{"id":"cb9f03b6-ef9f-48dc-b3ec-2f4324f0ced3","arxiv_id":"2507.16493","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":2.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Black hole quasinormal mode sums can display long transient plateaus and transient energy growth due to non-normality of the evolution operator, and this review systematically collects the frequency- and time-domain evidence.","lead":"This review explains how sums of black hole quasinormal modes can stay almost constant for a long time, or even grow briefly, because the underlying wave operator is non-normal. It maps out a growing subfield of black hole perturbation theory with implications for gravitational wave ringdown analysis and holographic models.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The energy-norm transient plateau in Schwarzschild is computed for a finite QNM subspace and is not shown to survive the continuum limit; the review's own H^0 grid results are non-convergent, so 'arbitrarily long-lived' perturbations remain unestablished.","rationale":"The reader's weakest assumption identifies precisely the finite-M/grid representativeness problem, and the strongest_claim depends on the reality of the transient plateau. I agree that this is the load-bearing point. The review's own admission that the H^0 plateau is non-convergent with grid resolution is a direct red flag for the analogous energy-norm plateau in Schwarzschild. Because the plateau duration grows as log M with no demonstrated continuum limit, the phrase 'arbitrarily long-lived' is not supported by the evidence presented. The H^p and RN-AdS transient growth results are less affected by this particular concern, but they are either norm-dependent with an open physical interpretation or also exhibit M-dependence without established convergence. The footnote-4 correction of [23] is peripheral to the main argument but is asserted without derivation, adding to the review's overall reliance on unpublished or self-cited results. A concrete full-PDE evolution of the optimal perturbation would settle whether the plateau is physical. I do not recommend changing the reader's CONDITIONAL verdict; the concern warrants the condition that the continuum limit be demonstrated, which is exactly the current verdict.","tokens_in":9116,"tokens_out":4990,"duration_ms":57884,"concrete_test":"Evolve the M=39 optimal perturbation from Fig. 1 under the full hyperboloidal PDE (without projecting onto the QNM subspace) using a high-resolution code, and compute E[u(τ)]/E[u(0)]. If the energy decays at the fundamental-mode rate e^{-2|Im ω_0|τ} from early times with no plateau, the plateau is a truncation artifact. Repeat for M=10, 19, 39 to check whether any plateau duration grows with M in the full evolution. For the Pöschl-Teller H^0 case, fix a target time τ* and verify convergence of G(τ*) as N→∞; if G(τ*) diverges or the plateau duration scales as log N, the energy-norm observable is ill-defined in the continuum limit.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that linear black hole perturbations can be arbitrarily long-lived rests on the energy-norm plateau computed in [22] for Schwarzschild, where G_W(τ) exhibits a plateau of duration ~ log M (Section 3, Fig. 1). This is a statement about the restriction H_W of the evolution operator to W = span of the first M QNMs; the optimal initial data are M-dependent (Eqs. 11-12). No argument is given that the sequence of optimal perturbations converges in the full Hilbert space to an initial datum whose exact evolution has a plateau, nor that the log M scaling survives the continuum limit. The review itself provides a direct warning: in the Pöschl-Teller model, the energy-norm (H^0) plateau computed on the full grid has duration ~ log N and is not convergent as N increases (Section 3, H^p discussion). Since H^0 is the same energy norm used for the Schwarzschild plateau, this acknowledged non-convergence makes it plausible that the log M plateau is likewise an artifact of the QNM truncation rather than a property of the continuum dynamics. Moreover, the eventual 'modal decay' is enforced by the QNM-sum ansatz (12); full asymptotically flat evolutions generically exhibit power-law tails, so the finite-M calculation cannot distinguish a transient plateau from truncation-induced behavior. In addition, transient growth in H^p norms is norm-dependent (G(τ_max) ~ p) and the paper states its physical interpretation is open, so it cannot independently support a physical claim of transient growth. The footnote-4 correction of [23] is asserted without derivation; while not central, it is another unsupported load-bearing element of the review's narrative.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is a review article on transient (non-modal) phenomena in linear black hole perturbation theory. It argues that the dissipative nature of black hole horizons renders the evolution operator non-normal, so that quasinormal modes (QNMs) are not orthogonal under standard inner products such as the energy product; this non-orthogonality gives rise to transient plateaus (arbitrarily long-lived sums of QNMs) and transient growth as measured in suitable norms. The review covers pseudospectral methods (Kreiss constant, numerical abscissa), time-domain studies of optimal perturbations in a truncated QNM subspace, and higher-derivative Sobolev norms, with examples from Schwarzschild, RN-AdS4, and a Pöschl-Teller toy model. It also corrects a transient-growth result in a previous paper [23].","tokens_in":9450,"tokens_out":4721,"duration_ms":50250,"significance":"If the summarized results are correct, the review highlights a genuine gap in the standard QNM picture: linear perturbations can exhibit long-lived transient behavior even when the mode spectrum is stable. The review is clearly written and brings together recent developments, including several from the authors' own work, with useful figures and a coherent narrative. It also explicitly notes a convergence caveat for the energy-norm plateau in the toy model, which is a useful caution. The correction of [23] is a strong claim that would be impactful if substantiated, but it is currently asserted without the promised calculation.","major_comments":[{"comment":"The review claims that [22] demonstrates \"arbitrarily long-lived linear black hole perturbations\" based on a plateau in GW(τ) whose duration scales as log M within a finite QNM subspace W of dimension M. However, later in the same section, the review states that in the Pöschl-Teller model the energy-norm (H^0) plateau computed on the full grid has duration scaling as log N and is non-convergent as N grows. Since H^0 is the same energy norm, this raises the real possibility that the log M plateau for Schwarzschild is a truncation artifact. The review does not reconcile these observations; it merely says the non-convergence \"further motivates the use of H^p norms.\" To sustain the headline claim of arbitrarily long-lived perturbations, the authors should either provide evidence of convergence (e.g., show that the optimal initial data converge in the full Hilbert space as M increases, or compare against a full-grid evolution) or carefully qualify the claim as a statement about the finite-dimensional truncated system rather than about the continuum evolution. As written, the abstract's \"arbitrarily long-lived\" is not established.","section":"Section 3, H^p discussion"},{"comment":"The statement that \"(3.19) there can be written as a total derivative\" in [23] is a mathematical assertion that is not demonstrated. Since the review explicitly concludes that the reported transient growth in [23] is \"incorrect,\" this correction is a load-bearing claim and should be backed by an explicit calculation or by a reference to a paper that provides one. Without this, the wording should be softened to indicate that the result appears inconsistent or that the total-derivative claim is made without proof.","section":"Footnote 4"},{"comment":"The review repeatedly states that after the transient plateau the evolution \"conforms to modal decay\" with the fundamental mode decay rate. This is, however, an artifact of the QNM-sum ansatz (12), which restricts the dynamics to a finite subspace of QNMs. Full asymptotically flat evolutions generally exhibit power-law tails at late times, not pure modal decay. The review does not warn the reader that the claimed modal-decay behavior holds only within the truncation and is not a property of the exact evolution. This should be clarified explicitly, as it bears on the physical interpretation of the plateau phenomenon.","section":"Section 3, Eq. (12); Section 4"}],"minor_comments":[{"comment":"The phrase \"arbitrarily long-lived sums of quasinormal modes\" in the abstract is too strong without the caveat that this holds within a truncated QNM subspace and that the continuum limit is not established; consider qualifying it, e.g., \"arbitrarily long-lived within the truncated QNM model.\"","section":"Abstract"},{"comment":"The discussion of the pseudospectrum and the bounds (6)-(7) would benefit from explicitly stating which norm is used for the operator norm, since the values of w(H) and K(H) depend crucially on the chosen norm (energy, Sobolev, etc.). The reader is left to infer that the energy norm is used for the Schwarzschild result in [20].","section":"Section 2"},{"comment":"There are several formatting inconsistencies in the references, such as \"V ,\" and \"Toomani V ,\" with reversed comma placement; these should be corrected in the final version.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The review is heavily self-referential: the key results on plateaus [22], transient growth [29], and Sobolev-norm growth [26] are co-authored by the current authors. This is not a problem per se, but it means the review functions partly as a summary of the authors' own work. The correction of [23] is a strong negative claim against another group; given the visibility of a review article, it should be airtight. I am also concerned that the non-convergence of the energy-norm plateau in the toy model is mentioned only in passing and not connected to the Schwarzschild claim; this weakens the review's reliability for readers who may take the abstract at face value."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a well-written review of work on transient effects in black hole perturbation theory, mostly summarizing results by the same group. It is a good entry point to the non-normal/QNM framing, but the headline claim of arbitrarily long-lived perturbations is not established beyond a finite-mode truncation, and the paper gives its own warning that the energy-norm plateau in a toy model is non-convergent with resolution. I'd send it to peer review, but with clear requests: state what is known in the continuum limit, and back up the footnote-4 correction.\n\nWhat it does well: the exposition of the pseudospectrum, Kreiss constant, and optimal perturbation machinery is clean. The figures from [22], [26], and [29] are informative, and the paper is honest about the open physical interpretation of the H^p norms and about the non-convergent H^0 plateau in the Pöschl-Teller model. That transparency counts. The footnote-4 correction of [23] is a real attempt at engaging with the broader literature, and if correct it is important; it also makes the review more than a pure restatement of the group's papers.\n\nThe soft spots, in proportion: first, the novelty is genuinely low—this is a review, and it reads like a guided tour of work by the same authors. That is acceptable if marketed as a review, but the abstract's phrase 'arbitrarily long-lived sums of quasinormal modes' is a far stronger claim than what is shown. The plateau in the Schwarzschild case is built from the first M QNMs, and the optimal initial data depend on M. No argument is given that the sequence of optimal initial data converges to a continuum solution, or that the log M plateau is not a truncation artifact. The review's own H^0 example, computed on a full grid, explicitly does not converge in the grid size N, so the energy-norm plateau is at least sometimes a numerical artifact. That does not disprove the Schwarzschild result, but it places a burden on the authors to say why the QNM-subspace truncation is the physically relevant one.\n\nSecond, the transient growth in H^p norms is presented as a phenomenon, but it is norm-dependent and the physical interpretation remains open—the authors admit this. It is a mathematical observation about a chosen measure, not yet a physical effect. Third, the correction of [23] is asserted in a footnote with a one-line 'can be written as a total derivative.' For a correction aimed at another group's paper, that deserves a worked calculation or a reference to one; as it stands it is unsupported.\n\nWho should read this: anyone wanting a quick map of non-normal transient phenomena in BHPT, and especially people interested in ringdown modeling or holographic thermalization. For those readers it is a good starting point, but they should go back to the primary papers for evidence.\n\nMy recommendation: do send to peer review. The topic is timely and the review is competent, but the referees should insist on a statement of what is actually established in the continuum limit and on substantiating the footnote-4 claim.","headline":"A clear and useful review of transients from non-normal QNM operators, but the 'arbitrarily long-lived' plateau claim is only demonstrated in a finite-mode truncation, and the review's own toy-model result warns that the energy-norm plateau may not survive the continuum limit.","tokens_in":10006,"tokens_out":3137,"would_cite":false,"duration_ms":36718,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C35"],"pacs":["04.70.-s","04.30.-w"],"model":"deepseek-v4-flash","headline":"This review argues that the dissipative nature of black holes makes the linear perturbation Hamiltonian non-normal, so quasinormal modes are non-orthogonal and produce long-lived or growing transients before modal decay sets in.","keywords":["quasinormal modes","black hole perturbation theory","non-normal operators","transients","pseudospectra","ringdown","transient growth","Sobolev norms"],"falsifier":"Take the Schwarzschild Regge-Wheeler problem and compute the energy growth curve $G(\\tau)$ at increasing spectral resolution $N$; if the transient plateau duration continues to scale as $\\log N$ without converging as $N\\to\\infty$, the continuum existence of arbitrarily long-lived linear perturbations is falsified. A complementary check is to evaluate the pseudospectral abscissa $\\alpha_\\epsilon(H)$ of the untruncated operator and test whether the Kreiss constant $K(H)$ exceeds 1 in the continuum limit.","tokens_in":8935,"feed_emoji":"🕳️","tokens_out":5968,"duration_ms":59677,"temperature":0.7,"pith_summary":"This review argues that the dissipative nature of black holes makes the linear perturbation Hamiltonian non-normal, so quasinormal modes are neither orthogonal nor complete under standard inner products. A direct consequence is that sums of quasinormal modes can display transient behaviour, either long-lived plateaus or actual growth, before settling into the familiar exponential modal decay. The paper assembles evidence from frequency-domain pseudospectra and from time-domain optimal perturbation constructions in Schwarzschild, Reissner-Nordström anti-de Sitter, and the Pöschl-Teller de Sitter toy model. A sympathetic reader should care because these transients are in principle observable, potentially contaminating black hole spectroscopy and holographic thermalization predictions.","feed_headline":"Black hole ringdown sums can outlive the expected decay","feed_subtitle":"Non-orthogonal quasinormal modes let energy linger near the horizon or grow transiently before modal decay.","key_machinery":"The central object is the non-normal evolution operator $H$ in $i\\partial_\\tau u = Hu$ on hyperboloidal slices, whose eigenfunctions are the quasinormal modes $\\chi_n$ with complex frequencies $\\omega_n$. Non-normality means the quasinormal modes fail orthogonality under the energy inner product $\\langle\\cdot,\\cdot\\rangle_E$, so the energy of a superposition is not the sum of individual mode energies: cross-terms appear. The paper uses two complementary tools to expose the consequences: the pseudospectrum $\\sigma_\\epsilon(H)$ with the Kreiss constant $K(H)$, which bounds possible transient growth, and time-domain optimal perturbations constructed by singular value decomposition of the finite-rank evolution operator, which realise the maximum of the energy growth curve $G(\\tau)$. In the Sobolev $H^p$ norms the truncation to a mode subspace is replaced by higher-derivative regularity, shifting the transient peak to shorter times.","core_discovery":"The central claim is that non-normality of the Hamiltonian, caused by energy flowing through the horizon and out to null infinity, is not a technical nuisance but the origin of a class of linear phenomena. Because quasinormal modes are non-orthogonal with respect to the energy inner product, the energy of a multimode perturbation contains cross-terms that can cancel or amplify. These cross-terms sustain energy packets near the future horizon and null infinity for times scaling with the number of modes, or produce genuine transient energy growth when the perturbation can borrow from a coupled field or when measured in higher-derivative Sobolev norms. The review presents the strongest established cases: arbitrarily long-lived perturbations in Schwarzschild, transient superradiance in Reissner-Nordström anti-de Sitter branes, and $H^p$ growth in the de Sitter static patch, all culminating in late-time modal decay.","pith_inferences":["If the transients survive in the continuum limit, gravitational-wave ringdown templates that assume early modal decay could misattribute a portion of the signal, so one could look for plateau-like persistence in numerical relativity waveforms.","The non-convergence of the $H^0$ plateau duration with grid size suggests that only higher-derivative norms may capture a true continuum effect, and a fully resolved continuum computation would be the test.","The open nonlinear question points to a concrete numerical experiment: evolve the optimal perturbations identified here beyond linear order to see whether transient linear energy becomes a seed for nonlinear instability or turbulence-like behaviour."],"forward_implications":["In Schwarzschild, optimal sums of the first $M$ Regge-Wheeler quasinormal modes keep their energy constant for a duration $\\sim \\log M$ before decaying at the fundamental rate, so linear theory contains arbitrarily long-lived packets for sufficiently large $M$.","In the Reissner-Nordström anti-de Sitter black brane, a stable spectrum still allows the scalar-field energy to grow transiently by borrowing from the gauge field through a transient form of superradiance.","In Sobolev $H^p$ norms the transient peak grows as $G(\\tau_{\\max})\\sim p$ and occurs at $\\tau_{\\max}\\sim 1/p$, with the peak dominated by the $(p+1)$-th quasinormal-mode pair.","Pseudospectral protrusions into the unstable half-plane, quantified by a Kreiss constant $K>1$, predict such growth; for Schwarzschild in the energy norm $w(H)=0$ and $K=1$, consistent with the observed lack of energy growth."],"supporting_citations":[{"why":"Supplies the time-domain optimal perturbation construction and the analytic and numerical demonstration of transient plateaus in Schwarzschild.","marker":"[22]"},{"why":"Establishes the first transient energy growth for Reissner-Nordström anti-de Sitter charged scalar perturbations via transient superradiance.","marker":"[29]"},{"why":"Introduces Sobolev $H^p$ norms and demonstrates $H^p$ transient growth in the Pöschl-Teller de Sitter model.","marker":"[26]"},{"why":"Computes the numerical abscissa and Kreiss constant for Schwarzschild in the energy norm, showing no energy growth.","marker":"[20]"},{"why":"Introduces pseudospectrum analysis for black hole quasinormal-mode instability, motivating the frequency-domain perspective.","marker":"[17]"},{"why":"Provides the definitions and bounds for pseudospectra, the Kreiss constant, and the numerical abscissa used throughout.","marker":"[15]"},{"why":"Establishes the role of the scalar product and defines the energy inner product used in the transient analysis.","marker":"[16]"}],"fun_headline_variants":["Black hole ringdown energy can linger near horizon","Quasinormal modes non-orthogonal: energy can grow transiently","Transient plateaus: black hole modes outlive expected decay","Non-normal black hole Hamiltonians yield transient growth","Energy packets from black hole modes can exceed decay times"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that a computation with finitely many quasinormal modes or finitely many grid points faithfully represents the true continuum dynamics, and the evidence for arbitrarily long-lived perturbations would collapse if the plateau duration keeps growing with resolution instead of settling down.","fun_headline_variants_meta":{"raw":{"variants":["Black hole ringdown energy can linger near horizon","Quasinormal modes non-orthogonal: energy can grow transiently","Transient plateaus: black hole modes outlive expected decay","Non-normal black hole Hamiltonians yield transient growth","Energy packets from black hole modes can exceed decay times"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000911,"raw_usage":{"total_tokens":3856,"prompt_tokens":826,"completion_tokens":3030,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":442,"completion_tokens_details":{"reasoning_tokens":2950}},"tokens_in":442,"tokens_out":3030,"duration_ms":24104,"temperature":1.0,"reasoning_tokens":2950,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:07:55.689351+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the Schwarzschild Regge-Wheeler problem and compute the energy growth curve $G(\\tau)$ at increasing spectral resolution $N$; if the transient plateau duration continues to scale as $\\log N$ without converging as $N\\to\\infty$, the continuum existence of arbitrarily long-lived linear perturbations is falsified. A complementary check is to evaluate the pseudospectral abscissa $\\alpha_\\epsilon(H)$ of the untruncated operator and test whether the Kreiss constant $K(H)$ exceeds 1 in the continuum limit.","supporting_citations":[{"cited_title":"Transient dynamics of quasinormal mode sums","cited_arxiv_id":null,"evidence_quote":"Supplies the time-domain optimal perturbation construction and the analytic and numerical demonstration of transient plateaus in Schwarzschild."},{"cited_title":"Pseudospectrum and binary black hole merger transients","cited_arxiv_id":null,"evidence_quote":"Computes the numerical abscissa and Kreiss constant for Schwarzschild in the energy norm, showing no energy growth."},{"cited_title":"Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators (Princeton University Press) (2005)","cited_arxiv_id":null,"evidence_quote":"Provides the definitions and bounds for pseudospectra, the Kreiss constant, and the numerical abscissa used throughout."}],"review_version":1}