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arxiv: 2509.23318 · v2 · submitted 2025-09-27 · 🌀 gr-qc

Gravitational waveforms from periodic orbits around a novel regular black hole

Huajie Gong , Sheng Long , Xi-Jing Wang , Zhongwu Xia , Jian-Pin Wu , Qiyuan Pan This is my paper

Pith reviewed 2026-05-18 12:32 UTC · model grok-4.3

classification 🌀 gr-qc
keywords regular black holeMinkowski coreperiodic orbitsgravitational waveformsquantum gravity effectsphase shiftsnumerical kludgedistinguishability
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The pith

Periodic orbits around a regular black hole with a Minkowski core generate gravitational waveforms showing phase shifts and amplitude modulations from quantum gravity effects that differ from Schwarzschild predictions.

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

This paper studies periodic orbits around a novel regular black hole with a Minkowski core, classified using a rational number q, to identify potential quantum gravity signatures in gravitational wave emissions. The deviation parameter α0 reshapes the bound-orbit region while preserving zoom-whirl structures. Numerical kludge waveforms reveal that quantum gravity effects produce detectable phase shifts and amplitude modulations, with radiation reaction breaking orbital periodicity. Faithfulness analysis indicates that larger α0 and q values increase distinguishability from the Schwarzschild case, and comparisons with Hayward and quantum Oppenheimer-Snyder black holes show similar large-scale orbital and waveform behaviors.

Core claim

The paper establishes that periodic orbits around the novel regular black hole with Minkowski core, parameterized by α0 and classified by rational q, produce gravitational waveforms via numerical kludge methods that exhibit quantum gravity-induced phase shifts and amplitude modulations. Radiation reaction disrupts periodicity, and larger α0 and q enhance distinguishability from Schwarzschild waveforms in faithfulness tests, while large-scale behaviors remain similar to those of Hayward and quantum Oppenheimer-Snyder black holes.

What carries the argument

The deviation parameter α0 in the regular black hole metric with Minkowski core, which modifies the spacetime and thereby alters bound orbits and the resulting numerical kludge gravitational waveforms.

If this is right

  • The bound-orbit region reshapes under α0 while zoom-whirl structures remain intact.
  • Radiation reaction from gravitational wave emission breaks the periodicity of the orbits.
  • Larger α0 and q values reduce faithfulness to Schwarzschild waveforms and improve distinguishability.
  • Large-scale orbits and waveforms remain macroscopically similar to those of Hayward and quantum Oppenheimer-Snyder black holes.

Where Pith is reading between the lines

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

  • Observations with future gravitational wave detectors could place bounds on α0 by searching for or excluding the predicted phase and amplitude deviations.
  • The approach of classifying orbits by rational q and computing kludge waveforms could be applied to other regular black hole models for comparative tests of quantum signatures.
  • If phase shifts are absent in real data at expected scales, it would constrain the physical relevance of Minkowski-core regular black hole metrics.
  • The persistence of zoom-whirl features under quantum corrections suggests that certain orbital qualitative traits may be robust across different spacetime modifications.

Load-bearing premise

The novel regular black hole metric with Minkowski core is assumed to provide a physically relevant description of quantum gravity effects at the scales relevant for periodic orbits and gravitational wave emission.

What would settle it

A high-precision gravitational wave observation from a periodic orbit around an astrophysical black hole that matches Schwarzschild predictions exactly, with no detectable phase shift or amplitude modulation at parameter values where α0 effects are predicted to appear.

Figures

Figures reproduced from arXiv: 2509.23318 by Huajie Gong, Jian-Pin Wu, Qiyuan Pan, Sheng Long, Xi-Jing Wang, Zhongwu Xia.

Figure 1
Figure 1. Figure 1: FIG. 1: Left plot: The effective potential [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The radius [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows that as the deviation parameter α0 increases, the radius, energy, and angular 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The allowed ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Left plot: The rational number [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Periodic orbits of different ( [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Periodic orbits of different ( [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: , we present the GW polarizations h+ and h× corresponding to several periodic orbits with different sets of (z, w, v). The resulting waveforms exhibit typical zoom and whirl features within one orbital period, reflecting the complexity of the orbits of the small object around the RBH. It is evident that as the zoom number increases, the waveform develops 14 [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: GW polarisations [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Left plot: The faithfulness of the GW signal between the Schwarzschild BH and RBH [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

We explore potential quantum gravity signatures by studying periodic orbits and their GW emissions around a novel regular black hole (BH) featuring a Minkowski core. Using a rational number $q$, periodic orbits are classified, revealing that the deviation parameter $\alpha_0$ reshapes the bound-orbit region while preserving characteristic ``zoom-whirl" structures. Numerical kludge waveforms reveal detectable phase shifts and amplitude modulations induced by quantum gravity effects with radiation reaction breaking orbital periodicity. Faithfulness analysis demonstrates that larger $\alpha_{0}$ and $q$ enhance distinguishability from the Schwarzschild case, and a comparison with Hayward and quantum Oppenheimer-Snyder BHs shows their similar large-scale behaviors yield macroscopically indistinguishable orbits and waveforms.

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

1 major / 3 minor

Summary. The paper explores potential quantum gravity signatures by studying periodic orbits and their GW emissions around a novel regular black hole featuring a Minkowski core. Using a rational number q, periodic orbits are classified, revealing that the deviation parameter α0 reshapes the bound-orbit region while preserving characteristic zoom-whirl structures. Numerical kludge waveforms reveal detectable phase shifts and amplitude modulations induced by quantum gravity effects with radiation reaction breaking orbital periodicity. Faithfulness analysis demonstrates that larger α0 and q enhance distinguishability from the Schwarzschild case, and a comparison with Hayward and quantum Oppenheimer-Snyder BHs shows their similar large-scale behaviors yield macroscopically indistinguishable orbits and waveforms.

Significance. If the numerical results hold, this manuscript contributes to probing quantum gravity effects in gravitational wave signals from strong-field periodic orbits. The demonstration of phase shifts and enhanced distinguishability for larger deviation parameters provides a concrete example of how regular black hole metrics can lead to observable differences. Credit is given for the numerical computations supporting phase shifts and the comparative analysis with other regular black hole models, which helps contextualize the findings at large scales.

major comments (1)
  1. [Waveform generation and numerical methods] The central results depend on numerical kludge waveforms. Standard kludge methods combine geodesic motion with approximate wave generation and radiation reaction often tuned to Schwarzschild. For the novel metric with Minkowski core, the geometry near the center and effective potential differ, affecting orbital frequencies and multipoles. The manuscript does not detail the adaptation of the kludge (e.g., re-derivation of source terms or validation against the new geodesic equations), which risks uncontrolled errors in the reported phase shifts and faithfulness. This is a load-bearing issue for the claim that the effects are due to quantum gravity parameter α0.
minor comments (3)
  1. [Abstract] The abstract is clear but could include a brief mention of the specific metric form or key equations for better context.
  2. [Notation] Ensure consistent use of α_0 versus α0 throughout the text and figures.
  3. [References] Consider adding references to prior works on numerical kludge waveforms in modified gravity to strengthen the methodological foundation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful review and for recognizing the potential of our numerical results to probe quantum gravity effects. We address the major comment on waveform generation and numerical methods below, providing clarification on our approach while agreeing to enhance the manuscript's detail.

read point-by-point responses

  1. Referee: The central results depend on numerical kludge waveforms. Standard kludge methods combine geodesic motion with approximate wave generation and radiation reaction often tuned to Schwarzschild. For the novel metric with Minkowski core, the geometry near the center and effective potential differ, affecting orbital frequencies and multipoles. The manuscript does not detail the adaptation of the kludge (e.g., re-derivation of source terms or validation against the new geodesic equations), which risks uncontrolled errors in the reported phase shifts and faithfulness. This is a load-bearing issue for the claim that the effects are due to quantum gravity parameter α0.

    Authors: We appreciate this substantive point regarding the need for explicit documentation of our numerical implementation. In our work, the kludge waveforms are constructed by first solving the geodesic equations derived directly from the novel regular black hole metric, which incorporates the deviation parameter α0 into the effective potential and orbital frequencies. The particle trajectories are obtained via numerical integration of these metric-specific equations rather than Schwarzschild-tuned ones. Wave generation proceeds via the quadrupole formula, with multipole moments evaluated along the computed trajectory in the modified geometry; radiation reaction is included through energy and angular momentum loss rates computed consistently from the metric's Killing vectors and conserved quantities. We performed internal validations by cross-checking orbital periods and zoom-whirl frequencies against the analytic effective potential for varying α0. That said, we acknowledge that the manuscript would benefit from expanded exposition of these adaptations and validation steps to reduce any ambiguity. We will revise the methods section accordingly in the next version. revision: yes

Circularity Check

0 steps flagged

No circularity: results from direct numerical integration of orbits and kludge waveforms

full rationale

The paper classifies periodic orbits via rational q and integrates geodesics in the novel regular BH metric with parameter α0, then generates numerical kludge waveforms and computes faithfulness distances to Schwarzschild. These steps are computational outputs from the metric and radiation-reaction model rather than any reduction by definition, fitted-parameter renaming, or load-bearing self-citation. The central distinguishability claims follow from the simulated phase shifts and amplitude modulations; no equation or result is shown to equal its own input by construction. The derivation chain remains self-contained against the stated numerical procedure.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The central claim depends on the validity of the regular black hole metric as a quantum-gravity-inspired background and on the accuracy of the kludge waveform approximation for capturing phase and amplitude effects.

free parameters (1)
  • α0
    Deviation parameter introduced to control the strength of quantum gravity corrections in the metric; its specific values are chosen to explore deviations from Schwarzschild.
axioms (2)
  • domain assumption The spacetime is described by the novel regular black hole metric featuring a Minkowski core.
    This metric is taken as the fixed background for all orbit and waveform calculations.
  • domain assumption The kludge approximation provides a sufficiently accurate representation of gravitational waveforms for the purpose of detecting phase shifts.
    Invoked to generate the reported waveforms without full numerical relativity.
invented entities (1)
  • Regular black hole with Minkowski core no independent evidence
    purpose: To replace the central singularity with a flat core motivated by quantum gravity.
    Postulated as the novel spacetime geometry under study.

pith-pipeline@v0.9.0 · 5659 in / 1479 out tokens · 56095 ms · 2026-05-18T12:32:33.572684+00:00 · methodology

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Forward citations

Cited by 7 Pith papers

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  3. Periodic orbits and their gravitational wave radiations in $\gamma$-metric

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    Deviations from γ=1 in the Zipoy-Voorhees metric shift the (z,w,v) classification of periodic orbits and induce phase shifts plus amplitude modulations in their gravitational-wave signals.

  4. Probing Gravitational Wave Signatures from Periodic Orbits of Regular Black Holes in Asymptotically Safe Gravity

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  5. Gravitational waves of extreme-mass-ratio inspirals in a rotating black hole with Dehnen dark matter halo

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  6. Assessing EMRI Detectability of the Rotating Quantum Oppenheimer-Snyder Black Hole

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  7. Taxonomy of periodic orbits and gravitational waves in a non-rotating Destounis-Suvorov-Kokkotas black hole spacetime

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