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REVIEW 3 major objections 5 minor 22 references

Orbits of Massive Particles in a Spherically Symmetric Gravitational Field in View of Cosmological Constant

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims a complete classification of massive-particle trajectories in the Kottler metric for both signs of the cosmological constant, obtained by Puiseux-series root expansions, plus an upper bound on negative Lambda from galaxy…

desk verdict A novel Puiseux-root approach to Kottler orbits is undercut by a false root-at-rg expansion and a dimensionally inconsistent rotation-curve bound; the rg=0 closed orbit is worth a look, but the paper is not publishable as is. read the letter →

arxiv 2412.02455 v1 pith:L7TXZQJK submitted 2024-12-03 gr-qc astro-ph.GA

Ruslan Nakibov , Andrey Ursulov This is my paper · ORCID
classification gr-qcastro-ph.GA PACS 95.10.Eg98.80.Es
keywords KottlermetriccosmologicalconstantmassiveparticleorbitsPuiseuxseriesNewtonpolygonellipticintegralsgalaxyrotationcurvesgeneralrelativity
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

This paper tries to establish how the cosmological constant $\Lambda$ changes every possible trajectory of a massive test particle orbiting a spherically symmetric central body. Working in the Kottler metric, the authors reduce the motion to a quintic polynomial $p(r)$ whose positive roots are turning points, and claim that expanding the associated algebraic curve in Puiseux series—with Newton-polygon inequalities—gives a complete classification of orbits for both signs of $\Lambda$. The classification assigns each parameter set to one of eight root configurations and determines which configurations allow finite, bound, spiral, or unbound motion, with explicit trajectory formulas built from incomplete elliptic integrals. It also turns galaxy rotation-curve data into an upper bound $|\Lambda|\lesssim 10^{-13}\, v_{\min}^6/M^2$ (SI units) for negative $\Lambda$. If the classification is right, orbit observations become a direct probe of the sign and magnitude of the cosmological constant.

What carries the argument

The object that carries the argument is the trajectory polynomial $p(r)$: its positive roots are the turning points of the orbit, and the number, reality, and ordering of those roots decide whether the orbit falls inward, escapes, spirals, or oscillates between two radii. The paper treats $p(r)$ as an algebraic curve in its coefficients and analyzes it with the Newton-Puiseux method: each monomial contributes a point to a Newton polygon, the slopes of the polygon edges give the leading Puiseux exponents of the roots, and the inequalities defining each convex hull become the parameter conditions in Table 2. The same expansion isolates the factor $(r-r_g)^{-1/2}$, which is expanded as $1/\sqrt{r}+(r_g/2)r^{-3/2}+\dots$ so that the orbit integral reduces to a sum of incomplete elliptic integrals of the first, second, and third kinds.

What would settle it

Evaluate $p(r)$ for the parameter set $\Lambda=-1$, $r_g=1$, $j=2$, $K=4/3$ in the paper's units: $p(1)=2\neq 0$, so the root $r=r_g$ is absent there, contradicting the assumption that $r_g$ is one of the positive roots in the classification. A denser check is to compute the positive roots of $p(r)$ over a grid of the Table 2 conditions and compare their count and ordering with the claimed configurations.

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Extended reading notes

Core claim

The central claim is that the quintic $p(r)=(\Lambda/3)r^5+K r^3+r_g r^2-j^2 r+j^2 r_g$ (with $\Lambda$ replaced by $-|\Lambda|$ when it is negative) determines all non-circular orbits of massive particles in the Kottler metric, and that Puiseux expansion of this polynomial's algebraic curve yields every possible configuration of its positive roots. For $\Lambda<0$ and $\Lambda>0$ the paper lists five numbered and three lettered configurations (Table 2), with inequalities that decide how many roots are real, positive, and ordered. Motion is possible only where $p(r)>0$; in configurations A, B, and C a particle is confined between two turning radii, and the trajectory integral is evaluated as a linear combination of incomplete elliptic integrals, with explicit formulas (4.6) and (4.7). In the limit $r_g\to 0$ the paper derives a closed precessing orbit (4.10)–(4.12), and for $\Lambda<0$ it derives from the rotation curve the bound $|\Lambda|\lesssim 10^{-13}\, v_{\min}^6/M^2$ in SI units, consistent with earlier estimates for a $10^5\,M_\odot$ black hole.

Load-bearing premise

The classification hangs on the unproved premise that the leading Puiseux-root approximations and the Newton-polygon inequalities in Table 2 always give the true number, reality, and ordering of the positive roots of the quintic—and that one of those roots is exactly $r=r_g$ so the $(r-r_g)^{-1/2}$ expansion can be used.

Editorial extensions

If this is right

  • For negative $\Lambda$, every rotation curve has a minimum circular speed $v_{\min}$ at a radius near $(3r_g/(4|\Lambda|))^{1/3}$, and observed speeds above $v_{\min}$ translate into the bound $|\Lambda|\lesssim 10^{-13}\, v_{\min}^6/M^2$ (SI units).
  • For negative $\Lambda$, a stable circular orbit switches from the Schwarzschild-like radius $r\approx 2j^2/r_g$ to a cosmological-constant-dominated radius $r\approx (3j^2/|\Lambda|)^{1/4}$ once the dimensionless angular momentum exceeds a critical value.
  • For positive $\Lambda$, unbound trajectories outside $r_{\max}=\sqrt{3|K|/\Lambda}$ are modified hyperbolic spirals when $K>0$ and modified hyperbolas when $K<0$, both covered by a single formula.
  • In the confined configurations A, B, and C, trajectories between $r_{\min}$ and $r_{\max}$ are explicit linear combinations of incomplete elliptic integrals, so any concrete parameter set can be integrated without numerical ODE solving.
  • With $r_g=0$, negative $\Lambda$, and positive $K$, the motion is a closed bound orbit; a nonzero $r_g$ adds perihelion precession whose magnitude is given by an elliptic integral.

Reading between the lines

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

  • Beyond the paper, the Table 2 conditions could be stress-tested by a random numerical scan of the positive roots of $p(r)$; any mismatch between the computed roots and the claimed configuration would reveal a missing or mislabeled case.
  • Beyond the paper, the bound implies a mass–velocity relation $v_{\min}\propto M^{1/3}$ for galaxies if $\Lambda$ is universal, so the lower envelope of a $v_{\min}$ versus $M$ plot from real rotation-curve catalogs is a direct observational test.
  • Beyond the paper, the Newton-polygon approach is not tied to this particular metric and could classify orbits for other spherically symmetric line elements whose trajectory polynomial has higher degree.
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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 / 5 minor

Summary. The paper studies massive-particle orbits in the Kottler metric with a cosmological constant Λ. Starting from the Hamilton–Jacobi equation and the Binet equation, it derives a trajectory equation whose radicand is the polynomial F(r) in Eq. (1.11), equivalently the quintic p(r)=F(r)/r in Eq. (4.1). For both signs of Λ it proposes a classification of non-circular trajectories based on leading-order Puiseux expansions of the roots of p(r) (Table 2), singles out three configurations A, B, C that permit finite motion, and writes the corresponding orbits as combinations of incomplete elliptic integrals. For negative Λ it also proposes a circular-velocity formula (2.2) and an upper bound |Λ| ≲ 10^-13 v_min^6 / M^2 from galaxy rotation curves.

Significance. The topic is of continuing interest, and the paper has some useful ingredients: the reduction to a single quintic is standard and correctly carried out, the Newton-polygon/Puiseux viewpoint is a legitimate heuristic for organizing parameter regimes, and the explicit elliptic-integral formulas, if valid, would be convenient. The authors also provide a promising heatmap tool for scanning the (j, K) parameter plane. However, the main results as stated are not reliable: the factorization underlying the non-circular orbit integrals assumes a root at r = r_g that is absent in the very configurations to which it is applied, the rotational-curve formula has a wrong Λ = 0 limit, and the claimed completeness of the classification is not demonstrated. These are load-bearing defects, not presentation issues.

major comments (3)
  1. [Sec. 4, Eqs. (4.3)–(4.7) and Table 2] The assertion after Eq. (4.3) that “the root r = r_g is present only once in each of those configurations and it’s the smallest positive root” is false for the Λ < 0 finite-motion configurations to which the paragraph refers. Substituting r = r_g into Eq. (4.1) with Λ = -|Λ| and K = k^2 - 1 + |Λ| j^2 / 3 gives p(r_g) = r_g^3 [k^2 + (|Λ|/3)(j^2 - r_g^2)]. Configurations A and C explicitly require j > r_g, so p(r_g) > 0 there; configuration B, while not requiring j > r_g explicitly, lists r_g as a root in Table 2 without any condition that would make p(r_g) vanish. Therefore r_g is not a root of p(r) in the cases where the text asserts it is, the factor (r - r_g)^(-1/2) in Eq. (4.4) is absent, and the elliptic-integral formulas (4.5)–(4.7) do not follow from the trajectory equation.
  2. [Sec. 2, Eqs. (2.2)–(2.5)] Eq. (2.2) cannot be the classical circular-speed formula. For Λ = 0 it reduces to v^2/c^2 = (1/2)(r_g/r)^2 - (3/2)(r_g/r), which is negative for every r > r_g, rather than the Keplerian circular speed. The derivation of Eq. (2.2) from Eq. (2.1) is not shown, and the subsequent minimum-speed expressions (2.3)–(2.4) and the bound (2.5) inherit this problem. Moreover, substituting r_g = 2GM/c^2 into the preceding formula in Eq. (2.5) gives a prefactor 32/9 with a factor 1/(G^2 M^2 c^2), not 1/(G^2 c^4), and the simplified form “≈10^-13 v_min^6/M^2” is dimensionally inconsistent unless additional factors are intended.
  3. [Sec. 3, Table 2, and Appendix] The completeness of the proposed classification is not established. Table 2 lists only the first term of each Puiseux expansion, and the conditions in the table are Newton-polygon inequalities for term dominance. These determine the leading asymptotic form of roots, not the exact number, reality, or ordering of the positive roots of the quintic (4.1) for all parameter values. The Appendix states that the method is intended for “extracting valuable information about all possible sets of roots” and not for precise formulas, yet the paper’s central claim is a complete classification. No theorem or exhaustive numerical verification is supplied to close this gap. Because the finite-motion regions and the orbit formulas are read off from the root configurations, this gap is load-bearing.
minor comments (5)
  1. [Eq. (1.12)] The horizon equation is written as 1 - r_g/r - Λ r^2 = 0, while the metric component (1.10) contains Λ r^2 / 3; the relation between r_g, the mass parameter, and the actual event horizon should be stated consistently.
  2. [Table 2] The layout is extremely dense because conditions and roots for four sign combinations share one row; separating the four cases into distinct columns or tables and marking which listed roots are real and positive would make the classification checkable.
  3. [Fig. 7] The heatmap lacks axis labels, numerical scales, and the fixed values of Λ and r_g; without these the claimed parameter-scanning tool cannot be reproduced.
  4. [Eqs. (4.6)–(4.7)] The symbols X_s, R_n, V_n, α, α_1, and the elliptic-integral arguments are introduced abruptly; please define every symbol and state which roots correspond to a, b, c, d.
  5. [Throughout] The manuscript contains several typographical and grammatical errors (e.g., “rotatonal curves”, inconsistent capitalization, duplicated Russian/English abstracts); a thorough language edit is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning found: the trajectory classification and the Lambda bound follow from the stated metric and equations, with no fit or self-citation used as the load-bearing premise.

full rationale

The derivation chain is self-contained: the trajectory integral (1.5) follows from the Hamilton-Jacobi equation with the Kottler metric (1.10), and the radicand F(r) in (1.11) is obtained by direct substitution. The orbit classification in Table 2 is constructed from the Puiseux/Newton-polygon analysis of p(r)=0 in (4.1), which is presented as a prescribed root-finding method rather than as an imported result. The elliptic-integral expressions in Section 4 use standard integral formulas from Byrd and Friedman [19], an external reference, and are not used to define the premises. The bound on |Lambda| in Eq. (2.5) is derived algebraically from the rotation-curve formula (2.2) with externally adopted galaxy parameters, so it is an output of the calculation rather than a fitted input. No load-bearing self-citation or imported uniqueness theorem is invoked; the references to [5], [6], and [19] supply background or standard mathematical material. The possible issue that r=rg is asserted as a root in configurations A, B, and C in Section 4 is a correctness objection, not circularity: that assertion is not assumed in deriving the trajectory polynomial, and if it were false it would invalidate the expansion rather than make the derivation equivalent to its input by construction.

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

The central derivation rests on standard GR geodesic equations and the Puiseux theorem, plus an unproved domain assumption that first-term root approximations and the stated inequalities capture all real root configurations. The rotation-curve bound additionally assumes the classical limit formula and a clean separation between central mass and Lambda effects.

assumptions (4)
  • standard math Standard GR geodesic equations and the Binet equation apply to massive test particles in a static spherically symmetric metric.
    Invoked in Sec. 1 via the Hamilton-Jacobi equation (1.2) and the trajectory integral (1.5).
  • standard math The Newton-Puiseux theorem and Descartes' rule of signs can be applied to the polynomial p(r) over real positive r.
    Used in Sec. 4 and the Appendix to count roots and construct root expansions.
  • ad hoc to paper The first-term Puiseux root approximations and the inequalities in Table 2 are exact enough to identify all real positive roots and their ordering for every parameter regime.
    Claimed implicitly by the classification in Table 2; no proof of exhaustiveness or truncation error is given, and a spot check contradicts the root-at-rg assumption.
  • domain assumption The classical limit v^2 = (r/m) dU/dr yields the correct galaxy rotation curve and the upper bound (2.5).
    Used in Sec. 2 to derive Eq. (2.2) and the bound; the resulting formula appears to have sign and dimensional problems.

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Cite this review

Pith. "Pith review of Orbits of Massive Particles in a Spherically Symmetric Gravitational Field in View of Cosmological Constant." pith.science (2026). https://pith.science/paper/L7TXZQJK

@misc{pith2026241202455,
  author       = {Pith},
  title        = {Pith review of: Orbits of Massive Particles in a Spherically Symmetric Gravitational Field in View of Cosmological Constant},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L7TXZQJK}},
  note         = {Machine review of arXiv:2412.02455}
}
read the original abstract

In this paper we present the results of a theoretical study of the trajectories of massive particles in the K\"ottler metric in view of the cosmological constant {\Lambda}. For both negative and positive signs of {\Lambda} a classification of trajectories is proposed, with entries based on different solutions of the trajectory equation, obtained by the expansion of the corresponding algebraic curve in Puiseux series. We also provide some specific types of trajectories which correspond to different values of the cosmological constant. In the case of negative values of the cosmological constant its upper limit is estimated from the galaxy rotation curves

Figures

Figures reproduced from arXiv: 2412.02455 by the authors.

Figure 1
Figure 1. Stability diagram for circular orbits in dimensionless coordinates. The upper branch corresponds to stable roots, the lower branch to unstable. 4. Non-circular motion. Classification and example orbits All possible trajectories of non-circular orbits are determined by the equation (1.4) where F(r) is provided by (1.11). Trajectories depend on the number and value of positive roots of the p(r) = F(r)/r, (r ̸= 0). p(r… view at source ↗
Figure 2
Figure 2. Unbound orbit with Λ > 0, K > 0 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 6
Figure 6. Bound orbit with perihelion precession with Λ < 0, K > 0 due to accounting for rg. 5. A study of trajectories Let us fix rg and Λ and vary parameters j and K. Then for each pair {j, K} one can calculate the value of any quantity P, thus obtaining the dependence P = P(j, K) as a dataset. The function P = P(j, K) then can be represented as a heatmap (e.g [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figures from the paper (1 more)
Figure 7
Figure 7. Figure 7: An example of a heatmap: for any j, K with fixed Λ, rg a logarithm of the ratio of outer and inner radii is calculated. Larger values correspond to the spiral trajectories, smaller values to bound orbits with precession. curve F(x, y) = 0 is a polynomial function of ea…

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