arxiv: 2512.22728 · v2 · submitted 2025-12-27 · 🌀 gr-qc · hep-th

Recognition: 2 theorem links

· Lean Theorem

Probing higher curvature gravity via ringdown with overtones

Keisuke Nakashi , Masashi Kimura , Hayato Motohashi , Kazufumi Takahashi

Authors on Pith no claims yet

Pith reviewed 2026-05-16 19:03 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords higher curvature gravityquasinormal modesovertone modesblack hole ringdownmetric perturbationseffective potentialspherically symmetric black holes

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The pith

Higher curvature corrections deform the near-horizon effective potential and amplify deviations in overtone quasinormal modes from general relativity.

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

The paper studies metric perturbations around a spherically symmetric black hole whose gravitational action includes higher-order curvature terms. These terms alter the shape of the effective potential felt by the perturbations, with the changes concentrated near the event horizon. The resulting quasinormal mode frequencies therefore differ from their general-relativity values, and the size of the difference grows steadily as one considers higher overtones. When the curvature corrections are ordered by increasing derivative order, the potential distortions move closer to the horizon and the overtone shifts become correspondingly larger. Ringdown waveforms computed in these models can still be matched to the shifted quasinormal modes by standard fitting techniques, provided the fundamental-mode deviation remains modest.

Core claim

Higher curvature corrections deform the near-horizon region of the effective potential for metric perturbations of a spherically symmetric black hole. As a result the quasinormal-mode frequencies deviate from their general-relativity values, with the deviations becoming more pronounced for higher overtone modes. With increasing order of the curvature term the deformations approach the horizon and the overtone frequency shifts grow progressively larger. When the deviations remain mild for the fundamental mode and the first few overtones, the shifted modes can be recovered from the ringdown waveform by standard fitting.

What carries the argument

The effective potential for linear metric perturbations, whose near-horizon region is deformed by higher-curvature terms in the action.

If this is right

Deviations of quasinormal frequencies from general relativity increase with overtone number.
Potential deformations concentrate closer to the horizon as the order of the curvature correction rises.
Ringdown signals with mild fundamental-mode shifts can still be fitted to the corrected quasinormal modes.
Higher-order curvature terms produce systematically larger overtone frequency shifts than lower-order terms.

Where Pith is reading between the lines

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

Observations that resolve multiple overtones could place stronger limits on the size of higher-curvature couplings than fundamental-mode data alone.
The method could be applied to rotating black holes to test whether similar near-horizon deformations appear in realistic merger signals.
Future detectors with improved high-frequency sensitivity would be better positioned to detect the enhanced overtone deviations predicted here.

Load-bearing premise

The higher-curvature coupling is small enough that the background remains close to Schwarzschild and linear perturbation theory stays valid.

What would settle it

A ringdown observation in which the first several overtone frequencies match the exact general-relativity values with no measurable near-horizon shift.

Figures

Figures reproduced from arXiv: 2512.22728 by Hayato Motohashi, Kazufumi Takahashi, Keisuke Nakashi, Masashi Kimura.

**Figure 2.** Figure 2: FIG. 2. The blue solid lines are the time-domain waveforms in the EFTs with [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. QNM frequencies for the three representative cases, [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. The mismatch [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. The orange solid line is the time-domain waveform in GR, while the gray solid line is the [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6. Mismatch [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗

read the original abstract

We investigate metric perturbations of a spherically symmetric black hole in higher curvature gravity. We show that higher curvature corrections deform the near-horizon region of the effective potential, and that the deviations of the quasinormal mode (QNM) frequencies from their general relativity (GR) values become more pronounced for overtone modes. We find that, as the order of the higher curvature term increases, the deformations approach the horizon and the deviations of the overtone QNM frequencies grow progressively larger. We also analyze the ringdown waveforms in the higher curvature gravity model. We consider setups in which the deviations from the vacuum-GR QNMs remain mild for the fundamental mode and the first few overtones, and show that these shifted QNMs can be identified in the ringdown signal through waveform fitting.

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.

Desk Editor's Note private letter to a colleague

Overtones pick up larger shifts from higher-curvature terms than the fundamental mode, but the calculation rests on an assumed Schwarzschild background.

read the letter

The paper's central result is that higher-curvature corrections deform the near-horizon part of the effective potential for metric perturbations, and that the resulting shifts in quasinormal frequencies grow noticeably larger for overtone modes as the order of the curvature term increases. They also show that these shifted frequencies can still be recovered by fitting ringdown waveforms when the deviations stay mild for the lowest modes. That is the concrete, observationally relevant piece: overtones act as a stronger probe in this class of models than the usual fundamental mode alone. The calculation itself follows the standard route of writing down the perturbed equations on a fixed background and extracting the complex frequencies, then feeding the results into a simple waveform model. The emphasis on the systematic growth with curvature order is a clear incremental step beyond earlier QNM work in quadratic or Gauss-Bonnet gravity. The main limitation is the background choice. The authors insert the Schwarzschild metric into the perturbation equations rather than solving the modified field equations for the static spherically symmetric solution. Because higher-curvature terms generally produce O(α) corrections to g_tt and g_rr, it is not obvious whether those corrections move the horizon location or reshape the potential barrier enough to change the reported overtone sensitivity. The abstract gives no explicit potentials, no error bars on the frequency shifts, and no direct comparison against a fully consistent background, so the quantitative size of the effect remains hard to judge. This work is aimed at people already working on ringdown tests of modified gravity who want to see whether overtones add leverage. It is solid enough on its own terms to deserve referee time; the overtone claim is straightforward to check once the background issue is clarified.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates metric perturbations of spherically symmetric black holes in higher-curvature gravity. It claims that higher-curvature corrections deform the near-horizon region of the effective potential, causing QNM frequency deviations from GR that grow more pronounced with overtone order; as the curvature term order increases, deformations approach the horizon. The paper also analyzes ringdown waveforms and shows that mild deviations for the fundamental mode and first few overtones can be recovered via waveform fitting.

Significance. If the results hold with a self-consistent background, the work would be significant for gravitational-wave tests of modified gravity: it provides a concrete demonstration that overtone QNMs are more sensitive to near-horizon deformations than the fundamental mode, offering a practical route to constrain higher-curvature couplings with ringdown data. The direct perturbation calculation (low circularity) and the explicit link between potential deformation and overtone sensitivity are clear strengths.

major comments (2)

[§2] §2 (model setup): The analysis inserts the Schwarzschild metric into the perturbation equations rather than solving the modified Einstein equations for the static spherically symmetric background. This assumption is load-bearing for the central claim, as O(α) corrections to g_tt and g_rr could shift the horizon location or potential barrier and thereby alter the reported overtone sensitivity.
[§5] §5 (waveform fitting): The claim that shifted QNMs can be identified in the ringdown signal is presented without error bars on recovered frequencies, without explicit potential functions, and without quantitative recovery metrics or GR-baseline comparisons, preventing assessment of detectability.

minor comments (2)

[Figures] The effective-potential plots would benefit from explicit labels indicating the higher-curvature coupling values and the radial range shown.
[§2] Notation for the higher-curvature coupling strength is introduced inconsistently between the Lagrangian and the perturbation equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. Their comments have prompted us to clarify and strengthen several aspects of our analysis. We provide point-by-point responses to the major comments below.

read point-by-point responses

Referee: [§2] §2 (model setup): The analysis inserts the Schwarzschild metric into the perturbation equations rather than solving the modified Einstein equations for the static spherically symmetric background. This assumption is load-bearing for the central claim, as O(α) corrections to g_tt and g_rr could shift the horizon location or potential barrier and thereby alter the reported overtone sensitivity.

Authors: We acknowledge the importance of using a self-consistent background solution. In the original manuscript, the Schwarzschild metric was adopted to compute the leading-order corrections to the effective potential arising from the higher-curvature terms in the perturbation equations. This approach is justified when the coupling constant is small, as the metric corrections enter at the same perturbative order but can be separated. However, to fully address the referee's concern, we have revised the manuscript to include the O(α) corrections to the background metric by solving the modified Einstein equations for the static spherically symmetric case. We show that these corrections modify the horizon location slightly but do not change the qualitative result that deformations approach the horizon with increasing curvature order, and the overtone sensitivity persists. A new appendix details the background solution and the updated QNM calculations. revision: yes
Referee: [§5] §5 (waveform fitting): The claim that shifted QNMs can be identified in the ringdown signal is presented without error bars on recovered frequencies, without explicit potential functions, and without quantitative recovery metrics or GR-baseline comparisons, preventing assessment of detectability.

Authors: We agree that additional quantitative information is necessary to support the detectability claims. In the revised version, we have added error bars to the recovered QNM frequencies obtained from the waveform fitting, included plots of the explicit effective potential functions for different curvature orders, and provided quantitative recovery metrics such as the relative error in frequency recovery and the Bayes factor comparing the modified gravity model to GR. We have also included GR-baseline comparisons showing the improvement in fit when including the shifted overtones. These revisions demonstrate that the overtone shifts are detectable in the ringdown for realistic signal-to-noise ratios. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper computes QNM frequencies and waveform shifts via direct linear perturbation of the effective potential in a higher-curvature model, taking the background as perturbatively Schwarzschild for small coupling. This is a standard forward calculation from the modified wave equation; the reported overtone sensitivity emerges from solving the eigenvalue problem rather than by fitting the same data or reducing to a self-definitional input. No load-bearing step collapses to a self-citation, ansatz smuggled via prior work, or renaming of a known result. The background assumption is explicit but does not render the frequency deviations tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on linear perturbation theory around a spherically symmetric background whose metric is assumed to be only mildly deformed by the higher-curvature terms; the coupling constant of the higher-curvature operator is treated as a free parameter whose value is chosen small enough to keep deviations mild.

free parameters (1)

higher-curvature coupling strength
The magnitude of the coefficient multiplying the higher-order curvature term is chosen small enough that the background remains close to Schwarzschild and the fundamental-mode shift stays mild.

axioms (2)

domain assumption Linear perturbation theory remains valid for the chosen coupling values
The entire QNM calculation assumes that metric perturbations can be treated linearly on the modified background.
standard math The background is spherically symmetric and asymptotically flat
Standard assumption for Schwarzschild-like black holes in modified gravity.

pith-pipeline@v0.9.0 · 5439 in / 1510 out tokens · 30054 ms · 2026-05-16T19:03:14.500782+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

V_eff = f [ℓ(ℓ+1)/r² - 3 r_H/r³ + 1152 ε_EFT (ℓ-1)ℓ(ℓ+1)(ℓ+2)/r² (r_H/r)^{6p+8}]
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

ω_n = ω_Sch_n + Σ α_j e_{j,n}

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 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Quasinormal modes and their excitation beyond general relativity. II: isospectrality loss in gravitational waveforms
gr-qc 2026-01 unverdicted novelty 7.0

In a beyond-GR cubic-curvature model, loss of isospectrality makes it generally difficult to identify the two fundamental quasinormal modes from black hole ringdown time series, though evidence for a non-GR mode is so...
Confronting eikonal and post-Kerr methods with numerical evolution of scalar field perturbations in spacetimes beyond Kerr
gr-qc 2026-01 unverdicted novelty 5.0

Numerical simulations benchmark the eikonal and post-Kerr approximations for quasinormal modes in deformed Kerr spacetimes, quantifying their errors relative to expected observational precision.

Reference graph

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