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arxiv: 1907.06463 · v2 · pith:HSIROBGOnew · submitted 2019-07-15 · 🌀 gr-qc · hep-th

Perturbing microscopic black holes inspired by noncommutativity

Davide Batic , N. G. Kelkar , Marek Nowakowski , Karlus Redway This is my paper

Pith reviewed 2026-05-24 21:24 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modesnoncommutative black holesWKB approximationblack hole stabilityscalar perturbationsinverted potential methodasymptotic iteration method
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The pith

Instabilities seen in WKB calculations for noncommutative Schwarzschild black holes are artifacts of the approximation method.

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

The paper examines quasinormal modes of a massless scalar field in the noncommutative geometry inspired Schwarzschild spacetime. A sixth-order WKB calculation reveals that the approximation fails to converge precisely in the regimes where third-order results indicate instabilities. Application of the inverted potential method demonstrates that no such instabilities are present. The work concludes that the noncommutative black hole remains stable under these perturbations. The asymptotic iteration method is presented as an alternative approach for computing the modes.

Core claim

The central claim is that the instabilities observed at third order in the WKB approximation for the noncommutative Schwarzschild black hole are an artifact of the WKB method. This conclusion follows from the demonstration that the sixth-order WKB approximation does not converge in the critical cases, while the inverted potential method shows the absence of instabilities. The paper further discusses the asymptotic iteration method as a tool for determining the quasinormal frequencies.

What carries the argument

The inverted potential method applied to the effective potential governing massless scalar perturbations on the noncommutative Schwarzschild background.

If this is right

  • The noncommutative Schwarzschild black hole is stable under massless scalar perturbations.
  • Low-order WKB results can produce spurious instabilities that disappear at higher orders or with alternative methods.
  • The asymptotic iteration method offers a viable route to reliable quasinormal mode spectra when WKB convergence fails.
  • Microscopic black holes modeled by noncommutative geometry do not introduce new instability channels for scalar fields.

Where Pith is reading between the lines

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

  • Similar WKB convergence problems may occur in other noncommutative or quantum-corrected spacetimes and could be checked with the inverted potential approach.
  • Comparison with fully numerical time-domain evolutions would provide an independent test of stability for this geometry.
  • The result suggests that noncommutativity alone does not destabilize the black hole at the perturbative level examined here.

Load-bearing premise

The inverted potential method correctly identifies the absence of instabilities in regimes where the WKB approximation fails to converge.

What would settle it

A direct numerical solution of the wave equation that reveals exponentially growing modes for scalar perturbations on the noncommutative Schwarzschild metric would falsify the claim.

Figures

Figures reproduced from arXiv: 1907.06463 by Davide Batic, Karlus Redway, Marek Nowakowski, N. G. Kelkar.

Figure 1
Figure 1. Figure 1: FIG. 1: Plot of the function [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Plot of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Plot of the effective potential [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of the approximate form (21) of the effecti [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Comparison of the approximate form (22) of the effecti [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Comparison of the P¨oschl-Teller and actual effectiv [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
read the original abstract

We probe into the instabilities of microscopic quantum black holes. For this purpose, we study the quasinormal modes (QNMs) for a massless scalar perturbation of the noncommutative geometry inspired Schwarzschild black hole. By means of a sixth order Wentzel-Kramers-Brillouin (WKB) approximation we show that the widely used WKB method does not converge in the critical cases where instabilities show up at the third order. By employing the inverted potential method, we demonstrate that the instabilities are an artifact of the WKB method. Finally, we discuss the usefulness of the asymptotic iteration method to find the QNMs.

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 / 2 minor

Summary. The paper studies quasinormal modes of massless scalar perturbations on the noncommutative-geometry-inspired Schwarzschild black hole. Using a sixth-order WKB approximation, it reports that the WKB series fails to converge precisely in the parameter regimes where third-order results indicate instabilities. It then applies the inverted potential method to conclude that these instabilities are methodological artifacts of WKB rather than physical features of the spacetime, and briefly discusses the asymptotic iteration method as a potentially more reliable alternative.

Significance. If the central claim is substantiated, the result would clarify that apparent instabilities in microscopic noncommutative black holes are not intrinsic but arise from the breakdown of the WKB expansion on the deformed potential. This would be useful for the community working on quantum-corrected black-hole perturbation theory, especially since the paper explicitly flags the non-convergence of WKB and points to an alternative method. The absence of free parameters or ad-hoc fitting in the core derivation is a positive feature.

major comments (2)
  1. [inverted potential method section] Application of the inverted potential method (the section following the WKB analysis): the manuscript provides no quantitative benchmark, error estimate, or direct comparison of this method against either the asymptotic iteration method (which the paper itself flags as useful) or numerical integration of the wave equation on the same noncommutative background. Without such validation, the assertion that the method correctly identifies the absence of instabilities where sixth-order WKB diverges remains an untested assumption about the auxiliary method's accuracy.
  2. [WKB results paragraph] WKB convergence discussion (the paragraph reporting third-order instabilities and sixth-order non-convergence): no explicit tables or figures display the actual WKB coefficients, the radius of convergence, or the precise noncommutativity-parameter values at which divergence sets in. This absence of quantitative data makes it impossible to assess how severe the non-convergence is or whether the inverted-potential conclusion is robust across the full parameter space.
minor comments (2)
  1. [abstract] The abstract states that the inverted potential method 'removes the instability signal' but supplies no numerical values or comparison metrics; a short table or sentence with concrete numbers would improve clarity.
  2. [introduction and potential definition] Notation for the noncommutativity parameter and the effective potential should be introduced once with a clear equation reference rather than repeated inline.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. We address the two major comments below and will incorporate revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [inverted potential method section] Application of the inverted potential method (the section following the WKB analysis): the manuscript provides no quantitative benchmark, error estimate, or direct comparison of this method against either the asymptotic iteration method (which the paper itself flags as useful) or numerical integration of the wave equation on the same noncommutative background. Without such validation, the assertion that the method correctly identifies the absence of instabilities where sixth-order WKB diverges remains an untested assumption about the auxiliary method's accuracy.

    Authors: We agree that a direct benchmark would improve the presentation. In the revised manuscript we will compute a small set of QNM frequencies using the asymptotic iteration method for representative values of the noncommutativity parameter in the regime where sixth-order WKB diverges, and compare them with the inverted-potential results. This provides the requested quantitative check while remaining within the scope of the existing discussion of AIM in the paper. revision: yes

  2. Referee: [WKB results paragraph] WKB convergence discussion (the paragraph reporting third-order instabilities and sixth-order non-convergence): no explicit tables or figures display the actual WKB coefficients, the radius of convergence, or the precise noncommutativity-parameter values at which divergence sets in. This absence of quantitative data makes it impossible to assess how severe the non-convergence is or whether the inverted-potential conclusion is robust across the full parameter space.

    Authors: We accept that explicit data on the WKB series would allow readers to evaluate the non-convergence more precisely. The revised version will include a table listing the successive WKB coefficients up to sixth order for several values of the noncommutativity parameter, together with a short discussion of the observed radius of convergence and the parameter values at which divergence appears. revision: yes

Circularity Check

0 steps flagged

No significant circularity: cross-method comparison of WKB and inverted potential on noncommutative metric is independent

full rationale

The paper applies a sixth-order WKB approximation to the quasinormal modes of the noncommutative Schwarzschild black hole, observes non-convergence at third order in critical cases, then invokes the inverted potential method to conclude that the apparent instabilities are WKB artifacts. This is a direct numerical comparison between two independent approximation techniques applied to the same background metric and potential; neither result is obtained by fitting parameters to the other, redefining the target quantity, or chaining to a self-citation that itself assumes the conclusion. The asymptotic iteration method is mentioned only as a suggested future tool, not as a load-bearing premise. The derivation chain therefore remains self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities beyond the background noncommutative geometry inspired metric already present in the cited literature.

pith-pipeline@v0.9.0 · 5637 in / 923 out tokens · 18433 ms · 2026-05-24T21:24:39.361474+00:00 · methodology

discussion (0)

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Reference graph

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