arxiv: 2601.12848 · v3 · submitted 2026-01-19 · ❄️ cond-mat.quant-gas · astro-ph.GA· cond-mat.stat-mech· quant-ph

Recognition: 2 theorem links

· Lean Theorem

Anisotropic dispersion relation of ultralight Bose gases in modified Newtonian dynamics

Ning Liu

Authors on Pith no claims yet

Pith reviewed 2026-05-16 13:47 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas astro-ph.GAcond-mat.stat-mechquant-ph

keywords ultralight Bose gasMONDdispersion relationanisotropyJeans instabilityGross-Pitaevskii equationmodified gravity

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The pith

Ultralight Bose gases under modified Newtonian dynamics exhibit an anisotropic dispersion relation that depends on the angle between perturbation wavevector and background gravitational field.

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

The paper couples the Gross-Pitaevskii equation describing the ultralight Bose gas to the nonlinear MOND Poisson equation and linearizes around a uniform background. The resulting dispersion relation for collective modes contains an explicit angular dependence on the direction of the wavevector relative to the gravitational field. This produces a Jeans instability whose critical wavelength and mass threshold both vary with orientation. A reader would care because the anisotropy supplies a concrete, direction-sensitive signature that could distinguish MOND from Newtonian gravity in quantum astrophysical systems such as ultralight dark-matter condensates.

Core claim

Starting from the coupled Gross-Pitaevskii and MOND Poisson equations, we derive an anisotropic dispersion relation that depends on the angle between the perturbation wavevector and the background gravitational field. This anisotropy arises directly from the nonlinear structure of the MOND field equation and leads to a direction-dependent Jeans instability, with critical wavelengths and masses varying with orientation.

What carries the argument

The nonlinear MOND Poisson equation coupled to the Gross-Pitaevskii equation, whose linearization around a uniform state yields explicit angular dependence in the dispersion relation.

If this is right

The critical wavelength for Jeans instability changes with the angle between the wavevector and the gravitational field.
The critical mass for instability is therefore orientation-dependent.
Ultralight Bose gases furnish a distinctive directional signature of MOND effects.
These condensates can function as probes of modified gravity in quantum astrophysical regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The directional dependence may select preferred axes for structure formation inside MOND-governed ultralight-dark-matter halos.
Stability calculations for boson stars or galactic condensates would need to incorporate the angle between local gravity and density perturbations.
Laboratory analogs using trapped Bose-Einstein condensates under effective modified-gravity potentials could test the predicted anisotropy.

Load-bearing premise

The MOND Poisson equation can be directly coupled to the Gross-Pitaevskii equation without additional relativistic or quantum corrections to the gravitational sector.

What would settle it

A numerical integration of the coupled equations in which the dispersion relation is recomputed after replacing the MOND Poisson operator with the standard Newtonian Laplacian, checking whether the angular dependence disappears.

Figures

Figures reproduced from arXiv: 2601.12848 by Ning Liu.

**Figure 1.** Figure 1: Squared frequency ˜ω 2 versus angle θ for fixed wavenumbers ˜k = 0.5 (dashed), ˜k = 1.0 (dot-dashed), and ˜k = 1.5 (solid), with χ = 1 and η = 0.1. The full anisotropic landscape is displayed in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

**Figure 2.** Figure 2: Contour plot of the squared frequency ˜ω [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: Polar plot of the normalized Jeans mass MeJ (θ). The radial coordinate is MeJ (θ); the dashed circle indicates the isotropic Newtonian limit (normalized to 1). Parameters are the same as in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

We investigate the dispersion relation of collective modes in ultralight Bose gases under Modified Newtonian Dynamics (MOND). Starting from the coupled Gross-Pitaevskii and MOND Poisson equations, we derive an anisotropic dispersion relation that depends on the angle between the perturbation wavevector and the background gravitational field. This anisotropy arises directly from the nonlinear structure of the MOND field equation and leads to a direction-dependent Jeans instability, with critical wavelengths and masses varying with orientation. Our results provide a distinctive signature of MOND in a quantum astrophysical context and suggest that ultralight Bose gases can serve as novel probes of modified gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives an anisotropic dispersion relation for ultralight Bose gases by coupling the Gross-Pitaevskii equation to the MOND Poisson equation, but the direct coupling lacks clear justification at macroscopic de Broglie scales.

read the letter

The central result is a dispersion relation whose frequency depends on the angle between the wavevector and the background gravitational field, which then produces an orientation-dependent Jeans scale. This comes from linearizing the coupled system around a uniform background density and potential obtained from the nonlinear MOND equation. The angular dependence is a direct consequence of how the MOND mu function responds differently parallel and perpendicular to the acceleration vector. That specific combination with ultralight bosons has not appeared in the earlier literature, so the directional instability signature is new on its own terms. The derivation follows the usual Bogoliubov-style steps once the potential is inserted, and the final expression for the critical wavelength tracks the angle in a straightforward way. If the algebra holds, the paper supplies a concrete, falsifiable prediction that could be checked in simulations of ultralight dark matter or in analog gravity setups. The main limitation is the assumption that the MOND gravitational potential can be treated as an external classical field inside the Gross-Pitaevskii equation without further quantum or relativistic corrections. At the accelerations and wavelengths relevant to ultralight bosons, the same scalar degree of freedom sources both the density and the phase dynamics, so it is not automatic that the effective mu function remains unmodified or that no extra gradient terms arise. The paper does not estimate the size of those corrections or show why they can be dropped. This is a real gap rather than a minor technicality, because altering the gravitational sector would change the angular dependence itself. The work is aimed at people already working on ultralight scalar dark matter or on possible quantum signatures of modified gravity. A reader who knows both the Gross-Pitaevskii framework and the MOND field equation will find the proposed signature useful to think about, even while wanting to verify the coupling step. It is worth sending to a serious referee because the claim is specific enough to be checked and the potential observational handle is clear, though the authors should expect pointed questions on the semiclassical consistency.

Referee Report

1 major / 1 minor

Summary. The manuscript derives an anisotropic dispersion relation for collective modes in ultralight Bose gases by coupling the Gross-Pitaevskii equation to the nonlinear MOND Poisson equation. The resulting relation depends on the angle between the perturbation wavevector and the background gravitational field, producing a direction-dependent Jeans instability with orientation-dependent critical wavelengths and masses.

Significance. If the central derivation is valid, the result supplies a concrete, angle-dependent signature of MOND that could be probed in ultralight Bose-Einstein condensates or ultralight dark-matter models. It connects quantum many-body dynamics to modified gravity without introducing new free parameters beyond the standard MOND acceleration scale.

major comments (1)

[§3] §3 (linearized dispersion derivation): the step that treats the MOND potential obtained from ∇·[μ(|∇Φ|/a0)∇Φ]=4πGρ as an external classical field in the Gross-Pitaevskii equation lacks explicit justification for the absence of quantum or relativistic corrections at the macroscopic de Broglie scale; if such corrections modify the effective μ or add gradient terms, the claimed angular dependence of the dispersion (and the direction-dependent Jeans scale) would change.

minor comments (1)

[Abstract] The abstract and introduction should explicitly state the background density and acceleration regime assumed for the uniform MOND field to allow readers to assess the validity of the linearization.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and the constructive comment on the justification of our effective-field treatment. We address the point below and have incorporated additional discussion to strengthen the manuscript.

read point-by-point responses

Referee: [§3] §3 (linearized dispersion derivation): the step that treats the MOND potential obtained from ∇·[μ(|∇Φ|/a0)∇Φ]=4πGρ as an external classical field in the Gross-Pitaevskii equation lacks explicit justification for the absence of quantum or relativistic corrections at the macroscopic de Broglie scale; if such corrections modify the effective μ or add gradient terms, the claimed angular dependence of the dispersion (and the direction-dependent Jeans scale) would change.

Authors: We acknowledge the referee’s concern and have added an explicit paragraph in the revised §3 clarifying the effective-theory assumptions. The MOND Poisson equation is solved classically to obtain the gravitational potential Φ, which then enters the Gross-Pitaevskii equation as an external potential; this is the standard mean-field procedure used for self-gravitating Bose gases under Newtonian gravity. At the macroscopic de Broglie wavelengths characteristic of ultralight bosons (typically astrophysical scales), gravitational quantum corrections are expected to be negligible, and no microscopic completion of MOND exists that would systematically alter the interpolating function μ or introduce new gradient terms at leading order. Relativistic corrections are outside the non-relativistic scope of the model, as is conventional for Gross-Pitaevskii treatments. Within this phenomenological framework the nonlinear structure of the MOND field equation produces the reported angular dependence of the dispersion relation, and the direction-dependent Jeans scales remain unchanged. We therefore view the derivation as robust for the intended regime. revision: partial

Circularity Check

0 steps flagged

Derivation proceeds directly from coupled equations without reduction to inputs or self-citations

full rationale

The paper begins with the standard coupled Gross-Pitaevskii equation and the MOND Poisson equation, then performs linearization around a background field to obtain the dispersion relation. No steps reduce by construction to fitted parameters, self-citations, or ansatzes imported from prior author work; the anisotropy emerges from the nonlinear structure of the MOND term as stated. The derivation chain remains self-contained against the input equations, with no evidence of renaming known results or load-bearing self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review limits visibility into parameters and axioms; the central claim rests on the validity of coupling the standard Gross-Pitaevskii equation to the nonlinear MOND Poisson equation.

axioms (1)

domain assumption The MOND field equation can be directly coupled to the Gross-Pitaevskii equation for ultralight Bose gases
Explicitly stated as the starting point of the derivation in the abstract.

pith-pipeline@v0.9.0 · 5400 in / 1092 out tokens · 28855 ms · 2026-05-16T13:47:06.665539+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

the factor D(k, θ) embodies the directional dependence emerging from the nonlinear MOND field equation... D(k, θ)=μ0 k² + μ'0 G0/a0 (k·ê)²
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel contradicts

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contradicts
CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

In the Newtonian limit (G0 ≫ a0), μ0 →1 and μ'0 →0, so D(k, θ)→k² and Eq. (19) reduces to the isotropic quantum Jeans dispersion

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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