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arxiv: 2604.10348 · v1 · submitted 2026-04-11 · 🌀 gr-qc

Entropy covector field and macroscopic observables for rotating and non-rotating relativistic kinetic gases around a Schwarzschild black hole

Carlos Gabarrete , Daniela Montoya , Roger Raudales This is my paper

Pith reviewed 2026-05-10 15:22 UTC · model grok-4.3

classification 🌀 gr-qc
keywords relativistic kinetic gasSchwarzschild black holeentropy covector fieldrotating and non-rotating gasesanisotropy parameterkinetic temperaturemacroscopic observablescollisionless particles
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The pith

Angular momentum produces distinct asymptotic behaviors in macroscopic observables of relativistic kinetic gases near Schwarzschild black holes.

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

The paper derives the entropy covector field for relativistic kinetic gases consisting of collisionless massive particles on bound orbits in curved spacetime. It introduces two models for such gases around a Schwarzschild black hole, one rotating and one non-rotating, distinguished by their dependence on the inclination angle of particle orbits. From this field the authors construct observables including particle density, energy density, principal pressures, anisotropy parameter, and kinetic temperature. The work finds that rotation leads to notable differences in the behavior of these observables at large distances from the black hole. This matters for understanding how angular momentum affects the structure of matter in strong gravitational fields.

Core claim

We derive the components of the entropy covector field for a relativistic kinetic gas composed of collisionless, spinless, massive, and uncharged particles following bound orbits. By assuming a dependence on the inclination angle of the particle orbits, we consider two distinct models that describe a rotating and a non-rotating relativistic kinetic gas around a Schwarzschild black hole. Analysis of the anisotropy parameter, kinetic temperature, and average pressure constructed from the particle density, energy density, and principal pressures reveals significant differences between the rotating and non-rotating cases, particularly in their asymptotic behavior.

What carries the argument

The entropy covector field derived from the particle distribution function on bound orbits, which serves as the basis for calculating the macroscopic observables that distinguish the rotating and non-rotating gas models.

If this is right

  • The rotating model exhibits a different asymptotic value for the anisotropy parameter compared to the non-rotating model.
  • Kinetic temperature and average pressure display distinct behaviors at large distances in the presence of angular momentum.
  • The morphology of the gas configurations is shaped by the inclusion of rotation.
  • These differences underscore the role of angular momentum in collisionless gases in strong gravitational fields.

Where Pith is reading between the lines

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

  • Similar distinctions might appear in other black hole spacetimes if angular momentum is conserved.
  • Simulations of particle ensembles could verify the predicted asymptotic differences.
  • The framework might inform models of matter distributions in astrophysical environments near black holes.

Load-bearing premise

The models rely on the assumption that the gas distribution depends on orbital inclination angle in a specific way that separates rotating and non-rotating cases.

What would settle it

A calculation showing that the asymptotic anisotropy parameter, kinetic temperature, and average pressure are the same for both the rotating and non-rotating models would contradict the reported significant differences.

Figures

Figures reproduced from arXiv: 2604.10348 by Carlos Gabarrete, Daniela Montoya, Roger Raudales.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: shows the behavior of the anisotropy parameter as a function of the dimensionless radius ξ in the equatorial plane for the non-rotating gas model (zero angular momentum). In the left panel, it can be noticed that the anisotropy parameter exhibits a significant dependence on the value of k, especially in the intermediate ξ region. For a fixed ξ, the anisotropy decreases as k increases, indicating that, even… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p014_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png] view at source ↗
read the original abstract

In this article, we derive the components of the entropy covector field for a relativistic kinetic gas composed of collisionless, spinless, massive, and uncharged particles following bound orbits in a curved spacetime background. By assuming a dependence on the inclination angle of the particle orbits, we consider two distinct models that describe a rotating and a non-rotating relativistic kinetic gas around a Schwarzschild black hole. We analyze the behavior of key macroscopic observables (including the anisotropy parameter and the kinetic temperature) which are constructed from the particle density, energy density, and principal pressures. We aim to characterize and compare the morphology of the resulting configurations, thereby extending and complementing a previous work. The results reveal significant differences between the rotating and non-rotating cases, particularly in the asymptotic behavior of the anisotropy parameter, kinetic temperature, and average pressure, highlighting the role of angular momentum in shaping the macroscopic properties of collisionless gases in strong gravitational fields.

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 derives the components of the entropy covector field for a relativistic kinetic gas of collisionless, spinless, massive, uncharged particles on bound orbits in curved spacetime. By assuming a dependence on the inclination angle of the particle orbits, it constructs two models for a rotating and a non-rotating gas around a Schwarzschild black hole. Macroscopic observables (anisotropy parameter, kinetic temperature, average pressure) are built from the particle density, energy density, and principal pressures; their morphology and asymptotic behavior are compared between the two cases, with the central claim being that angular momentum produces significant differences, especially at large radii.

Significance. If the derivations hold and the inclination-angle assumption is physically justified, the work would extend prior analyses of collisionless gases in strong gravity by incorporating rotation and quantifying its effect on thermodynamic-like observables. This could be relevant for modeling stationary axisymmetric configurations near black holes, though the significance is limited by the lack of independent benchmarks or falsifiable predictions shown in the provided material.

major comments (2)
  1. [Abstract and model-construction section] Abstract and model-construction section: the distinction between the rotating and non-rotating cases rests entirely on positing an inclination-angle dependence in the one-particle distribution function (or entropy covector). No derivation from the geodesic constants of motion (energy, angular momentum, Carter constant) or from an explicit phase-space measure is supplied to show that this functional form encodes net angular momentum while preserving stationarity and axisymmetry; alternative weightings with the same cutoff could alter or reverse the reported asymptotic trends.
  2. [Results section on macroscopic observables] Results section on macroscopic observables: the claimed significant differences in the asymptotic behavior of the anisotropy parameter, kinetic temperature, and average pressure are presented without the explicit functional forms, integration steps, or cross-checks against the inclination assumption itself. It is therefore impossible to verify whether these differences are load-bearing consequences of angular momentum or artifacts of the model definition.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a concise statement of the precise functional form chosen for the inclination dependence and how it reduces to the non-rotating case.
  2. Notation for the entropy covector components and the principal pressures should be introduced with a clear table or equation reference before the numerical analysis begins.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments below with clarifications on the model assumptions and the presentation of results. Where appropriate, we indicate revisions that will be incorporated to improve transparency and verifiability.

read point-by-point responses

  1. Referee: [Abstract and model-construction section] Abstract and model-construction section: the distinction between the rotating and non-rotating cases rests entirely on positing an inclination-angle dependence in the one-particle distribution function (or entropy covector). No derivation from the geodesic constants of motion (energy, angular momentum, Carter constant) or from an explicit phase-space measure is supplied to show that this functional form encodes net angular momentum while preserving stationarity and axisymmetry; alternative weightings with the same cutoff could alter or reverse the reported asymptotic trends.

    Authors: We agree that the rotating versus non-rotating distinction is introduced via an assumed inclination-angle dependence in the distribution function (and thus the entropy covector). This functional form is selected because it is invariant under time translations and rotations about the axis, thereby preserving stationarity and axisymmetry while allowing a net angular momentum through preferential weighting of prograde or inclined orbits. The assumption is motivated by physical considerations for collisionless gases on bound orbits, where the inclination angle serves as a proxy for the Carter constant's effect on orbital planes. However, we did not supply an explicit derivation from the full set of geodesic constants or phase-space measure in the manuscript. We will revise the model-construction section to include a clearer statement of this motivation, a discussion of why this particular dependence encodes net rotation without violating the symmetries, and an explicit acknowledgment that alternative weightings could modify the quantitative trends (though we maintain that the qualitative distinction between rotating and non-rotating cases remains robust for the chosen cutoff). revision: yes

  2. Referee: [Results section on macroscopic observables] Results section on macroscopic observables: the claimed significant differences in the asymptotic behavior of the anisotropy parameter, kinetic temperature, and average pressure are presented without the explicit functional forms, integration steps, or cross-checks against the inclination assumption itself. It is therefore impossible to verify whether these differences are load-bearing consequences of angular momentum or artifacts of the model definition.

    Authors: We acknowledge that the results section presents the asymptotic behaviors without reproducing the full integration steps or explicit expressions for the observables. The anisotropy parameter, kinetic temperature, and average pressure are obtained by integrating the entropy covector components over the allowed phase-space region defined by the bound-orbit conditions and the inclination-dependent distribution. In the revised manuscript we will add an appendix containing the explicit functional forms for the particle density, energy density, and principal pressures, together with the principal integration steps leading to the macroscopic observables. We will also include a brief cross-check subsection that recomputes the asymptotic trends under a modest variation of the inclination weighting to confirm that the reported differences (particularly the slower decay of anisotropy and temperature at large radii in the rotating case) are driven by the presence of net angular momentum rather than by the specific cutoff choice. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit modeling assumption drives distinction between cases

full rationale

The paper states it assumes inclination-angle dependence to define two distinct models (rotating vs non-rotating). Macroscopic observables are then computed from the resulting distribution function, particle density, energy density and pressures. No equation reduces a claimed prediction back to the input by construction; the differences are the direct output of the chosen ansatz rather than a hidden redefinition or self-citation chain. The derivation chain is therefore self-contained once the modeling choice is granted.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central construction rests on the assumption that the entropy covector depends on orbit inclination angle to separate rotating and non-rotating cases; this is introduced without derivation in the abstract and functions as an ad-hoc modeling choice.

free parameters (1)
  • inclination-angle dependence
    Used to define distinct rotating and non-rotating models; no explicit functional form or fitting procedure is given in the abstract.
axioms (2)
  • domain assumption Particles are collisionless, spinless, massive, and uncharged and follow bound orbits in Schwarzschild spacetime
    Stated directly in the abstract as the setup for the kinetic gas.
  • domain assumption Macroscopic observables are constructed from particle density, energy density, and principal pressures
    Standard kinetic-theory relation invoked without further justification in the abstract.

pith-pipeline@v0.9.0 · 5469 in / 1493 out tokens · 84257 ms · 2026-05-10T15:22:58.981728+00:00 · methodology

discussion (0)

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Reference graph

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