{"id":"49187027-9e8b-4068-a6c4-ae7c39e5b937","arxiv_id":"2412.02727","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Stationary Gross-Pitaevskii solutions on a funnel-like metric produce sonic black/white hole and wormhole analogues with closed-form Hawking temperatures for uniform-density cases.","lead":"This paper constructs stationary, non-singular flows of a two-dimensional Bose-Einstein condensate that behave like black holes, white holes, and wormholes for sound, and it derives the analogue Hawking temperature for the uniform-density cases. It shows that the same flow profile realized with a curved spatial metric and with a flat metric plus tailored coupling and potential give different fluctuation behavior.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"In Section 3, the local speed of sound is defined as sqrt(2ρ)/sqrt(f) with ρ the amplitude, but the linearized equations imply c_s = sqrt(2)ρ/sqrt(f); the resulting crossing condition differs from the acoustic-metric horizon by a factor of ρ, so the claimed non-uniform wormhole horizons may be…","rationale":"The reader's weakest assumption concerns the uncontrolled hydrodynamic approximation. My stress-test identifies a more concrete and directly testable flaw: the definition of the local speed of sound in Section 3 is inconsistent with the linearized equations and with the acoustic metric (3.11). The wave equation derived from (3.9)-(3.10) in the hydrodynamic limit has speed squared 2ρ0^2/f, whereas Eq. (3.1) states 2ρ0/f. This is not a matter of approximation quality; it is an algebraic inconsistency. For uniform-density solutions (ρ0 = 1) the factor cancels, so the Hawking temperature claims in Sections 4 and 6 remain intact, and the paper's analytic uniform-density wormhole construction is a legitimate regular solution with a well-defined acoustic metric. But the non-uniform wormhole claim (Section 3, abstract, title) rests on crossing of C and V with the wrong C. If the corrected C does not cross V for the presented parameters, the wormhole configuration would not have acoustic horizons, undermining a central advertised result. The concrete test is decisive: recompute the crossing with c_s = sqrt(2) ρ0 / sqrt(f) and compare with the g_00 = 0 roots of Eq. (3.11). This is a quick check that does not require new experiments or heavy numerics. The reader's CONDITIONAL verdict remains appropriate: the paper should correct (or justify) the speed-of-sound definition and verify the non-uniform crossing before the wormhole claim is accepted. I therefore keep the verdict at CONDITIONAL/UNCHANGED, but with a different and sharper concern than the reader's.","tokens_in":12472,"tokens_out":46849,"duration_ms":388463,"concrete_test":"Recompute the sound speed as c_s = sqrt(2) ρ0(R) / sqrt(f(R)) and the flow speed V = 2|B|/(f R ρ0^2) for the numerical solution in Section 3.1 (|B| = 1, R0 = 1.4827). Check whether c_s = V has real roots and compare those roots with the solutions of ρ0(R)^6 f(R) R^2 = 2B^2 obtained from g_00 = 0 in Eq. (3.11). If the roots differ or do not exist, the claimed wormhole horizons are misidentified. Independently, solve the linearized Equations (3.9)-(3.10) numerically without the hydrodynamic approximation for a localized wave packet and measure the local propagation speed in the comoving frame; it should equal sqrt(2) ρ0 / sqrt(f), not sqrt(2ρ0)/sqrt(f).","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.1, Eq. (3.1) defines the scaled local speed of sound as C = sqrt(2ρ(R))/sqrt(f(R)), with Φ = ρ(R) e^{iθ(R)} so that ρ is the wave-function amplitude and the density is n0 = ρ^2. However, the linearized equations (3.9)-(3.10) combined under the hydrodynamic approximation ∇·(ρ0^2 ∇ N1) ≈ 0 give the wave equation D^2 θ1 = (2/f) ∇·(ρ0^2 ∇ θ1), whose local phonon speed is c_s = sqrt(2) ρ0 / sqrt(f), not sqrt(2ρ0)/sqrt(f). Consequently, the condition C = |V| used in Figures 3-4 to identify acoustic horizons reduces to ρ0^5 f R^2 = 2B^2, whereas the acoustic metric (3.11) has g_00 = 0 (the actual horizon) at ρ0^6 f R^2 = 2B^2. These differ by a factor ρ0. For the non-uniform wormhole solution (ρ0 ≠ 1 in the throat), the plotted crossing does not coincide with the horizon of the derived phonon metric. The uniform-density cases (Sections 4 and 6) have ρ0 = 1, so the Hawking temperature formulas are unaffected, but the central claim of acoustic wormhole horizons in the non-uniform funnel geometry rests on the incorrect speed-of-sound definition.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies stationary solutions of the two-dimensional Gross-Pitaevskii equation with a conformally flat background metric or, equivalently, with position-dependent coupling and external potential. For a funnel-like metric f(R)=1+(R0/R)^4, it constructs non-singular radial flows connecting two asymptotically flat regions and interprets them as acoustic black/white hole and one-way wormhole configurations. It then linearizes density and phase fluctuations under a hydrodynamic approximation to derive acoustic metrics, identifies horizons in closed form for uniform-density solutions, and computes Hawking temperatures via analytic continuation, Euclidean periodicity, and tunneling arguments. The main quantitative results are the temperature formulas (4.8) and (6.8) and their R0→0 limit TH=1/(2π|B|), which reproduces the photon-gas result of Ref. [22].","tokens_in":12796,"tokens_out":19606,"duration_ms":158705,"significance":"If the claims hold, the paper provides a concrete two-dimensional BEC setting for stationary sonic black/white hole and one-way wormhole configurations, with the notable advantage that the uniform-density cases admit closed-form horizon locations and Hawking temperatures. The temperature derivations are internally cross-checked by three methods (analytic continuation, Wick-rotated periodicity, and tunneling coefficient), and the R0→0 limit correctly reduces to a known result. The paper also uses two independent numerical methods, Chebyshev collocation Newton iteration and a physics-informed neural network, which mutually corroborate the stationary profiles. However, the non-uniform wormhole claim currently rests on an incorrect definition of the local speed of sound, and the numerical evidence lacks convergence diagnostics, so the central existence claim needs revision before the paper can be accepted.","major_comments":[{"comment":"The local speed of sound is defined in Eq. (3.1) as C = sqrt(2ρ(R))/sqrt(f(R)), with ρ the wave-function amplitude and n=ρ^2 the density. The linearized system (3.9)-(3.10), when combined under the hydrodynamic approximation, implies a phonon speed c_s = sqrt(2)ρ0/sqrt(f), and the acoustic metric (3.11) is consistent with that: its g00 vanishes at ρ0^6 f R^2 = 2B^2. The crossing condition C=|V| used in Fig. 3 therefore reduces to ρ0^5 f R^2 = 2B^2, which differs from the true horizon condition by a factor ρ0. For the non-uniform wormhole solution, where ρ0 is less than 1 in the throat region, the plotted crossings do not coincide with the horizons of the derived phonon metric, and it is not established that any true crossing exists for the presented parameters. The same factor-ρ error appears in Section 5, where the variable-coupling speed is written as C=√(2fρ) instead of C=ρ√(2f). Because the Conclusions claim that all configurations exhibit a crossing and hence wormhole configurations, this issue is load-bearing for the central claim of the paper.","section":"§3.2, Eqs. (3.9)-(3.10)"},{"comment":"The acoustic metric derivation relies on the approximation ∇·(ρ0^2∇N1)≈0, but no quantitative estimate of its validity is provided. For the non-uniform wormhole solution, the background density varies substantially through the throat, and the same approximation is used for the uniform-density cases, where it is justified only at long wavelengths. The manuscript should either estimate the neglected terms on the actual numerical solutions or explicitly state the low-frequency regime in which the acoustic-metric description applies. Without this, the Hawking-temperature predictions (4.8) and (6.8) remain uncontrolled approximations.","section":"§3.2, Eqs. (3.9)-(3.10)"},{"comment":"The existence of the non-uniform wormhole solution is established numerically, but the numerical evidence is incomplete. The Chebyshev-Newton method is described with the map (3.5) and boundary conditions, yet the number of collocation points N, the Newton tolerance, the maximum residual, and the sensitivity to the mapping parameter A=10 are not reported. The neural-network comparison in Fig. 2 is only qualitative. Please report the convergence in N and the residual norms for the solutions used in Figs. 3-4, so that the reader can judge whether the claimed horizon crossings are reliably resolved.","section":"§3.1, Eqs. (3.5)-(3.7)"}],"minor_comments":[{"comment":"The corrupted glyph sequence “Leftr⮯g⊸tl⮯ne⇒” appears in Eqs. (4.7), (6.7), and Appendix A; these should be replaced by the intended arrows or words.","section":"§4.3, §6.3, Appendix A"},{"comment":"The diagonalized metrics are central to the temperature derivation, but the line-broken equations make the denominators ambiguous; please typeset (4.4) and (4.6) unambiguously, especially the numerator in (4.6).","section":"§4.2-4.3"},{"comment":"The statement that two sets of functions “have the exact same set of solutions for ρ(R)” should add the obvious qualifications that the integration constant B and the boundary conditions must also match; as written it is slightly too broad.","section":"§2"},{"comment":"Reference [24] is incomplete: “S W Hawking. ‘Quantum gravity and path integrals’. In: Phys. Rev., D; (United States) (Sept. 1978)” lacks volume, issue, and page numbers.","section":"References"},{"comment":"The statement that the variable-coupling uniform-density solutions are “perhaps the solutions that are easier to reproduce in an experimental setting” is too strong, given that the required coupling g0(R)=1+(R0/R)^4 is singular at R=0 and the construction requires an unmodeled sink at the center; the text should soften or justify this experimental-readiness claim.","section":"§6"},{"comment":"The transition from the linearized equations (3.9)-(3.10) to the acoustic metric (3.11) is not shown; displaying the resulting wave equation for θ1 and the explicit identification of the Painlevé-Gullstrand form would make the derivation easier to verify.","section":"§3.2 to §4.1"}],"recommendation":"major_revision","confidential_remarks":"Major Comment 1 is the decisive issue. The uniform-density temperature results (Sections 4 and 6) appear internally consistent and are unaffected by the speed-of-sound factor, since ρ0=1 there. But the paper's advertised wormhole configuration in Section 3 is currently supported by a horizon criterion that disagrees with the derived acoustic metric by a factor ρ0. The authors should be asked to recompute the crossing with c_s=√(2)ρ0/√f, to report whether a true supersonic region exists for any (B,R0), and to update Figs. 3-4 and the Conclusions accordingly. If no such region exists, the wormhole claim should be retracted or substantially qualified. Strengthening the numerical convergence reporting is also needed for the existence claim."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me skip the formalities. The paper has a genuinely useful core and one load-bearing soft spot.\n\nThe useful core is the uniform-density construction in Sections 4 and 6. For ρ=1 the paper finds closed-form horizon radii Rp,m = sqrt(B^2 ± sqrt(B^4 − R0^4)) and derives the Hawking temperature of the outer horizon by three independent methods — Bogoliubov transform, analytic continuation, and Wick-rotated periodicity — that all agree (Eqs. 4.8 and 6.8). The R0→0 limit reduces to the photon-gas system of [22], which is a solid consistency check. The correspondence principle between a curved-metric description and a flat metric with position-dependent coupling is also a nice observation.\n\nThe soft spot is the non-uniform wormhole of Section 3, which the title advertises. The paper locates acoustic horizons by the crossing of the local speed of sound C and flow speed |V|, but C is defined in Eq. (3.1) as sqrt(2ρ)/sqrt(f). The linearized equations (3.9)–(3.10), combined under the hydrodynamic approximation, actually give a phonon speed of sqrt(2)ρ/sqrt(f). For ρ≠1 these differ by a factor sqrt(ρ). The true horizon condition from the acoustic metric (3.11) is ρ^6 (R^4+R0^4) = 2B^2 R^2, while the plotted crossing satisfies ρ^5 (R^4+R0^4) = 2B^2 R^2. In the wormhole solution the density is not constant in the throat, so the crossing shown in Figure 3 does not coincide with the actual phonon horizon. The wormhole claim therefore does not currently have supporting evidence. The uniform-density cases are immune to this because ρ=1.\n\nSecondary concerns: the hydrodynamic approximation ∇·(ρ0^2 ∇N1) ≈ 0 is adopted without a quantitative error estimate; the numerical BVP solutions lack convergence data and code; and the flat-space variable-coupling realization needs a singular coupling and an unmodeled sink at R=0, so the 'easier to reproduce experimentally' claim is overstated.\n\nNet: the uniform-density part deserves serious attention, but the central wormhole result needs to be redone with the correct speed of sound. I would send this to peer review — the flaw is specific, fixable, and confined to one section, and the uniform-density results are valuable enough to justify referee time. But a referee should demand a corrected crossing analysis for the non-uniform solution before acceptance.","headline":"A solid uniform-density analogue black-hole result is paired with a speed-of-sound error that undermines the non-uniform wormhole claim.","tokens_in":13318,"tokens_out":5636,"would_cite":true,"duration_ms":45916,"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":"Smooth 2D condensate flows can form acoustic black holes, white holes, and a one-way wormhole, with a calculable Hawking temperature in uniform-density cases.","keywords":["acoustic black holes","Bose-Einstein condensates","Gross-Pitaevskii equation","analogue gravity","acoustic wormhole","Hawking temperature","supersonic flow","hydrodynamic approximation"],"falsifier":"Evolve the full linearized density-phase equations (the uniform-density analogues of (3.9)-(3.10)) around the uniform-density funnel solution without dropping $\\nabla\\cdot(\\rho_0^2\\nabla N_1)$; if the radial transmission amplitude across the outer horizon differs from $\\exp(-2\\pi |B|^3 \\omega/\\sqrt{B^4-R_0^4})$, the derived Hawking temperature (4.8) fails.","tokens_in":12268,"feed_emoji":"🕳️","tokens_out":10783,"duration_ms":81749,"temperature":0.7,"pith_summary":"Stationary flows of a two-dimensional Bose-Einstein condensate can realize sonic analogues of black holes, white holes, and a one-way wormhole without the density singularities that plagued earlier ideal solutions. The paper solves the Gross-Pitaevskii equation numerically for a conformally flat funnel metric and, equivalently, for flat space with position-dependent coupling and potential; in every configuration the local speed of sound and the radial flow velocity cross, creating a supersonic region with acoustic horizons. For uniform-density solutions the horizon locations are known in closed form, and fluctuations behave as a massless scalar in a Painlevé-Gullstrand-like acoustic metric, giving a Hawking temperature $T_H = \\sqrt{B^4 - R_0^4}/(2\\pi |B|^3)$ at the outer horizon. If correct, these are concrete, non-singular analog-gravity systems whose horizon temperature is calculable and potentially measurable.","feed_headline":"Smooth condensate flows realize sonic black holes and wormholes","feed_subtitle":"In 2D condensates, flow can outrun sound at a horizon, with a computable Hawking temperature for uniform-density setups.","key_machinery":"The load-bearing object is the reduced radial equation for the condensate amplitude $\\rho(R)$ in a conformally flat two-dimensional metric, Eq. (2.9): $\\rho'' + \\rho'/R - B^2/(\\rho^3 R^2) + f(R)[(1-V(R))\\rho - g_0(R)\\rho^3]=0$, where $B$ fixes the radial flow. From a solution one computes the flow velocity $v = 2B/(f(R) R \\rho^2)$ and the local sound speed $c \\simeq \\sqrt{n g_0/m}$; an acoustic horizon is a radius where $|v|=c$. A second central identity is the correspondence principle, Eqs. (2.10)-(2.11): two triples $(f,V,g_0)$ share the same stationary solutions when $f(1-V)$ and $f g_0$ match, which lets the funnel-metric wormhole be redrawn as a flat-space variable-coupling black/white hole. The fluctuation analysis uses the hydrodynamic approximation $\\nabla\\cdot(\\rho_0^2 \\nabla N_1)\\approx 0$ to turn the coupled linearized equations into a massless scalar in the acoustic metric.","core_discovery":"The central claim is that non-singular stationary solutions of the two-dimensional Gross-Pitaevskii equation exist in which the condensate flows from one asymptotically flat region to another through a funnel-like metric $f(R)=1+(R_0/R)^4$, with the flow becoming supersonic near $R_0$; this is an acoustic one-way wormhole, with a black-hole horizon on one side and a white-hole horizon on the other. The same stationary profiles can be reinterpreted in flat space with a radially varying coupling $g_0(R)=1+(R_0/R)^4$ and a tuned external potential, yielding acoustic black/white holes without the wormhole's second asymptotic region. In the hydrodynamic approximation, linearized density and phase fluctuations combine into a single massless scalar wave equation in an acoustic metric of Painlevé-Gullstrand type. For the uniform-density subclass the acoustic horizons are $R_{p,m}=\\sqrt{B^2 \\pm \\sqrt{B^4-R_0^4}}$, and the Hawking temperature of the outer horizon is given by Eq. (4.8).","pith_inferences":["Because Eq. (4.8) depends only on the ratio $R_0/B$, a uniform-density experiment could in principle scan the horizon temperature by controlling the flow parameter $B$ at fixed $R_0$; the paper does not discuss this tuning.","The correspondence principle is not limited to the funnel profile: any two triples satisfying (2.10)-(2.11) yield identical stationary condensates, so the method may extend to other curved metrics.","The flat-space realization needs a singular coupling and an unmodeled sink at $R=0$, so a realistic test would require a finite system with a drain or a bounded domain; the paper leaves this as an open practical step.","The same hydrodynamic-limit acoustic-metric construction could be applied to anisotropic or non-conformally-flat backgrounds in two dimensions, where the metric is not simply $f(R)(dR^2+R^2 d\\varphi^2)$."],"forward_implications":["The funnel-metric solution regularizes the earlier singular acoustic black hole: fluid flows through a second asymptotic region instead of accumulating, so no sink or density divergence is required.","The flat-space analog with coupling $g_0(R)=1+(R_0/R)^4$ and a tuned external potential reproduces the same stationary profiles, giving a route to laboratory experiments without curved surfaces.","Uniform-density configurations have horizons at $R_{p,m}=\\sqrt{B^2\\pm\\sqrt{B^4-R_0^4}}$ (extremal when $|B|=R_0$), so horizon positions are known exactly.","Phase fluctuations in the hydrodynamic limit propagate as massless scalars in an acoustic metric of Painlevé-Gullstrand type, allowing a semiclassical Hawking-temperature and tunneling-coefficient computation."],"supporting_citations":[{"why":"Proposes acoustic black holes, establishing the analog-gravity concept the paper builds on.","marker":"[1]"},{"why":"Provides the previous singular stationary solutions and the numerical BVP methods (Chebyshev collocation, neural networks) that this work extends and regularizes.","marker":"[11]"},{"why":"Source of the Bose-Einstein condensate analogue-gravity framework and the hydrodynamic approximation used to combine fluctuation equations.","marker":"[4]"},{"why":"Gives the acoustic-metric derivation and hydrodynamic-limit treatment used for the massless scalar description.","marker":"[5]"},{"why":"Derives the acoustic metric for sonic black holes in dilute BECs, the background used for fluctuations.","marker":"[20]"},{"why":"Supplies the $R_0\\approx 0$ Hawking-temperature result and the Bogoliubov/tunneling calculation methods applied here.","marker":"[22]"},{"why":"The river model of black holes, giving the Painlevé-Gullstrand form of the acoustic metric and coordinate transformations.","marker":"[21]"},{"why":"Provides the tortoise-coordinate tunneling treatment used to interpret the horizon transmission coefficient.","marker":"[23]"},{"why":"The Euclidean path-integral periodicity argument used to derive the Hawking temperature from Wick-rotated time.","marker":"[24]"}],"fun_headline_variants":["2D condensate flows host sonic black holes and wormholes","Stationary BEC solutions reveal acoustic wormhole horizons","Sonic Hawking radiation predicted in 2D condensate horizons","Acoustic black/white holes from funnel-shaped 2D condensates","Flat-space mapping yields wormhole and black hole analogues"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The acoustic metric and Hawking temperature rest on the hydrodynamic approximation $\\nabla\\cdot(\\rho_0^2 \\nabla N_1)\\approx 0$ used to combine the linearized density and phase equations; the paper gives no quantitative check of this approximation for the constructed flow profiles, and for the non-uniform wormhole the background density varies significantly.","fun_headline_variants_meta":{"raw":{"variants":["2D condensate flows host sonic black holes and wormholes","Stationary BEC solutions reveal acoustic wormhole horizons","Sonic Hawking radiation predicted in 2D condensate horizons","Acoustic black/white holes from funnel-shaped 2D condensates","Flat-space mapping yields wormhole and black hole analogues"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000292,"raw_usage":{"total_tokens":1718,"prompt_tokens":973,"completion_tokens":745,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":660}},"tokens_in":589,"tokens_out":745,"duration_ms":7794,"temperature":1.0,"reasoning_tokens":660,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T23:51:03.236544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evolve the full linearized density-phase equations (the uniform-density analogues of (3.9)-(3.10)) around the uniform-density funnel solution without dropping $\\nabla\\cdot(\\rho_0^2\\nabla N_1)$; if the radial transmission amplitude across the outer horizon differs from $\\exp(-2\\pi |B|^3 \\omega/\\sqrt{B^4-R_0^4})$, the derived Hawking temperature (4.8) fails.","supporting_citations":[{"cited_title":"Experimental Black-Hole Evaporation?","cited_arxiv_id":null,"evidence_quote":"Proposes acoustic black holes, establishing the analog-gravity concept the paper builds on."},{"cited_title":"Stationary acoustic black hole solutions in Bose-Einstein condensates and their Borel analysis","cited_arxiv_id":"2411.06678","evidence_quote":"Provides the previous singular stationary solutions and the numerical BVP methods (Chebyshev collocation, neural networks) that this work extends and regularizes."},{"cited_title":"Analogue gravity from Bose-Einstein condensates","cited_arxiv_id":null,"evidence_quote":"Source of the Bose-Einstein condensate analogue-gravity framework and the hydrodynamic approximation used to combine fluctuation equations."},{"cited_title":"Analogue Models of and for Gravity","cited_arxiv_id":null,"evidence_quote":"Gives the acoustic-metric derivation and hydrodynamic-limit treatment used for the massless scalar description."},{"cited_title":"Sonic black holes in dilute Bose-Einstein conden- sates","cited_arxiv_id":null,"evidence_quote":"Derives the acoustic metric for sonic black holes in dilute BECs, the background used for fluctuations."},{"cited_title":"Proposal for an analog Schwarzschild black hole in con- densates of light","cited_arxiv_id":null,"evidence_quote":"Supplies the $R_0\\approx 0$ Hawking-temperature result and the Bogoliubov/tunneling calculation methods applied here."},{"cited_title":"The river model of black holes","cited_arxiv_id":null,"evidence_quote":"The river model of black holes, giving the Painlevé-Gullstrand form of the acoustic metric and coordinate transformations."},{"cited_title":"TORTOISE COORDINATE TRANSFORMATION ON APPARENT HORIZON OF A DYNAMICAL BLACK HOLE","cited_arxiv_id":null,"evidence_quote":"Provides the tortoise-coordinate tunneling treatment used to interpret the horizon transmission coefficient."},{"cited_title":"Quantum gravity and path integrals","cited_arxiv_id":null,"evidence_quote":"The Euclidean path-integral periodicity argument used to derive the Hawking temperature from Wick-rotated time."}],"review_version":1}