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arxiv: 2506.01035 · v2 · pith:Y3GK7QPWnew · submitted 2025-06-01 · 🌀 gr-qc · hep-th

Finite Curvature Construction of Regular Black Holes and Quasinormal Mode Analysis

Chen Lan , Zhen-Xiao Zhang , Hao Yang This is my paper

Pith reviewed 2026-05-22 01:44 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords regular black holescurvature invariantsquasinormal modesstability analysissingularity avoidancegravitational perturbationsmass functioneffective potential
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The pith

Prescribing finite curvature invariants constructs regular black holes whose perturbation potentials control stability through their peak-to-valley ratio.

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

The paper develops regular black holes by choosing analytic profiles for finite Ricci or Weyl scalars and reconstructing the metric from them. This yields mass functions that produce geometries without curvature singularities while remaining asymptotically flat and compatible with energy conditions. Quasinormal mode calculations under axial perturbations show that the effective potential's shape determines the waveform behavior. A large peak relative to any valley produces clean exponential decay, whereas a small ratio permits late-time growth. The work therefore ties the choice of curvature profile directly to both geometric regularity and dynamical stability.

Core claim

Prescribing analytic forms for the Ricci scalar or Weyl scalar with Gaussian, hyperbolic secant, or rational profiles allows reconstruction of the spacetime metric to obtain regular black hole solutions free of curvature singularities. The resulting mass functions depend on free parameters and set the horizon structure. Quasinormal mode spectra for axial gravitational perturbations are computed from the associated effective potentials; models whose potentials exhibit a large peak-to-valley ratio produce stable, exponentially decaying ringdown signals, while those with small ratios can develop late-time instabilities.

What carries the argument

Reconstruction of the metric from prescribed analytic profiles of finite curvature invariants (Ricci scalar or Weyl scalar).

If this is right

  • The constructed spacetimes contain no curvature singularities at finite radius.
  • Horizon existence and location vary with the model parameters.
  • Large peak-to-valley ratios in the effective potential guarantee exponentially decaying quasinormal modes.
  • Small peak-to-valley ratios can produce late-time instabilities in the waveforms.
  • The approach supplies concrete examples of regular black holes usable in modified-gravity and quantum-gravity settings.

Where Pith is reading between the lines

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

  • Observational gravitational-wave ringdown data could constrain which curvature profiles remain viable.
  • The same reconstruction technique might be applied to rotating or charged regular solutions.
  • Different curvature profiles could produce distinguishable tidal or lensing signatures around the horizon.
  • Stability criteria based on potential shape may generalize to other classes of modified black-hole metrics.

Load-bearing premise

The chosen analytic profiles for the curvature invariants can be integrated into a metric that satisfies asymptotic flatness and the dominant energy condition for the selected parameter ranges.

What would settle it

An explicit computation showing a curvature singularity at the center or a growing late-time tail in the axial perturbation waveform for any of the constructed mass functions would refute the central construction.

Figures

Figures reproduced from arXiv: 2506.01035 by Chen Lan, Hao Yang, Zhen-Xiao Zhang.

Figure 1
Figure 1. Figure 1: Schematic of the β functions. The black line is a normal Gaus￾sian function C1e ´px´x0q 2 , the gray dashed line is an asymmetric Gaussian function C1e ´px´x0q 2 ` C2rtanhp´xq ` 1s, the gray dot-dashed line is a Skew Gaussian function C1e ´px´x0q 2 ş dx e ´px´x0q 2 , and the gray solid line is a combination of multiple Gaussian functions C1e ´px´x0q 2 ´ C2e ´px´x1q 2 ` C3rtanhpx1 ´ xq ` 1s. Except for the … view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the sigma functions. The black line is an asymmetric Gaussian [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Horizons with different parameters: A model based on the Gaussian function. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Horizons with different parameters: A model based on the hyperbolic secant [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The energy conditions of regular black holes constructed by taking the hy [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Horizons with different parameters: A model based on the fuzzy logic function. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Effective potential for the models constructed by the Ricci-Scalar approach. [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: QNMs of models constructed by Ricci-scalar approach. [PITH_FULL_IMAGE:figures/full_fig_p020_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Effective potential for the models constructed by the Weyl-scalar approach. [PITH_FULL_IMAGE:figures/full_fig_p021_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: QNMs of models constructed by Weyl-scalar approach. [PITH_FULL_IMAGE:figures/full_fig_p022_10.png] view at source ↗
read the original abstract

We develop a class of regular black holes by prescribing finite curvature invariants and reconstructing the corresponding spacetime geometry. Two distinct approaches are employed: one based on the Ricci scalar and the other on the Weyl scalar. In each case, we explore a variety of analytic profiles for the curvature functions, including Gaussian, hyperbolic secant, and rational forms, ensuring regularity, asymptotic flatness, and compatibility with dominant energy conditions. The resulting mass functions yield spacetime geometries free from curvature singularities and exhibit horizons depending on model parameters. To assess the stability of these solutions, we perform a detailed analysis of quasinormal modes (QNMs) under axial gravitational perturbations. We show that the shape of the effective potential, particularly its width and the presence of potential valleys, plays a critical role in determining the QNMs. Models with a large peak-to-valley ratio in the potential barrier exhibit stable, exponentially decaying waveforms, while a small ratio may induce late-time instabilities. Our results highlight the significance of potential design in constructing physically viable and dynamically stable regular black holes, offering potential observational implications in modified gravity and quantum gravity scenarios.

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 manuscript develops regular black holes by prescribing finite analytic profiles (Gaussian, sech, rational) for the Ricci scalar or Weyl scalar and reconstructing the metric function or mass m(r) to achieve no curvature singularities. It claims the resulting geometries are asymptotically flat and satisfy the dominant energy condition for chosen parameters, then analyzes axial gravitational quasinormal modes, concluding that a large peak-to-valley ratio in the effective potential yields stable exponentially decaying waveforms while a small ratio may produce late-time instabilities.

Significance. If the reconstructions are shown to satisfy the dominant energy condition and asymptotic flatness, and if the QNM stability conclusions are confirmed by explicit calculations, the work offers a systematic curvature-based route to regular black holes with potential implications for modified gravity and stability criteria. The link between potential shape and waveform behavior is a useful observation for constructing viable models.

major comments (2)
  1. [Abstract / metric reconstruction] Abstract and reconstruction procedure: the claim that regularity, asymptotic flatness, and dominant energy conditions are ensured for the Gaussian, sech, and rational profiles lacks explicit derivations of the metric components, mass function m(r), or numerical verification that ρ ≥ 0, ρ ≥ |p_r|, ρ ≥ |p_t| holds everywhere for the reported parameter ranges; without these the physical viability of the spacetimes cannot be confirmed.
  2. [Quasinormal mode analysis] Quasinormal mode section: the statement that 'models with a large peak-to-valley ratio exhibit stable, exponentially decaying waveforms, while a small ratio may induce late-time instabilities' requires concrete QNM frequency tables or time-domain evolution results for at least two contrasting profiles; the effective potential is mentioned but its explicit form and the quantitative ratio threshold are not provided.
minor comments (2)
  1. Clarify the notation for the curvature invariants (Ricci vs. Weyl) and ensure the same symbols are used consistently when switching between the two approaches.
  2. Add a brief comparison table of the resulting horizon radii or ADM masses across the different analytic profiles to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment point by point below and have revised the manuscript accordingly to provide the requested explicit details and supporting calculations.

read point-by-point responses

  1. Referee: [Abstract / metric reconstruction] Abstract and reconstruction procedure: the claim that regularity, asymptotic flatness, and dominant energy conditions are ensured for the Gaussian, sech, and rational profiles lacks explicit derivations of the metric components, mass function m(r), or numerical verification that ρ ≥ 0, ρ ≥ |p_r|, ρ ≥ |p_t| holds everywhere for the reported parameter ranges; without these the physical viability of the spacetimes cannot be confirmed.

    Authors: We appreciate the referee highlighting the need for greater explicitness. The reconstruction procedure is described in Section II of the manuscript, where the metric function and mass function m(r) are derived from the prescribed curvature profiles. In the revised manuscript we have expanded this section to include the full analytic expressions for the metric components and m(r) for the Gaussian, sech, and rational cases. We have also added numerical verification plots and tabulated checks demonstrating that the dominant energy condition (ρ ≥ 0, ρ ≥ |p_r|, ρ ≥ |p_t|) is satisfied everywhere for the parameter ranges used in the paper. These additions are placed in the main text and a new appendix for easy reference. revision: yes

  2. Referee: [Quasinormal mode analysis] Quasinormal mode section: the statement that 'models with a large peak-to-valley ratio exhibit stable, exponentially decaying waveforms, while a small ratio may induce late-time instabilities' requires concrete QNM frequency tables or time-domain evolution results for at least two contrasting profiles; the effective potential is mentioned but its explicit form and the quantitative ratio threshold are not provided.

    Authors: We thank the referee for this request for concreteness. The manuscript already derives the effective potential for axial gravitational perturbations and analyzes its peak-to-valley structure in Section IV. To strengthen the presentation, the revised version now includes the explicit functional form of the effective potential, a table of computed quasinormal frequencies for two representative profiles (one with large and one with small peak-to-valley ratio), and time-domain evolution plots showing the corresponding waveform decay or late-time behavior. A brief quantitative discussion of the ratio threshold separating stable and potentially unstable regimes, based on these calculations, has also been added. revision: yes

Circularity Check

0 steps flagged

Direct prescription of finite curvature profiles yields self-contained construction with no reduction to inputs by definition.

full rationale

The derivation begins by choosing analytic finite profiles (Gaussian, sech, rational) for Ricci or Weyl scalars, then algebraically solves for the mass function m(r) so the invariants match the chosen forms exactly. Regularity follows immediately from the finiteness of the prescribed scalars; asymptotic flatness and DEC compliance are verified by direct substitution for selected parameter ranges rather than being derived as independent predictions. The QNM analysis computes the effective potential from the resulting metric and examines its peak-to-valley ratio numerically, producing stability statements that depend on the explicit shape of that potential rather than re-expressing the input profiles. No self-citation chain, fitted parameter renamed as prediction, or uniqueness theorem is invoked to close the argument; the entire chain remains an explicit construction whose outputs are checked against external conditions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central construction rests on a small number of free parameters inside the chosen curvature profiles and on standard domain assumptions of spherical symmetry and asymptotic flatness; no new particles or forces are postulated.

free parameters (1)
  • Parameters inside curvature profiles (width, amplitude, etc.)
    Chosen by hand to enforce regularity, asymptotic flatness and dominant energy condition compliance for each analytic form.
axioms (1)
  • domain assumption The spacetime is spherically symmetric and asymptotically flat.
    Required to reconstruct a unique metric from the prescribed curvature invariants.

pith-pipeline@v0.9.0 · 5723 in / 1364 out tokens · 67267 ms · 2026-05-22T01:44:57.636418+00:00 · methodology

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Reference graph

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