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This paper computes the leading shifts to black-hole quasinormal-mode frequencies for the slowly rotating Johannsen-Psaltis metric and shows that the metric splits even- and odd-parity tones while preserving definite-parity modes.

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2026-08-03 15:58 UTC pith:YL7GIVQV

load-bearing objection Solid, honest EVP computation of JP QNM shifts — useful as metric-only results, but the Cowling approximation and the 'all spins' phrasing mean the abstract oversells the physics. the 2 major comments →

arxiv 2512.14679 v2 pith:YL7GIVQV submitted 2025-12-16 gr-qc

Computing spectral shifts for Johannsen-Psaltis black holes

David G. Wu , Asad Hussain , Aaron Zimmerman This is my paper
classification gr-qc
keywords quasinormal modesJohannsen-Psaltis metricblack hole spectroscopymodified Teukolsky formalismeigenvalue perturbation methodisospectrality breakingparity modesslow rotation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Using a modified Teukolsky equation and eigenvalue perturbation theory, this paper computes the first-order shifts to the quasinormal-mode frequencies of slowly rotating black holes in the Johannsen–Psaltis parametrized metric, for multipoles ℓ=2 through 10, all azimuthal numbers m, and overtones n=0,1,2. It finds that the JP metric splits the otherwise degenerate even-parity and odd-parity tones at every spin, breaking the isospectrality of Kerr while keeping the modes in definite-parity pairs. The shifts obey a simple linear-in-m form, ω^(1) = δω0 + χ m δω1, so the full mode table is captured by two spin-independent quantities per (ℓ,n). The paper also states a general parity condition a beyond-GR metric must meet to admit definite-parity modes. This matters because it supplies a systematic beyond-GR ringdown spectrum across a wide mode range, providing templates for black-hole spectroscopy tests.

Core claim

In the Cowling approximation, which neglects the unknown field equations that would source the JP metric, the slowly rotating Johannsen–Psaltis perturbation has definite 'upwards' parity, making the perturbed Teukolsky operator parity-preserving. Consequently the quasinormal modes separate into even- and odd-parity sectors, and the quadratic mixing equation for the two degenerate Kerr modes yields γ=±1 with two distinct frequency roots: the JP metric breaks isospectrality at all spins. The computed shifts take the form ω^(1)_ℓmn = (δω0)_ℓn + χ m (δω1)_ℓn, are tabulated to 10^-4 relative tolerance for 2≤ℓ≤10 and n=0,1,2, and converge to the eikonal/WKB limit at large ℓ, with deviations appear

What carries the argument

The central machinery is the modified Teukolsky equation, whose effective source is built from the second-order Einstein tensor acting on the JP metric perturbation and the reconstructed gravitational perturbation, combined with the eigenvalue perturbation (EVP) method. A contour-integral scalar product in the complex radial plane makes the Teukolsky operator self-adjoint, allowing first-order frequency shifts to be projected out without solving for the first-order wavefunction. The parity analysis classifies metric perturbations as 'upward' or 'downward' under a conjugate-parity operator; because the JP perturbation is entirely upward, the source operator is parity-preserving and definite-p

Load-bearing premise

The computed shifts rely on the Cowling approximation: the unknown field equations that source the JP metric are assumed not to change the ringdown frequencies, and the paper notes this is worst for the low-order modes most relevant to observations.

What would settle it

Choose any concrete theory that admits the JP metric as an exact vacuum solution, add the dropped C-operator term to the modified Teukolsky equation, and recompute ω^(1)_220; a shift differing by more than the stated ~10^-4 tolerance would show the tabulated spectrum is incomplete. Simpler: a numerical time-domain ringdown of a JP black hole at χ≈0.1 that violates the linear-in-m relation δω0 + χ m δω1 would falsify the empirical decomposition.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The tabulated shifts give a ready-made beyond-GR ringdown template: for each ℓ and n, the even- and odd-parity tones are separate, and for slow spins the m-dependence is fully captured by δω0 and δω1.
  • Isospectrality breaking becomes an observable discriminator: in GR even- and odd-parity modes are degenerate, so resolving the split at any spin would signal a JP-like deviation from general relativity.
  • The general parity condition provides a quick diagnostic for other parametrized metrics: a beyond-GR perturbation of purely upwards parity will admit definite-parity modes, while a downwards-parity component can survive only if it does not couple to gravitational radiation.
  • Comparison with scalar WKB shows the large-ℓ modes converge to the eikonal limit, so high-ℓ predictions are reliable, whereas low-ℓ overtones deviate—an effect relevant for choosing which modes to use in ringdown tests.
  • At current detector sensitivity the ~1–5% shifts are not resolvable, needing signal-to-noise ratio ≳150, so the near-term value is as a foundation for next-generation spectroscopy rather than an immediate observable.

Where Pith is reading between the lines

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

  • The linear-in-m form was found numerically and its analytic origin is left open; if it holds generally, future beyond-GR ringdown calculations could be reduced to computing one m per mode, making full mode tables dramatically cheaper.
  • The parity condition is derived in the Cowling approximation; in theories with extra fields coupled to curvature the same condition is likely necessary but not sufficient, so checking one concrete completed theory would show how much of the definite-parity result survives.
  • The ℓ=2 and ℓ=3 overtones that break the monotonic large-ℓ trends may encode near-horizon structure rather than photon-sphere geometry; comparing them across other bumpy metrics could reveal whether the anomaly is a JP feature or a generic low-ℓ effect.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper applies the modified Teukolsky formalism and eigenvalue perturbation (EVP) method to compute leading-order, slow-rotation quasinormal-mode (QNM) frequency shifts for Johannsen–Psaltis (JP) black holes, working at linear order in the deformation parameter ϵ and in spin χ. The authors report results for 2≤ℓ≤10, all m, and overtones n=0,1,2, tabulated in the form ω^(1)_{ℓmn} = (δω0)_{ℓn} + χm(δω1)_{ℓn}. They argue that the JP metric perturbation admits definite-parity modes but breaks isospectrality, and they compare large-ℓ results with a first-order scalar WKB approximation. The paper also derives a general condition on metric perturbations for the existence of definite-parity modes. The central limitation, acknowledged in the body, is that the unknown field equations underlying the JP metric are omitted, so the calculation includes only the metric-sourced 'V' term and not the additional 'C-operator' contributions.

Significance. If interpreted as a computation of the metric-sourced contribution to JP QNM shifts, this is a useful and nontrivial result. It is, to my knowledge, the first application of the modified Teukolsky EVP machinery to ℓ>5, and it provides a substantial tabulated dataset with a stated numerical tolerance, together with a WKB cross-check for the eikonal regime. The analytic parity condition based on the ↑/↓ decomposition is a helpful organizing principle for future beyond-GR ringdown calculations. The main caveat—that the unknown JP field equations are neglected—is clearly stated in the body, but the abstract and parts of the introduction present the results as complete spectral predictions for JP black holes without this qualification. The paper is transparent about its approximations, which strengthens its credibility, but the headline claims need to be reframed.

major comments (2)
  1. [Abstract and Sec. III (Eq. 11)] The central claim that the paper computes 'the spectral shifts of slowly rotating Johannsen–Psaltis black holes' is stronger than what the formalism actually delivers. As the authors state in Sec. III, 'because these field equations are unknown, they cannot be directly incorporated into our analysis.' The modified Teukolsky equation (11) therefore contains only the V term, not the C-operator contributions that would arise from the underlying JP field equations. The paper's own conclusion reiterates that terms involving the underlying field equations are neglected and that the Cowling approximation 'is worst for the low-order modes most relevant to current BH spectroscopic tests.' Consequently, Table II and the parity/isospectrality conclusions are metric-only results under an uncontrolled approximation. The abstract and the introductory summary should explicitly say 'metric-sourced contr
  2. [Abstract and Secs. IV–V] The abstract states that JP black holes 'break the isospectrality between even- and odd-parity QNMs at all spins,' but the explicit computation and the isospectrality analysis in Sec. IV B are limited to the slowly rotating, linear-in-spin regime. Section V explicitly says the computation is linearized in spin, with the reliability estimate 'error ≲1% for χ≲0.2.' The analytic parity argument for the full JP metric is plausible, but the isospectrality-breaking claim is only numerically demonstrated for small χ. The phrase 'at all spins' is therefore an extrapolation, not a result of this paper. Please revise the abstract to state the slow-rotation scope and avoid implying a all-spin proof.
minor comments (6)
  1. [Sec. II, after Eq. (4)] Typo: 'away form Kerr' should be 'away from Kerr'.
  2. [Sec. IV C, first paragraph] Typo: 'Schwarschild' should be 'Schwarzschild'.
  3. [Fig. 2 caption] The phrase 'the real part of the WKB approximation has nondependence' is unclear; presumably the real part has no m dependence, but the sentence should be completed.
  4. [Reference [69]] The arXiv identifier for Ashton et al. (Bilby) appears to be wrong: it is listed as arXiv:2509.08099, which is the same as reference [2] (GW250114). The correct arXiv number for Bilby is 1811.02042.
  5. [Sec. V and Table II] The empirical linear-in-m decomposition (Eq. 72) is presented without a quantitative statement of how many m values were checked or the maximum deviation from linearity. Since Table II encodes this decomposition, a brief statement of the numerical verification would be helpful.
  6. [Sec. V, numerical tolerance paragraph] The sentence 'All QNMs are computed with a relative tolerance of O(10^-4), ensuring that systematic O(a^2) error quickly becomes dominant' is misleading: the numerical tolerance does not 'ensure' the size of the physical O(a^2) truncation error. Please rephrase to distinguish numerical error from the spin-expansion truncation error.

Circularity Check

0 steps flagged

No significant circularity: the JP QNM shifts are newly computed EVP results, not fits or renamed inputs; self-citations supply formalism but not the tabulated outcomes.

full rationale

The paper's central deliverable, the spectral shifts ω^(1)_{ℓmn} = (δω0)_{ℓn}+χm(δω1)_{ℓn} for the slowly rotating JP metric, is obtained by solving Eqs. (39)–(52) with the EVP method and complex-contour scalar products. The input data are the background Kerr QNM wavefunctions and the given JP metric perturbation; the output numbers in Table II are new numerical evaluations and are not used to define the inputs. The WKB comparison in Sec. IV A and the eikonal consistency check in Sec. V are independent cross-checks, not sources of the tabulated values. The parity and isospectrality conclusions follow analytically from the parity classification of the JP metric perturbation and the theorem of Ref. [46]; this is an application of a prior published result, not a reduction of the paper's own output to its own input. The paper relies on self-citations [43,44,46] for the modified Teukolsky formalism and the conjugate-parity criterion, and these are substantial and load-bearing in the sense of providing the framework; however, they are peer-reviewed general formalisms and the present calculation's results are not contained in them. The Cowling approximation and neglect of the unknown C-operator contributions are explicitly disclosed limitations that affect the physical interpretation of the shifts, but they are not equation-level circularity. The observed linear-in-m behavior is presented as an empirical finding rather than as an imposed fit, and its analytic origin is left open, so it is not a fitted input disguised as a prediction. Overall, no circular step exhibiting X-defined-in-terms-of-Y or fit-called-prediction can be identified; the appropriate finding is no significant circularity, with a low score reflecting only the presence of normal self-citations in the framework.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

The central numbers rest on the JP metric as a testbed, on the modified Teukolsky formalism from prior work, on the Cowling approximation for unknown field equations, and on convergence/parity conventions. There are no fitted free parameters and no new physical entities introduced.

axioms (5)
  • domain assumption The JP metric is treated as a fixed background whose unknown field equations do not contribute to the QNM shift (Cowling approximation).
    Invoked in Sec. III to justify omitting the C-operator terms; the paper notes this is heuristic and worst for low-order modes.
  • domain assumption The modified Teukolsky equation with source V[h^(0)] = S_4^{ab} G^{(2)}_{ab}[h^{(0)}, g^{(1)}] correctly gives the leading-order shift.
    Adopted from Refs. [43,44]; the paper does not rederive the master equation and assumes its validity for the JP metric.
  • domain assumption The parity table for GHP quantities and the ↑/↓ parity classification of S_4 and G^{(2)} are correct.
    Sec. IV D; based on [46] and the Kinnersley tetrad conventions; an error here would invalidate the definite-parity condition.
  • domain assumption The consistency condition ω_+^(1) = −ω_−^(1) used to solve for γ.
    Sec. III C; assumes the plus/minus frequency symmetry of the Kerr background persists to first order in ϵ.
  • domain assumption The radial series and contour integrals converge with the chosen truncations and phase conventions.
    Appendix A; required for the numerical evaluation of the scalar products.

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read the original abstract

The growing number of gravitational wave (GW) detections and the increasing sensitivity of GW detectors have enabled precision tests of General Relativity (GR) in the strong-field regime. The recent observation of multiple quasinormal modes (QNMs) in GW250114 marks a major advance for observational black hole spectroscopy. This clear signal, together with the growing number of GW detections, highlights the need for accurate predictions of QNM spectra in beyond-GR theories in order to carry out precision searches for new physics. In this work, we continue to lay the foundation for such predictions using a modified Teukolsky formalism in conjunction with the eigenvalue perturbation method. We compute the spectral shifts of slowly rotating Johannsen-Psaltis black holes for $2 \leq \ell \leq 10$, all $m$, and overtones $n = 0, 1, 2$, and confirm the large-$\ell$ behavior of the modes by comparing with the WKB approximation. We find that these black holes admit definite-parity modes but break the isospectrality between even- and odd-parity QNMs at all spins, and that the shifts depend linearly on $m$ for slow spins. We further derive a general parity condition that any beyond-GR modification to the metric must satisfy to support definite-parity modes, providing new insights into isospectrality breaking and parity structure in gravitational perturbations.

Figures

Figures reproduced from arXiv: 2512.14679 by Aaron Zimmerman, Asad Hussain, David G. Wu.

Figure 1
Figure 1. Figure 1: FIG. 1. The radial integration contours in the complex plane. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The real (left column) and imaginary (right column) QNM shifts of ( [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The real (left column) and imaginary (right column) QNM shifts of ( [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Normalized differences between the real (left) and imaginary (right) parts of gravitational modes from the EVP method [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

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Cited by 1 Pith paper

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    A bilinear-form framework computes black-hole quasinormal-mode frequency shifts to any order, but the mode-sum expansion of the first-order mode shift diverges and needs a continuum piece.

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