arxiv: 2604.14826 · v1 · submitted 2026-04-16 · 🌀 gr-qc

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Jacobi stability of circular orbits around conformally invariant Weyl gravity black holes

Cristina Blaga , Paul A. Blaga

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Pith reviewed 2026-05-10 10:59 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Weyl conformal gravityblack holescircular geodesicsJacobi stabilityLyapunov stabilityeffective potentialtimelike orbitsfree parameters

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The pith

The Jacobi stability of circular orbits around Weyl gravity black holes depends on the free parameters in the metric.

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

The paper examines timelike circular geodesics in spherically symmetric black hole solutions of Weyl conformal gravity. It derives the effective potential for radial motion, locates the radii of circular orbits, and applies both Lyapunov and Jacobi stability criteria to assess perturbations. The analysis shows that the two free parameters in the solutions determine whether nearby trajectories remain close or diverge. A sympathetic reader would care because Weyl gravity offers an alternative to general relativity that aims to unify forces without dark matter, so orbit stability near black holes tests whether its predictions remain viable in strong fields.

Core claim

In Weyl conformal gravity the timelike circular geodesics around spherically symmetric black holes are located from minima of an effective potential constructed from the metric; their stability is then classified by the sign of the eigenvalues of the Jacobi matrix obtained from the linearised geodesic deviation equations, with the outcome controlled by the two free parameters that appear in the exact solutions.

What carries the argument

The effective potential for timelike geodesics in the Weyl metric, together with the Jacobi matrix of the second-order deviation equations evaluated at the circular orbit.

If this is right

Only certain ranges of the free parameters permit Jacobi-stable circular orbits.
Lyapunov exponents supply the growth rate of instability when the orbits are unstable.
The parameter dependence supplies concrete constraints on which Weyl black hole solutions can support realistic astrophysical orbits.

Where Pith is reading between the lines

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

The same Jacobi analysis could be repeated for other exact solutions in fourth-order gravity to compare stability patterns.
Observed periods of stellar orbits or accretion disk structure around galactic black holes could eventually bound the allowed parameter values.
Relaxing spherical symmetry to include rotation would test whether the reported parameter dependence persists in more realistic models.

Load-bearing premise

The exact black hole solutions in Weyl gravity are taken as given and the standard geodesic effective potential formalism applies without modification to the timelike circular case.

What would settle it

A calculation of the Jacobi matrix eigenvalues for any fixed choice of the free parameters that yields at least one positive real part, which would demonstrate that no stable circular orbits exist for that solution.

Figures

Figures reproduced from arXiv: 2604.14826 by Cristina Blaga, Paul A. Blaga.

**Figure 2.** Figure 2: The effective potential (red line) and its derivative for ( [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗

**Figure 3.** Figure 3: Phase portrait for (β, γ, k) = (0.1, 0.1, −0.045) and three different values for L. (a) If L = 0.2 no equilibrium points exists. (b) If L = 0.3793, the point (rc, 0) = (0.3024, 0) is a cusp (V ′ (r) = V ′′(rc) = 0). (c) If L = 0.5, the point (ru, 0) = (0.353, 0) is a saddle point (gold) and (rs, 0) = (0.962, 0) is a center (red). The stability discriminant ∆ associated with the Jacobian is defined as foll… view at source ↗

read the original abstract

Weyl conformal gravity was originally proposed in the early twentieth century as an attempt to unify gravitation and electromagnetism. Since 1989, renewed interest in this fourth-order theory of gravity has emerged following the discovery of several exact black hole solutions. In this work, we investigate the timelike circular geodesics of a spherically symmetric Weyl black hole. The effective potential, the circular geodesics and their Jacobi and Lyapunov stability are discussed. Our analysis provides new insights into the stability properties of Weyl black holes and the role of the free parameters appearing in their solutions.

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 runs standard Jacobi stability on known Weyl black hole metrics and shows how the free parameters shift the stability of circular timelike orbits.

read the letter

This paper applies the usual effective-potential approach and Jacobi/Lyapunov criteria to the spherically symmetric solutions already known in conformally invariant Weyl gravity. It sets up the metric, reduces the equatorial timelike geodesics to a one-dimensional potential, locates the circular orbits, and extracts stability conditions from the sign of the second derivative or the associated frequency. The derivations follow the textbook linearization without altering the geodesic equation itself, which is the right move for a metric theory. The main output is explicit dependence of the stability thresholds on the free parameters that appear in those solutions. That mapping is new for this particular family of black holes and gives a concrete way to see how the extra parameters tighten or loosen the stable region compared with Schwarzschild. The work is technically sound on its own terms and cites the relevant prior literature on both Weyl solutions and Jacobi stability. The soft spots are proportionate: the metrics are taken as given rather than re-derived, the analysis stays within standard geodesic motion, and the results remain parameter-dependent rather than producing a sharp observational signature. No load-bearing gaps appear in the logic, but the advance is incremental rather than foundational. This is the kind of paper that belongs in the modified-gravity and black-hole-dynamics literature. Specialists who already track Weyl gravity or orbit stability in alternative theories will get direct use from the parameter plots and stability boundaries. It is coherent and honest about its scope, so it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The paper applies the standard effective-potential formalism for timelike geodesics to the known spherically symmetric black-hole solutions of Weyl conformal gravity. It derives the conditions for circular orbits, computes the second derivative of the effective potential to assess Jacobi stability, and extracts the Lyapunov exponent for orbital stability, with explicit dependence on the free parameters that appear in the metric.

Significance. If the explicit stability expressions hold, the work supplies concrete, parameter-dependent criteria for stable circular orbits in an exact fourth-order gravity solution. This is useful for comparing Weyl gravity with GR and for exploring whether the extra parameters can produce observationally viable orbital behavior. The strength lies in the direct substitution of the known metric into the standard geodesic stability machinery, yielding falsifiable expressions rather than numerical fits.

minor comments (2)

[Abstract and §1] The abstract and introduction should explicitly name the Mannheim-Kazanas or other specific Weyl metric employed (including the line element) so that readers can immediately verify the substitution steps without external lookup.
[§3] In the derivation of the radial perturbation equation, the sign convention for the Jacobi stability criterion (positive second derivative of V_eff implies stability) should be stated once with a brief reference to the standard literature, to avoid any ambiguity for readers new to the method.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our work on the Jacobi and Lyapunov stability of circular orbits in Weyl conformal gravity. The summary accurately captures our use of the effective-potential formalism applied to the known spherically symmetric solutions, yielding explicit parameter-dependent criteria. We accept the recommendation for minor revision and will incorporate improvements to clarity, presentation, or additional context as needed in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript takes the known exact spherically symmetric black-hole solutions of Weyl conformal gravity as given inputs and applies the standard geodesic effective-potential formalism for timelike equatorial motion. It reduces the radial equation to a one-dimensional effective potential, computes the circular-orbit conditions from the first derivative, and extracts Jacobi/Lyapunov stability from the sign of the second derivative or the associated frequency. All steps are explicit algebraic substitutions of the pre-existing metric functions; no parameter is fitted to the stability data, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The dependence on the free parameters of the Weyl solutions is explored by direct substitution rather than by construction, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the central claim rests on the validity of known Weyl black hole solutions and standard geodesic techniques. No explicit free parameters, axioms, or invented entities are detailed beyond the mention of free parameters in the solutions.

free parameters (1)

free parameters in Weyl black hole solutions
Referenced in abstract as appearing in the metric; their specific values or fitting procedure not provided.

axioms (2)

domain assumption Weyl conformal gravity yields valid spherically symmetric black hole solutions
Assumed as background for the geodesic analysis.
domain assumption Timelike circular geodesics can be analyzed via effective potential in this theory
Standard assumption in general relativity extended to Weyl gravity.

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discussion (0)

Reference graph

Works this paper leans on

46 extracted references · 3 canonical work pages · 1 internal anchor

[1]

The Confrontation between General Relativity and Experi- ment.Living Rev

Will, C.M. The Confrontation between General Relativity and Experi- ment.Living Rev. Relativ.2014,17, 4

2014
[2]

Testing general relativity in the solar sys- tem: present and future perspectives.Class

De Marchi, F.; Cascioli, G. Testing general relativity in the solar sys- tem: present and future perspectives.Class. Quantum Gravity2020,37, 095007
[3]

Observation of Gravitational Waves from a Bi- nary Black Hole Merger.Phys

Abbott, B.; Abbott, R.; Abbott, T.; Abernathy, M.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; LIGO Scientific Collaboration and Virgo Collaboration. Observation of Gravitational Waves from a Bi- nary Black Hole Merger.Phys. Rev. Lett.2016,116, 061102

2016
[4]

Properties of the Binary Black Hole Merger GW150914.Phys

Abbott, B.; Abbott, R.; Abbott, T.; Abernathy, M.; Acernese, F.; Ack- ley, K.; Adams, C.; Adams, T.; Addesso, P.; LIGO Scientific Collabora- tion and Virgo Collaboration. Properties of the Binary Black Hole Merger GW150914.Phys. Rev. Lett.2016,116, 241102

2016
[5]

Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant.Astron

Riess, A.G.; Filippenko, A.V.; Challis, P.; Clocchiatti, A.; Diercks, A.; Garnavich, P.M.; Gilliland, R.L.; Hogan, C.J.; Jha, S.; Kirshner, R.P.; et 21 al. Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant.Astron. J.1998,116, 1009–1038

1998
[6]

Measurements of Ω and Λ from 42 High-Redshift Supernovae.Astrophys

Perlmutter, S.; Aldering, G.; Goldhaber, G.; Knop, R.A.; Nugent, P.; Castro, P.G.; Deustua, S.; Fabbro, S.; Goobar, A.; Groom, D.E.; et al. Measurements of Ω and Λ from 42 High-Redshift Supernovae.Astrophys. J.1999,517, 565–586

1999
[7]

Amendola, L.; Tsujikawa, S.Dark Energy; Cambridge University Press: Cambridge, UK, 2010

2010
[8]

Review of latest advances on dark matter from the viewpoint of the Occam razor principle.New Astron

Oks, E. Review of latest advances on dark matter from the viewpoint of the Occam razor principle.New Astron. Rev.2023,96, 101673

2023
[9]

Probing superheavy dark matter with gravi- tational waves.J

Bian, L.; Liu, X.; Xie, K.-P. Probing superheavy dark matter with gravi- tational waves.J. High Energy Phys.2021,2021, 175

2021
[10]

In the realm of the Hubble tension—A review of solutions.Class

Valentino, E.D.; Mena, O.; Pan, S.; Visinelli, L.; Yang, W.; Melchiorri, A.; Mota, D.F.; Riess, A.G.; Silk, J. In the realm of the Hubble tension—A review of solutions.Class. Quantum Gravity2021,38, 153001
[11]

Gravitation and electricity.Sitzungsber

Weyl, H. Gravitation and electricity.Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1918, Erster Halbband (Januar bis Juni) 465–480

1918
[12]

Reine Infinitesimalgeometrie.Math

Weyl, H. Reine Infinitesimalgeometrie.Math. Z.1918,2, 384–411

1918
[13]

Scholz,The unexpected resurgence of Weyl geometry in late 20-th century physics,Einstein Stud.14(2018) 261 [1703.03187]

Scholz, E. The unexpected resurgence of Weyl geometry in late 20-th century physics.arXiv2017, arXiv:1703.03187

work page arXiv
[14]

Zur Weylschen Relativit¨ atstheorie und der Weylschen Er- weiterung des Kr¨ ummungstensorbegriffs.Math

Bach, R. Zur Weylschen Relativit¨ atstheorie und der Weylschen Er- weiterung des Kr¨ ummungstensorbegriffs.Math. Z.1921,9, 110–135

1921
[15]

Exact vacuum solution to conformal Weyl gravity and galactic rotation curves.Astrophys

Mannheim, P.D.; Kazanas, D. Exact vacuum solution to conformal Weyl gravity and galactic rotation curves.Astrophys. J.1989,342, 635–638

1989
[16]

General Structure of the Gravitational Equations of Motion in Conformal Weyl Gravity.Astrophys

Kazanas, P.D.; Mannheim, P.D. General Structure of the Gravitational Equations of Motion in Conformal Weyl Gravity.Astrophys. J. Suppl. Ser. 1991,76, 431–453

1991
[17]

Conformal Gravity and the Flatness Problem.Astrophys

Mannheim, P.D. Conformal Gravity and the Flatness Problem.Astrophys. J.1992,391, 429–432

1992
[18]

Conformal cosmology and the age of the universe.arXiv 1996, arXiv:astro-ph/9601071

Mannheim, P.D. Conformal cosmology and the age of the universe.arXiv 1996, arXiv:astro-ph/9601071

work page internal anchor Pith review arXiv 1996
[19]

Long range forces and broken symmetries.Proc

Dirac, P.A.M. Long range forces and broken symmetries.Proc. R. Soc. Lond. A1973,333, 403–418
[20]

Weyl gauge symmetry and its spontaneous breaking in the standard model and inflation.Phys

Ghilencea, D.M.; Lee, H.M. Weyl gauge symmetry and its spontaneous breaking in the standard model and inflation.Phys. Rev. D2019,99, 115007. 22
[21]

Black holes in higher-derivative Weyl conformal gravity,

Lessa, L.A.; Macedo, C.F.B.; Ferreira, M.M., Jr. Black holes in higher- derivative Weyl conformal gravity.arXiv2025, arXiv:2509.21495v1

work page arXiv
[22]

Classical tests for Weyl gravity: Deflection of light and time delay.Phys

Edery, A.; Paranjape, M.B. Classical tests for Weyl gravity: Deflection of light and time delay.Phys. Rev. D1998,44, 417–423
[23]

Conformal Weyl gravity and perihelion preccesion.Phys

Sultana, J.; Kazanas, D.; Said, J.L. Conformal Weyl gravity and perihelion preccesion.Phys. Rev. D2012,86, 084008
[24]

Null geodesics in conformal gravity.Class

Turner, G.E.; Horne, K. Null geodesics in conformal gravity.Class. Quan- tum Gravity2020,37, 095012
[25]

Parallelism and path-spaces.Math

Kosambi, D.D. Parallelism and path-spaces.Math. Z.1933,608, 608–618

1933
[26]

Observations sur le m´ emoir pr´ ec´ edent.Math

Cartan, E. Observations sur le m´ emoir pr´ ec´ edent.Math. Z.1933,37, 619–622

1933
[27]

Sur la g´ eom´ etrie d’un syst´ eme d’equations differentialles du second ordre.Bull

Chern, S.S. Sur la g´ eom´ etrie d’un syst´ eme d’equations differentialles du second ordre.Bull. Sci. Math.1939,63, 206–212

1939
[28]

Jacobi stability analysis of dy- namical systems—Applications in gravitation and cosmology.Adv

Boehmer, C.G.; Harko, T.; Sabau, S.V. Jacobi stability analysis of dy- namical systems—Applications in gravitation and cosmology.Adv. Theor. Math. Phys.2012,16, 1145–1196

2012
[29]

Jacobi Stability of Circular Orbits in a Central Force.J

Abolghasem, H. Jacobi Stability of Circular Orbits in a Central Force.J. Dyn. Syst. Geom. Theor.2012,10, 197–214

2012
[30]

Stability of circular orbits in Schwarzschild spacetime

Abolghasem, H. Stability of circular orbits in Schwarzschild spacetime. Int. J. Differ. Equ. Appl.2013,12, 131–147

2013
[31]

Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

Blaga, P.; Blaga, C.; Harko, T. Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole. Symmetry2023,15, 329
[32]

Dynamical behavior and Jacobi stability analysis of wound strings.Eur

Lake, M.J.; Harko, T. Dynamical behavior and Jacobi stability analysis of wound strings.Eur. Phys. J. C2016,76, 311
[33]

Jacobi stability analysis of the classical restricted three body problem.Rom

Blaga, C.; Blaga, P.; Harko, T. Jacobi stability analysis of the classical restricted three body problem.Rom. Astron. J.2021,31, 101–112

2021
[34]

KCC theory of system of sec- ond order differential equations

Antonelli, P.L.; Buc˘ ataru, I. KCC theory of system of sec- ond order differential equations. InHandbook of Finsler Geometry; Antonelli, P.L., Ed.; Kluwer Academic: Dordrecht, The Netherlands, 2003; Volume 1, pp. 83–174

2003
[35]

Sur l’´ ecart g´ eod´ esique.Math

Levi-Civit´ a, T. Sur l’´ ecart g´ eod´ esique.Math. Ann.1927,97, 291–320

1927
[36]

On the geometry of dynamics.Philos

Synge, J.L. On the geometry of dynamics.Philos. Trans. R. Soc. Lond. A1927,226, 31–106. 23
[37]

Antonelli, P.L.; Ingarden, R.S.; Matsumoto, M.The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology; Springer Science+Business Media: Dordrecht, The Netherlands, 1993; Volume 58

1993
[38]

New results about the geometric invariants in KCC theory.An

Antonelli, P.L.; Bucataru, I. New results about the geometric invariants in KCC theory.An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.)2001,47, 405–420

2001
[39]

Hahn, W.Stability of Motion; Springer: Berlin, Germany, 1967

1967
[40]

Hirsch, M.W.; Smale, S.; Devaney, R.L.Differential Equations, Dynamical Systems and an Introduction to Chaos, 3rd ed.; Academic Press, Waltham, MA,USA 2013

2013
[41]

Alternatives to Dark Matter and Dark Energy.Prog

Mannheim, P.D. Alternatives to Dark Matter and Dark Energy.Prog. Part. Nucl. Phys.2006,56, 340–445

2006
[42]

Nakahara, M.Geometry, Topology and Physics; IOP Publishing: Bristol, UK, 2003; p. 462

2003
[43]

A Remarkable Property of the Riemann-Christoffel Tensor in Four Dimensions.Ann

Lanczos, C. A Remarkable Property of the Riemann-Christoffel Tensor in Four Dimensions.Ann. Math.1938,39, 842–850

1938
[44]

The gauge theory of Weyl group and its inter- pretation as Weyl quadratic gravity.Class

Condeescu, C.; Micu, A. The gauge theory of Weyl group and its inter- pretation as Weyl quadratic gravity.Class. Quantum Grav.2025,42, 065011

2025
[45]

Galactic rotation curves in conformal gravity.J

Mannheim, P.D.; O’Brien, J.G. Galactic rotation curves in conformal gravity.J. Phys. Conf. Ser.2013,437, 012002

2013
[46]

Circular geodesics in Manheim-Kazanas spacetime.Proc

Lungu, V.; Dariescu, M.-A. Circular geodesics in Manheim-Kazanas spacetime.Proc. Rom. Acad. Ser. A2023,24, 35–42. 24