{"id":"35c280a9-4aac-4b74-a353-361cf23d0681","arxiv_id":"2412.08205","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For slowly rotating Johannsen black holes, the deformation parameter alpha_13 dominates the quasi-normal mode shifts, and ringdown data from GW170104 bound it to be consistent with zero.","lead":"This paper computes the ringdown frequencies of a parametrically deformed black hole metric and reports that one deformation parameter, alpha_13, shifts them more than the other leading parameters. It uses the gravitational wave event GW170104 to place a weak constraint on alpha_13, which is consistent with general relativity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The frozen effective-source assumption in Sec. III imposes a linearized-Bianchi constraint on a non-Einstein background that the paper never verifies; if it fails, the computed QNMs are not linearized Einstein-matter perturbations and the GW170104 constraint inherits the inconsistency.","rationale":"The reader's weakest-assumption identification is exactly the right target: the paper's 'no perturbations on the right hand side' is doing enormous work, because the Johannsen metric is not a vacuum solution. My stress-test sharpens this from a question of physical plausibility to a concrete consistency condition. For a non-Einstein background, the linearized Bianchi identity is not simply the divergence of δG; it contains terms in which the perturbed connection contracts with the background Einstein tensor. If δG=0, those terms must vanish, which is an additional constraint on the metric perturbation. The paper derives a single master equation from the linearized field equations but does not demonstrate that the QNM solutions satisfy this constraint, nor does it specify a matter model whose perturbation vanishes while its covariant divergence remains zero with respect to the perturbed metric. This is not merely an interpretive caveat: if the constraint fails, the modes are not linearized solutions of any Einstein-matter system with fixed source, and the mapping from ringdown measurements to α13 is not supported by the stated assumptions. I do not recommend changing the reader's conditional verdict, because the authors explicitly flag the assumption and the paper is framed as a null test; however, the Bianchi check should be performed before the fitting formula is used for parameter estimation. The numerical infrastructure already exists, so the test is inexpensive and decisive.","tokens_in":19856,"tokens_out":22542,"duration_ms":255739,"concrete_test":"Use the already available numerical solutions for a representative mode (e.g., {l,m,n}={2,2,1} at a=0.2, α13=0.1) and evaluate (i) all components of the linearized Einstein tensor δG_{μν} and (ii) the first-order Bianchi combination ∇_μ δG^{μν} + δΓ·G0 (equivalently δ∇_μ T0^{μν}) on the Johannsen background. If (ii) vanishes to numerical tolerance wherever (i) does, the frozen-source assumption is internally consistent for these modes; if (ii) is nonzero at order α13, the master equation is not equivalent to δG=0 and the quoted QNM frequencies, and therefore the GW170104 α13 constraint, do not follow from the stated Einstein-equations-plus-unperturbed-source setup.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III assumes the Einstein equations but takes the Johannsen background to have a non-vanishing effective stress-energy tensor T0 = G[g0]/8π, then sets δT=0 and solves δG[h]=0. This is not a self-evident approximation: expanding the Bianchi identity ∇_μ G^{μν}=0 to first order gives ∇0_μ δG^{μν} + δΓ^μ_{μρ}G0^{ρν} + δΓ^ν_{μρ}G0^{μρ}=0. When δG=0, the leftover δΓ·G0 terms are an additional constraint on h. The derivation of the Regge-Wheeler/Zerilli master equations (Eq. 12 and Appendix A) neither displays nor enforces this constraint, and no matter model that would keep T0 fixed under the perturbed connection is specified. If the constraint is violated by the direct-integration modes, those modes solve only the reduced master equation and are not linearized solutions of the Einstein-matter system used to justify the constraint on α13. This is the load-bearing weak point: it affects the physical interpretation of every frequency, not just the extrapolated tails.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript computes the quasi-normal modes (QNMs) of gravitational perturbations of slowly rotating Johannsen black holes, a parametrized non-Kerr metric designed for agnostic tests of GR. The authors linearize the Einstein equations about the Johannsen background to first order in the spin a and in the leading deformation parameters alpha_13, alpha_22, alpha_52, and epsilon_3, assume that the effective stress-energy supporting the background has no perturbations, and derive deformed Regge-Wheeler and Zerilli master equations whose effective potentials are given in Eq. (12) and Appendix A. Using the direct integration method, they compute the fundamental l = 2, m = 2 Kerr QNMs as a function of spin, introduce a correction factor calibrated against Teukolsky frequencies (Eqs. 18-19), and find that alpha_13 produces the largest frequency shifts among the four parameters (alpha_22 does not enter at the retained order). They fit the alpha_13 dependence with the linear formula Eq. (20), tabulate coefficients for all m of l = 2 and l = 3, and apply the l = 2, m = 2 formula to the ringdown of GW170104 using pyRing, finding alpha_13 = -0.5 +2.1 -4.0 at 90% CL, consistent with general relativity. The central claims are the alpha_13-dominance result, the fitting formula, and the GW170104 constraint; the central assumptions are the frozen effective-source approximation and the additivity of the Kerr correction factor.","tokens_in":3068,"tokens_out":3276,"duration_ms":404701,"significance":"If the leading-order approximations and the frozen-effective-source assumption are accepted, the paper is a useful and transparent contribution to agnostic strong-field tests of GR: it delivers the first QNM fitting formula for slowly rotating Johannsen black holes, with explicit master equations (Eq. 12 and Appendix A), a calibrated Kerr-Teukolsky comparison (Fig. 1), and a complete demonstration pipeline from theory to a ringdown constraint on alpha_13 using public LVK data and pyRing. The paper practices good epistemic hygiene: the main assumptions are stated in the text (Section III; Concluding Remarks), the Kerr limit is validated separately, and the headline constraint (consistent with GR) is falsifiable. The significance is nevertheless qualified by the correctness risks detailed below: the physical interpretation of the modes relies on an unverified linearized-consistency condition, the correction-factor additivity is untested for non-Kerr shifts, the l = 3 coefficient table contains a factor-of-10 error, and the astronomical constraint extrapolates the fit beyond its calibration domain.","major_comments":[{"comment":"The perturbation scheme rests on the statement in Section III that, within the authors' agnostic approach, 'we assume that there are no perturbations on the right hand side,' i.e., delta-T = 0 and delta-G[h] = 0 on a background with nonvanishing G0. For such a background, the linearized contracted Bianchi identity reads nabla0_mu delta-G^{mu nu} + delta-Gamma^mu_{mu rho} G0^{rho nu} + delta-Gamma^nu_{mu rho} G0^{mu rho} = 0, so a solution of the reduced master equations (12) and (A1) can be a linearized solution of the full Einstein-matter system only if it additionally satisfies delta-Gamma dot G0 = 0. No check of this constraint is reported, and no matter model that would keep the effective T0 fixed under the perturbed connection is specified. Because G0 begins at first order in the deformation parameters, the unverified constraint is at the same order as the QNM shifts reported here. At a = 0 the axial sector of a spherical non-vacuum background does decouple, but the polar sector in general couples to matter perturbations, and the slow-rotation axial/polar separation relies on the same assumption away from spherical symmetry. The computed frequencies are therefore, strictly, frequencies of a reduced scalar equation on a fixed background; their status as gravitational-wave QNMs of an Einstein-matter system, and the interpretation of the Section VI constraint on alpha_13, depends on the frozen-source assumption holding. I recommend either numerically verifying that the direct-integration solutions satisfy the full set of linearized Einstein equations and the Bianchi constraint, or reframing the results as a phenomenological fixed-background computation and stating that limitation prominently in the abstract and conclusions.","section":"Section III, Eq. (12), and Appendix A"},{"comment":"The correction-factor construction omega_corr = omega_Non-Kerr + (C - 1) omega_Kerr is exact by construction in the Kerr limit, so the validation in Fig. 1 tests the Kerr baseline but not the additivity assumption that the slow-rotation error of the Kerr part is independent of the deformation shift. The shift delta-omega_alpha = omega_Non-Kerr - omega_Kerr is computed with the same scheme whose uncorrected Im(omega) deviates from Teukolsky by about 10% at a = 0.3 and 20% at a = 0.4 (Fig. 1), whereas the alpha_13-induced change in Im(omega) is only about 4% of the base value at alpha_13 = 0.3 (Table I). The neglected O(a^2 alpha_13) and O(alpha_13^2) terms in the shift are not removed by the correction factor, so the expectation in Section V.B that the corrected non-Kerr frequencies are more accurate is an unvalidated premise. A concrete check would be a comparison of the corrected l = 2, m = 2 frequencies against one of the Teukolsky-like methods cited in Refs. [60-62] at a few points in the (a, alpha_13) plane; without such a comparison, or a quantitative bound on the neglected terms, the fitting coefficients and the GW170104 constraint carry an unquantified systematic error.","section":"Section V.B, Eqs. (18)-(19)"},{"comment":"Table VII is internally inconsistent with the numerical data in Table VI by a factor of approximately 10 in the P0 column. For example, at a = 0 and m = 0, Table VI gives Re(omega) = 0.5994 at alpha_13 = 0 and 0.5970 at alpha_13 = 0.1, i.e., dRe/dalpha_13 is about -0.024, whereas Table VII lists P0 = -0.0024; the l = 2 tables are mutually consistent (Table IV's P0 approximately -0.0195 for m = -2 reproduces Table III), so the discrepancy is not a convention issue. If Eq. (20) is used with the published l = 3 coefficients, the predicted alpha_13 shift is an order of magnitude too small. The P1, Q0, and Q1 columns of Tables VII and VIII appear consistent with Table VI, so the error seems localized to P0, but as printed the l = 3 real-part fitting formula, which is part of the paper's advertised product for l = 3 fundamental modes, is wrong and must be corrected.","section":"Appendix D, Table VII"},{"comment":"The GW170104 analysis extrapolates the fitting formula beyond its calibrated domain in two ways. First, the posterior final spin a = 0.31 +0.45 -0.28 reaches values up to about 0.76, while the fit is calibrated for a in [0, 0.3] and the abstract claims validity only for a* < 0.4; the paper acknowledges this only with the phrase 'exceeds slightly the range allowed with our methods.' Second, the reported 90% interval alpha_13 in [-4.46, 1.66] extends far outside the extended fit range alpha_13 in [-1, 1], where the O(alpha_13^2) and O(a^2 alpha_13) terms dropped in Eq. (20) are no longer small. The leakage is visible within the paper's own numbers: P1 changes from 0.0025 to 0.0166 when the alpha_13 range is widened from [-0.3, 0.3] to [-1, 1], showing that the coefficients are sensitive to the fitting window. Because the quoted uncertainty on alpha_13 does not include the truncation systematic, the constraint should be reported as an exploratory estimate with an explicit error budget for the extrapolation, or the posterior should be restricted to the calibrated region.","section":"Section VI and Table I"}],"minor_comments":[{"comment":"Typos and wording: 'Zerlini' should be 'Zerilli' in Section I; the Section III heading reads 'MASTER EQUA TIONS'; Section IV.B uses 'DNSolve' instead of 'NDSolve'; Section V.C has 'The errors is not more than 0.2%'; Appendix C has 'relatee' instead of 'relates'.","section":"Throughout"},{"comment":"In the a = 0 row at alpha_13 = 0.2, the m = -2 entry is 0.3798 while the other four m entries are 0.3698 or 0.3699; the 0.3798 value breaks the monotonic trend and appears to be a typo.","section":"Table III"},{"comment":"The abstract states that the fitting formula is valid for a* < 0.4, but Section V.C calibrates the fit for a in [0, 0.3], and Fig. 1 shows the uncorrected imaginary part has about 20% error already at a = 0.4; the claimed validity range should be reconciled with the calibration range.","section":"Abstract and Section V.C"},{"comment":"The abstract's claim that alpha_13 has a stronger impact on the QNMs than alpha_52 and epsilon_3 is supported only by visual comparison of Figs. 2-4; please provide quantitative values (for example, dRe/dalpha and dIm/dalpha at a = 0 for each parameter) so that the headline claim is numerically documented.","section":"Abstract and Section V.B"},{"comment":"The overtone numbering should be clarified: the paper labels the fundamental mode {l, m, 1}, while the LVK 'Kerr 220' convention of Ref. [4] is (l, m, n) = (2, 2, 0), and Section V.C states that Appendix D reports coefficients for the first overtone mode although the tables contain the fundamental-mode data; please align the two conventions.","section":"Section V.C and Appendix D"},{"comment":"Ref. [45] lists arXiv:1501.02809 for T. Johannsen, Phys. Rev. D 88, 044002 (2013); that arXiv identifier dates from January 2015 and appears to be incorrect, so please verify it. Several references (e.g., [3] and [33]) are missing titles or author initials in the bibliography.","section":"References"},{"comment":"The mapping from delta-f and delta-tau to alpha_13 in Eq. (22) neglects O(a^2) and O(alpha_13^2) corrections, but the posterior on a is broad and enters through (P0 + P1 a) and (Q0 + Q1 a); a sentence stating that these denominator corrections are neglected relative to P0 and Q0 would make the approximation explicit.","section":"Section VI, Eq. (22)"},{"comment":"The extended fitting coefficients for alpha_13 in [-1, 1] (P0 = -0.0205, P1 = 0.0166, Q0 = 0.0136, Q1 = 0.0021) are quoted without fit residuals; please report the maximum error of the extended fit, since the stated 0.2% (l = 2) figure refers to the restricted calibration range.","section":"Section VI"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is honest and technically standard, and I found no circularity: the fit coefficients are fitted to the paper's own numerical QNM data, the correction factor is anchored to external Teukolsky results, and the GW constraint uses independent pyRing posteriors. The main review risks are (i) the frozen-source perturbation scheme, which needs either a consistency check or a prominent limitation statement; (ii) the factor-of-10 error in Table VII, which suggests that all coefficient tables should be machine-checked; and (iii) the extrapolation used for the headline GW170104 constraint. I would ask the authors for a comparison of their corrected non-Kerr frequencies against one of the Teukolsky-like methods cited in Refs. [60-62] at a small number of parameter points; since those works are already cited, this is a feasible request. The citation record otherwise looks clean apart from the apparent arXiv identifier error in Ref. [45], and the paper's claims of novelty ('first QNM fitting formula for slowly rotating Johannsen black holes') are appropriately scoped."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"What you should know: this is the paper that finally gives a usable mapping between the Johannsen α13 deformation and ringdown frequencies. The deformed Regge-Wheeler/Zerilli equations at first order in spin and deformation are new, the α13 fitting formula for the l=2 and l=3 fundamental modes is new, and the observation that α22 drops out of the effective potential at this order is the sort of thing people will quote. It is an incremental contribution, not a paradigm shift, but it is a real one.\n\nThe reader's conditional verdict is about right. The paper is transparent about its approximations: the frozen effective source, the axial-for-polar assumption, the spin range of the GW170104 analysis exceeding the validated band. The frequency tables in the appendix are reproducible output, and the reported α13 constraint being consistent with the inspiral-based constraint from Shashank and Bambi is a sensible cross-check.\n\nThe stress-test note tries to sharpen the frozen-source issue into an inconsistency: with δG=0 and a non-vanishing background G0, the first-order Bianchi identity leaves a leftover δΓ·G0 term that the derivation never enforces. I think the sharp version does not land. The first-order Bianchi identity is an identity, not a constraint: it holds for every perturbation. Any h that actually solves the full linearized Einstein equations automatically satisfies the leftover condition. What the note has correctly identified underneath is a narrower and real problem: the reduction to a single master equation is not shown in enough detail to verify that all components of δG[h]=0 are captured. The potentials are final expressions with no machine check, and the derivation is condensed to a paragraph. That is the load-bearing soft spot — not a demonstrated inconsistency, but an unverified reduction.\n\nThe other soft spots are modest. The correction factor assumes the slow-rotation error is independent of the deformation shift; plausible at first order, but unvalidated. The GW170104 constraint extrapolates the fit to α13 in [-1,1] and spins above 0.4; the authors say so themselves, which makes it an illustration, not a precision statement.\n\nThis is for people doing agnostic ringdown parametrization and parametric-metric tests of GR. It deserves a serious referee — send it out, and ask the referee to check the master equation derivation or request an independent verification of the potentials. Conditional acceptance is the right call.","headline":"Solid incremental contribution: new deformed Regge-Wheeler/Zerilli equations and an α13 ringdown mapping worth having, held back mainly by an unverified master-equation reduction and an extrapolated data constraint.","tokens_in":20657,"tokens_out":10535,"would_cite":true,"duration_ms":110453,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C35","83C25"],"pacs":["04.25.Nx","04.70.Bw","04.30.-w","95.30.Sf"],"model":"deepseek-v4-flash","headline":"This paper claims that, under the assumptions of the Einstein equations, slow rotation, and small deformation parameters, the quasi-normal modes of a Johannsen black hole are dominated by a single deformation parameter, $\\alpha_{13}$…","keywords":["quasi-normal modes","Johannsen metric","black hole perturbation theory","slow-rotation approximation","ringdown","tests of general relativity","deformation parameters","GW170104"],"falsifier":"Recompute the axial and polar quasi-normal modes at $a_*=0.4$ with a method that does not use the slow-rotation expansion, such as a Teukolsky-like equation for beyond-Kerr spacetimes, and check whether the $\\alpha_{13}$ shift follows $P_0+P_1a$ to within the paper's quoted 0.2%–0.3% error; alternatively, repeat the perturbation calculation while keeping the perturbation of the effective stress–energy tensor that the Johannsen background requires and look for frequency shifts comparable to the $\\alpha_{13}$ contribution.","tokens_in":19669,"feed_emoji":"🕳️","tokens_out":9821,"duration_ms":88331,"temperature":0.7,"pith_summary":"Gravitational-wave ringdowns promise a clean test of general relativity, but to interpret them one needs the vibration spectrum of the black hole that emits them. This paper computes the quasi-normal modes of slowly rotating black holes in the parametric Johannsen metric, a theory-agnostic family of spacetimes that reduces to Kerr when all deformation parameters vanish. Working to first order in the spin and in the leading deformation parameters, and assuming the Einstein equations hold for the perturbations, the authors find that one parameter, $\\alpha_{13}$, shifts the modes far more than the others ($\\alpha_{22}$, $\\alpha_{52}$, and $\\epsilon_3$), and they provide a two-term fitting formula, $\\mathrm{Re}(\\omega)=\\mathrm{Re}(\\omega_{\\mathrm{Teu}})+P_0\\alpha_{13}+P_1\\alpha_{13}a$ plus the analogous imaginary part, valid for $a_*<0.4$. Applied to the LIGO–Virgo event GW170104, the formula turns measured ringdown frequency and damping deviations into the bound $\\alpha_{13}=-0.5^{+2.1}_{-4.0}$ at 90% confidence, which is consistent with general relativity.","feed_headline":"One deformation parameter dominates deformed black-hole ringdowns","feed_subtitle":"From the ringdown of GW170104, α13 comes to −0.5 +2.1 −4.0, consistent with general relativity.","key_machinery":"The machinery is a pair of one-dimensional master equations of Regge–Wheeler and Zerilli type, $\\partial^2\\Psi/\\partial r_*^2+V\\Psi=0$, with effective potentials computed to first order in the spin $a$ and in the deformation parameters $\\alpha_{13}$, $\\alpha_{52}$, and $\\epsilon_3$; the potentials are obtained by linearizing the Einstein equations about the slow-rotating Johannsen metric in the Regge–Wheeler gauge. The modes are found by the direct integration method, which matches an ingoing solution at the horizon to an outgoing solution at infinity by a Wronskian determinant condition, retaining the full horizon frequency $\\omega_H$ even though the equations themselves are truncated. A correction factor $C=\\omega_{\\rm Teu}/\\omega_{\\rm Kerr}$ removes the slow-rotation error from the Kerr baseline, and the fitting formula $\\mathrm{Re}(\\omega)=\\mathrm{Re}(\\omega_{\\rm Teu})+P_0\\alpha_{13}+P_1\\alpha_{13}a$ with tabulated coefficients $P_0$, $P_1$, $Q_0$, $Q_1$ converts the corrected frequencies into a direct mapping from ringdown data to $\\alpha_{13}$.","core_discovery":"The central claim is that the quasi-normal modes of a Johannsen black hole, computed from the Einstein equations in the slow-rotation, small-deformation limit, are dominated by the deformation parameter $\\alpha_{13}$. The axial and polar perturbation problems separate in this limit, giving deformed Regge–Wheeler and Zerilli master equations whose effective potentials contain $\\alpha_{13}$, $\\alpha_{52}$, and $\\epsilon_3$ but not $\\alpha_{22}$ at leading order. Because the slow-rotation approximation degrades the Kerr part of the answer, the authors calibrate the Kerr baseline against the exact Teukolsky result and fold the mismatch in as a correction factor, so the final frequency is $\\omega_{\\rm corr}=\\omega_{\\rm Non\\text{-}Kerr}+(C-1)\\omega_{\\rm Kerr}$. For the fundamental $l=2$ and $l=3$ modes the result takes the form of a fitting formula with the reported coefficients, and the paper argues that, in this regime, a ringdown observation translates directly into a measurement of $\\alpha_{13}$.","pith_inferences":["If the additivity of the correction factor in $\\omega_{\\rm corr}=\\omega_{\\rm Non\\text{-}Kerr}+(C-1)\\omega_{\\rm Kerr}$ is not exact near $a_*=0.4$, the inferred $\\alpha_{13}$ from an event at the edge of the validity range would carry a systematic bias; recomputing the modes without factorizing the slow-rotation error would quantify that bias.","The no-source assumption means the computed modes describe the effective Einstein vacuum of the metric rather than any concrete theory; in a theory that really produces the Johannsen background, perturbations of its matter sector would generically shift the modes by an amount comparable to the $\\alpha_{13}$ effect, so the quoted constraint should be read as a null test of the parametric family.","The paper leaves the polar sector covered only by an assumption of near-isospectrality; because deformations generically break the equality of axial and polar spectra, measuring a difference between them in a loud ringdown would be a distinctive signature that this formalism is built to predict but has not yet computed."],"forward_implications":["A ringdown detection from any low-spin remnant ($a_*<0.4$) can be converted directly into a measurement of $\\alpha_{13}$ through $\\alpha_{13}=f^{\\rm GR}_{lmn}\\delta f_{lmn}/(P_0+P_1 a)$, giving a theory-agnostic test of the Kerr hypothesis from one gravitational-wave event.","The {2,2,1} ringdown of GW170104 yields $\\alpha_{13}=-0.5^{+2.1}_{-4.0}$ at 90% confidence, consistent with general relativity and with the inspiral-based constraint on the same event.","At leading order in the slow-rotation and small-deformation expansion, $\\alpha_{22}$ drops out of the perturbation equations and $\\alpha_{52}$, $\\epsilon_3$ have weaker effects, so the ringdown test is effectively one-parameter in this regime.","The tabulated coefficients for the fundamental and first-overtone modes with $l=2$ and $l=3$ keep fit errors below 0.3%, so the same formulas can be dropped directly into future ringdown templates."],"supporting_citations":[{"why":"Defines the parametric Johannsen metric whose quasi-normal modes are the object of the paper.","marker":"[45]"},{"why":"Supplies the direct integration method used to compute the quasi-normal mode frequencies.","marker":"[28]"},{"why":"Provides the LIGO–Virgo ringdown data and final-spin estimate of GW170104 used for the constraint.","marker":"[4]"},{"why":"Gives the inspiral-based constraint on deformation parameters for GW170104 that the ringdown result is compared against.","marker":"[6]"},{"why":"Provides the pyRing time-domain ringdown analysis that extracts the frequency and damping-time deviation posteriors.","marker":"[52]"},{"why":"Establishes the overtone ({2,2,1}) extraction and analysis framework used to read the ringdown of GW170104.","marker":"[53]"},{"why":"Supplies the Regge–Wheeler gauge and axial perturbation formalism that the master-equation derivation extends.","marker":"[46]"},{"why":"Sets the standard approach for linearized perturbations on a background metric that the derivation follows.","marker":"[39]"}],"fun_headline_variants":["One parameter rules deformed black-hole ringdowns","α13 dominates Johannsen black-hole quasinormal modes","Slow-spin Johannsen ringdowns point to α13","Johannsen black holes: α13 drives ringdown, matches GR"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the Johannsen metric, which is not a vacuum solution of the Einstein equations, can still be treated as a vacuum background whose gravitational perturbations obey the source-free Einstein equations, so that perturbations of the effective stress–energy tensor are neglected.","fun_headline_variants_meta":{"raw":{"variants":["One parameter rules deformed black-hole ringdowns","α13 dominates Johannsen black-hole quasinormal modes","Slow-spin Johannsen ringdowns point to α13","Johannsen black holes: α13 drives ringdown, matches GR"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"total_tokens":1279,"prompt_tokens":959,"completion_tokens":320,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":575,"completion_tokens_details":{"reasoning_tokens":251}},"tokens_in":575,"tokens_out":320,"duration_ms":3716,"temperature":1.0,"reasoning_tokens":251,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:06:11.123881+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the axial and polar quasi-normal modes at $a_*=0.4$ with a method that does not use the slow-rotation expansion, such as a Teukolsky-like equation for beyond-Kerr spacetimes, and check whether the $\\alpha_{13}$ shift follows $P_0+P_1a$ to within the paper's quoted 0.2%–0.3% error; alternatively, repeat the perturbation calculation while keeping the perturbation of the effective stress–energy tensor that the Johannsen background requires and look for frequency shifts comparable to the $\\alpha_{13}$ contribution.","supporting_citations":[{"cited_title":"Pani, International Journal of Modern Physics A 28, 1340018 (2013)","cited_arxiv_id":null,"evidence_quote":"Supplies the direct integration method used to compute the quasi-normal mode frequencies."},{"cited_title":"Wagle, N","cited_arxiv_id":null,"evidence_quote":"Sets the standard approach for linearized perturbations on a background metric that the derivation follows."}],"review_version":1}