, electromagnetic and phantom-scalar degrees of freedom, and compute the corresponding spectra by means of a Chebyshev spectral method. The uncharged limit reproduces the known Ellis--Bronnikov spectrum and exhibits the expected electromagnetic isospectrality. For charged configurations we track the axial and polar branches across the three families of solutions and identify the effect of the charge on the damping times and oscillation frequencies. In particular, we find that charge can substantially reduce damping rates as the extremal Reissner--Nordstr\"om limit is approached. We also uncover a nonradial polar instability, most clearly visible in the fundamental $l=2$ branch for sufficiently large wormhole masses. This instability is distinct from the familiar radial Ellis--Bronnikov instability and shows that the nonradial sector imposes additional constraints on the dynamical viability of charged wormholes.

Figures

Figures reproduced from arXiv: 2607.18399 by the authors.

Figure 1
Figure 1. Domain of existence of the static charged wormhole [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Another representation of the domain of existence [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. l = 2, 3 axial and polar modes for EB wormholes. From top to bottom the rows correspond to: l = 2 axial, l = 2 polar, l = 3 axial, l = 3 polar modes. Left panels show the real part of the QNMs. Right panels exhibit the imaginary part. In each panel we show, with different colors, the fundamental modes (n = 0), and a few excitations (n > 0). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (16 more)
Figure 4
Figure 4. Figure 4: Axial modes for EB wormholes. Im(ω) vs Re(ω) for l = 2 (left) and l = 3 (right). -2 -1 0 0 1 2 l=2 polar ωI rT ωR rT -4 -2 0 0 1 2 3 l=3 polar ωI rT ωR rT [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: Polar modes for EB wormholes. Im(ω) vs Re(ω) for l = 2 (left) and l = 3 (right). the isospectrality of the EM modes (note that both dashed curves overlap perfectly, see also Tables 1 and 2). As noted above, the axial and two polar QNMs coincide when M/rT = 0 [43]. But …
Figure 6
Figure 6. Figure 6: Isospectrality of l = 2 axial and polar modes for EB wormholes. The EM (dashed) curves overlap for these uncharged configurations. -0.6 -0.5 -0.4 -0.3 0 0.1 0.2 0.3 stable l=0 ωI rT M/rT 0.8 1 1.2 0 0.1 0.2 0.3 unstable l=0 ωI rT M/rT [PITH_FULL_IMAGE:figures/full_fig…
Figure 7
Figure 7. Figure 7: Radial modes for EB wormholes. Stable (left) and un [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: l = 2 and l = 3 axial and polar modes for critical wormholes. From top to bottom the rows correspond to: l = 2 axial, l = 2 polar, l = 3 axial, l = 3 polar modes. Left panels show the real part of the QNMs. Right panels exhibit the imaginary part. The gray vertical lin…
Figure 9
Figure 9. Figure 9: Axial modes for critical wormholes. Im(ω) vs Re(ω) for l = 2 (left) and l = 3 (right). -4 -2 0 0 1 2 l=2 polar ωI rT ωR rT -4 -2 0 0 1 2 3 l=3 polar ωI rT ωR rT [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]
Figure 10
Figure 10. Figure 10: Polar modes for critical wormholes. Im(ω) vs Re(ω) for l = 2 (left) and l = 3 (right). -4 -2 0 1 2 l=2 n=0 n=1 n=2 n=3 Axial Polar ωI rT ωR rT -4 -2 0 1 2 3 l=3 n=0 n=1 n=2 n=3 n=4 Axial Polar ωI rT ωR rT [PITH_FULL_IMAGE:figures/full_fig_p019_10.png]
Figure 11
Figure 11. Figure 11: Comparison of EM modes for critical wormholes. So [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: Radial modes for critical wormholes. Stable (left [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: l = 2 axial and polar modes for subcritical wormholes with Λ = 0.1. From top to bottom the rows correspond to: l = 2 axial, l = 2 polar. Left panels show the real part of the QNMs. Right panels exhibit the imaginary part. 0 0.05 0.1 0.4 0.5 0.6 0.7 0.8 0.9 unstable l=…
Figure 14
Figure 14. Figure 14: l = 2 polar unstable b2 branch for subcritical wormholes, for which Λ is fixed to 0.75. Left: Real part of the QNMs. Right: Imaginary part. real and imaginary parts rapidly approach zero as the mass increases. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]
Figure 15
Figure 15. Figure 15: Radial unstable branch for subcritical wormhole [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]
Figure 16
Figure 16. Figure 16: l = 2 and l = 3 axial and polar modes for supercritical wormholes with γ1 = 0. From top to bottom the rows correspond to: l = 2 axial, l = 2 polar, l = 3 axial, l = 3 polar modes. Left panels show the real part of the QNMs. Right panels exhibit the imaginary part. The…
Figure 17
Figure 17. Figure 17: Axial modes for supercritical wormholes. Im( [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: Polar modes for supercritical wormholes. Im( [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]
Figure 19
Figure 19. Figure 19: Unstable radial branch of supercritical wormhol [PITH_FULL_IMAGE:figures/full_fig_p025_19.png]

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