Long-lived quasinormal modes, shadows and particle motion in four-dimensional quasi-topological gravity
Pith reviewed 2026-05-15 13:11 UTC · model grok-4.3
The pith
Massive scalar fields around quasi-topological regular black holes approach long-lived modes with rising mass.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
As the mass of the scalar field increases, the damping rate of its quasinormal modes around regular black holes in four-dimensional quasi-topological gravity decreases substantially, indicating the approach to the quasi-resonant regime of long-lived modes, while photon motion and circular geodesics exhibit only moderate deviations from Schwarzschild values.
What carries the argument
Regular black hole spacetime in four-dimensional quasi-topological gravity, supporting both massive scalar field perturbations computed via WKB and time-domain methods, and analysis of photon spheres and geodesics.
If this is right
- As the mass increases, the damping rate decreases substantially, indicating the approach to the quasi-resonant regime of long-lived modes.
- For sufficiently large masses, the late-time signal becomes dominated by oscillatory power-law tails.
- Photon-sphere radius, shadow size, Lyapunov exponent, and ISCO characteristics show only moderate deviations from Schwarzschild.
- Hawking temperature of the black hole exhibits larger deviations compared to the geometric quantities.
Where Pith is reading between the lines
- Long-lived modes in these spacetimes might lead to prolonged ringdown signals in potential gravitational wave detections.
- Moderate shadow deviations imply that very high-resolution imaging would be needed to distinguish these black holes from standard ones.
- Power-law tails at high masses suggest that quasi-resonant modes could be observable only in a narrow mass window for the scalar field.
Load-bearing premise
The high-order WKB approximation with Padé resummation remains accurate for large scalar masses even as time-domain profiles indicate power-law tails beginning to dominate the signal.
What would settle it
Direct comparison of WKB-predicted quasinormal frequencies with full time-domain evolution for scalar masses where the damping is predicted to be very small, to see if the mode is indeed visible before tails overwhelm it.
Figures
read the original abstract
We investigate massive scalar perturbations and several characteristics of particle motion in the spacetime of regular black holes arising in four-dimensional quasi-topological gravity. Quasinormal modes are computed using high-order WKB approximations with Pad\'e resummation and verified through time-domain integration. For moderate values of the scalar-field mass, the time-domain profiles confirm the WKB results with excellent accuracy. As the mass increases, the damping rate decreases substantially, indicating the approach to the quasi-resonant regime of long-lived modes. For sufficiently large masses, the late-time signal becomes dominated by oscillatory power-law tails, which mask the quasi-resonant mode in the time-domain profile. In addition, we analyze photon motion and circular geodesics, including the photon-sphere radius, shadow size, Lyapunov exponent, and ISCO characteristics. These quantities exhibit only moderate deviations from their Schwarzschild values, unlike the Hawking temperature of the black hole.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims to study massive scalar quasinormal modes in the spacetime of regular black holes in four-dimensional quasi-topological gravity. It employs high-order WKB approximations with Padé resummation, verified by time-domain integration for moderate scalar masses, showing that the damping rate decreases substantially with increasing mass, approaching long-lived quasi-resonant modes. For large masses, power-law tails dominate the late-time signal. It further examines photon motion and circular geodesics, reporting moderate deviations from Schwarzschild in shadow size, Lyapunov exponent, and ISCO, contrasting with the Hawking temperature.
Significance. If the results hold, particularly the approach to long-lived modes with increasing mass, this would highlight interesting perturbative behavior in quasi-topological gravity black holes, potentially relevant for stability and observational signatures. The geodesic analysis provides a solid comparative study. The cross-verification for moderate masses is a positive aspect, though the large-mass regime requires additional validation to fully support the claims.
major comments (1)
- [Quasinormal modes computation and time-domain integration sections] The central claim regarding the substantial decrease in damping rate and approach to long-lived modes for large scalar masses rests on WKB results, but the time-domain verification is only provided for moderate masses. Given that the paper acknowledges power-law tails dominating for large masses, masking the mode, this extrapolation is load-bearing and would benefit from an independent method like continued-fraction expansion to confirm accuracy in that regime.
minor comments (2)
- [Abstract] The abstract lacks explicit metric functions, specific coupling values, or error estimates, making the central claims harder to verify without the full text.
- [Results presentation] Clarify the precise mass ranges where time-domain and WKB agree versus where tails dominate, perhaps with a dedicated table of damping rates.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the constructive comments. We appreciate the positive assessment of the cross-verification for moderate masses and the geodesic analysis. We address the major comment below.
read point-by-point responses
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Referee: [Quasinormal modes computation and time-domain integration sections] The central claim regarding the substantial decrease in damping rate and approach to long-lived modes for large scalar masses rests on WKB results, but the time-domain verification is only provided for moderate masses. Given that the paper acknowledges power-law tails dominating for large masses, masking the mode, this extrapolation is load-bearing and would benefit from an independent method like continued-fraction expansion to confirm accuracy in that regime.
Authors: We agree that the time-domain verification is necessarily limited to moderate masses, as the power-law tails dominate and mask the quasinormal ringing for large scalar masses; this is a physical feature of massive perturbations rather than a numerical limitation. The high-order WKB approximation with Padé resummation is well-established for capturing the fundamental mode in this regime, and our results are consistent with the expected suppression of the damping rate as the mass increases. While continued-fraction expansion would provide valuable independent confirmation, its implementation for the quasi-topological metric involves deriving and solving a non-standard recurrence relation, which is computationally intensive and beyond the scope of the current work. In the revised manuscript we will add an expanded discussion of the WKB method's reliability for large masses, including comparisons with lower-order approximations and the expected asymptotic behavior, to better support the extrapolation. revision: partial
Circularity Check
No circularity: standard WKB and time-domain methods applied directly to given spacetime
full rationale
The paper computes quasinormal modes via high-order WKB with Padé resummation and verifies via time-domain integration on the quasi-topological black hole background. Moderate-mass agreement is shown explicitly; large-mass damping decrease is reported from the same WKB procedure with explicit caveat on power-law tails. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing self-citations appear, and no ansatz or uniqueness result is smuggled in. The derivation chain is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption The four-dimensional quasi-topological gravity black-hole solution is regular and satisfies the field equations for the chosen coupling parameters.
- standard math Linear scalar perturbations around this background obey the standard wave equation with an effective potential determined by the metric.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Quasinormal modes are computed using high-order WKB approximations with Padé resummation and verified through time-domain integration... effective potential V(r) = f(r)(ℓ(ℓ+1)/r² + f'(r)/r + μ²)
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking echoes?
echoesECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.
ds² = −f(r)dt² + dr²/f(r) + r²dΩ² (standard 4D spherically symmetric ansatz)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Forward citations
Cited by 10 Pith papers
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Reference graph
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7 0 . 647280 − 0. 032298i 0. 647238 − 0. 032325i 0. 00761 TABLE I. Quasinormal modes of the scalar perturbations of the regular black-hole I ( M = 1 ) calculated using the WKB method at different orders (8th and 10th orders) with Padé approximants for 4l4 = 9. 4 and n = 0. time-domain integration. For moderate values of the scalar-field mass the time-domain...
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5 1 . 385769 − 0. 012815i 1. 387581 − 0. 010364i 0. 220 TABLE II. Fundamental (n = 0) QNMs of the scalar potential for the regular black-hole II with M = 1 and l = 0. 51. Results are obtained using WKB formulas with Padé approximants at different orders. The last column shows the relative deviati on between the two WKB orders. 8 4l4 r0 TH rm Rs λ Ω ISCO BE
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discussion (0)
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