{"id":"1084f46c-d7d7-4c07-b170-002e4f3f6972","arxiv_id":"2610.13612","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The non-linear evolution of the Gregory-Laflamme instability in Einstein-Gauss-Bonnet gravity terminates through a loss of hyperbolicity, with the type (Tricomi or Keldysh) depending on the sign of the Gauss-Bonnet coupling.","lead":"This paper studies how the Gregory-Laflamme instability of black strings behaves in Einstein-Gauss-Bonnet gravity, a modification of general relativity with higher-derivative corrections. It finds that the instability grows faster for positive coupling but the evolution breaks down before reaching the endpoint due to loss of hyperbolicity, indicating when the effective theory ceases to be predictive.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's weakest_assumption (symmetry reduction) is the most significant constraint, but it is acknowledged and does not undermine the internal logic of the paper. The central claim is correctly derived from that assumption and supported by simulations. The verdict remains ACCEPT with high confidence.","tokens_in":34338,"tokens_out":3149,"duration_ms":29369,"concrete_test":"Independently recompute the eigenvalues F_a of the effective metric on the UBS backgrounds of Section II using a separate numerical code (e.g., a direct spectral solver for the static EGB equations), verifying that the threshold λ_GB/r_H^2 = (√2−1)/4 ≈ 0.1036 is reproduced. If the threshold matches, the central claim is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that for any positive Gauss-Bonnet coupling an open set of uniform black string solutions exhibits a degenerate effective metric and thus cannot be attained dynamically—is supported by a clear analytic derivation of the effective metric (Section IV.B), an explicit diagnostic (Section IV.C), and consistent non-linear simulations (Section V). The main limitation, the SO(3) symmetry reduction, is explicitly acknowledged by the authors (Section VI), and it provides a sufficient condition for loss of hyperbolicity; even if the full theory breaks hyperbolicity at a different (possibly smaller) coupling, the existence of a degenerate polarization in the symmetric sector already invalidates the Cauchy problem for those backgrounds. No internal inconsistency or unaccounted gap was found. The non-linear simulations, though at very small couplings, agree with the picture qualitatively and quantitatively (breakdown at local coupling ≈0.08 vs. threshold ≈0.104). The paper is thorough and honest about its scope.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"This paper studies the Gregory-Laflamme (GL) instability of uniform black strings (UBS) in five-dimensional Einstein-Gauss-Bonnet (EGB) gravity. The authors first perform a linear stability analysis, computing zero modes and time-dependent growth rates as functions of the Gauss-Bonnet coupling λ_GB. They find that positive λ_GB makes the string more unstable (larger growth rates, shorter unstable wavelengths), while negative λ_GB initially stabilises it, though the trend reverses near the singular limit λ_GB/r_H^2 → −1/4. In the second part, they specialise to SO(3)-symmetric perturbations (the cartoon method) and derive an effective metric g^eff_{ab} that governs the propagation of the single physical degree of freedom. By analysing this effective metric on the horizon of UBS backgrounds, they show that for 1/4(√2−1) < λ_GB/r_H^2 ≤ 1/8 the effective metric becomes degenerate (Euclidean signature) in a neighbourhood of the horizon. They argue that such UBS solutions, though stationary and regular, cannot be reached by Cauchy evolution from weakly coupled initial data, and that linearised modes around them lack dynamical meaning. The paper complements these analytic results with non-linear simulations (at very small couplings, |λ_GB|/r_0^2 = 10^{-4}, 10^{-5}) that terminate through loss of hyperbolicity: a Tricomi-type transition for λ_GB > 0 and a Keldysh-type transition for λ_GB < 0.","tokens_in":34594,"tokens_out":3704,"duration_ms":38082,"significance":"If the central claim holds — that for any positive λ_GB an open set of regular UBS solutions lie outside the regime where EGB gravity has a well-posed initial-value problem — it has important implications for the interpretation of stationary solutions in effective field theories of gravity. It shows that regularity of a solution of the covariant field equations does not guarantee dynamical attainability, and that linear perturbation analyses around such backgrounds can be misleading. The paper provides a concrete, analytically derived diagnostic (the effective metric and its generalised eigenvalues Δ_±) that can be applied to other backgrounds and theories. The non-linear simulations, though at very weak coupling, demonstrate the predicted breakdown mechanism and correlate the local coupling at breakdown with the analytic threshold. The work is self-consistent, careful about its assumptions (especially the SO(3) symmetry reduction, which is explicitly acknowledged as providing an upper bound on the hyperbolicity threshold), and offers a clear conceptual distinction between physical instabilities and artefacts of ill-posedness.","major_comments":[],"minor_comments":[{"comment":"The paper states that strong hyperbolicity of the full mCCZ4 system is established in [23], which is listed as '(2026)' with no journal reference. Since this result is used to justify the well-posedness of the formulation in the weakly coupled regime, the authors should either ensure that [23] is published or provide a brief summary of the relevant part to allow the reader to verify the claim without chasing an unpublished source.","section":"Section IV.A (and reference [23])","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The diagnostics Δ_± are defined as generalised eigenvalues of Δ^ij with respect to Δ^{ij}_{GR}, and the text claims they are 'invariant under spatial coordinate changes'. While this is plausible because they are eigenvalues of a mixed tensor, a brief justification or reference to a known fact would be helpful. The current phrasing might leave some readers wondering about the volume factor or conformal invariance.","section":"Section IV.C (Eq. (60))","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The footnote states that N^{w1 w2}_{w1 w2} > 0 on UBS horizons but also that it is positive wherever ∇_a R ∇^a R ≤ 1. This condition appears to hold in the exterior of a black string but may fail inside the horizon. Since the effective metric is evaluated in the exterior for the main argument, this is not a problem, but a brief remark on where the assumption could break down would increase clarity.","section":"Section IV.B (footnote 9)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The comparison between the non-linear breakdown (local coupling ≈0.08) and the UBS threshold (≈0.104) is described as 'heuristic' and the authors note that the string segments are not uniform. This disclaimer is appropriate, but the paragraph would benefit from a one-sentence recapitulation of why the comparison is still informative (e.g., consistency with the bound being an upper limit).","section":"Section V.C","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The phrase 'the effective field theory becomes strongly coupled' is used to describe when the effective metric degenerates. A brief reminder in the text (or a cross-reference to Section IV.C) that 'strongly coupled' here means the dimensionless combination λ_GB/r_H^2 exceeds the hyperbolicity threshold, rather than a Wilsonian RG notion, would avoid ambiguity for readers unfamiliar with the EFT-in-gravity literature.","section":"Abstract and Section I","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"minor_revision","confidential_remarks":"The paper is thorough, honest about its limitations, and the central claim is convincingly supported by the analytic derivation and numerical checks. The main reason for 'minor revision' rather than 'accept' is the reliance on reference [23] (an unpublished work) for the strong hyperbolicity of the full formulation. Since the paper's main conclusion about the effective metric is independent of [23], this is not a fatal issue, but the journal may require that all cited results be publicly available or at least clearly summarised. The paper otherwise meets the standard for a well-executed analysis in numerical relativity / EFT gravity. The authors have been careful to separate what is proven from what is conjectured, and the discussion of the symmetry-reduction limitation is exemplary."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"This is a thorough paper. The central result—that for any positive Gauss-Bonnet coupling an open set of uniform black string solutions cannot be reached by Cauchy evolution because the effective metric degenerates—is well-supported by an analytic derivation, an explicit diagnostic, and consistent non-linear simulations. The time-dependent growth rates and the Tricomi/Keldysh classification of the breakdown are genuinely new.\n\nWhat I like most is the clean derivation of the effective metric in the SO(3)-symmetric sector and the eigenvalue diagnostics Δ± that are gauge-invariant. The linear stability analysis is numerically sound (cross-checked via zero modes and time-dependent perturbations), and the non-linear simulations, though at very small couplings (10⁻⁴, 10⁻⁵), match the analytic picture. The authors are explicit about the limitations: the symmetry reduction may give only an upper bound on the coupling where hyperbolicity fails, and the comparison with the UBS threshold is heuristic.\n\nThe soft spots are proportionally minor. The SO(3) reduction is the relevant symmetry for the black-string evolution, but the full theory has more polarizations, so the quoted bound (λ/r²_H ≈ 0.104) is a sufficient condition, not a necessary one—the authors say this themselves. The non-linear simulations use very small couplings and a gradual switch-on, so the quantitative comparison with the threshold is suggestive rather than definitive. These are not flaws; they are honest scope limitations.\n\nWho is this for? People working on higher-dimensional black holes, EFTs of gravity, and well-posedness in modified gravity. It is a solid, self-contained contribution that complements the companion paper [19].\n\nA serious editor should send this to peer review. The math is sound, the evidence supports the claims, and the paper is upfront about what it does and does not prove.","headline":"A clean, careful paper that shows when and how the EGB evolution breaks down for black strings, with an explicit effective-metric diagnostic. The central claim is well-supported, and the limitations are open and acknowledged.","tokens_in":35106,"tokens_out":1280,"would_cite":false,"duration_ms":21201,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["83C57","83C75","83E15"],"pacs":["04.50.Kd","04.70.Bw","04.25.D-"],"model":"deepseek/deepseek-v4-flash","headline":"Effective metric degeneracy rules out a class of black string solutions in Einstein-Gauss-Bonnet gravity","keywords":["Gregory-Laflamme instability","Einstein-Gauss-Bonnet gravity","black strings","hyperbolicity","effective metric","loss of hyperbolicity","Cauchy evolution","effective field theory"],"falsifier":"A fully non-linear simulation without SO(3) symmetry that reaches a uniform black string with $\\lambda_{\\rm GB}/r_H^2 > (\\sqrt{2}-1)/4$ and continues to evolve stably past the point where the effective metric would become degenerate would falsify the paper's central claim.","tokens_in":35832,"feed_emoji":"🕳️","tokens_out":4288,"duration_ms":46317,"temperature":0.7,"pith_summary":"This paper argues that in Einstein-Gauss-Bonnet gravity—the leading higher-curvature correction to general relativity—a range of uniform black string solutions, including the one of minimal thickness, cannot be reached by time evolution from any admissible initial data. The authors show that the propagation of the single physical degree of freedom under the relevant symmetry is governed by an effective metric that depends on the Gauss-Bonnet coupling and spacetime curvature. For any positive coupling, this effective metric becomes Euclidean on and near the horizon of those solutions, making the Cauchy problem ill-posed. Consequently, the linearized perturbations around these backgrounds are not approximations to nearby non-linear evolutions, and the non-linear Gregory-Laflamme instability terminates in a loss of hyperbolicity before the string approaches the putative endpoint.","feed_headline":"No Cauchy path to thinnest black strings in EGB gravity","feed_subtitle":"Even regular solutions above a hyperbolicity threshold cannot be dynamically reached, making linear instability results formal.","key_machinery":"The key is the effective metric $g^{ab}_{\\rm eff}$ derived from the principal symbol of the symmetry-reduced Einstein-Gauss-Bonnet equations under $SO(3)$ symmetry. This metric differs from the spacetime metric by curvature terms proportional to $\\lambda_{\\rm GB}$. When it ceases to be Lorentzian—specifically when its spatially reduced eigenvalues become negative—the characteristic speeds are imaginary and the initial value problem becomes ill-posed. The paper derives a threshold $\\lambda_{\\rm GB}/r_H^2 > (\\sqrt{2}-1)/4 \\approx 0.104$ for uniform black strings, beyond which the effective metric degenerates on and near the horizon.","core_discovery":"The central discovery is that the condition for well-posed Cauchy evolution in Einstein-Gauss-Bonnet gravity—Lorentzian signature of an effective metric derived from the principal symbol—fails on a non-empty set of regular uniform black string solutions for any positive Gauss-Bonnet coupling. This set includes the uniform black string of minimal thickness that would otherwise be a plausible endpoint of the Gregory-Laflamme instability. Consequently, this endpoint cannot be reached by time evolution from generic initial data; the solutions of the linearized equations around these backgrounds are formal and do not approximate non-linear evolutions. The non-linear simulations corroborate this: they terminate through a Tricomi-type loss of hyperbolicity for positive coupling and a Keldysh-type for negative coupling, both inside the apparent horizon.","pith_inferences":["The bound $\\lambda_{\\rm GB}/r_H^2 > (\\sqrt{2}-1)/4$ is derived within the $SO(3)$-symmetric sector; perturbations breaking this symmetry may cause loss of hyperbolicity at smaller couplings, as the paper acknowledges. Therefore the actual range of well-posed evolution for generic black strings may be even narrower.","This analysis provides a template for diagnosing the dynamical relevance of stationary solutions in effective field theories of gravity: one can test whether the effective metric of the symmetry-reduced system remains Lorentzian on and outside the horizon.","The results suggest that if string theory or a UV completion resolves the Gregory-Laflamme endpoint, it must do so before the effective metric degenerates, i.e., at curvatures well below the string scale for positive Gauss-Bonnet coupling."],"forward_implications":["Any uniform black string with $\\lambda_{\\rm GB}/r_H^2 > (\\sqrt{2}-1)/4$ is dynamically inaccessible; thus the limiting string of minimal thickness $r_H = \\sqrt{8\\lambda_{\\rm GB}}$ is not an attainable endpoint of the Gregory-Laflamme instability.","Linear perturbation analyses on such backgrounds yield formal solutions that do not correspond to nearby non-linear evolutions; growth rates computed for couplings above the threshold are not physically meaningful.","The non-linear Gregory-Laflamme instability in Einstein-Gauss-Bonnet gravity always terminates through a loss of hyperbolicity before the string reaches the thinness of the limiting static solution.","For $\\lambda_{\\rm GB} < 0$, the effective metric remains Lorentzian on and outside the horizon, so the evolution may in principle continue closer to the singular limit, but practical simulations break down due to stiffness."],"supporting_citations":[{"why":"Provides the non-linear numerical evidence that black string simulations in EGB gravity terminate via loss of hyperbolicity, supporting the paper's main conclusion.","marker":"[19]"},{"why":"First constructed uniform black string solutions in EGB gravity, establishing the existence of the branch up to $\\lambda/r_H^2=1/8$.","marker":"[26]"},{"why":"Computes zero modes and entropy of uniform black strings in EGB gravity, confirming the critical wavenumber behavior that the paper extends.","marker":"[27]"},{"why":"Extends the study to non-uniform strings in EGB gravity, relevant for the endpoint of the instability.","marker":"[28]"},{"why":"Previously argued that loss of hyperbolicity implies stationary solutions are not dynamically accessible, which the paper builds upon for its claim.","marker":"[63]"},{"why":"Proves strong hyperbolicity of the mCCZ4 formulation for EGB gravity in the weakly coupled regime, establishing the baseline for the analysis.","marker":"[23]"}],"fun_headline_variants":["Black strings in EGB: linear instabilities, no real evolution","Effective metric breakdown halts black string thinning","No Cauchy path to thinnest EGB black strings","Hyperbolicity loss blocks black string endpoint in EGB gravity"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analysis assumes SO(3) symmetry in the asymptotically flat directions, which reduces the dynamics to a single physical degree of freedom; the derived bound on hyperbolicity may only be an upper bound because perturbations breaking this symmetry could lose hyperbolicity at smaller couplings.","fun_headline_variants_meta":{"raw":{"variants":["Black strings in EGB: linear instabilities, no real evolution","Effective metric breakdown halts black string thinning","No Cauchy path to thinnest EGB black strings","Hyperbolicity loss blocks black string endpoint in EGB gravity"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000148,"raw_usage":{"cost":0.000147574812,"is_byok":false,"cost_details":{"upstream_inference_cost":0.000147574812,"upstream_inference_prompt_cost":7.3796492e-05,"upstream_inference_completions_cost":7.377832e-05},"total_tokens":1423,"prompt_tokens":1017,"completion_tokens":406,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":256,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":338}},"tokens_in":1017,"tokens_out":406,"cost_source":"openrouter_usage","duration_ms":5344,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":338,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-12T13:28:30.294283+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"A fully non-linear simulation without SO(3) symmetry that reaches a uniform black string with $\\lambda_{\\rm GB}/r_H^2 > (\\sqrt{2}-1)/4$ and continues to evolve stably past the point where the effective metric would become degenerate would falsify the paper's central claim.","supporting_citations":[{"cited_title":"Non-linear evolution of five-dimensional black strings in effective field theory","cited_arxiv_id":"2606.05319","evidence_quote":"Provides the non-linear numerical evidence that black string simulations in EGB gravity terminate via loss of hyperbolicity, supporting the paper's main conclusion."},{"cited_title":"Five-dimensional black strings in Einstein-Gauss-Bonnet gravity","cited_arxiv_id":"gr-qc/0412139","evidence_quote":"First constructed uniform black string solutions in EGB gravity, establishing the existence of the branch up to $\\lambda/r_H^2=1/8$."},{"cited_title":"The fate of black branes in Einstein-Gauss-Bonnet gravity","cited_arxiv_id":"0810.0525","evidence_quote":"Computes zero modes and entropy of uniform black strings in EGB gravity, confirming the critical wavenumber behavior that the paper extends."},{"cited_title":"Einstein-Gauss-Bonnet black strings","cited_arxiv_id":"1004.2164","evidence_quote":"Extends the study to non-uniform strings in EGB gravity, relevant for the endpoint of the instability."},{"cited_title":"Causality and Hyperbolicity of Lovelock Theories","cited_arxiv_id":"1406.3379","evidence_quote":"Previously argued that loss of hyperbolicity implies stationary solutions are not dynamically accessible, which the paper builds upon for its claim."},{"cited_title":"high- frequency gauge transformations","cited_arxiv_id":null,"evidence_quote":"Proves strong hyperbolicity of the mCCZ4 formulation for EGB gravity in the weakly coupled regime, establishing the baseline for the analysis."}],"review_version":1}