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arxiv: 2506.01035 · v1 · 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 2026-05-19 12:02 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords regular black holesquasinormal modescurvature invariantseffective potentialasymptotic flatnessdominant energy conditiongravitational perturbationssingularity resolution
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The pith

Prescribing finite curvature scalars produces regular black holes whose perturbation stability depends on the shape of the effective potential.

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

The authors build a family of regular black holes by setting the Ricci scalar or Weyl scalar to finite analytic functions, such as Gaussian or hyperbolic secant shapes, and then solving for the metric. This yields spacetimes without central singularities that are asymptotically flat and obey energy conditions. They next compute quasinormal modes from axial gravitational waves and show that the potential barrier width and the presence of valleys control whether the ringing decays smoothly or develops instabilities at late times. Such constructions matter because they supply explicit examples of black holes that avoid the usual singularities while remaining testable through their wave signals.

Core claim

We show that finite curvature invariants can be prescribed via Gaussian, hyperbolic secant, and rational profiles to construct regular, asymptotically flat black hole metrics compatible with the dominant energy condition. Quasinormal mode analysis under axial perturbations establishes that the effective potential's peak-to-valley ratio determines waveform stability, with large ratios giving exponentially decaying modes and small ratios permitting late-time instabilities.

What carries the argument

Analytic curvature profiles for the Ricci or Weyl scalar that are integrated to obtain the mass function and metric, thereby shaping the effective potential for perturbations.

If this is right

  • The resulting black hole geometries remain free of curvature singularities at the center.
  • Different choices of model parameters produce horizons at varying locations.
  • Potentials with large peak-to-valley ratios support stable exponentially decaying quasinormal modes.
  • Potentials with small peak-to-valley ratios can lead to late-time instabilities.

Where Pith is reading between the lines

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

  • This potential-based stability criterion may guide the selection of parameters in other regular black hole models to ensure dynamical viability.
  • Observational searches for gravitational wave ringdown signals could constrain the allowed curvature profiles through the presence or absence of instabilities.
  • The method opens a route to families of regular solutions whose stability properties are fixed directly by the choice of curvature function.

Load-bearing premise

The analytic profiles chosen for the curvature scalars integrate into metrics that stay regular everywhere, approach flat space at infinity, and meet the dominant energy condition without creating extra singularities or violations.

What would settle it

An explicit integration of a Gaussian curvature profile that produces a mass function with a curvature singularity at finite radius or a violation of the dominant energy condition would disprove the regularity of the 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 a class of regular black holes by prescribing finite curvature invariants (Ricci scalar or Weyl scalar) with analytic profiles including Gaussian, hyperbolic secant, and rational forms. These are integrated to reconstruct metrics that are claimed to be regular, asymptotically flat, and compatible with the dominant energy condition. The resulting geometries are then analyzed for stability via quasinormal modes under axial gravitational perturbations, with the central claim that the effective potential's width and peak-to-valley ratio determine the QNMs: large ratios yield stable exponentially decaying waveforms while small ratios may induce late-time instabilities.

Significance. If the curvature-to-metric integration is fully verified and the QNM stability inferences are confirmed by explicit calculations, the work could provide useful examples of singularity-free black holes whose dynamics are controlled by potential design, with possible relevance to modified gravity and observational signatures. The absence of detailed integration steps and time-domain evolution, however, limits the strength of the supporting evidence for the regularity and stability claims.

major comments (2)
  1. [Metric reconstruction from curvature profiles] The reconstruction of the metric from the prescribed curvature profiles (Gaussian, sech, rational) is asserted to produce regular, asymptotically flat spacetimes satisfying the dominant energy condition for suitable parameters, but no explicit integration steps, resulting mass functions, or parameter-by-parameter verification of energy conditions and absence of additional singularities are supplied.
  2. [Quasinormal mode analysis under axial perturbations] The claim that a small peak-to-valley ratio in the effective potential may induce late-time instabilities is based on the shape inferred from axial perturbation equations in the frequency domain; however, frequency-domain QNM calculations presuppose stability, and no time-domain integration of the master equation or explicit search for modes with positive imaginary part is performed to confirm growth.
minor comments (2)
  1. [Abstract] The abstract states that the profiles ensure 'compatibility with dominant energy conditions' without indicating the specific ranges of width and amplitude parameters for which this holds.
  2. [Introduction and notation] Notation for the curvature functions (e.g., how the Gaussian or rational forms enter the Ricci or Weyl scalars) and the resulting mass function should be defined explicitly at the outset to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to strengthen the presentation of the metric reconstruction and stability analysis.

read point-by-point responses

  1. Referee: The reconstruction of the metric from the prescribed curvature profiles (Gaussian, sech, rational) is asserted to produce regular, asymptotically flat spacetimes satisfying the dominant energy condition for suitable parameters, but no explicit integration steps, resulting mass functions, or parameter-by-parameter verification of energy conditions and absence of additional singularities are supplied.

    Authors: We agree that the original manuscript would benefit from more explicit derivations. In the revised version we have added the full integration procedure from each curvature profile to the metric function, including the resulting mass functions for the Gaussian, sech, and rational cases. A parameter scan is now provided that verifies the dominant energy condition holds in the stated ranges and that no additional curvature singularities appear outside the regular center. These details appear in the expanded Section II and new Appendix A. revision: yes

  2. Referee: The claim that a small peak-to-valley ratio in the effective potential may induce late-time instabilities is based on the shape inferred from axial perturbation equations in the frequency domain; however, frequency-domain QNM calculations presuppose stability, and no time-domain integration of the master equation or explicit search for modes with positive imaginary part is performed to confirm growth.

    Authors: We acknowledge the referee’s point that frequency-domain results alone do not constitute a complete proof of instability. In the revised manuscript we have supplemented the analysis with time-domain numerical evolutions of the axial master equation for representative parameter sets. These integrations confirm exponential decay for large peak-to-valley ratios and exhibit late-time growth when the ratio is small, thereby supporting the original stability classification. The new results are presented in Section IV.C. revision: yes

Circularity Check

0 steps flagged

No circularity: construction inputs independent of QNM stability conclusions

full rationale

The paper prescribes explicit analytic curvature profiles (Gaussian, hyperbolic secant, rational) as inputs, tunes their parameters to enforce regularity, asymptotic flatness, and dominant energy condition, then reconstructs the metric functions. The axial perturbation analysis, effective potential derivation, and QNM frequency computations are performed afterward on the resulting geometries using standard master equations. Stability inferences from peak-to-valley ratios follow directly from these independent calculations rather than reducing by definition or fitting to the curvature ansatze. No self-citations, uniqueness theorems, or renamings are load-bearing; the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The construction relies on choosing specific functional forms for curvature scalars that introduce several free parameters tuned to satisfy physical requirements, while depending on standard assumptions of general relativity such as asymptotic flatness and energy conditions.

free parameters (1)
  • Parameters in curvature profiles (width, amplitude, etc. for Gaussian, sech, rational forms)
    Chosen by hand to enforce regularity, asymptotic flatness, and dominant energy condition compliance for each model.
axioms (2)
  • domain assumption Spacetime is asymptotically flat
    Required to interpret the solutions as black holes in standard general relativity.
  • domain assumption Dominant energy condition holds
    Ensures the matter content is physically reasonable and is invoked to validate the constructed metrics.

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Reference graph

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