{"id":"78424ab5-d1c4-4864-a0e7-14183e1dcbb0","arxiv_id":"2602.05766","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A probe scattering off a Schwarzschild black hole transfers a definite leading-order post-Minkowskian angular momentum to the horizon, given by new closed formulas (3.24b), (3.29), (3.36).","lead":"A light particle scattering past a Schwarzschild black hole makes the horizon absorb energy and angular momentum; this paper computes both to leading post-Minkowskian order. The energy matches earlier calculations, and the angular-momentum result — the spin the black hole gains — is new.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Velocity-resummed formulas (3.20a-c) are extrapolated from an O(p_infty^20) check; the new absorbed-angular-momentum results (3.24b), (3.29), (3.36) hinge on this unproved all-order pattern.","rationale":"The reader's weakest-assumption identification is correct and matches the most load-bearing concern. The paper's new results for angular momentum absorption all flow through the velocity-resummed integrals (3.20a)-(3.20c). The paper checks these only to O(p_infty^20) and offers no proof of the all-order pattern; the final integrals over u then convert these into the total J_abs. Without that proof, the new PM predictions are conditional. I considered other potential concerns: the horizon supertranslation ambiguity and the possible static contributions to J_abs are explicitly discussed in Secs. 2.6 and 3.3, and the absence of a 1/omega pole in W_H makes the argument plausible. The gauge/falloff derivations in Appendix C are detailed and standard. The agreement of the energy results with [68,69] provides a nontrivial independent check, and the nonrelativistic check in Sec. 3.6 verifies the leading small-velocity coefficient of the angular momentum, but neither validates the all-orders resummation. Therefore I do not see grounds to reject the paper; the appropriate verdict remains CONDITIONAL, pending a check of the resummation. The proposed numerical test is decisive and inexpensive: the exact leading-PM R_up in (3.3) is a closed hypergeometric expression, so the integral can be evaluated without any expansion in p_infty. Agreement would substantially raise confidence; disagreement would invalidate the new J_abs results. Thus the reader's verdict should stand unchanged.","tokens_in":35133,"tokens_out":8175,"duration_ms":83888,"concrete_test":"Evaluate the kappa-integral (3.15) numerically for a few representative velocities, e.g., p_infty = 0.5, 1, 2, and a set of u values (e.g., 0.1, 1, 5), using the exact leading-PM solution R_up in (3.3) (with x->0 at fixed y, no expansion in p_infty), and compare against the claimed resummed forms (3.20a)-(3.20c) (or (3.19) for the m=2 mode). Agreement to numerical precision (say <1e-8 relative) across these points would provide strong evidence for the extrapolation; any discrepancy would falsify the new J_abs results. This check exploits the fact that (3.3) is exact at leading PM order and requires only high-precision numerical integration.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new results—J_abs in (3.24b), (3.29), (3.36)—are derived from the velocity-resummed horizon source integrals (3.20a)-(3.20c) (with (3.19) for the m=2 example). These closed forms are asserted to be exact in sigma at leading PM order, but the evidence provided is the statement in Sec. 3.3: 'We checked the resummed expressions explicitly up to relative O(p_infty^20).' No proof of the all-order pattern is given. The expansion in p_infty is performed at fixed u, and the master integral (3.17) yields Bessel K functions; the proposed resummation collapses the infinite series into a specific linear combination of K0 and K1 with polynomial coefficients. If this pattern breaks at some higher order (e.g., a term proportional to K2(u) or a different u-dependence appears), the spectral fluxes (3.22)-(3.23) and the integrated J_abs would change. The energy results (3.24a), (3.29), (3.33) agree with independent EFT calculations [68,69], which is a strong check on the framework and on the energy sector, but it does not test the angular momentum sector, where the only check is the nonrelativistic (small-p_infty) limit in Sec. 3.6. That check fixes only the leading coefficient in the p_infty expansion and cannot validate the all-orders resummation. Thus the novel claim of a new PM result for absorbed angular momentum rests on an unproved extrapolation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the energy and angular momentum absorbed by a Schwarzschild black hole when a light probe (scalar, electromagnetic, or gravitational) scatters off it, working to leading order in the post-Minkowskian expansion and to leading order in the probe mass ratio. Using the confluent-Heun/Seiberg–Witten technology of [67], the authors derive horizon waveforms in terms of velocity-resummed expressions, from which they obtain spectral and total absorbed fluxes. The gravitational energy result (3.24a) matches the independent EFT computations of [68,69]; the electromagnetic and scalar energy results similarly match [69]. The new claims are the absorbed angular momentum formulas (3.24b), (3.29), and (3.36), which are stated as exact in the velocity at leading PM order. The gravitational angular momentum is checked in the nonrelativistic limit against the dissipative force of [68].","tokens_in":35586,"tokens_out":2739,"duration_ms":34038,"significance":"If the angular-momentum formulas are correct, this is a substantive new set of PM results: they give the leading-order spin acquired by a Schwarzschild black hole in a scattering event and provide a target for future amplitude/EFT calculations. The energy-sector agreement with independent EFT computations is a strong validation of the overall framework, including the connection formulae and source treatment. Deriving the aborbed fluxes from horizon waveforms rather than from worldline EFT is a conceptually useful cross-check. The appendices supply detailed derivations of the source terms and flux formulas, which increases confidence in the setup. However, the angular-momentum results rest entirely on an unproved all-order resummation of the velocity expansion, and the only non-velocity check (the PN limit) tests just the leading term in that expansion. The paper's central novelty therefore sits on an extrapolation that is not demonstrated.","major_comments":[{"comment":"The closed-form velocity-resummed horizon amplitudes (3.20a)–(3.20c) are obtained by extrapolating an expansion in p_infty that, as stated in Sec. 3.3, was verified only up to relative O(p_infty^20). The new angular-momentum result (3.24b) is derived entirely from these expressions, and the electromagnetic and scalar analogues (3.29), (3.36) similarly depend on the same kind of extrapolation. No proof is given that the pattern persists to all orders, e.g. that no K2(u) terms or different u-dependence appear at higher order. This is the load-bearing step for the 'new PM result' claim. The PN check in Sec. 3.6 only fixes the leading small-p_infty coefficient and cannot validate the all-orders resummation. I ask the authors to either supply a proof of the resummation (for instance from an integral representation of the geodesic integral) or explicitly downgrade the claim to a conjecture ver","section":"Sec. 3.3, Eqs. (3.20a)–(3.20c) and (3.24b)"},{"comment":"The statement that 'only the first line of (3.3) gives nonzero contributions' and that 'the same pattern continues also for the other Z^H_{ℓm,s≤0}' is not demonstrated. Since the second line of (3.3) carries different hypergeometric terms, its vanishing after κ-integration at every order is a nontrivial input. This is part of the same extrapolation issue as the previous comment and should be proved rather than asserted, or else the resulting formulas should be presented as conditional.","section":"Sec. 3.3, after Eq. (3.19)"},{"comment":"The cross-check of the angular momentum result (3.24b) in the nonrelativistic limit is a check of only the first term in a p_infty expansion of the resummed expression. It does not test the finite-velocity structure that distinguishes (3.20a)–(3.20c) from any other resummation with the same leading PN term. The energy agreement with [68,69] exercises the same framework but not the m-weighted combination that defines J_abs. Thus the new angular-momentum sector lacks an independent finite-velocity confirmation; I would treat this as an open point rather than a closed validation.","section":"Sec. 3.6, Eqs. (3.41)–(3.43)"}],"minor_comments":[{"comment":"The notation Z^H_{2(±2),-2}, etc., is compact but potentially confusing; please state explicitly that the parenthesized index refers to the magnetic quantum number m=±2, ±1, 0.","section":"Sec. 3.3, notation in (3.20a)–(3.20c)"},{"comment":"The discussion of peeling violations and static δ(ω) terms is interesting but somewhat orthogonal to the main derivation; consider moving part of it to a footnote or an appendix to improve readability.","section":"Sec. 2.6, peeling discussion"},{"comment":"The sentence 'We checked the resummed expressions explicitly up to relative O(p^20_infty)' should specify which expressions were checked and whether the check was performed for all m and for each helicity sign, since later formulas rely on this statement.","section":"Sec. 3.3, sentence introducing (3.19)"},{"comment":"The relative normalization factor of 2 between vector and scalar cases is stated correctly, but it would help to spell out the two polarizations explicitly at first use.","section":"Sec. 2.8, Eqs. (2.93)–(2.95)"}],"recommendation":"major_revision","confidential_remarks":"The core issue is the unproved all-order resummation behind the new J_abs formulas. If the authors can prove the resummation or at least recast the central claim as a conjecture with the O(p^20) verification, the paper would be acceptable for publication; in its current form, the new PM result is not fully established. The energy-sector checks are strong and speak in favor of the framework. I do not see a fatal inconsistency, but the main novelty needs either a proof or an honest downgrade."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a technically careful Teukolsky calculation that reproduces the known leading-PM absorbed energy and adds a new observable, the absorbed angular momentum. The main caveat is that the J_abs results depend on a velocity resummation that the authors checked to O(p_infty^20) but did not prove to all orders. That is a real soft spot, though not a disqualifying one.\n\nWhat is actually new: the horizon-side waveform decomposition with G^{ell+1} harmonic weighting, which makes the lowest multipole capture the leading PM result for arbitrary velocity; the closed-form velocity-exact expressions (3.20a-c); and the J_abs formulas for gravity, electromagnetism, and a scalar. The energy results agree with the independent EFT calculations of [68,69], which is a strong check on the framework and the source terms. The nonrelativistic limit of J_abs also matches the dissipative force of [68], so the angular momentum sector is not completely unchecked.\n\nThe soft spot is the resummation. The master integral (3.17) yields Bessel K functions, and the authors assert that the infinite p_infty series collapses into a specific combination of K0 and K1, verified up to O(p_infty^20). That is a lot of orders, and I would bet the formulas are correct, but it is still an extrapolation. If a K2(u) term or a different u-dependence appears at higher order, the spectral fluxes and the integrated J_abs change. The PN check only probes the leading small-velocity coefficient, not the all-order structure. So the headline new result rests on shakier ground than the energy result, which has independent confirmation over the full velocity range.\n\nWorth noting: the central input R_up is imported from the authors' own [67]. This is not itself a problem—the paper is published and appears to be refereed—but it means the new results inherit any issues in that construction. The paper is transparent about what was checked and what was not, and the appendices give full derivations.\n\nWho this is for: people working on PM scattering, black hole absorption, and EFT matching. I would send it to a serious referee, with an explicit request to focus on the resummation claim and, if possible, to find an independent check of J_abs beyond the PN limit. If the resummation holds, this is a solid contribution that fills a concrete gap in the two-body scattering program.","headline":"Solid Teukolsky calculation that reproduces known absorbed energy and adds a new leading-PM absorbed angular momentum, but the new result leans on an unproved all-order velocity resummation that deserves a close look in review.","tokens_in":36013,"tokens_out":2009,"would_cite":true,"duration_ms":23813,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"When a light particle scatters off a Schwarzschild black hole, the black hole absorbs a calculable amount of angular momentum — a new leading-order post-Minkowskian result that fixes the final spin of the black hole.","keywords":["post-Minkowskian","black hole perturbation theory","Schwarzschild black hole","horizon absorption","Teukolsky equation","confluent Heun equation","absorbed angular momentum","gravitational waveform"],"falsifier":"Compute the next term in the p_infty expansion of the horizon coefficient Z^H_{22,-2} (relative order p_infty^22) and check it against the resummed expression; alternatively, solve the Teukolsky equation numerically at small but nonzero Mω for a relativistic probe and compare the integrated absorbed angular momentum with (3.24b).","tokens_in":35065,"feed_emoji":"🕳️","tokens_out":6587,"duration_ms":70035,"temperature":0.7,"pith_summary":"This paper tries to establish a new leading-order post-Minkowskian (weak-field, arbitrary-velocity) result: how much angular momentum a Schwarzschild black hole absorbs when a light particle scatters past it. Using black-hole perturbation theory, the authors compute the field the probe induces on the horizon, then integrate the Noether fluxes of energy and angular momentum. For gravity, electromagnetism and a scalar field, the absorbed energy reproduces previous results, while the absorbed angular momentum is new: for gravity, J_abs = π G^7 μ^2 M_BH^6 σ(7σ^2−3)/(2b^6). A sympathetic reader should care because, if correct, the formula fixes the spin a non-rotating black hole acquires from a single scattering event, turning it into a Kerr black hole in the final state, and gives a target that independent methods can be checked against.","feed_headline":"New formula: how a flyby spins up a black hole","feed_subtitle":"The leading-order result gives the angular momentum a Schwarzschild black hole absorbs when a light particle scatters past it.","key_machinery":"The Teukolsky master equation for spin s = ±2, ±1, 0 perturbations, solved through the confluent Heun equation. Its solutions are expressed as hypergeometric functions in an expansion in the small parameter x = 4iMω, with the upgoing radial solution R_up scaling as G^{ℓ+1+s} for s≤0. Because of that scaling, only the lowest spherical-harmonic multipole (ℓ=2 for gravity, ℓ=1 for electromagnetism, ℓ=0/1 for scalars) contributes at leading PM order. From these radial functions the horizon waveform coefficients W^H_{ℓm,s} are assembled, and the absorbed energy and angular momentum follow from Noether-flux integrals that reduce to summing |W^H|^2 with Bessel-function master integrals.","core_discovery":"The central claim: the leading post-Minkowskian (large-impact-parameter) flux of energy and angular momentum into a Schwarzschild horizon is governed by the lowest multipoles, since each horizon waveform harmonic is suppressed by G^{ℓ+1}. The authors build the horizon waveforms from Teukolsky solutions resummed in the small-frequency parameter, then integrate the Noether fluxes. They reproduce the known absorbed energy for gravity, electromagnetism and scalars, and give new absorbed angular momentum: for gravity, J_abs = π G^7 μ^2 M_BH^6 σ(7σ^2−3)/(2b^6); for electromagnetism, 4G^4 M_BH^4 π^2 q_e^2 σ/b^4; for a scalar, 2G^4 M_BH^4 π^2 q^2 σ/b^4. In the nonrelativistic limit the gravitational","pith_inferences":["Because the horizon waveform coefficients lack the 1/ω soft pole, the absorbed angular momentum may be free of the zero-frequency ambiguities that complicate the radiated angular momentum, making it a cleaner target for comparing independent calculational methods.","The G^{ℓ+1} suppression of horizon multipoles suggests a general organizational rule: at each PM order only finitely many low harmonics enter near the horizon, which could let future higher-PM computations work from a handful of multipoles.","If the resummed velocity expressions are exact, taking the ultrarelativistic limit σ→∞ forces a partial resummation of the PM series; the paper identifies a parametric bound beyond which the naive result breaks down, and a natural next step would be to test this by computing the next PM correction.","Because the confluent-Heun dictionary extends to Kerr, the same method should yield spin-dependent absorbed fluxes; this is a stated possible extension of the authors and would give a direct check of the Schwarzschild limit."],"forward_implications":["If correct, the gravitational formula fixes the final spin of a Schwarzschild black hole after a single scattering: the absorbed angular momentum J_abs converts it into a Kerr black hole.","The absorbed energy formulas reproduce previous results, confirming that the gravitational energy entering the horizon equals the change in black-hole mass; the same balance is expected to hold for angular momentum.","Because the leading PM results are monomials in the masses, the same formulas apply, after exchanging the roles of the objects, to the absorption by the lighter body in a two-body scattering at leading order.","The horizon waveforms are exponentially suppressed at high frequency and lack the 1/ω soft pole that the waveforms at infinity have, so the absorbed angular momentum is insensitive to static zero-frequency contributions that complicate radiated angular momentum."],"fun_headline_variants":["Flyby spins black hole: new PM formula","Black hole absorbs spin from passing particle","New leading-order law for black hole spin-up","How a flyby twists a Schwarzschild horizon","Particle flyby: new spin absorption formula"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The closed-form velocity-resummed waveforms are obtained by extrapolating a small-velocity expansion that the authors checked only up to relative O(p_infty^20); if the pattern stops there, the new angular-momentum formulas would not be the true leading-order PM results.","fun_headline_variants_meta":{"raw":{"variants":["Flyby spins black hole: new PM formula","Black hole absorbs spin from passing particle","New leading-order law for black hole spin-up","How a flyby twists a Schwarzschild horizon","Particle flyby: new spin absorption formula"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00042,"raw_usage":{"total_tokens":1959,"prompt_tokens":668,"completion_tokens":1291,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":412,"completion_tokens_details":{"reasoning_tokens":1232}},"tokens_in":412,"tokens_out":1291,"duration_ms":10470,"temperature":1.0,"reasoning_tokens":1232,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T04:06:54.652610+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the next term in the p_infty expansion of the horizon coefficient Z^H_{22,-2} (relative order p_infty^22) and check it against the resummed expression; alternatively, solve the Teukolsky equation numerically at small but nonzero Mω for a relativistic probe and compare the integrated absorbed angular momentum with (3.24b).","supporting_citations":[],"review_version":1}