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arxiv: 2603.23931 · v3 · pith:BGHNDS4Lnew · submitted 2026-03-25 · 🌀 gr-qc · quant-ph

Energy Balance of a Boson Gas at Zero Temperature in Curved Spacetime

Jorge Meza-Dom\'inguez , Tonatiuh Matos , Pierre-Henri Chavanis This is my paper

Pith reviewed 2026-05-19 17:45 UTC · model grok-4.3

classification 🌀 gr-qc quant-ph
keywords boson gaszero temperaturecurved spacetimeenergy balanceFisher entropyMadelung representationADM formalismboson stars
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The pith

Zero-temperature boson gas in curved spacetime obeys a spacetime-derived energy balance equation alongside a Fisher entropy constraint.

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

The paper develops a thermodynamic description for a zero-temperature boson gas in curved spacetime using the hydrodynamic Madelung representation in the ADM formalism. It establishes an energy balance equation that represents the first law of thermodynamics from a spacetime perspective and an information-theoretic constraint linking Fisher entropy to the evolution of the boson density. This approach separates energy transport from information conservation, revealing how quantum information persists in gravitational fields. The framework is shown to be consistent for systems in Minkowski and Schwarzschild spacetimes and connects to models of boson stars and scalar dark matter.

Core claim

Using the Madelung hydrodynamic representation within the ADM formalism, the authors derive an energy balance equation for the zero-temperature boson gas that encodes the first law of thermodynamics in curved spacetime, together with a constraint that ties the Fisher entropy to the dynamical evolution of the boson density. This dual structure separates the transport of energy from the conservation of quantum information, with a stochastic velocity introduced to relate quantum potential effects to underlying spacetime fluctuations.

What carries the argument

The hydrodynamic Madelung representation applied in the ADM formalism, which enables derivation of the energy balance and Fisher entropy constraint.

If this is right

  • The first law of thermodynamics arises directly from the spacetime geometry for the boson fluid.
  • Fisher entropy provides an independent constraint on the density dynamics that preserves quantum information.
  • The approach applies consistently to boson stars and scalar field dark matter in curved backgrounds.
  • A stochastic velocity bridges quantum potential to metric fluctuations.
  • Verification holds for both flat Minkowski and Schwarzschild spacetimes.

Where Pith is reading between the lines

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

  • This formulation may offer a way to incorporate quantum information considerations into relativistic fluid models without full quantum gravity.
  • It suggests possible extensions to other scalar field systems in general relativity.
  • Numerical checks in dynamical spacetimes could test the separation of energy and information flows.

Load-bearing premise

The hydrodynamic Madelung representation remains valid when applied within the ADM formalism to a zero-temperature boson gas in curved spacetime.

What would settle it

A detailed calculation or simulation of a boson star in Schwarzschild spacetime where the energy balance equation is violated would disprove the central relationships.

Figures

Figures reproduced from arXiv: 2603.23931 by Jorge Meza-Dom\'inguez, Pierre-Henri Chavanis, Tonatiuh Matos.

Figure 1
Figure 1. Figure 1: (Color online) One-dimensional harmonic oscillator. Left: Boson density [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (Color online) Hydrogen atom with fixed ℓ = 0, m = 0 (s-states). Left: Density n(r) for principal quantum numbers ν = 1, 2, 3, 4, 5. Right: Fisher entropy IF (r). Higher ν states extend farther from the nucleus and show more radial structure [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (Color online) Hydrogen atom with fixed ν = 3, m = 0. Left: Radial density n(r) for angular momentum quantum numbers ℓ = 0, 1, 2. Right: Fisher entropy IF (r). Radial coordinate is in units of Bohr radius a0. Third, for non-stationary superpositions (A, B ̸= 0), the function F(t) = |ϕ(t)| 2 modulates IF in time, demonstrating a dynamic redistribution of quantum information due to interference between forwa… view at source ↗
Figure 4
Figure 4. Figure 4: (Color online) Hydrogen atom with fixed ν = 4, ℓ = 3. Left: Density n(r) for magnetic quantum numbers m = −3, −2, . . . , 3. Right: Fisher entropy IF (r). The angular dependence modulates the radial profiles through spherical harmonics Yℓm(θ, ϕ) evaluated at θ = π/4. 7.2 Schwarzschild Metric The Schwarzschild metric describes the spacetime ex￾terior to a static, spherically symmetric, black hole and serves… view at source ↗
Figure 5
Figure 5. Figure 5: (Color online) Klein-Gordon field in Schwarzschild geometry with fixed [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: (Color online) Klein-Gordon field in Schwarzschild geometry with fixed [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
read the original abstract

We develop a comprehensive thermodynamic description for a zero-temperature boson gas in curved spacetime, integrating energy conservation with information-theoretic principles. Using the hydrodynamic Madelung representation within the ADM formalism, we establish two fundamental relationships: an energy balance equation representing the first law of thermodynamics from a spacetime perspective, and an information-theoretic constraint connecting Fisher entropy to the dynamical evolution of the boson density. This dual formulation clearly separates energy transport from information conservation while revealing how quantum information is preserved in curved backgrounds. The introduction of a stochastic velocity provides a bridge between quantum potential effects and underlying spacetime fluctuations. We demonstrate the consistency of our framework through detailed analyses of quantum systems in both Minkowski and Schwarzschild spacetimes. This work provides a unified foundation for studying relativistic bosonic systems, with direct relevance to boson stars and scalar field dark matter models.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a thermodynamic framework for a zero-temperature boson gas in curved spacetime by combining the hydrodynamic Madelung representation with the ADM 3+1 formalism. It claims to derive an energy balance equation that constitutes the first law of thermodynamics from a spacetime perspective and an information-theoretic constraint linking Fisher entropy to the dynamical evolution of the boson density. A stochastic velocity is introduced to bridge quantum potential effects with spacetime fluctuations. Consistency is asserted through analyses in Minkowski and Schwarzschild backgrounds, with the framework positioned as relevant to boson stars and scalar-field dark matter.

Significance. If the central derivations hold without residual curvature contributions in the energy balance, the work could provide a useful bridge between quantum hydrodynamics and relativistic thermodynamics for bosonic systems. The explicit separation of energy transport from information conservation via Fisher entropy is a potentially interesting angle, though its novelty and applicability rest on whether the ADM decomposition cleanly isolates these aspects as claimed.

major comments (2)
  1. [§4] §4 (ADM decomposition and energy balance derivation): the quantum potential term, after the 3+1 split, acquires contributions from the spatial metric determinant, extrinsic curvature, and lapse gradients in the Laplacian acting on the amplitude. The manuscript does not explicitly compute or cancel these terms in the resulting continuity and Euler equations, leaving open whether they mix into the energy flux and prevent the balance equation from being exactly the first law with no residual curvature sources.
  2. [§5] §5 (Fisher entropy constraint and stochastic velocity): the information-theoretic relation is presented as independent of the energy transport, but the stochastic velocity is introduced without a derivation showing it absorbs or decouples the curvature-induced quantum-potential corrections identified in the energy sector. This separation is load-bearing for the dual-formulation claim.
minor comments (2)
  1. [§3] Notation for the stochastic velocity and its relation to the quantum potential should be defined more explicitly, including any averaging procedure used to connect it to spacetime fluctuations.
  2. [§6] The consistency checks in Minkowski and Schwarzschild sections would benefit from explicit comparison tables or plots showing the magnitude of any residual terms after the claimed cancellations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us strengthen the presentation of the derivations. We address each major comment below and have revised the manuscript to provide the requested explicit calculations and clarifications.

read point-by-point responses

  1. Referee: [§4] §4 (ADM decomposition and energy balance derivation): the quantum potential term, after the 3+1 split, acquires contributions from the spatial metric determinant, extrinsic curvature, and lapse gradients in the Laplacian acting on the amplitude. The manuscript does not explicitly compute or cancel these terms in the resulting continuity and Euler equations, leaving open whether they mix into the energy flux and prevent the balance equation from being exactly the first law with no residual curvature sources.

    Authors: We thank the referee for this observation. Upon re-examining the 3+1 decomposition of the quantum potential in the Madelung representation, the contributions arising from the spatial metric determinant, extrinsic curvature, and lapse gradients in the Laplacian do cancel exactly when forming the continuity and Euler equations. This cancellation follows from the divergence-free property of the probability current and the structure of the ADM constraints, leaving the energy balance equation free of residual curvature sources and equivalent to the first law. To make this transparent, we have added an appendix containing the full term-by-term expansion and cancellation. revision: yes

  2. Referee: [§5] §5 (Fisher entropy constraint and stochastic velocity): the information-theoretic relation is presented as independent of the energy transport, but the stochastic velocity is introduced without a derivation showing it absorbs or decouples the curvature-induced quantum-potential corrections identified in the energy sector. This separation is load-bearing for the dual-formulation claim.

    Authors: The stochastic velocity is defined by projecting the curvature-corrected quantum potential onto the fluid velocity field in such a way that it exactly absorbs the residual terms identified in the energy sector. This construction ensures that the Fisher entropy evolution equation remains independent of the energy flux. We have expanded the derivation in §5 to include the explicit projection and absorption steps, thereby confirming the separation required for the dual formulation. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation remains self-contained

full rationale

The paper presents a framework integrating the hydrodynamic Madelung representation with the ADM formalism to derive an energy balance equation (first law) and a Fisher entropy constraint for a zero-temperature boson gas in curved spacetime. No quoted equations or steps in the abstract or description reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations that presuppose the target result. The stochastic velocity is introduced as a conceptual bridge rather than a fitted input renamed as prediction, and consistency checks in Minkowski and Schwarzschild backgrounds are presented as external validations. The central separation of energy transport from information conservation follows from the stated assumptions without evident reduction to prior inputs by definition. This is the expected non-finding for a paper whose claims rest on explicit derivations rather than circular renaming or imported uniqueness theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the applicability of the Madelung representation in curved spacetime and introduces a stochastic velocity without independent evidence provided in the abstract.

axioms (1)
  • domain assumption The hydrodynamic Madelung representation is valid for describing a zero-temperature boson gas within the ADM formalism in curved spacetime.
    Invoked to derive the energy balance equation and information-theoretic constraint.
invented entities (1)
  • stochastic velocity no independent evidence
    purpose: Bridge between quantum potential effects and underlying spacetime fluctuations.
    Introduced to connect quantum and gravitational aspects in the framework.

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Reference graph

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