{"id":"d72a04b0-ab1c-43e3-862d-14fb5d841752","arxiv_id":"2506.18991","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Gravitational perturbations of Schwarzschild-AdS5 with analytic SO(3)-symmetric data decay at late times as v^{-2α/C}, with subleading oscillations periodic in log v with period C(y+).","lead":"This paper studies gravitational perturbations of a Schwarzschild black hole in five-dimensional anti-de Sitter spacetime and finds that smooth, symmetric perturbations decay at late times as a power law, modulated by a slow oscillation in the logarithmic time. The result bears on whether these black holes are stable, which matters for the AdS/CFT duality linking gravity to quantum field theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The prediction assumes n=0 QNMs dominate, but Fig. 6 shows overtones are not small; the peak-offset argument only shifts the peaks, and the overtone tail slopes alpha_n needed to establish subdominance are never measured.","rationale":"The reader's weakest assumption is the right one: linear QNM dominance with the fundamental n=0 modes controlling the late-time tail. My reading of Sections 2.3 and 3.1 sharpens this into a concrete, checkable gap. The paper admits in Section 3.1 that 'this assumption does not hold' ('we see more and more overtones contributing to the spectrum as l increases, and the overtone contributions are not small compared to the n=0 mode'). The rebuttal in Section 2.4, that each overtone peak is shifted by a fixed number of modes, only relocates each overtone's saddle point; it does not reduce the amplitude of that saddle, which decays as v^{-2 alpha_n / C}. For the n=0 exponent to be correct, one needs alpha_n > alpha_0 for every overtone family carrying non-negligible initial amplitude, or at least no alpha_n < alpha_0, and the finite-time contribution of overtones with alpha_n approximately equal to alpha_0 must be quantified. The paper does not report alpha_n, and its overtone-inclusive extrapolation is limited to resolvable overtones at l <= 32 and log v <= 42. This is a genuine gap in the central quantitative claim, not merely a presentation issue. The numerical method and QNM fitting are otherwise careful; the overtone resolution in Figs. 4-5 and the dealiasing discussion are real strengths. A secondary concern is that the WKB asymptotic regime (l >= 58 for y+ = 0.5) is not reached by the runs, but the overtone spectral-slope issue is more directly load-bearing because it affects the exponent even in the asymptotic limit. The conditional verdict remains appropriate.","tokens_in":27272,"tokens_out":23615,"duration_ms":265822,"concrete_test":"Using the spectra shown in Fig. 6 (or the QNM decomposition used for the Fig. 7 extrapolation), fit the logarithmic tail slope alpha_n for each resolvable overtone n in the same l-window used for the n=0 fit. Compute each overtone family's late-time contribution v^{-2 alpha_n / C} using its fitted alpha_n and the shifted-peak offset (kappa_n - log(alpha_n/C))/C. If any n>=1 overtone has alpha_n < alpha_0 within the fit uncertainty, or if the sum over resolved overtones changes d log||V4||^2/d log v by more than about 0.05 at log v = 42, then the n=0-only prediction (2.28) is not the correct late-time exponent. If all resolved overtones have alpha_n > alpha_0 with a clear margin and their summed contribution is subdominant, the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The derivation of the late-time exponent (2.28) explicitly assumes that the norm is controlled by the fundamental (n=0) quasinormal modes: Section 2.3 says 'We ignore higher overtones n>=1 for each l as these decay faster than the fundamental n=0 modes.' Section 3.1 then states that this assumption does not hold in the actual data: Fig. 6 shows overtone contributions are not small compared to the n=0 mode at high l. The rescue argument in Section 2.4 is that each overtone's spectral peak sits at a fixed offset from the n=0 peak, but this only shifts the peak; the amplitude of overtone n at its own peak decays as v^{-2 alpha_n / C}, where alpha_n is the tail slope of that overtone's spectrum. The n=0 exponent survives only if alpha_n > alpha_0 for every n (or alpha_n = alpha_0, in which case the exponent is unchanged but the prefactor is). No alpha_n values are reported, and the overtone-inclusive QNM extrapolation is limited to resolvable overtones at l <= 32 and to log v <= 42. The central quantitative claim therefore depends on an unverified spectral property of the overtones.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies gravitational perturbations of five-dimensional Schwarzschild-AdS black holes restricted to an SO(3)-symmetric sector, using ingoing Bondi-Sachs coordinates and fully nonlinear numerical evolutions. From two assumptions—exponentially decaying large-ℓ mode amplitudes for analytic initial data and exponentially decaying quasinormal-mode (QNM) decay rates at large ℓ—the authors derive a late-time power law for the boundary quantity ||V4||^2, with exponent -2α/C and subleading oscillations periodic in log v with period C(y_+). They test this prediction for two horizon sizes, y_+ = 0.5 and y_+ = 1.0, fitting the spectral slope α from the numerical data and reporting agreement with the predicted late-time slope, while finding no sign of instability over 4,000 crossing times (y_+ = 0.5) and 550 crossing times (y_+ = 1.0). The paper also presents a global bulk norm ||I1||^2 as a secondary diagnostic and discusses the slow approach to the asymptotic regime.","tokens_in":27495,"tokens_out":7022,"duration_ms":69740,"significance":"If the main claim holds, the paper would replace the inverse-logarithmic decay picture for smooth SO(3)-symmetric perturbations of Schwarzschild-AdS_5 with a computable power law and would provide evidence for nonlinear stability in this sector, in line with the arguments of Dias et al. The derivation of the power-law functional form, the prediction of the log-periodic oscillations, and the careful treatment of the numerical approach are valuable contributions. The paper is also unusually candid about its limitations, including the finite evolution time, the limited angular resolution, and the reliance on extrapolation for the smaller black hole. The main factor determining the significance is whether the overtone contributions, which the paper itself shows are not negligible, can be properly controlled; as it stands, the quantitative prediction rests on an unverified spectral property of the overtones.","major_comments":[{"comment":"The derivation of Eq. (2.28) begins with the assumption that higher overtones n≥1 decay faster than the fundamental n=0 mode and can be ignored. However, Fig. 6 (and Fig. 5, which resolves overtones up to n=7 at ℓ=32) shows that overtone contributions to the V4 spectrum are not small compared with the n=0 mode at high ℓ. The argument in Sec. 2.4 that each overtone's peak sits at a fixed offset from the n=0 peak only shifts the peak; for each overtone n, the contribution to the norm decays as v^{-2α_n/C} with its own tail slope α_n, so the n=0 exponent survives only if α_n ≥ α_0 for every n. No α_n values are measured or reported, and the overtone-inclusive extrapolation is restricted to ℓ≤32 and log v≤42. The central quantitative claim therefore depends on an unverified spectral property of the overtones, and the manuscript should either establish α_n ≥ α_0 or provide a direct measurement of the overtone tail slopes.","section":"Sec. 2.3–2.4 and Fig. 6"},{"comment":"For y_+ = 0.5, the late-time log-log slope in the actual nonlinear evolution has not converged to the predicted value -2α/C within the 4,000 crossing times shown; the agreement is achieved through a QNM-decomposition extrapolation to log v = 42. Since this extrapolation uses the same linear-QNM and n=0-dominance assumptions that underlie the derivation of (2.28), it cannot independently validate the power-law exponent. The manuscript's statement that the nonlinear evolutions 'support the prediction' should be correspondingly qualified, and the separate status of the extrapolated and directly observed slopes should be made explicit in the main text.","section":"Sec. 3.1, Fig. 7"},{"comment":"The exponent -2α/C is evaluated using α fitted from the tail of the n=0 spectrum of the same numerical evolution (Fig. 6 at v=8 and v=80; Fig. 9), rather than from an independent characterization of the initial data. As a result, the agreement between the late-time slope and Eq. (2.28) is partially a self-consistency check between two quantities extracted from the same run. This does not invalidate the result, but it weakens the force of the word 'predict' in the abstract and introduction. The authors should state explicitly whether α can be obtained directly from the initial data (for instance from its analyticity domain) and, if so, whether the resulting prediction matches the observed slope.","section":"Secs. 3.1–3.2, Eq. (2.28)"}],"minor_comments":[{"comment":"The estimate for the time v_ℓ at which mode ℓ begins to dominate is stated without derivation; a short explanation of the formula log(v_ℓ) = Cℓ - κ + log(α) - log(e^C - 1) would improve readability.","section":"Sec. 2.4"},{"comment":"The notation ||I_1||^2 is potentially confusing because I_1 is itself defined as the squared Weyl scalar C_abcd C^abcd; consider using a different symbol or explicitly stating that this is the square of the L^2 norm of I_1 over the hypersurface.","section":"Eq. (2.36)"},{"comment":"The panels showing the derivative of log||V4||^2 with respect to log v lack explicit axis labels and legends for the shaded prediction regions; please add them for clarity.","section":"Figs. 7 and 10"},{"comment":"The parameter y_+ is used in the abstract and throughout without definition; please define it at first occurrence, for example as the horizon radius in AdS units.","section":"Abstract and Sec. 2.1"},{"comment":"There are several typographical issues, including 'invis given byπy +' in Sec. 3.1 and an incomplete reference entry for [111] (missing year). Please correct these.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a serious numerical study with a clear and honest presentation, and the functional-form derivation is a useful contribution. The main obstacle to acceptance is the overtone question: the paper's own figures show that overtones are not negligible in the spectra, yet the prediction (2.28) assumes they are subdominant in a way that is never quantitatively checked. The extrapolation argument for y_+ = 0.5 also shares the same assumptions as the prediction, so the claimed agreement is not as strong as the abstract suggests. If the authors can supply a direct measurement of α_n or a convincing argument that α_n ≥ α_0, the paper would likely meet the standard for publication. There is no indication of any incorrectness in the numerical implementation or misrepresentation of the cited literature; the issue is the gap between the strength of the claim and the evidence provided."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe paper you'd want to know about: Crump and Santos claim smooth, SO(3)-symmetric gravitational perturbations of Schwarzschild-AdS5 decay polynomially — not 1/log t — with a universal log-periodic modulation, and they back it with fully nonlinear evolutions past 4,000 AdS crossing times. On reading, I think the claim mostly holds, with qualifications.\n\nWhat's genuinely new: the explicit Poisson-summation formula for the subleading oscillations (Eq. 2.33-2.35), and the numerical campaign. The numerics are careful: overtone fits up to n=7 at l=32, dealiasing in the natural basis, two independent norms, two horizon sizes. The appendix gives enough to reproduce the method, though no code or data is released. The paper is honest about what it cannot see: it says plainly that a later instability is not excluded and that the asymptotic regime may take exponentially long to reach.\n\nThe soft spots are real but manageable. The exponent -2α/C is not predicted from first principles: α is fitted from the same evolution, so the quantitative law is partly a scaled fit. For y+=0.5 the late-time slope has not converged; the reach to log v = 42 comes from an extrapolation using the early-time QNM decomposition. That extrapolation includes the resolvable overtones and is indistinguishable from n=0-only, which addresses the 'overtones are not small' concern in Fig. 6 more than the stress-test note credits. What remains unproven is the spectral property alpha_n > alpha_0 for all n beyond l=32, so the n=0-dominance argument is plausible but not closed. The y+=0.5 alpha fit also has no quoted error, which should be a straightforward fix.\n\nThe central physical claim — power law rather than 1/log t for smooth data in this sector — is well supported. The disagreement with Figueras & Rossi is not fully settled, since that work uses different data and less symmetry, but the contradiction for smooth data is addressed head-on.\n\nWho is this for? Anyone working on AdS stability, QNM tails, or holographic thermalization. It deserves a serious referee. I'd send it out with a request for data/code and error bars, and ask that the overtone-tail slope question be either answered explicitly or the claim softened. This is a solid, useful paper, not a complete proof.","headline":"A careful numerical study that makes a credible case for power-law decay with log-periodic modulation in Schwarzschild-AdS5, but with an exponent that is partly fitted and a late-time approach that is slower than the runs themselves.","tokens_in":28095,"tokens_out":2271,"would_cite":true,"duration_ms":25753,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C57","83C05","35B40"],"pacs":["04.70.-s"],"model":"deepseek-v4-flash","headline":"This paper claims that smooth gravitational perturbations of five-dimensional Schwarzschild–AdS black holes decay as a power law rather than 1/log t, and that nonlinear evolutions support the prediction over 4,000 bounces.","keywords":["gravitational perturbations","Schwarzschild-AdS black holes","quasinormal modes","power-law decay","stable trapping","nonlinear stability","AdS/CFT correspondence","eikonal approximation"],"falsifier":"Evolve the $y_+=0.5$ initial data to the times where the paper's QNM-decomposition extrapolation claims the power law becomes clean (about $\\log v\\simeq 42$, or $10^{18}$ crossing times) and compare the log-log slope with $-2\\alpha/C\\simeq -1.11$; any sustained deviation, or any divergence between the nonlinear run and the QNM extrapolation, would falsify the linear-QNM tail picture. A more accessible test: repeat the evolution with non-analytic (compactly supported) initial data and measure whether the decay becomes $1/\\log v$ rather than a power law.","tokens_in":27010,"feed_emoji":"🕳️","tokens_out":11091,"duration_ms":101357,"temperature":0.7,"pith_summary":"This paper claims that smooth, SO(3)-symmetric gravitational perturbations of five-dimensional Schwarzschild–anti-de Sitter (AdS) black holes do not settle down through the widely quoted 1/log t tail. Instead, the late-time decay is a power law, $\\|V_4\\|^2(v) \\propto v^{-2\\alpha/C}$, where $\\alpha$ measures the exponential falloff of the initial data's angular-momentum spectrum and $C(y_+)$ is a constant set by the black hole radius. The same argument predicts subleading oscillations that are periodic in $\\log v$ with period $C$, a fingerprint of the large-angular-momentum (eikonal) sector of the quasinormal-mode spectrum. Fully nonlinear evolutions of two representative black holes, one small and one large, run for over 4,000 AdS wall-crossing times, approach the predicted exponent and show no instability. If correct, the result replaces an inverse-log decay picture with a computable power law and strengthens the case that a sizeable sector of small, smooth perturbations of Schwarzschild-AdS$_5$ is nonlinearly stable.","feed_headline":"Black hole ripples decay by a power law, not 1/log t","feed_subtitle":"Full nonlinear runs over 4,000 bounces match the predicted tail, pointing to stability.","key_machinery":"The load-bearing object is the quasinormal-mode tail of the boundary energy-density perturbation $V_4$, measured by the norm $\\|V_4\\|^2(v)$. The argument combines a WKB/eikonal result for the large-$\\ell$ decay rates, $\\mathrm{Im}\\,\\omega_\\ell \\sim -e^{-C\\ell+\\kappa}$ with $C(y_+)$ given by (2.23), with exponential analyticity of the initial data, $|\\tilde V_{4\\ell}|\\sim e^{-\\alpha\\ell+\\beta}$. Laplace's method converts the mode sum to the power law (2.28), and Poisson summation converts the discrete sum into the log-periodic correction (2.34). The mechanism is that the dominant contribution at time $v$ comes from the modes near the moving peak $\\ell_{\\max}\\sim C^{-1}\\log v$, so the late-time tail is a property of the spectrum's slope $\\alpha$ and the eikonal constant $C$, not of any single low mode.","core_discovery":"On the paper's own terms, the central discovery is that, within the SO(3)-symmetric sector, the late-time gravitational signal of a Schwarzschild-AdS$_5$ black hole is controlled by a moving peak in angular-momentum space. Because high-$\\ell$ quasinormal-mode decay rates degenerate as $\\mathrm{Im}\\,\\omega_\\ell \\sim -e^{-C\\ell+\\kappa}$, and because analytic initial data have exponentially decaying mode amplitudes $|\\tilde V_{4\\ell}|\\sim e^{-\\alpha\\ell+\\beta}$, a saddle-point evaluation of the mode sum gives $\\|V_4\\|^2(v)\\propto v^{-2\\alpha/C}$, with the peak mode moving as $\\ell_{\\max}\\sim C^{-1}\\log v$. Poisson summation adds a universal subleading modulation periodic in $\\log v$ with period $C(y_+)$. The paper verifies this behaviour in fully nonlinear evolutions for $y_+=0.5$ and $y_+=1.0$, including the predicted oscillation period and amplitude in the small-black-hole case, and finds that the decay continues for more than 4,000 bounces without sign of turbulent instability. The same power law is seen in the bulk Weyl-curvature norm $\\|I_1\\|^2$, so the effect is not confined to the boundary quantity $V_4$.","pith_inferences":["The exponent's linear dependence on the initial tail slope $\\alpha$ is a sharp testable signature: if one ran the same black hole with initial data engineered to have a different $\\alpha$, the log-log slope should shift accordingly; failure of that shift would expose the linear-QNM assumption.","The mechanism should be generic to asymptotically AdS spacetimes with stable trapping: any setting where high-$\\ell$ decay rates vanish exponentially in $\\ell$ and initial data are analytic should exhibit a power-law tail with log-periodic modulation, so analogous tails may appear for scalar fields, AdS$_4$, and slowly rotating black holes.","The paper's own extrapolation implies the true power law can be invisible for practically inaccessible times (next $10^{18}$ crossing times for $y_+=0.5$), which may explain why prior finite-time numerics reported $1/\\log t$ decay; distinguishing the two pictures may require the log-periodic phase signature rather than a single slope measurement.","If rough, non-analytic initial data are admitted, the exponential tail $\\alpha$ is replaced by a slower decay and the power-law prediction should break down; this is the natural place to look for the weak-turbulence instability conjectured for lower-regularity data."],"forward_implications":["The decay is predictable from two inputs — the analytic tail slope $\\alpha$ of the initial perturbation and the black-hole-radius-dependent constant $C(y_+)$ — so different initial profiles should yield different but computable power-law exponents.","Because the dominant mode number grows as $\\ell_{\\max}\\sim C^{-1}\\log v$, observing the true power law requires exponentially long runs for larger black holes; the paper's extrapolation for $y_+=0.5$ indicates the regime only becomes clean after roughly $10^{18}$ crossing times.","Higher radial overtones, though long-lived and visible in the spectrum, do not change the exponent because their spectral peaks trail the fundamental-mode peak by a fixed $\\ell$ offset; the paper's extrapolations with and without overtones agree.","No instability develops within 4,000 bounces in either case, supporting the conjecture that smooth perturbations of Schwarzschild-AdS$_5$ in the SO(3)-symmetric sector are nonlinearly stable, in tension with earlier numerical claims of a $1/\\log t$ tail and instability.","The same power law appears in the bulk Weyl invariant $\\|I_1\\|^2$, so the late-time tail is a global spacetime property, not an artefact of the boundary extraction."],"supporting_citations":[{"why":"Argues that asymptotically AdS solutions with non-resonant spectra, including Schwarzschild-AdS, should be nonlinearly stable and predicts power-law decay for smooth data; this paper's prediction is a quantitative version of that claim.","marker":"[29]"},{"why":"Reports a 1/log t decay and numerical evidence of nonlinear instability in Kerr-AdS; the paper's evolutions are constructed to test and contradict that picture for Schwarzschild-AdS.","marker":"[88]"},{"why":"Attributes 1/log t decay to stable trapping; this paper's power-law plus log-periodic prediction is the alternative that supersedes it.","marker":"[87]"},{"why":"Provides the WKB/eikonal computation of C(y+), the constant governing exponential large-ell QNM degeneracy and hence the decay exponent.","marker":"[105]"},{"why":"Shows the high-ell QNM decay rates approach zero exponentially in ell, the property at the base of the power-law tail.","marker":"[104]"},{"why":"Establishes that quasinormal modes are not complete, which frames the paper's stated assumption that they still dominate late-time behaviour.","marker":"[100]"},{"why":"Justifies using the linearised Einstein equation to model the late-time perturbation once the mode is small.","marker":"[98]"}],"fun_headline_variants":["AdS5 black hole tail decays as power law with moving peak","Universal log-periodic oscillation in AdS5 tail decay","Gravitational tail in AdS5: power law, not 1/log t","4000 bounces confirm power-law tail in AdS5 black hole","Moving peak in angular momentum sets AdS5 tail power law"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, once the perturbation is small enough, the late-time dynamics are governed by the linearised Einstein equation and by the fundamental ($n=0$) quasinormal modes, so that nonlinear resonant mode coupling and higher radial overtones do not change the decay exponent even at small amplitude.","fun_headline_variants_meta":{"raw":{"variants":["AdS5 black hole tail decays as power law with moving peak","Universal log-periodic oscillation in AdS5 tail decay","Gravitational tail in AdS5: power law, not 1/log t","4000 bounces confirm power-law tail in AdS5 black hole","Moving peak in angular momentum sets AdS5 tail power law"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001314,"raw_usage":{"total_tokens":5338,"prompt_tokens":914,"completion_tokens":4424,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":530,"completion_tokens_details":{"reasoning_tokens":4331}},"tokens_in":530,"tokens_out":4424,"duration_ms":27090,"temperature":1.0,"reasoning_tokens":4331,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:40:56.164963+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evolve the $y_+=0.5$ initial data to the times where the paper's QNM-decomposition extrapolation claims the power law becomes clean (about $\\log v\\simeq 42$, or $10^{18}$ crossing times) and compare the log-log slope with $-2\\alpha/C\\simeq -1.11$; any sustained deviation, or any divergence between the nonlinear run and the QNM extrapolation, would falsify the linear-QNM tail picture. A more accessible test: repeat the evolution with non-analytic (compactly supported) initial data and measure whether the decay becomes $1/\\log v$ rather than a power law.","supporting_citations":[],"review_version":1}