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REVIEW 3 major objections 6 minor 26 references

Acoustic black holes, white holes, and wormholes in Bose-Einstein condensates in two dimensions

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict A solid uniform-density analogue black-hole result is paired with a speed-of-sound error that undermines the non-uniform wormhole claim. read the letter →

arxiv 2412.02727 v2 pith:6UHCIMWR submitted 2024-12-03 cond-mat.quant-gas gr-qcquant-ph

Sachin Vaidya , Martin Kruczenski This is my paper · ORCID
classification cond-mat.quant-gasgr-qcquant-ph
keywords acousticblackholesBose-EinsteincondensatesGross-PitaevskiiequationanaloguegravitywormholeHawkingtemperaturesupersonicflowhydrodynamicapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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).

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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 $f(R)(dR^2+R^2 d\varphi^2)$.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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].

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 (3)
  1. [§3.2, Eqs. (3.9)-(3.10)] 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.
  2. [§3.2, Eqs. (3.9)-(3.10)] 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.
  3. [§3.1, Eqs. (3.5)-(3.7)] 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.
minor comments (6)
  1. [§4.3, §6.3, Appendix A] 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.
  2. [§4.2-4.3] 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).
  3. [§2] 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.
  4. [References] 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.
  5. [§6] 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.
  6. [§3.2 to §4.1] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: Hawking temperatures and acoustic metrics are derived from the linearized GPE without fitting; the self-citation to [11] is only for numerical methods and comparison.

full rationale

The paper's derivation chain is self-contained. Static backgrounds are obtained by solving the GPE (Eq. 2.9) numerically; no parameter is fitted to any target Hawking temperature. The acoustic metrics (3.11), (4.3), (5.5), and (6.5) are obtained from the linearized density/phase equations (3.9)-(3.10) under the standard hydrodynamic approximation, cited to Refs. [4,5] (not the authors' own work). The Hawking temperatures (4.8) and (6.8) are computed from the resulting metrics by analytic continuation and by Wick-rotated Euclidean time periodicity; the R0 -> 0 reductions reproduce the independent photon-gas result [22] as a consistency check, not as the derivation. The only self-citation, Ref. [11], supplies numerical BVP techniques and singular solutions used for comparison; it is not load-bearing for the central claims. A separate correctness caveat, not a circularity: the 'scaled local speed of sound' in Eq. (3.1) is amplitude-based, and for the non-uniform density wormhole the plotted crossing with |V| differs from the acoustic-metric horizon condition (g00 = 0) by a factor of rho0; this concerns the validity of the horizon identification, but it is not a reduction of the prediction to its inputs. The flat-space implementation also assumes a singular coupling and an unmodeled sink at R = 0, which weakens experimental claims but is not circular.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the input parameters B and R0, the GPE as the dynamical equation, the conformally flat funnel metric, and the hydrodynamic approximation used to define the acoustic metric. No new particles, forces, or fields are introduced. The acoustic wormhole is an emergent description of phonons, not a new fundamental entity.

free parameters (2)
  • R0 = 1.4827 in Fig. 2/3 for |B| = 1; 0.1 and 10.0 in Fig. 4
    Throat scale of the funnel metric f(R) = 1 + (R0/R)^4. It is chosen by hand as the metric parameter and the Hawking temperature depends on it. Not fitted to data, but it is an input parameter of the model.
  • B = |B| = 1 in sample solutions; |B| = 50 in Fig. 4(b)
    Integration constant for the radial flow, B = R rho^2 d_theta/dR. It sets the flow strength and appears in the Hawking temperature TH. It is an input parameter, not derived.
assumptions (4)
  • domain assumption Gross-Pitaevskii equation is the correct mean-field description of the BEC dynamics
    Used throughout, e.g., Eq. (1.1); standard for dilute BECs at zero temperature in the regime considered.
  • domain assumption Spatial metric can be taken conformally flat f(R)(dR^2 + R^2 dphi^2)
    Any 2D metric is conformally flat; the funnel form is a choice of coordinates and surface shape, introduced in Section 2, Eq. (2.4).
  • domain assumption Hydrodynamic approximation div(rho0^2 grad N1) = 0
    Used in Sections 3.2, 4.1, 5.1, 6.1 to combine linearized equations into a scalar wave equation and derive the acoustic metric; cited to [4,5] but no validity estimate for these solutions.
  • standard math Standard semi-classical methods for Hawking temperature from a metric (analytic continuation, Euclidean periodicity, Bogoliubov coefficients)
    Used in Sections 4.2, 4.3, 6.2, 6.3; these are standard results in analogue gravity.

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Cite this review

Pith. "Pith review of Acoustic black holes, white holes, and wormholes in Bose-Einstein condensates in two dimensions." pith.science (2026). https://pith.science/paper/6UHCIMWR

@misc{pith2026241202727,
  author       = {Pith},
  title        = {Pith review of: Acoustic black holes, white holes, and wormholes in Bose-Einstein condensates in two dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6UHCIMWR}},
  note         = {Machine review of arXiv:2412.02727}
}
read the original abstract

In a previous article, we studied stationary solutions to the dynamics of a Bose-Einstein condensate (BEC) corresponding to acoustic (or Unruh) black/white holes, namely configurations where the flow becomes supersonic creating a horizon for phonons. In this paper, we consider again the Gross-Pitaevskii Equation (GPE) but looking for stationary numerical solutions in the case where the couplings are position dependent in a prescribed manner. Initially we consider a 2D quantum gas in a funnel-like spatial metric. We then reinterpret this solution as a solution in a flat metric but with spatially dependent coupling and external potential. In these solutions the local speed of sound and magnitude of flow velocity cross, indicating the existence of a supersonic region and therefore of sonic analogues of black/white holes and wormholes. We discuss the numerical techniques used. We also study phase (and density) fluctuations in these solutions and derive approximate acoustic metric tensors. For certain external potentials, we find uniform density acoustic black hole configurations and obtain their Hawking temperature.

Figures

Figures reproduced from arXiv: 2412.02727 by the authors.

Figure 1
Figure 1. Theoretical wormhole configuration. The fluid moves along a non-trivial surface [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Sample solution with background metric compared with logarithmic singular [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Speed of sound (dashed line) and magnitude of flow velocity (solid line) for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Speed of sound (dashed line) and magnitude of flow velocity (solid line) for [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

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