{"id":"b836ba1e-c9d9-4579-b0a3-37dc4644877c","arxiv_id":"1909.01628","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Charged AdS black holes with a dipolar differential rotation boundary are constructed numerically, and pairs of small solutions with different horizon radii are found to share identical horizon geometry, entropy, and quasinormal modes.","lead":"This paper computes new solutions for black holes in anti-de Sitter space that carry electric charge and have a distorted, rotating boundary. The authors report that in some temperature ranges two distinct-looking small black holes share the same horizon shape, entropy, and vibration frequencies.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (3.1b) as printed has a sign inconsistency: its Δ(0) gives a temperature with +q^2/y_+^2, contradicting Eq. (3.6) and the claimed RN-AdS limit; if real, the three-branch region and the two-small-branch degeneracy are consequences of the wrong temperature formula.","rationale":"The reader's weakest assumption was numerical convergence, but a more basic, analytically checkable inconsistency appears in the ansatz itself. Equation (3.1b) as printed gives Δ(0)=1+3y_+^2+q^2/y_+^2, whereas the standard RN-AdS limit and the paper's own temperature formula (3.6) require 1+3y_+^2-q^2/y_+^2. This sign discrepancy changes the number of temperature extrema: with the printed plus sign, T(y_+) has only one extremum for q>0, eliminating the three-horizon region on which the paper's central degeneracy claim rests. The authors may have a typo in the manuscript and the code may use the correct sign; even so, the manuscript as written is internally inconsistent and the central claim cannot be assessed until the sign is fixed and the horizon counting is redone. I therefore keep a conditional verdict, but for a different reason than the reader's numerical-convergence concern. The proposed test is a direct algebraic/numerical check of the RN limit and the temperature formula, which settles whether the three-branch structure is real or an artifact.","tokens_in":12735,"tokens_out":25056,"duration_ms":230373,"concrete_test":"Set U_i=1, δ=1 for a nonzero charge, e.g., q=0.1 and y_+=1, and plug the line element (3.1a) with Δ from Eq. (3.1b) into the Einstein-Maxwell equations; compute the surface gravity at y=0 symbolically or numerically. If Δ(0)=1+3y_+^2+q^2/y_+^2, the metric does not solve the equations and the temperature differs from Eq. (3.6). Alternatively, derive the temperature directly from the ansatz using T=Δ̃(0)√(U1/U2)/(4πy_+) and compare the q^2 term with Eq. (3.6). Then check which sign is actually implemented in the numerical code and rerun the horizon count; if the printed plus sign is used, the three-branch regions and the two-small-branch degeneracy disappear.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the stated RN-AdS limit (U_i=1, δ=1), the ansatz (3.1a)-(3.1b) should reduce to the RN metric f(r)=1-2M/r+q^2/r^2+r^2 with r=y_+/(1-y^2), M=(r_h+q^2/r_h+r_h^3)/2. Near the horizon y=0, standard algebra gives f ≈ y^2(1+3r_h^2-q^2/r_h^2), so the surface gravity yields T=(1+3y_+^2-q^2/y_+^2)/(4πy_+), matching Eq. (3.6). But Eq. (3.1b) as printed has Δ(0)=1+3r_h^2+q^2/r_h^2, which would give T=(1+3y_+^2+q^2/y_+^2)/(4πy_+). With this sign, T(y_+) has only one stationary point for q>0, so the regime 0<q<1/6 with T_min<T<T_max and three horizon radii (Sec. 3, Figs. 1, 4, 6-8) does not exist. The central claim that two small branches have identical horizon geometry, entropy, and quasinormal modes depends entirely on this three-root temperature structure. Unless the minus sign in Eq. (3.6) is accompanied by a minus sign in Eq. (3.1b), or the code actually uses a different Δ, the numerical solutions are not charged AdS black holes with the claimed thermodynamics. This internal inconsistency is more fundamental than the absence of convergence tests and must be resolved before the numerical claims can be evaluated.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper numerically constructs charged, asymptotically AdS_4 black holes with a dipolar differential rotation on the boundary, using the DeTurck method in Einstein-Maxwell theory. The authors study how the number of black hole horizons depends on the temperature T and charge q, and identify a temperature window for 0<q<1/6 in which three horizon radii exist for a fixed T. They report that in this window the two smaller-horizon solutions share the same horizon geometry, entropy, and quasinormal-mode frequencies despite having different horizon radii. The paper also studies isometric embeddings of the horizon into hyperbolic space and the entropy as a function of the boundary rotation parameter, and discusses the stability of the small branches under massless scalar perturbations.","tokens_in":13156,"tokens_out":17782,"duration_ms":153834,"significance":"If the main claims hold, the most notable result is the apparent degeneracy between the two small-horizon branches: different horizon radii would lead to identical geometric and dynamical properties. This would be a new phenomenon in the study of deformed AdS black holes and could have implications for holographic models with deformed boundaries. The paper follows an established numerical approach (the DeTurck method) and extends previous work on uncharged deforming black holes to the charged case. However, the significance is conditional: the internal sign inconsistency in the temperature formula and the absence of numerical convergence tests prevent a reliable assessment of the central claims.","major_comments":[{"comment":"There is a sign inconsistency in the q^2 term. The ansatz in Eq. (3.1b) contains Δ(y) = q^2(1-y^2)^2/(L^2 y_+^2) + (1-y^2)^2 + y_+^2(3-3y^2+y^4), so Δ(0)=1+3y_+^2+q^2/y_+^2 (with L=1). The standard surface gravity calculation in the RN-AdS limit then gives T=(1+3y_+^2+q^2/y_+^2)/(4π y_+), in conflict with Eq. (3.6), whose minus sign is required to reproduce the RN-AdS temperature. With the printed plus sign, T(y_+) has only a single extremum (a minimum) for q>0, so the three-horizon region for 0<q<1/6, on which the central degeneracy claim rests, would not exist. Please correct the sign in Eq. (3.1b) or explicitly state that the code uses a different Δ, and confirm that Eq. (3.6) is the temperature of the solutions actually constructed.","section":"3 (Eq. (3.1b) vs. Eq. (3.6))"},{"comment":"The analytic expressions for T_min and T_max in Eq. (3.7) do not reproduce the values stated in the text. For q=0.07057, substituting s=√(1−36q^2)≈0.9059 into the printed formula gives T≈0.193, while the text and Fig. 1 quote T_min≈0.2735 and T_max≈0.4635 for this charge. The correct RN-AdS extrema obtained from T(y_+)=(1+3y_+^2−q^2/y_+^2)/(4πy_+) are T_min=(2+s)√6/(6π√(1+s)) and T_max=(2−s)√6/(6π√(1−s)), which do match the quoted numbers. The equations should be corrected and the derivation shown.","section":"3 (Eq. (3.7))"},{"comment":"The paper reports no convergence tests, no residual or constraint-violation measures, and no error estimates for any of the numerical quantities (metric functions, entropy, quasinormal frequencies). This is especially serious for the central claim that the two small branches have identical horizon geometry, entropy, and QNMs: if the DeTurck solutions are not fully converged, the apparent degeneracy could be a numerical artifact. Please provide the numerical grid sizes and a representative convergence study (e.g., the DeTurck vector norm as a function of resolution) for the cases in Figs. 2, 4, 6, and 9.","section":"2 and 3"},{"comment":"The stability analysis is based on the sign of Re(ω): the text states that 'Re ω would appear a negative value ... which means we could obtain a stable deforming charged black hole solution with scalar condensation.' For the perturbation convention Φ∼e^{-iωt} in Eq. (3.3.2), an instability corresponds to Im(ω)>0; Re(ω)<0 is not a standard instability criterion, and 'stable ... with scalar condensation' is internally contradictory. The paper never reports Im(ω). The stability conclusions in this section should be reworked: compute Im(ω) for the modes in Fig. 9, or otherwise justify the criterion, and interpret the onset of the unstable mode.","section":"3.3"},{"comment":"The paper states repeatedly that the two small branches, e.g., y_+=0.0992 and y_+=0.1773, have the same horizon embedding, entropy, and quasinormal frequencies. Since y_+ is a coordinate parameter tied to the horizon scale and the boundary metric is fixed, it is not obvious how two solutions with different y_+ can be physically identical. No direct comparison of the full metric functions U_i(x,y) for the two branches is given, and no symmetry or coordinate transformation is identified to explain the degeneracy. Please provide evidence that the two solutions represent the same physical geometry, or clarify the precise sense in which they are degenerate.","section":"3.1-3.3"}],"minor_comments":[{"comment":"The caption and the text disagree on the values of q in the bottom-right panel: the text lists q=0, 1.7068, 2.2684, 3.4299 for five colored lines, while the caption includes q=2.8363 as well.","section":"Figure 2"},{"comment":"The text describes the line for q=1/6 as 'orange' while the caption calls it 'red'; these labels should be made consistent.","section":"Figure 1"},{"comment":"The boundary condition A_t(x,1)=μ and A_t(1,y)=0 conflict at the corner (x=1,y=1); the treatment of corners in the numerical scheme should be described.","section":"3 (after Eq. (3.5))"},{"comment":"The statement 'we also find another family of small black hole solutions' is vague; the authors should specify the boundary conditions or parameter range in which this second family exists and how it relates to the first family.","section":"3.2"},{"comment":"The statement that there exists at least one horizon for an arbitrary temperature is only true for q≠0, since for q=0 there is no horizon for T<T_S; please qualify the statement accordingly.","section":"Abstract and Introduction"},{"comment":"Reference [18] is cited as arXiv:1906.06183 without a journal reference; if it has been published, please update the citation.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of JHEP and the main idea is interesting, but the internal inconsistency in the temperature/ansatz and the lack of numerical validation are serious. The authors should be asked to provide the exact ansatz used in the code, correct the analytic formulas, and supply convergence tests before the claims can be accepted. I would also urge the editor to request a proper stability analysis using Im(ω) rather than Re(ω)."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear Colleague,\n\nThe short version: this is a plausible but under-validated numerical construction of charged AdS black holes with a dipolar differential rotation boundary. The interesting finding — two small branches with different horizon radii but identical horizon geometry, entropy, and quasinormal frequencies — deserves attention, but the paper as written has a likely sign error that threatens the whole temperature branch structure, and the numerics are not backed by convergence or residual checks.\n\nWhat is genuinely new is the charged extension of the 'stirring a black hole' construction [16], and the first observation of that small-branch degeneracy in a charged setting. The plots are consistent with actual solutions having been found. The temperature-extremum analysis and horizon-counting logic are standard and clearly presented. The reference list is appropriate; the earlier self-citations are for the numerical method, not for this result.\n\nThe soft spots, in order of severity. First, Eq. (3.1b) as printed has Δ(0) = 1+3y_+^2+q^2/y_+^2. If you plug that into the standard surface-gravity calculation for the ansatz, you get T = (1+3y_+^2+q^2/y_+^2)/(4π y_+) in the RN limit, not the −q^2 that appears in Eq. (3.6) and that is required for RN-AdS. The three-branch region for 0<q<1/6 — and therefore the entire two-small-branch degeneracy — comes from the temperature formula with −q^2. So either the code uses a different Δ than the paper prints, in which case Eq. (3.1b) is a typo that must be fixed, or the solutions are not the charged AdS black holes the paper claims. This must be resolved before anything else.\n\nSecond, there are no convergence tests, no residuals, and no error estimates for the metric functions, entropies, or QNM frequencies. For a paper whose central claim is an exact degeneracy between two numerical branches, that is a serious omission. The inconsistent charge values in the Fig. 2 caption (q=0.07057 in the text vs q=1.7068 etc. in the caption) suggest the numerical logs were not carefully checked before submission.\n\nThird, the stability discussion conflates the sign of Re ω with instability. With the convention e^{-iωt}, stability is about Im ω; the real part is irrelevant. Their claim that negative Re ω means 'scalar condensation' is not backed by any standard criterion.\n\nWho should read this: people working on numerically constructed AdS black holes with deformed boundaries, and anyone interested in non-uniqueness of horizon data. If the sign error is repaired and convergence data supplied, the degeneracy claim would be worth publishing. As it stands, I would not cite it yet, and I'd want a referee to verify the sign and ask for numerics. But it is substantive enough to send to review — a good referee could sort out the sign in an hour.","headline":"Charged deforming black holes with an interesting degeneracy claim, but a sign inconsistency in the metric ansatz and absent numerical validation make it currently unconvincing.","tokens_in":13641,"tokens_out":11244,"would_cite":false,"duration_ms":89693,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","04.40.Nr"],"model":"deepseek-v4-flash","headline":"This paper claims that in charged, deforming anti-de Sitter black holes with a dipolar differential rotation boundary, two small-horizon branches with different radii have identical horizon geometry, entropy, and scalar quasinormal-mode…","keywords":["AdS/CFT duality","black holes","quasinormal modes","charged black holes","deforming horizon","differential rotation boundary","DeTurck method","horizon geometry"],"falsifier":"Recompute the two small branches at a representative point, for example $T=0.2585$, $q=0.07057$, $\\varepsilon=1.6$, on successively finer grids while monitoring the DeTurck residual; if their entropies and quasinormal frequencies cease to match as resolution improves, the two-branch degeneracy is a numerical artifact. An independent spectral code should reproduce the stated horizon radii $y_+=0.3110$ and $y_+=0.0859$ with identical embedding curves for the degeneracy claim to stand.","tokens_in":12553,"feed_emoji":"🕳️","tokens_out":5971,"duration_ms":59454,"temperature":0.7,"pith_summary":"The paper studies four-dimensional anti-de Sitter black holes whose boundary rotates differentially with the dipolar profile $\\Omega(\\theta)=\\varepsilon\\cos\\theta$, and adds electric charge to the bulk. It finds that charge removes the uncharged theory's minimum-temperature barrier: for any nonzero charge and any temperature, at least one horizon exists. The central discovery is a degeneracy: at temperatures admitting three horizon radii, the two smaller solutions have different radii yet the same horizon geometry, the same entropy, and the same quasinormal-mode frequencies. This means the boundary data and temperature do not fix the local size of a small horizon, even though they fix its observable features. The paper also maps how the charge $q$ controls temperature extrema, the entropy phase diagram, and the onset of scalar condensation.","feed_headline":"Two small black holes of different size can share entropy and geometry","feed_subtitle":"Adding charge to deforming AdS black holes guarantees a horizon at any temperature and creates degenerate small branches.","key_machinery":"The DeTurck method is the core mechanism: adding the gauge-fixing vector $\\xi^\\mu = g^{\\nu\\rho}(\\Gamma^\\mu{}_{\\nu\\rho}[g]-\\Gamma^\\mu{}_{\\nu\\rho}[\\tilde g])$ converts the Einstein equations into a determined elliptic system that can be solved numerically with a reference metric sharing the same boundary and horizon structure. The horizon-counting argument runs through the analytic temperature formula $T = [y_+^4 + \\delta(-q^2+y_+^2(1+2y_+^2))]/(4\\pi y_+^3)$, whose extrema in $y_+$ depend on $q$ and $\\delta$. The degeneracy of the two small branches is demonstrated through equal isometric embeddings into hyperbolic 3-space, equal entropy from $S=(2\\pi y_+^2 L^2/G_N)\\int_0^1 dx\\,(1-x^2)/\\sqrt{2-x^2}\\sqrt{U_3(x,0)U_5(x,0)}$, and equal quasinormal frequencies obtained from a massless scalar perturbation in Eddington-Finkelstein coordinates.","core_discovery":"The authors construct numerical solutions of the Einstein-Maxwell equations in AdS$_4$ with the conformal boundary metric $ds^2_\\partial = -dt^2 + d\\theta^2 + \\sin^2\\theta\\,(d\\varphi + \\varepsilon\\cos\\theta\\,dt)^2$, using the DeTurck method to turn the Einstein equations into elliptic equations. They find that the temperature-horizon relation depends sharply on the charge: for $q=0$ there is a minimum temperature below which no horizon exists; for $0<q<1/6$ there are two temperature extrema; for $q=1/6$ the extrema coalesce at the Reissner-Nordstr\\\"om-AdS value $T_{\\rm RN}=\\sqrt{6}/(3\\pi)$; and for $q>1/6$ there is no extremum, so a horizon exists at every temperature. In the temperature band with three horizon radii, the two small branches have horizon radii that differ by as much as a factor of four, yet their hyperbolic embeddings coincide, their entropy integrals give the same value, and their scalar quasinormal frequencies match for every azimuthal quantum number studied. The paper further reports that large-branch horizon deformation grows with the rotation parameter $\\varepsilon$, small-branch deformation shrinks with $\\varepsilon$, and scalar condensation appears when the azimuthal quantum number satisfies $m\\ge 13$.","pith_inferences":["If the two-branch degeneracy is exact rather than a numerical coincidence, it suggests an emergent symmetry or equivalence relating the two small solutions, possibly tied to the isometric embedding into hyperbolic space; the paper does not identify a mechanism.","A natural testable extension is to compute the full quasi-local stress tensor, angular momentum, and mass of the two small branches; if all charges and thermodynamic potentials also match, the degeneracy would be a genuine failure of uniqueness rather than a coincidence of entropy and geometry.","Since the paper reports no convergence tests or residual measures, a decisive check is to rerun the DeTurck solver at higher resolution and monitor the gauge-fixing residual; if the degeneracy persists to machine precision, it is robust.","Extending the construction to nonlinear electrodynamics or $f(R)$ gravity, as the authors propose to do, would test whether the degeneracy is special to Einstein-Maxwell theory or a generic feature of deformed AdS horizons."],"forward_implications":["For any nonzero charge $q$, a charged deforming AdS black hole exists at any temperature, unlike the uncharged case where temperatures below $T_S=\\sqrt{3}/(2\\pi)$ admit no horizon.","In the three-horizon temperature band, the two small branches are thermodynamically and spectroscopically degenerate despite having different horizon radii, so the boundary metric and temperature do not uniquely determine the local horizon size.","The entropy phase diagram splits into three temperature regions for $0<q<1/6$: the large-branch entropy increases with $\\varepsilon$, the small-branch entropy decreases, and the branches join when $T\\le T_{\\rm RN}$.","Scalar condensation, signalled by a negative real part of the quasinormal frequency, occurs for azimuthal quantum number $m\\ge 13$ at sufficiently large $\\varepsilon$.","Adjusting the parameter $\\delta$ below 1 produces three horizon radii even below the uncharged minimum temperature, broadening the region in which the small-branch degeneracy can be studied."],"supporting_citations":[{"why":"Supplies the dipolar differential rotation boundary $\\Omega(\\theta)=\\varepsilon\\cos\\theta$ and the DeTurck-based construction of deforming AdS$_4$ black holes that this paper extends to include charge.","marker":"[16]"},{"why":"Introduces the DeTurck method that turns the Einstein equations into elliptic equations through the gauge-fixing vector $\\xi^\\mu$.","marker":"[23]"},{"why":"Provides the numerical methods for stationary gravitational solutions used to discretize and solve the elliptic system of the paper.","marker":"[25]"},{"why":"Gives the hyperbolic 3-space isometric embedding technique used to characterize and compare the horizon geometries of the two small branches.","marker":"[31]"},{"why":"Supplies the Eddington-Finkelstein scalar perturbation method used to compute quasinormal modes of AdS black holes.","marker":"[36]"},{"why":"Provides the quasinormal-mode formalism and conventions for black holes and black branes used in the stability analysis.","marker":"[37]"}],"fun_headline_variants":["Charge lets deforming black holes pair up with identical entropy","Small black holes of different sizes share the same horizon shape and entropy","Charged deforming black holes: two radii, one entropy","Charge guarantees a horizon at any temperature for deforming black holes","Identical entropy from different-sized charged deforming black holes"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerical DeTurck solutions are fully converged and accurate, since the paper reports no convergence tests, residual norms, or error estimates for the computed metrics, entropies, or quasinormal frequencies.","fun_headline_variants_meta":{"raw":{"variants":["Charge lets deforming black holes pair up with identical entropy","Small black holes of different sizes share the same horizon shape and entropy","Charged deforming black holes: two radii, one entropy","Charge guarantees a horizon at any temperature for deforming black holes","Identical entropy from different-sized charged deforming black holes"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000598,"raw_usage":{"total_tokens":2817,"prompt_tokens":986,"completion_tokens":1831,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":1745}},"tokens_in":602,"tokens_out":1831,"duration_ms":13231,"temperature":1.0,"reasoning_tokens":1745,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:12:49.379353+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the two small branches at a representative point, for example $T=0.2585$, $q=0.07057$, $\\varepsilon=1.6$, on successively finer grids while monitoring the DeTurck residual; if their entropies and quasinormal frequencies cease to match as resolution improves, the two-branch degeneracy is a numerical artifact. An independent spectral code should reproduce the stated horizon radii $y_+=0.3110$ and $y_+=0.0859$ with identical embedding curves for the degeneracy claim to stand.","supporting_citations":[{"cited_title":"Global embedding of the Kerr black hole event horizon into hyperbolic 3-space","cited_arxiv_id":"0906.2768","evidence_quote":"Gives the hyperbolic 3-space isometric embedding technique used to characterize and compare the horizon geometries of the two small branches."}],"review_version":1}