{"id":"06373e67-3437-4699-84d4-815aa31da857","arxiv_id":"2501.01055","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Two modified CCZ4 schemes that propagate momentum constraint violations without damping suppress a late-time instability seen in long BSSN black hole simulations.","lead":"Long simulations of a single Schwarzschild black hole using a standard numerical relativity method develop a numerical instability after tens of thousands of black hole masses, which the authors trace to growing momentum constraint violations. They propose two modified schemes that let these violations propagate without damping, and report stable evolutions up to 100,000 black hole masses, including for charged and scalarized black holes.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper never directly measures the momentum constraint violations it claims cause the late-time instability, and its evidence is entirely 1D; the causal claim and its generalization to general black-hole spacetimes are not established.","rationale":"The reader's CONDITIONAL verdict is appropriate. My stress-test focuses on the paper's causal mechanism rather than only the 1D-to-3D extrapolation, but the two are linked: if the cause is misidentified, the proposed cure may not transfer. The paper's evidence is internally consistent for the spherical runs—parameter scans, comparisons among schemes, and matter-field examples—and the identification of CCZ3 as most robust in the scalarization case is honest. However, no constraint diagnostics and no convergence analysis mean 'stable' is demonstrated only for the apparent-horizon-area diagnostic at fixed resolution. The printed Eq. (17) omits damping and matter terms, which makes the interpretation of κΘ ambiguous. These issues are correctable with additional diagnostics and a 3D test, so the paper should not be rejected; it should remain conditional pending that evidence.","tokens_in":17184,"tokens_out":11827,"duration_ms":118558,"concrete_test":"Run a 3D (or axisymmetric 2D) Kerr spacetime with a/M=0.9 using BSSN and CCZ3 with the same moving-puncture gauge and boundary treatment, outputting ||Z^i||_2, ||H||_2 (or Θ), and Ah to t≈1e5 M. If CCZ3 remains stable, BSSN exhibits the same late-time instability, and the instability onset tracks growth of ||Z^i||_2, the central claim is supported; otherwise it is not.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section III explicitly restricts to spherically symmetric systems, and no figure or table reports the norm of Z^i, Θ, or the Hamiltonian/momentum constraints for any scheme. The abstract's causal statement—that the BSSN late-time instability arises from accumulated momentum-constraint violations and that undamped propagation of Z^i is the key to long-term stability—is inferred only from scheme-by-scheme comparisons in Figs. 2–3. This leaves open alternative explanations: gauge drift, outer-boundary noise at rmax=60000M, or resolution/dissipation effects (KO epsilon=0.2; no convergence study). Because Z^i has only a radial component in spherical symmetry, the 3D behavior of the proposed schemes is untested; the abstract's claim of resolution 'not only in Schwarzschild spacetimes but also in black hole spacetimes with matter fields' is supported only by spherical RN/EMS runs, and even there Fig. 6 shows CCZ4' does not converge in the scalarization case. A printed inconsistency adds to the concern: Eq. (17) for Θ lacks the κ1 damping and matter terms present in Eq. (6), yet CCZ4' is defined by placing κΘ in Eq. (15) and κΓ in Eq. (18).","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper investigates a late-time numerical instability observed in long-term black hole evolutions with the BSSN formulation, even for a Schwarzschild spacetime. The authors attribute this instability to accumulated violations of the momentum constraint and propose two modified versions of the conformal covariant Z4 system, CCZ4' and CCZ3, distinguished by propagating momentum-constraint violations Z^i without damping while allowing strong damping of the Hamiltonian-constraint violation Θ. Using the BlackHoles@Home platform in spherical symmetry, they show that CCZ4' and CCZ3 maintain apparent-horizon areas close to the expected Schwarzschild value for times up to 10^5 M, whereas BSSN and a comparison scheme CCZ0 (with Z^i set to zero) develop late-time instabilities. They extend the tests to Reissner-Nordström black holes and to spontaneous scalarization in the Einstein-Maxwell-scalar model, reporting that CCZ3 is the most robust, while CCZ4' fails to converge accurately in the scalarization case. The paper's central claim is that undamped propagation of momentum-constraint violations is the key to removing the late-time instability.","tokens_in":17366,"tokens_out":9322,"duration_ms":80599,"significance":"If the central claim holds, the paper offers a simple and practically useful modification for numerical relativity: turning off momentum-constraint damping in CCZ4-type systems, which could enable long-term simulations of weakly unstable black hole systems (e.g., superradiance, light-ring instabilities). The strength of the paper lies in the breadth of parameter variation (outer boundary, resolution, dissipation, CFL) and the use of a known benchmark, the Schwarzschild horizon area, so the stability result is not fitted to a target. However, the significance is currently tempered by three gaps: (i) the causal mechanism is inferred from scheme-by-scheme comparisons without direct measurement of constraint violations; (ii) the evidence is entirely one-dimensional, while the abstract claims resolution of the instability in 'black hole spacetimes with matter fields' in general; and (iii) no convergence study is reported, leaving open the possibility that the observed stability is partly a numerical artifact. The paper also contains a printed inconsistency in the evolution equation for Θ that affects the reproducibility of the CCZ4' scheme.","major_comments":[{"comment":"The paper's central causal claim—that the late-time instability stems from accumulated violations of the momentum constraint—is not directly evidenced. No figure or table reports the norm of Z^i, Θ, or the residuals of the Hamiltonian and momentum constraints for any scheme. The inference relies on comparing the stability of BSSN/CCZ0 (no Z^i propagation) with CCZ4'/CCZ3 (Z^i propagation without damping). Without direct diagnostic data, alternative explanations such as gauge drift, outer-boundary noise, or resolution/dissipation effects cannot be excluded. Please plot, for representative runs, the L2 norms of Z^i and Θ as functions of time for BSSN, CCZ4', CCZ3, and CCZ0, and show that the growth of Z^i correlates with the onset of the late-time instability.","section":"Section III.A, Figs. 1-3"},{"comment":"The evolution equation for Θ in the conformal CCZ4 system, Eq. (17), is inconsistent with the non-conformal version, Eq. (6). Equation (17) lacks the matter source term (-16παρ), the damping term (-ακ1(2+κ2)Θ), and the -Z^i∂_iα term present in Eq. (6). This is not a minor typo because CCZ4' is defined by replacing κ1 in Eq. (15) with κΘ and in Eq. (18) with κΓ, leaving the damping of Eq. (17) unspecified. As printed, κΘ does not damp Θ in its own evolution equation, yet the paper attributes the stability of CCZ4' to Hamiltonian constraint damping. The authors must clarify the actual damping structure used in their CCZ4' implementation and correct Eq. (17) so that the scheme is reproducible and the interpretation is sound.","section":"Section II, Eq. (17)"},{"comment":"No convergence study is reported. The resolution tests in Fig. 3 use NR=200, 300, and 400, but the plots do not show whether the error in the apparent horizon area decreases with resolution, and no convergence order is quoted. Given that the central evidence is numerical stability over 10^5 M, a convergence analysis is necessary to ensure the results are not dominated by the large Kreiss-Oliger dissipation (ϵKO=0.2) or other numerical artifacts. Please provide a convergence test, e.g., Richardson extrapolation of Ah at selected times for at least three resolutions.","section":"Section III, Fig. 3"},{"comment":"The abstract states that CCZ4' and CCZ3 'effectively resolve the late-time numerical instability not only in Schwarzschild spacetimes but also in black hole spacetimes with matter fields.' However, Fig. 6 (upper-left panel) and the text state that the CCZ4' scheme does not converge in the spontaneous scalarization case, at least as far as the apparent horizon is concerned. Thus the abstract overstates the success of CCZ4' in matter spacetimes. Please revise the abstract and conclusions to distinguish the performance of CCZ3 from that of CCZ4', or restrict the general claim to CCZ3, which is the only scheme that converges in the matter-field tests.","section":"Section IV.C, Fig. 6 and Abstract"},{"comment":"The numerical evidence is restricted to spherical symmetry: the paper states in Section III that 'we restrict our attention to spherically symmetric systems in this paper.' The abstract's claim that the schemes resolve the late-time instability 'not only in Schwarzschild spacetimes but also in black hole spacetimes with matter fields' goes beyond the tested domain. In spherical symmetry, Zi has only a radial component and the vector structure of the Einstein equations is degenerate. The paper should explicitly qualify the abstract and conclusions as applying to spherically symmetric spacetimes, or, if the general claim is intended, provide at least one non-spherical test (e.g., a Kerr or binary black hole run) to support it.","section":"Section III and Abstract"}],"minor_comments":[{"comment":"The benchmark parameter R0 is given as R0=0.00012 in Eq. (27) but as R0=0.0012 in the caption of Fig. 1 for the rmax=60000M case; please correct this typo.","section":"Section III, Eq. (27) vs Fig. 1"},{"comment":"The text says 'we use fourth-order finite differential on the spatial direction'; this should read 'fourth-order finite differences.'","section":"Section III"},{"comment":"The paper never explains how Zi is reconstructed from the evolved variable Λ̃i in the CCZ4' and CCZ3 implementations. Since Eq. (19) defines Λ̃i ≡ Λ̄i + 2γ̄ij Zj, please clarify the reconstruction step used in the code.","section":"Section II, Eq. (19)"},{"comment":"In the RN and scalarization figures, the BSSN and CCZ0 panels are plotted only up to t=1000M, whereas the CCZ4' and CCZ3 panels extend to 10^5M; using the same time range in all panels would make the comparison more straightforward.","section":"Figs. 4 and 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a numerical methods contribution; the editor may want to weigh whether the spherical-symmetry restriction is sufficient for the journal's readership, but the proposed schemes and the identification of the momentum-constraint damping issue are of general interest. The absence of code release (despite using a public platform) may raise reproducibility concerns, but the description of the equations is in principle sufficient once the typo in Eq. (17) is fixed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this if you care about long-term numerical relativity simulations. The new thing is simple: in spherical Schwarzschild evolution, BSSN shows a late-time instability around 10^4 to 10^5 M, and the authors argue it comes from momentum constraint violations. They then show that CCZ4-type schemes with undamped Z^i (CCZ4' and CCZ3) run stably to 10^5 M, including RN and scalarization cases. That is practically useful, and the parameter scans are thorough: outer boundary, resolution, dissipation, CFL, and gauge parameters are all checked against the horizon area benchmark. Credit where due: the schemes are not curve-fitted to the target result, and the horizon area stays near the known Schwarzschild value over long times. The paper is also honest that the tests are one-dimensional.\n\nThe soft spots are real. First, every simulation is spherically symmetric; the abstract's \"not only in Schwarzschild but also in black hole spacetimes with matter fields\" does not add dimensions, since the RN and EMS runs are also spherical. Rotating holes and binaries are untested, and that is where the scheme would need to work. Second, the causal claim about momentum constraint violations is inferred, not measured. No norm of Z^i or Theta is plotted anywhere. The scheme-by-scheme comparisons are consistent with the story, but they do not rule out gauge drift or outer-boundary noise, especially with rmax=60000M and a single grid. Third, there is a printed inconsistency: Eq. (17), the evolution equation for Theta, lacks the kappa1 damping term and the matter terms present in Eq. (6), and the CCZ4' definition only replaces kappa1 by kappa_Theta in Eq. (15) and by kappa_Gamma in Eq. (18). As printed, kappa_Theta does not appear in the Theta equation. This needs to be fixed before anyone can reproduce the scheme. Fourth, there is no convergence study and no code release. These are not fatal for the qualitative result, but they matter for a methods paper.\n\nWho is this for? People doing long-term black hole evolution, superradiance, or scalarization. If CCZ3 survives 3D tests, it will be cited. Right now it is a promising recipe with a plausible but not directly evidenced mechanism. I would send it to a serious referee, with the expectation of heavy revision: add constraint diagnostics, fix Eq. (17), report convergence, and at least attempt one non-spherical test or state clearly that the 3D extension is conjectural.","headline":"A useful, clearly reported numerical recipe for long-term black hole runs, but the mechanism is inferred rather than measured and the evidence is entirely 1D.","tokens_in":18008,"tokens_out":3842,"would_cite":false,"duration_ms":36433,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83C05","83C57","65M06","83-08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Undamped momentum-constraint evolution suppresses a late-time instability in black hole simulations, reaching $10^5 M$.","keywords":["numerical relativity","BSSN formulation","CCZ4","constraint violations","momentum constraint","late-time instability","black hole evolution","long-term stability"],"falsifier":"A three-dimensional evolution of a Kerr black hole using CCZ3 with $\\kappa_\\Gamma = 0$ that develops the same late-time blow-up in the apparent horizon area would falsify the general claim; alternatively, a spherical BSSN run with a different outer boundary condition that removes the instability would show the effect is a boundary artifact, not a formulation failure.","tokens_in":16874,"feed_emoji":"🕳️","tokens_out":5061,"duration_ms":44362,"temperature":0.7,"pith_summary":"This paper reports that the standard BSSN formulation of numerical relativity develops a late-time numerical instability even for the simplest black hole, Schwarzschild, and traces the culprit to accumulated violations of the momentum constraint. To fix it, the authors modify the conformal covariant Z4 scheme so that momentum constraint violations are evolved as dynamical variables but are not damped. Their two proposed schemes, CCZ4' and CCZ3, keep black hole simulations stable out to times of order $10^5 M$ for Schwarzschild and for charged black holes undergoing spontaneous scalarization. If correct, the result gives a practical recipe for studying weak, slow instabilities like superradiance that current codes cannot follow long enough.","feed_headline":"Undamped momentum violations stabilize black hole runs to 100,000 M","feed_subtitle":"Two CCZ4 variants that stop damping momentum-constraint errors keep Schwarzschild and charged black holes stable for times of order 10^5 M.","key_machinery":"The central object is the spatial momentum-constraint-violation vector $Z^i$, the projection of the Z4 four-vector onto the spatial hypersurface. The key move is to evolve $Z^i$ without damping: CCZ4' splits the single damping parameter $\\kappa_1$ into $\\kappa_\\Theta$ (damping $\\Theta$) and $\\kappa_\\Gamma$ (damping $Z^i$) and sets $\\kappa_\\Gamma = 0$; CCZ3 removes the evolution of $\\Theta$ entirely (sets $\\Theta = 0$) while keeping the $Z^i$-carrying equation for $\\tilde{\\Lambda}^i$. This separation lets the Hamiltonian constraint violation be damped strongly without inducing the nonlinear instabilities that damping of momentum violations causes.","core_discovery":"The central discovery is that the late-time instability seen in BSSN evolutions of black holes is driven by violations of the momentum constraint $Z^i$, not by the Hamiltonian constraint violation $\\Theta$, and that the cure is to let $Z^i$ propagate freely without damping. The paper introduces two schemes, CCZ4' and CCZ3, in which the momentum constraint violation is evolved through the conformal connection variable $\\tilde{\\Lambda}^i$ without a damping term. In these schemes the damping of the Hamiltonian constraint violation can be made strong, which keeps the Hamiltonian constraint under control, while the absence of momentum damping avoids the nonlinear instabilities that strong damping otherwise triggers. The schemes are demonstrated in spherical symmetry for a Schwarzschild black hole, a Reissner-Nordström black hole, and black hole spontaneous scalarization in the Einstein-Maxwell-scalar model, with stable evolutions reaching times of order $10^5 M$.","pith_inferences":["If the mechanism is generic, the 'no damping of $Z^i$' rule may also improve other free-evolution formulations, including generalized harmonic evolutions, wherever momentum constraint violations accumulate.","The same principle could be tested directly in full three-dimensional evolutions of Kerr black holes with superradiant scalar clouds; the predicted requirement is that simulations remain stable to times of order $10^5$ to $10^6 M$ without momentum-constraint damping.","The result suggests that the choice of which constraint to damp is more consequential than the overall strength of damping, which may inform future constraint-damping designs."],"forward_implications":["Long-term black hole simulations with the CCZ3 and CCZ4' schemes remain stable to at least $10^5 M$, a regime where BSSN and unmodified CCZ4 break down.","The momentum constraint violation, not the Hamiltonian one, is the driver of the late-time instability; disabling momentum propagation (CCZ0) reproduces the BSSN failure.","Strong damping of the Hamiltonian constraint violation is beneficial once momentum damping is removed, so constraint control can be improved without destabilizing the simulation.","In matter spacetimes, the CCZ3 scheme with non-propagating electromagnetic constraints gives the most accurate and robust evolution of spontaneous scalarization."],"supporting_citations":[{"why":"Introduces the BSSN formulation, the baseline scheme whose late-time instability is the paper's target.","marker":"[19,20]"},{"why":"Introduces the conformal covariant Z4 system with constraint-violation damping, the starting point for the proposed modifications.","marker":"[25]"},{"why":"Provides the fully covariant reference-metric CCZ4 implementation in spherical symmetry that the new schemes build on.","marker":"[29]"},{"why":"Documents that excessive constraint damping induces nonlinear instabilities, motivating the split-damping approach.","marker":"[34]"},{"why":"The public code platform used for all numerical evolutions in the paper.","marker":"[38]"},{"why":"Defines the Einstein-Maxwell-scalar spontaneous scalarization model used as the matter-field application.","marker":"[43]"},{"why":"Provides the constraint-violation extension of the Maxwell equations used in the charged black hole simulations.","marker":"[45]"}],"fun_headline_variants":["Damping momentum errors triggers black hole instability; undamped cures","CCZ4' and CCZ3: letting momentum violations propagate stabilizes runs","Stop damping momentum constraint: new schemes hold black holes for 10^5 M","Late-time instability from momentum damping; free propagation fixes it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical evidence is one-dimensional: every simulation is spherically symmetric, so the claim that these schemes resolve the instability in general black hole spacetimes rests on the assumption that the same behavior holds in full three-dimensional settings such as rotating or binary black holes.","fun_headline_variants_meta":{"raw":{"variants":["Damping momentum errors triggers black hole instability; undamped cures","CCZ4' and CCZ3: letting momentum violations propagate stabilizes runs","Stop damping momentum constraint: new schemes hold black holes for 10^5 M","Late-time instability from momentum damping; free propagation fixes it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000153,"raw_usage":{"total_tokens":1182,"prompt_tokens":897,"completion_tokens":285,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":206}},"tokens_in":513,"tokens_out":285,"duration_ms":3445,"temperature":1.0,"reasoning_tokens":206,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:36:28.909007+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A three-dimensional evolution of a Kerr black hole using CCZ3 with $\\kappa_\\Gamma = 0$ that develops the same late-time blow-up in the apparent horizon area would falsify the general claim; alternatively, a spherical BSSN run with a different outer boundary condition that removes the instability would show the effect is a boundary artifact, not a formulation failure.","supporting_citations":[{"cited_title":"Conformal and covariant formulation of the Z4 system with constraint-violation damping.Phys","cited_arxiv_id":null,"evidence_quote":"Introduces the conformal covariant Z4 system with constraint-violation damping, the starting point for the proposed modifications."},{"cited_title":"Constraint damping of the conformal and co- variant formulation of the Z4 system in simulations of binary neutron stars.Phys","cited_arxiv_id":null,"evidence_quote":"The public code platform used for all numerical evolutions in the paper."},{"cited_title":"Etienne and Ian Ruchlin et al","cited_arxiv_id":null,"evidence_quote":"Defines the Einstein-Maxwell-scalar spontaneous scalarization model used as the matter-field application."}],"review_version":1}