{"id":"6da41cca-d82e-4865-80b1-55262dd1eb26","arxiv_id":"2610.18521","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A linear response expansion expresses black hole observables as universal coefficients acting on an Einstein-tensor eigenvalue, turning measured observables into constraints on effective deformations.","lead":"This paper proposes a formalism that computes leading corrections to black hole observables from a small deformation function tied to the Einstein tensor. It offers a model-independent way to connect measurements such as shadows, ISCO radii, and quasinormal mode frequencies to effective physics beyond general relativity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek/deepseek-v4-flash","headline":"The global validity of (2.11)-(2.12) outside the horizon is asserted rather than proved; the numerical checks cover only fast-decaying power-law models, so the universal expansion (2.15) rests on an unproven uniformity assumption.","rationale":"The paper's advertised result is a functional expansion of observables valid everywhere outside the horizon. The unique input that carries this from a near-horizon statement to a global one is (2.11)/(2.12), and all u_n in (2.15) are computed from them. The author's own sentences [b109] and [b277] make the smallness conditions explicit, and in that sense the limitation is acknowledged; what is not acknowledged is that the identities themselves are imported from a series treatment [36] whose convergence is only established near the horizon. This is not an attack on the cited paper: it is a request for the analytic continuation or uniformity argument that the present paper claims but does not give. The numerical evidence is real support but limited: all three models share the same monotone power-law structure (2.32), so they cannot distinguish a true global identity from an approximation that happens to work for fast-decaying E. A direct derivation from E=F'(r)/r^2 and t(r)=∫√(u/F)du would settle the point and, if successful, would strengthen the paper. I therefore do not move the reader's verdict: conditional remains appropriate until that derivation, or a counterexample in a non-monotone regime, is supplied.","tokens_in":37211,"tokens_out":19731,"duration_ms":123395,"concrete_test":"Re-derive (2.11) and (2.12) directly from the exact equations (2.4)-(2.5), without the series (A.1): in r-coordinates the exact identities are E(r)=F'(r)/r^2 and t(r)=∫_{r_H}^r sqrt(u/F(u))du, with F(r)=r_H+∫_{r_H}^r s^2 E(s)ds. Expanding these in C and E should reproduce (2.11)-(2.12) with a remainder bounded uniformly in t by a constant times (C^2 + sup|E|^2 + sup|I_n[E]|^2). If that bound cannot be established, test instead a non-monotone E(t)=ε sin(t)/(1+t)^{5/2} with ε small: compute the exact t(r) and compare with (2.12) at large t; any deviation growing faster than O(ε^2) disproves the global-validity assertion.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim inherits all of its content from the global promotion of the linearised relations (2.11) and (2.12) in Section 2.3.2. Every subsequent expansion (2.15) and all quoted coefficient functions u_n for the shadow, ISCO and QNM observables are functionals of this step. The author states in [b109] and [b277] that the relations hold anywhere outside the horizon whenever C, E and its momenta are small, but the derivation displayed is a resummation of the near-horizon series (A.1)-(A.6) imported from [36]; no analytic continuation argument or remainder bound is supplied that would exclude secular growth of the nested integrals at large t. The numerical checks in Figures 5-14 use three models whose E(t) is of the special form (2.32) with fast decay and monotonic behaviour, so they do not test slowly decaying, oscillatory or sign-changing E, where pointwise smallness of E may not control I_1, I_2, or the O(E^2) remainder uniformly. If (2.11)-(2.12) fail in such a regime, the claimed universality of the coefficient functions and the proposed inversion of observables to constrain E would not hold.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek/deepseek-v4-flash","summary":"The paper proposes an effective linear-response description for static, spherically symmetric, asymptotically flat deformations of the Schwarzschild black hole. The deformation is parametrised by a single Einstein-tensor eigenvalue E(t) and a horizon-shift constant C, both assumed small. Starting from the two linearised relations (2.11)-(2.12), imported from the author's earlier EMD paper [36], the paper writes a generic observable in the form (2.15): O ~ O[0,0] + u_C C + sum_n u_n I_n[E], with coefficient functions u computed on the Schwarzschild background and I_n denoting derivatives or nested integrals of E. It then computes these coefficients for the metric, Ricci and Kretschmann scalars, Hawking temperature, photon sphere, shadow, Lyapunov exponent, ISCO quantities, the massless scalar potential, and scalar quasinormal-mode frequencies via third-order WKB. The results are compared with exact numerical computations for three models (Hayward, Simpson-Visser black bounce, and regular Simpson-Visser) at small gamma. The concluding claim is that measured observables can be inverted to constrain E and hence the effective matter source that deforms the geometry.","tokens_in":37609,"tokens_out":9206,"duration_ms":62383,"significance":"If the expansion (2.15) is valid throughout the exterior region, this is a useful and genuinely model-independent organising principle: the coefficient functions are computed once from Schwarzschild, are not fit to the example models, and the three examples serve as independent checks. The author is explicit about the scope (one Einstein-tensor eigenvalue, static spherical symmetry, small C and E) and about the effective rather than fundamental status of the source. The numerical comparisons in Figures 5-14 give concrete support to the algebraic computations and are a real strength of the paper. The value of the contribution depends, however, on closing the analytic gap identified below, because both the universality claim and the proposed inversion of observables rest on the global validity of the linearised relations (2.11)-(2.12).","major_comments":[{"comment":"The global validity of (2.11)-(2.12) is asserted, not proved. The derivation shown in Appendix A.1 is a resummation of the near-horizon series (A.1)-(A.6), and no remainder bound or analytic-continuation argument is supplied that controls the nested integrals I_n[E] uniformly for all t outside the horizon. Pointwise smallness of E does not by itself imply smallness of its integrals: for example E ~ alpha (1+t)^{-2} gives I_1[E](t) ~ const * log(1+t), which is unbounded. The numerical tests in Figures 5-14 cover only three monotone, rapidly decaying models of the special form (2.32), so they do not probe slowly decaying, oscillatory, or sign-changing E. Because (2.15) and all later coefficient functions for the shadow, ISCO, potential, and QNMs are functionals of (2.11)-(2.12), this gap is load-bearing for the central claim. Please supply a precise remainder estimate for the linearisation, or explicitly restrict the formalism to a class of E for which such an estimate can be proved.","section":"§2.3.2 and Appendix A.1","authors_acknowledge":"Due to convergence of the power series, this approach is typically confined to a limited region in space time (e.g.the close vicinity of the event horizon).","authors_acknowledge_p":0.65},{"comment":"The statement that the quasinormal-mode coefficients u_C(ell,n) and u_k(ell,n) in (4.12) 'can be computed explicitly' is not supported by the text: only the eikonal leading expansions (4.17)-(4.18) around t_ps,0 are displayed, and the finite-ell coefficient functions are neither given nor derived. The QNM response (4.12) and the proposed inversion of the spectrum to constrain E(t) near the photon sphere depend on those finite-ell coefficients, so this part of the universality claim is incomplete. The numerical checks in Figure 14 cannot replace the missing analytic coefficient functions. Please provide the explicit finite-ell expressions or a derivation from (4.8)-(4.9), or restrict the QNM section to the eikonal regime and adjust the conclusions accordingly.","section":"§4.2, Eqs. (4.12)-(4.18)","authors_acknowledge":null,"authors_acknowledge_p":null}],"minor_comments":[{"comment":"In (3.11) the coefficient displayed as a second u^{Vmassless}_1 should presumably be u^{Vmassless}_2, and in (3.16) the notation u^{(omega_c)}_n should be u^{(lambda)}_n.","section":"Eqs. (3.11), (3.16)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The sum in (3.12) runs over n=1,2 while the braces include a coefficient u^{(tps)}_0; either the sum should start at n=0 or the u_0 term is a typographical remnant.","section":"Eq. (3.12)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The first imaginary-part line of (4.18) has unbalanced parentheses and the constant term should read (n+1/2)/(6 sqrt(3)) rather than n+1/(2*6*sqrt(3)).","section":"Eq. (4.18)","authors_acknowledge":null,"authors_acknowledge_p":null},{"comment":"The inversion formula (4.15) is introduced without a derivation, and its regime of validity is not stated; a brief comment on where it follows from (4.10) would improve readability.","section":"§4.2, Eq. (4.15)","authors_acknowledge":null,"authors_acknowledge_p":null}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a reasonable and potentially useful contribution, and I see no circularity problem: the universal coefficients are computed from the Schwarzschild background, and the three models are used as independent checks. The main obstacle is the unproven global validity of the linearised relations (2.11)-(2.12), which is the foundation of the central claim; this needs either a genuine estimate or a clearly delimited class of admissible deformation functions. The quasinormal-mode section additionally needs explicit finite-ell coefficient functions or an explicit restriction to the eikonal regime. These are fixable within the manuscript's scope, so I support major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek/deepseek-v4-flash","letter":"Short version: useful framework paper, not a discovery paper. The genuinely new piece is writing leading corrections for deformed static spherical black holes as O ~ O0 + u_C C + sum_n u_n I_n[E], where the u's are computed once on Schwarzschild and the I_n's are derivatives or nested integrals of the Einstein-tensor eigenvalue E. The coefficient lists for photon sphere, shadow-adjacent quantities, ISCO, Lyapunov exponent, and scalar QNMs are the payload, and the parametrization-change relations are a practical addition. The shadow formula was already linearized in [30], and the paper says so. The algebra is internally consistent; the three model checks are honest, and at small gamma the linear response tracks the exact geometry. The paper also flags that model 3 remains sensitive to higher-order corrections.\n\nSoft spots, in proportion. The main one is the global step. Equations (2.11)-(2.12) come from a near-horizon series treatment in the author's [36] and are promoted to identities valid anywhere outside the horizon whenever E and its momenta are small. That promotion is asserted, not proved: no remainder bound, no analytic-continuation argument, no control of secular growth of the nested integrals for oscillatory or slowly decaying E. The numerical checks use three monotone, fast-decaying power-law models, so they do not test that regime. This is where the universality claim lives, so it needs either a proof or a clear downgrade to a working assumption. Second, the QNM section states the finite-l linear response but actually displays only the leading behavior near the photon-sphere limit; the full coefficient functions are not derived. Third, the static-spherically-symmetric single-eigenvalue restriction is real but the author acknowledges it; that is scope, not a flaw.\n\nCitations are fine; the main input is the author's own published paper and the dependence is explicit. No code or data files ship, but the analytic momenta and plots make the checks inspectable.\n\nI would send this to a serious referee. It deserves referee time. The referee should focus on the global-validity claim and on getting the QNM derivation out in the open.","headline":"Useful framework paper: universal linear-response coefficients for deformed Schwarzschild observables, with one load-bearing global-validity assumption that is asserted rather than proved.","tokens_in":38099,"tokens_out":4715,"would_cite":true,"duration_ms":35887,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"pith_extraction":{"msc":["83C57","83C25","83D05"],"pacs":["04.70.-s","04.70.Bw","04.50.Kd"],"model":"deepseek/deepseek-v4-flash","headline":"The paper claims every observable outside a slightly deformed static black hole reduces to its Schwarzschild value plus universal coefficients times derivatives and nested integrals of the Einstein-tensor eigenvalue.","keywords":["black hole deformations","linear response","Einstein tensor eigenvalue","black hole shadow","quasi-normal modes","innermost stable circular orbit","Hawking temperature","effective gravity beyond GR"],"falsifier":"Construct a static, spherically symmetric, asymptotically flat geometry whose Einstein-tensor eigenvalue $\\mathcal{E}$ is small at the horizon but develops a localised bump or large derivatives in the exterior, so the momenta $I_n[\\mathcal{E}]$ are not uniformly small; numerically compute the photon-sphere shift and a scalar quasi-normal mode frequency and compare with the linear-response formulae (3.12) and (4.16). If the discrepancy exceeds the order-$\\mathcal{E}^2$ error, the global validity of (2.11) and (2.12) is refuted. A complementary check: invert the linear-response quasi-normal mode expression for $\\mathcal{E}$ and verify that the reconstructed function reproduces the original metric function to first order in $\\mathcal{E}$.","tokens_in":38702,"feed_emoji":"🕳️","tokens_out":11232,"duration_ms":67295,"temperature":0.7,"pith_summary":"This paper proposes a linear-response treatment of black hole deformations: the leading corrections to any observable outside a static, spherically symmetric black hole horizon are written as a universal functional of a single function $\\mathcal{E}$, an eigenvalue of the Einstein tensor that vanishes for Schwarzschild and is interpreted as an external energy-momentum source. The central claim is that every such observable takes the form $O[\\mathcal{E}](t) \\sim O[0,0](t) + u_C(t)\\,C + \\sum_n u_n(t)\\,I_n[\\mathcal{E}](t)$, where the coefficients $u_n$ are fixed by the Schwarzschild geometry alone and the $I_n$ are derivatives, the function itself, or nested integrals of $\\mathcal{E}$. If correct, measurements of the black hole shadow, photon-sphere quantities, the ISCO, the Hawking temperature, and scalar quasi-normal mode frequencies become invertible constraints on $\\mathcal{E}$, giving a direct and model-uniform bridge between observations and effective physics beyond general relativity. The formalism is illustrated on three known deformed geometries and reproduces their exact numerical observables to within a few percent for small deformation parameters.","feed_headline":"Observables of deformed black holes reduce to one function","feed_subtitle":"Shadow, ISCO, and quasinormal modes all respond to the same Einstein-tensor eigenvalue.","key_machinery":"The load-bearing object is the linear-response decomposition (2.15) together with the momenta $I_n[\\mathcal{E}]$ of (2.16), which are derivatives, the function itself, or nested integrals of the deformation function. The decomposition rests on two functional identities, (2.11) and (2.12), imported from the author's earlier series-based effective-description paper [36], which express the metric function and the radial coordinate as linear functionals of $\\mathcal{E}$; the paper maintains that these relations hold globally outside the horizon whenever $\\mathcal{E}$ and all its momenta remain small. The choice of $t$ as the parametrisation is secondary: a change of coordinates merely shifts the coefficients $u_n$ in a way computable entirely within the Schwarzschild geometry.","core_discovery":"The discovery this paper argues for is that the response of a deformed black hole factorises: the observable-specific part is universal, and the model-dependent part is carried entirely by $\\mathcal{E}$ and its momenta $I_n[\\mathcal{E}]$, defined as ordinary derivatives for $n<0$, the function itself for $n=0$, and $n$-fold nested integrals weighted by $(3t+4)$ for $n>0$. Using as coordinate the normalised proper time $t$ of a radially infalling observer to reach the horizon, explicit coefficient functions are worked out for the surface gravity, photon-sphere position, shadow radius, circular-orbit frequency, Lyapunov exponent, ISCO position and angular momentum, and the scalar quasi-normal mode spectrum computed by third-order WKB. Because the coefficients depend only on undeformed Schwarzschild data, the same expression describes any small deformation in this class, and reversing the logic turns each observable into an integro-differential equation for $\\mathcal{E}$. In the eikonal regime the quasi-normal mode spectrum becomes a quasi-continuous family of such equations probing a region just inside the photon sphere.","pith_inferences":["A practical consequence the paper leaves implicit is a data-analysis strategy: fit the single function $\\mathcal{E}(r)$ to combined shadow, ringdown, and ISCO observations rather than fitting separate per-model parameters, since the linear structure would make this a linear inverse problem.","The coefficient functions $u_n$ play the role of a response kernel and the nested-integral momenta resemble the convolution structure of statistical linear response theory; making that analogy precise could yield a model-independent criterion for when the linear truncation is trustworthy.","A proof that every sufficiently regular small $\\mathcal{E}$ outside the horizon yields a metric satisfying (2.11) and (2.12) to first order would turn the numerical evidence into an existence theorem; the paper supplies the evidence but not that argument.","Extending the formalism to a second Einstein-tensor eigenvalue or to rotating geometries, which the paper lists as future work, would test whether one scalar function remains a sufficient descriptor of deformations."],"forward_implications":["Measured shadow radii, ISCO locations, and quasi-normal mode frequencies each constrain specific momenta $I_n[\\mathcal{E}]$ evaluated at specific exterior locations, so a handful of observations yield approximate integro-differential equations for $\\mathcal{E}$.","The coefficient functions $u_n(t)$ are model-independent, so any theory whose deformation produces the same $\\mathcal{E}$ gives the same linearised observables, making comparisons among models and with data uniform.","The eikonal relation $\\omega \\sim \\ell\\,M\\,\\omega_c - (n+\\tfrac12)|\\lambda|$ is recovered, and the linear response extends it to large but finite $\\ell$, where the spectrum traces a quasi-continuous family of points just inside the photon sphere.","Because a change of parametrisation shifts only the coefficients, observables computed in different coordinate systems agree to linear order once the change of variables is evaluated in the undeformed geometry."],"supporting_citations":[{"why":"Supplies the functional relations (2.11) and (2.12) expressing the metric function and radial coordinate as linear functionals of the energy density; the entire linear-response expansion rests on them.","marker":"[36]"},{"why":"Extends the effective metric description via Pade approximants and computes the black hole shadow; the shadow result here builds on and generalises that computation.","marker":"[30]"},{"why":"Introduces the effective metric description based on a physical distance to the horizon, whose observer-independent philosophy motivates choosing an Einstein-tensor eigenvalue as the deformation variable.","marker":"[28]"},{"why":"Establishes the eikonal relation between quasi-normal mode frequencies, photon-sphere frequency, and Lyapunov exponent, which the linear response recovers and extends beyond the eikonal limit.","marker":"[42]"},{"why":"The third-order WKB formalism used to compute the scalar quasi-normal mode spectrum and its linear-response coefficients.","marker":"[58]"},{"why":"Companion WKB paper supplying the potential-barrier machinery, including the maximum of the potential at $t_{\\mathrm{max},0}$, used in the quasi-normal mode calculation.","marker":"[59]"},{"why":"Provides the Hayward regular black hole geometry reproduced as Model 1, the first numerical test case for the linear response.","marker":"[16]"},{"why":"Provides the black-bounce geometry reproduced as Model 2, the second numerical test case.","marker":"[17]"},{"why":"Provides the asymptotically Minkowski-core geometry reproduced as Model 3, the third numerical test case.","marker":"[18]"},{"why":"The Hawking-Ellis energy-momentum classification used to interpret $\\mathcal{E}$ as the energy-density eigenvalue of a type I external source.","marker":"[38]"}],"fun_headline_variants":["Black hole deformations: one eigenvalue rules them all","Deformed black holes: universal response from one function","A single Einstein-tensor eigenvalue drives black hole observables","Black hole responses factorize through one eigenvalue","Tying deformed black hole observables to a single parameter"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The whole expansion stands or falls on the claim that the near-horizon linearised relations (2.11) and (2.12) remain valid as functional identities at every point outside the horizon whenever $\\mathcal{E}$ and all its derivatives and nested integrals stay small; the paper states this on the strength of those smallness conditions and supports it numerically for three small-parameter models, but gives no convergence or analytic-continuation argument that would justify the global extension.","fun_headline_variants_meta":{"raw":{"variants":["Black hole deformations: one eigenvalue rules them all","Deformed black holes: universal response from one function","A single Einstein-tensor eigenvalue drives black hole observables","Black hole responses factorize through one eigenvalue","Tying deformed black hole observables to a single parameter"]},"model":"deepseek/deepseek-v4-flash","effort":"low","cost_usd":0.000171,"raw_usage":{"cost":0.00017066,"is_byok":false,"cost_details":{"upstream_inference_cost":0.00017066,"upstream_inference_prompt_cost":0.00014126,"upstream_inference_completions_cost":2.94e-05},"total_tokens":1114,"prompt_tokens":1009,"completion_tokens":105,"prompt_tokens_details":{"audio_tokens":0,"video_tokens":0,"cached_tokens":0,"cache_write_tokens":0},"completion_tokens_details":{"audio_tokens":0,"image_tokens":0,"reasoning_tokens":28}},"tokens_in":1009,"tokens_out":105,"cost_source":"openrouter_usage","duration_ms":1993,"image_pages":[],"temperature":1.0,"billed_attempts":1,"reasoning_tokens":28,"transport_provider":"openrouter","cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-10-14T10:16:37.243074+00:00","model_set":{"reader":"deepseek/deepseek-v4-flash"},"falsifier":"Construct a static, spherically symmetric, asymptotically flat geometry whose Einstein-tensor eigenvalue $\\mathcal{E}$ is small at the horizon but develops a localised bump or large derivatives in the exterior, so the momenta $I_n[\\mathcal{E}]$ are not uniformly small; numerically compute the photon-sphere shift and a scalar quasi-normal mode frequency and compare with the linear-response formulae (3.12) and (4.16). If the discrepancy exceeds the order-$\\mathcal{E}^2$ error, the global validity of (2.11) and (2.12) is refuted. A complementary check: invert the linear-response quasi-normal mode expression for $\\mathcal{E}$ and verify that the reconstructed function reproduces the original metric function to first order in $\\mathcal{E}$.","supporting_citations":[{"cited_title":"Probing Effective Black Hole Deformations","cited_arxiv_id":"2508.17781","evidence_quote":"Supplies the functional relations (2.11) and (2.12) expressing the metric function and radial coordinate as linear functionals of the energy density; the entire linear-response expansion rests on them."},{"cited_title":"Black Hole Shadow and other Observables away from the Horizon: Extending the Effective Metric Descriptions","cited_arxiv_id":"2412.13673","evidence_quote":"Extends the effective metric description via Pade approximants and computes the black hole shadow; the shadow result here builds on and generalises that computation."},{"cited_title":"Living on the Edge: Quantum Black Hole Physics from the Event Horizon","cited_arxiv_id":"2307.13489","evidence_quote":"Introduces the effective metric description based on a physical distance to the horizon, whose observer-independent philosophy motivates choosing an Einstein-tensor eigenvalue as the deformation variable."},{"cited_title":"Geodesic stability, Lyapunov exponents and quasinormal modes","cited_arxiv_id":"0812.1806","evidence_quote":"Establishes the eikonal relation between quasi-normal mode frequencies, photon-sphere frequency, and Lyapunov exponent, which the linear response recovers and extends beyond the eikonal limit."},{"cited_title":"An Analytic representation for the quasi normal modes of Kerr black holes","cited_arxiv_id":null,"evidence_quote":"The third-order WKB formalism used to compute the scalar quasi-normal mode spectrum and its linear-response coefficients."},{"cited_title":"Iyer, Phys","cited_arxiv_id":null,"evidence_quote":"Companion WKB paper supplying the potential-barrier machinery, including the maximum of the potential at $t_{\\mathrm{max},0}$, used in the quasi-normal mode calculation."},{"cited_title":"Formation and evaporation of non-singular black holes","cited_arxiv_id":"gr-qc/0506126","evidence_quote":"Provides the Hayward regular black hole geometry reproduced as Model 1, the first numerical test case for the linear response."},{"cited_title":"Black-bounce to traversable wormhole","cited_arxiv_id":"1812.07114","evidence_quote":"Provides the black-bounce geometry reproduced as Model 2, the second numerical test case."},{"cited_title":"Regular black holes with asymptotically Minkowski cores","cited_arxiv_id":"1911.01020","evidence_quote":"Provides the asymptotically Minkowski-core geometry reproduced as Model 3, the third numerical test case."},{"cited_title":"and Ellis, George F","cited_arxiv_id":null,"evidence_quote":"The Hawking-Ellis energy-momentum classification used to interpret $\\mathcal{E}$ as the energy-density eigenvalue of a type I external source."}],"review_version":1}