{"id":"b3734e70-c33a-4ca4-b5df-2d4d6cb83da4","arxiv_id":"2502.05689","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a quantum-corrected spacetime, the fundamental quasinormal mode differs little from Schwarzschild, higher overtones deviate significantly, and wormhole states have extremely long-lived modes.","lead":"This paper calculates how a quantum-corrected black hole, which can morph into a wormhole, rings and emits radiation when disturbed. Its key finding is that the usual lowest tone stays close to Schwarzschild, while higher tones and the wormhole's very long-lived echoes differ strongly, offering possible gravitational wave signatures.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Long-lived-mode claim rests on pseudospectral eigenvalues at a single point (ξ=3.94) with no convergence study or time-domain check; the 'arbitrarily long-lived' threshold limit is extrapolated, not computed.","rationale":"The central claim has two load-bearing components: (i) the effective perturbation framework correctly represents the gravitational-sector response of the quantum-corrected spacetime, and (ii) the numerically computed near-threshold long-lived modes are genuine and not artifacts. I focus on (ii) because it is the most concrete and checkable, and because it is the quantitative basis for the paper's most striking statement—'arbitrarily long-lived modes' as ξ→ξ_c. The manuscript itself flags in Sec. III that the direct Hamiltonian perturbation analysis has not been carried out, and I do not make that the primary objection: the same drastic change at the transition appears in the scalar and electromagnetic test-field modes (Tables III and IV), which depend only on the fixed background metric and not on the effective-fluid prescription. The axial gravitational long-lived mode at ξ=4 is corroborated by both pseudospectral and time-domain methods, which is real independent support. The problem is the extrapolation: the closest tabulated point to threshold, ξ=3.94, has an imaginary part at the 10⁻⁹ level, a regime where spectral artifacts and conditioning issues are common, and no convergence study or time-domain cross-check is provided. Thus the appropriate outcome is unchanged from the reader's CONDITIONAL verdict: accept for the community as a data resource, but require either a dedicated convergence study of the near-threshold modes or a softening of the 'arbitrarily long-lived' and PTA-observability claims.","tokens_in":21119,"tokens_out":10787,"duration_ms":123154,"concrete_test":"Compute the axial ℓ=3 fundamental QNM for the wormhole at ξ=3.94 and additionally at ξ=3.938, 3.939, and 3.941 using Chebyshev pseudospectral grids with N=100, 200, 300, 400, and 500 in extended-precision arithmetic, and independently integrate the time-domain equation to t≥2×10⁴ M with Prony extraction. Require the complex frequency to stabilize to the reported digits and the time-domain fit to match the pseudospectral value. Additionally, derive a WKB/Bohr-Sommerfeld estimate of the double-barrier resonance width to provide an analytic cross-check. If Im(ω) drifts with N or is unresolved in the time domain, the 'arbitrarily long-lived at threshold' claim is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's most distinctive quantitative claim—that wormhole modes become arbitrarily long-lived as ξ approaches the transition threshold—is supported at the closest computed point (ξ=3.94) only by pseudospectral eigenvalues. Table II gives the ℓ=3 fundamental mode as Im(ω)=−6.6×10⁻⁹, roughly three orders of magnitude smaller than the value at ξ=4.0, yet no grid-convergence data, no independent time-domain confirmation, and no calculation closer to ξ_c≈3.937 are reported. The conclusion section nevertheless extrapolates to 'arbitrarily long-lived modes' and links them to PTA-visible low-frequency gravitational waves. Pseudospectral filtering by comparing two grid sizes can still retain spurious eigenvalues when imaginary parts are exponentially small, and near the transition the tortoise coordinate develops large stretches at the throat, so spectral conditioning is not guaranteed. The existence of long-lived modes at ξ=4 is independently supported by the time-domain profile in Fig. 3, so this concern is specific to the threshold limit; if the near-threshold eigenvalue is contaminated, the headline claim loses its quantitative force.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies axial gravitational, scalar, and electromagnetic perturbations of the quantum-corrected black hole and wormhole spacetimes derived by Zhang et al. from a Hamiltonian-constraint approach. Using Chebyshev pseudospectral and time-domain (Gundlach–Price–Pullin) methods, it computes quasinormal frequencies for the black-hole branch (ξ < ξ_c ≈ 3.937) and the wormhole branch (ξ > ξ_c), finds that the wormhole fundamental modes become extremely long-lived near the transition (Table II gives Im(ω) = −6.6×10⁻⁹ at ξ = 3.94 for ℓ = 3), and reports grey-body factors, absorption cross-sections, and Hawking emission rates. It also derives eikonal WKB expansions for the quasinormal modes to order ξ⁴ and 1/κ². The paper concludes that the black-hole-to-wormhole transition leaves an observable late-time signature with echoes and long-lived modes, potentially relevant for pulsar timing array experiments.","tokens_in":21333,"tokens_out":13334,"duration_ms":132276,"significance":"The background metric is motivated by a covariant effective quantum-gravity construction and interpolates between regular black holes and traversable wormholes, making the perturbation study timely. The main results—overtone deviations in the black-hole branch and long-lived wormhole modes near the transition—are physically interesting and, if confirmed, could serve as an observational discriminant for this class of quantum-corrected compact objects. The analytic WKB expansions and the explicit test of the quasinormal-mode/grey-body correspondence for black holes are useful additions. The numerical setup follows standard methods and the master equations appear correctly reduced. However, the headline claim of 'arbitrarily long-lived' modes rests on a single pseudospectral eigenvalue at ξ = 3.94 with no convergence study and no closer-to-threshold calculation, and there are data-quality issues in the appendix tables; the quantitative support is therefore incomplete.","major_comments":[{"comment":"The statement that wormhole modes become 'arbitrarily long-lived' as ξ approaches the critical threshold is an extrapolation from a single computation at ξ = 3.94, the closest value to ξ_c ≈ 3.937. At that point Table II gives the ℓ = 3 fundamental mode as Im(ω) = −6.6×10⁻⁹, more than three orders of magnitude smaller than the ℓ = 3 fundamental at ξ = 4.0, but no grid-convergence data, no time-domain integration, and no intermediate points (e.g., ξ = 3.95, 3.945) are reported. Because pseudospectral eigenvalues with exponentially small imaginary parts are susceptible to contamination, the numerical evidence does not yet establish the 'arbitrarily long-lived' limit. The authors should either provide a convergence study at ξ = 3.94, compute values closer to ξ_c, and confirm one near-threshold mode with an independent method (time-domain integration or a WKB estimate of the inter-barrier trapping time), or temper the conclusion to 'very long-lived at the closest computed point'.","section":"§V.B, Table II, Conclusions"},{"comment":"The pseudospectral section states that calculations are performed on two grids and only the overlapping eigenvalues are retained, but the manuscript never reports the grid sizes N, the differences between the two grids, or error estimates for the quoted frequencies. Given that the central quantitative claim rests on Table II entries with |Im(ω)| as small as 10⁻⁶–10⁻⁹, a convergence table (e.g., values of the ℓ = 3 fundamental at ξ = 3.94 and ξ = 4.0 for N = 200, 250, 300, 350) is necessary. Without such data, the reader cannot distinguish genuine long-lived modes from numerical artifacts near the transition.","section":"§IV.A, Appendix A"}],"minor_comments":[{"comment":"The phrase 'The of the axial gravitational perturbations' is missing a word; it should read 'The axial gravitational perturbations' or 'The perturbation of the axial gravitational field'.","section":"§III.A"},{"comment":"There are several typographical errors: 'domians' should be 'domains' (§IV), 'pertrubations' should be 'perturbations' (§V), and 'absorbtion' should be 'absorption' (§VI and figure captions).","section":"§IV, §V, §VI"},{"comment":"The caption of Table IV states 'ℓ = 0 and 1', but the table lists ℓ = 1 and 2; moreover, electromagnetic perturbations have no ℓ = 0 mode, so the caption should be corrected to 'ℓ = 1 and 2'.","section":"Appendix A, Table IV"},{"comment":"Several entries are duplicated across different ℓ values: in Table IV the ξ = 3.5 and ξ = 3.9 rows are identical for ℓ = 1 and ℓ = 2, and in Table V the ξ = 4.6 row is identical for ℓ = 2 and ℓ = 3. These appear to be copy-paste errors and must be corrected, as they undermine confidence in the time-domain data used to support the quasinormal-mode values.","section":"Appendix A, Tables IV and V"},{"comment":"Fig. 9 and Fig. 14 appear to be the same figure, with the same caption and the same panels. One should be removed and the figure numbering checked throughout the manuscript.","section":"Figures 9 and 14"},{"comment":"The correction term Σ(ω₀,ω₁) in Eq. (36) is not defined, and the symbol Γₗ(ω) is not used elsewhere in the text. Please define the notation and specify how many correction terms from Ref. [33] are included in the numerical comparison.","section":"§VI.B, Eq. (36)"},{"comment":"The expression for the throat radius r_m in Section II is typeset ambiguously; please rewrite it in standard notation so that the dependence on M and ξ is unambiguous.","section":"§II"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid numerical study of an interesting model, and the central physical claim—long-lived wormhole modes near the black-hole/wormhole transition—is plausible. The main obstacle to acceptance is the lack of numerical convergence evidence for the quantitative 'arbitrarily long-lived' claim; this is fixable with additional calculations. The duplicated figure and table entries suggest a hasty final assembly, and the appendix should be carefully verified. The effective-fluid perturbation approach is explicitly acknowledged by the authors and is a reasonable scope limitation rather than a fatal flaw."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a useful, competently executed QNM study of the Zhang et al. quantum-corrected spacetime, and the first to compute grey-body factors and Hawking radiation for it. The headline claim about ultra-long-lived wormhole modes near the transition is plausible but is pushed further than the numerics support, and a few tables contain obvious copy-paste errors.\n\nWhat's genuinely new: nobody had computed the axial gravitational, scalar, and electromagnetic QNMs for this specific metric before. The master equations look correctly reduced, the pseudospectral and time-domain methods are standard, and the paper is honest about the effective-fluid perturbation assumption (treating the metric as a solution of Einstein equations with anisotropic matter rather than deriving the perturbations from the Hamiltonian constraint theory). The WKB eikonal expansions at small xi are a nice addition, and the grey-body/Hawking results give the community a concrete dataset to compare against.\n\nCredit where due: the fundamental-mode deviations from Schwarzschild are mild, overtones deviate increasingly—this is the expected near-horizon effect and it's well presented. The wormhole double-barrier physics (echoes, quasi-resonances, trapped modes) is catalogued carefully. For xi=4 the long-lived mode is confirmed by time-domain integration, not just the spectral calculation.\n\nSoft spots. First, the 'arbitrarily long-lived' claim at the transition threshold is extrapolated from a single pseudospectral eigenvalue at xi=3.94 with Im(omega) around -6.6e-9. There is no convergence study, no time-domain check at that parameter value, and no calculation closer to the threshold xi_c approx 3.937. Near the throat the tortoise coordinate stretches, so spectral conditioning is not guaranteed. The qualitative statement that modes become very long-lived near the transition is fine; the quantitative 'arbitrarily long-lived' assertion needs either a dedicated computation or softer language. Second, no error bars or convergence tests are reported for any of the tables. Third, Tables IV and V contain identical entries for different l at the same xi (e.g., xi=3.5 and 3.9 in Table IV, xi=4.6 in Table V), which looks like copy-paste mistakes. The Table IV caption also mislabels the l values. These are fixable but need attention before the tables can be trusted as a reference dataset.\n\nBottom line: this deserves a serious referee. I would send it to review, but with major revision requested: fix the tables, add convergence checks or error estimates, and either compute closer to the threshold with a time-domain cross-check or pull back the 'arbitrarily long-lived' language. For someone working on quantum-corrected black holes or exotic compact object ringdowns, this is a citable resource.","headline":"Useful first QNM/grey-body computation for the Zhang et al. metric, but the 'arbitrarily long-lived' threshold claim outruns the numerics and several tables have copy-paste errors.","tokens_in":21867,"tokens_out":3787,"would_cite":true,"duration_ms":35909,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Quantum corrections to black holes predict ultra-long-lived wormhole ringdown modes near the black-hole–wormhole transition.","keywords":["quasinormal modes","regular black holes","traversable wormholes","quantum-corrected spacetimes","grey-body factors","Hawking radiation","ringdown echoes","Regge-Wheeler perturbations"],"falsifier":"Compute the axial gravitational quasinormal spectrum directly from the full Hamiltonian-constraint perturbation equations for the same metric family. If the fundamental $\\ell=2$ mode at $\\xi=3.94M$ has a damping rate far from $|\\mathrm{Im}\\,\\omega| \\approx 5.7\\times 10^{-6}$ (the effective-theory value), then the predicted long-lived wormhole modes are an artifact of the effective fluid treatment.","tokens_in":20917,"feed_emoji":"🕳️","tokens_out":7443,"duration_ms":70587,"temperature":0.7,"pith_summary":"Using a quantum-corrected spacetime family derived from Hamiltonian-constraint quantum gravity, this paper computes how black holes and wormholes ring when perturbed. For the black-hole branch, the fundamental quasinormal mode stays within about 10–15% of the Schwarzschild value while higher overtones deviate from Schwarzschild by more than 30%, encoding near-horizon quantum corrections. At the transition value $\\xi \\approx 3.93M$, the solution becomes a traversable wormhole and the spectrum changes abruptly: the fundamental mode becomes exceptionally long-lived, with its damping rate decreasing as $\\xi$ approaches the threshold, and time-domain signals show early echoes followed by a slowly decaying ringdown. The same calculation yields grey-body factors, absorption cross-sections, and Hawking radiation rates. If these features survive in the full quantum theory, they would give gravitational-wave observables that distinguish quantum-corrected compact objects from classical black holes.","feed_headline":"Black-hole ringdown modes get far longer-lived at wormhole threshold","feed_subtitle":"Near the black-hole-to-wormhole transition, the fundamental mode's damping rate falls toward zero.","key_machinery":"The central object is the quantum-corrected line element with metric functions $f(r)$ and $\\mu(r)$ controlled by the quantum parameter $\\xi$; for $\\xi/M < \\pi^{3/2}/\\sqrt{2} \\approx 3.937$ it describes a regular black hole, and for larger values a traversable wormhole with throat radius $r_m = \\sqrt[6]{2 M \\xi^2/3}$. The argument runs by reducing axial gravitational, scalar, and electromagnetic perturbations to a single master wave equation with the potentials of equations (13)–(15), then solving the boundary-value problem with a pseudospectral method and with time-domain integration. The mechanism that produces long-lived modes is the double-peaked effective potential in the wormhole regime: waves bounce between the two peaks, producing echoes and trapping slowly decaying modes, and as $\\xi$ approaches the critical value from above, the fundamental damping rate tends to zero. Analytic eikonal/WKB expansions provide control at large multipole number $\\ell$.","core_discovery":"On its own terms, the paper establishes that the quasinormal spectrum sharply encodes the quantum parameter $\\xi$ of the metric family given by equations (1)–(3). For regular black holes, the fundamental axial mode is close to Schwarzschild ($\\omega_0 \\approx 0.3737 - 0.0890i$ for $\\ell=2$ at small $\\xi$), but overtones drift increasingly, with the third overtone departing by more than 30% from its Schwarzschild counterpart. In the wormhole regime $\\xi > \\xi_{\\mathrm{cr}} \\approx 3.93M$, the spectrum is not a smooth continuation of the black-hole spectrum: it forms a new set of modes that are non-perturbative in $\\xi$, and near the threshold the fundamental mode becomes arbitrarily long-lived, for example $\\omega_0 \\approx 0.1330 - 5.7\\times 10^{-6} i$ for $\\ell=2$ at $\\xi=3.94M$. Time-domain evolution shows echoes at early times followed by a late-time ringdown controlled by these slowly decaying modes. The paper also reports that for wormholes the quasinormal-mode/grey-body-factor correspondence breaks down because the effective potential has two peaks, while for black holes it holds for larger $\\ell$; and it provides Hawking-temperature, grey-body, and energy-emission profiles that vary strongly with $\\xi$.","pith_inferences":["A natural extension is to test whether the long-lived modes survive direct Hamiltonian-constraint perturbation theory; if they do, they would provide a target for low-frequency gravitational-wave searches that is free of the tail-suppression problem noted for massive-field long-lived modes.","The abrupt spectral change and overtone reconnection at the transition suggest an analogy with level crossing in quantum systems; searches could look for a discontinuous shift in the ringdown frequency as a compact object's parameters evolve.","Because the wormhole potential is double-peaked and symmetric, the long-lived modes are symmetric or antisymmetric about the throat; a rotating generalization could break this symmetry and split or destabilize the modes, which would be a concrete next test."],"forward_implications":["Ringdown waveforms from a quantum-corrected black hole would look nearly Schwarzschild in the fundamental mode but show growing deviations in the first overtones, so searches that include overtones could expose near-horizon quantum structure.","Near the transition to a wormhole, the signal would contain early echoes followed by a very slowly decaying tone; detecting such a pattern in a gravitational-wave event would be evidence for a black-hole-to-wormhole transition rather than a classical remnant.","Wormhole grey-body factors are non-monotonic in frequency with narrow quasi-resonances, so the absorption and emission spectra carry a clear wormhole fingerprint.","The Hawking energy emission rate peaks at $\\xi \\approx 3.86M$, beyond the peak of the Hawking temperature at $\\xi \\approx 3.08M$, showing that grey-body factors, not just temperature, set the observed flux."],"supporting_citations":[{"why":"Supplies the Hamiltonian-constraint quantum-corrected metric family whose regular black-hole and wormhole branches are perturbed throughout the paper.","marker":"[1]"},{"why":"Provides the effective Hamiltonian constraint formalism and earlier quantum-corrected solutions that motivate treating the metric as an Einstein solution with a quantum-gravity-inspired source.","marker":"[4]"},{"why":"Gives the Regge–Wheeler gauge and axial gravitational perturbation equations used to derive the master wave equation.","marker":"[41]"},{"why":"Supplies the time-domain integration scheme used to obtain ringdown profiles and extract quasinormal frequencies from the waveform.","marker":"[46]"},{"why":"States the quasinormal-mode/grey-body-factor correspondence that the paper checks and finds to hold for black holes but break for the double-peaked wormhole potential.","marker":"[33]"}],"fun_headline_variants":["Wormhole threshold makes fundamental quasinormal mode nearly undamped","Black hole to wormhole: ringdown modes become ultra-long-lived at threshold","Quantum gravity transition: wormhole quasinormal modes have negligible damping","At wormhole limit, fundamental mode damping drops by four orders of magnitude"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The computation treats the quantum-corrected metric as a solution of Einstein's equations with an anisotropic fluid and perturbs it with standard Regge-Wheeler theory, because perturbing the underlying Hamiltonian constraint theory directly has not been done; if that effective description is not what the full quantum theory produces, the predicted overtones and long-lived modes would not be the real signal.","fun_headline_variants_meta":{"raw":{"variants":["Wormhole threshold makes fundamental quasinormal mode nearly undamped","Black hole to wormhole: ringdown modes become ultra-long-lived at threshold","Quantum gravity transition: wormhole quasinormal modes have negligible damping","At wormhole limit, fundamental mode damping drops by four orders of magnitude"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":3007,"prompt_tokens":1024,"completion_tokens":1983,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":640,"completion_tokens_details":{"reasoning_tokens":1905}},"tokens_in":640,"tokens_out":1983,"duration_ms":15187,"temperature":1.0,"reasoning_tokens":1905,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:21:58.527687+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the axial gravitational quasinormal spectrum directly from the full Hamiltonian-constraint perturbation equations for the same metric family. If the fundamental $\\ell=2$ mode at $\\xi=3.94M$ has a damping rate far from $|\\mathrm{Im}\\,\\omega| \\approx 5.7\\times 10^{-6}$ (the effective-theory value), then the predicted long-lived wormhole modes are an artifact of the effective fluid treatment.","supporting_citations":[],"review_version":1}