Stability analysis of geodesics in dynamical Chern-Simons black holes: a geometrical perspective

Benito Rodr\'iguez; Javier Chagoya; Tonatiuh Tiscare\~no

arxiv: 2412.14266 · v2 · submitted 2024-12-18 · 🌀 gr-qc · math.DS

Stability analysis of geodesics in dynamical Chern-Simons black holes: a geometrical perspective

Tonatiuh Tiscare\~no , Benito Rodr\'iguez , Javier Chagoya This is my paper

Pith reviewed 2026-05-23 06:24 UTC · model grok-4.3

classification 🌀 gr-qc math.DS

keywords dynamical Chern-Simons gravityJacobi stabilityKosambi-Cartan-Chern theorygeodesic deviationblack holesLyapunov stabilityrotating black holesgeodesics

0 comments

The pith

The Kosambi-Cartan-Chern theory supplies a geometrical Jacobi stability analysis of geodesics around dynamical Chern-Simons black holes that carries advantages over the Lyapunov method.

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

The paper applies the Kosambi-Cartan-Chern theory to the geodesic deviation equations of rotating black hole solutions in dynamical Chern-Simons gravity. It conducts a direct comparison between the resulting Jacobi stability and conventional Lyapunov stability, emphasizing the geometrical character of the former. A sympathetic reader would care because geodesic stability governs the behavior of orbits and light rays near black holes whose spacetime is altered by a non-minimally coupled scalar field. The work thereby offers a coordinate-independent route to classifying stable and unstable trajectories in this modified-gravity setting.

Core claim

The paper establishes that the Kosambi-Cartan-Chern theory can be applied to the geodesic deviation equations in dynamical Chern-Simons black hole spacetimes to determine Jacobi stability, and that this geometrical method possesses concrete advantages over the usual Lyapunov stability approach.

What carries the argument

Kosambi-Cartan-Chern theory applied to geodesic deviation equations, which converts the stability question into a geometrical problem involving curvature invariants of the deviation vector field.

If this is right

Geodesics in dynamical Chern-Simons black holes receive a stability classification expressed through geometrical invariants rather than auxiliary Lyapunov functions.
The comparison isolates regimes in which the two stability notions diverge, thereby clarifying when the geometrical method supplies additional information.
Orbital behavior near rotating solutions can be examined without choosing a particular coordinate chart or constructing a specific Lyapunov candidate.
The scalar-field modification to the spacetime enters the stability analysis only through its effect on the geodesic deviation curvature.

Where Pith is reading between the lines

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

The same KCC procedure could be repeated for other scalar-tensor or higher-curvature black hole families to test whether the reported geometrical advantage persists.
Stability maps obtained this way might be compared against numerical integrations of geodesic bundles to check consistency in strong-field regions.
If the method extends to timelike geodesics with non-zero angular momentum, it could constrain the existence of stable circular orbits that affect accretion-disk models.

Load-bearing premise

The Kosambi-Cartan-Chern theory applies directly and without modification to the geodesic deviation equations of dynamical Chern-Simons black hole spacetimes.

What would settle it

An explicit computation of the KCC curvature invariants for a circular geodesic in the Kerr limit (vanishing Chern-Simons coupling) that yields a stability classification contradicting the known Lyapunov result would falsify the direct applicability.

read the original abstract

We apply the Kosambi-Cartan-Chern theory to perform an extensive examination of Jacobi stability of geodesics around rotating black hole solutions to dynamical Chern-Simons gravity, a theory that introduces modifications to General Relativity via a scalar field non-minimally coupled to curvature scalars. We present a comparative study between Jacobi and Liapunov stability, pointing out the advantages of the more geometrical method over the usual Liapunov approach.

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

Straightforward extension of KCC theory to dCS rotating black hole geodesics with a side-by-side comparison to Lyapunov stability.

read the letter

The paper takes the Kosambi-Cartan-Chern framework and applies it to the geodesic deviation equations for rotating solutions in dynamical Chern-Simons gravity. The central move is to compute the associated curvature invariants and use them to assess Jacobi stability, then contrast that with the standard Lyapunov approach. The stress-test note is right: the geodesic equation stays in the required second-order autonomous form, so KCC carries over directly once the metric is fixed; differences from GR appear only through the numerical values of those invariants. That part looks clean on the abstract and the logic given. What the work does is supply a concrete example of the geometrical method in a modified-gravity setting that already has known rotating solutions. The comparison section is the part that could be useful to readers who already know Lyapunov analysis and want to see how the two pictures line up on the same backgrounds. The soft spot is that the abstract gives no explicit numbers, no error estimates, and no statement of how large the stability regions differ between the two methods. Without those details it is difficult to judge whether the geometrical route actually changes any conclusions or just repackages them. The paper also stays inside the existing literature on dCS black holes rather than deriving new solutions or making observational claims. This is for people already working on stability techniques in modified gravity or on KCC applications to GR. It is not the kind of result that would change how most relativists think about black hole geodesics. I would send it to peer review because the application is legitimate, the comparison is a reasonable thing to ask for, and the underlying formalism is standard. A referee could check the explicit invariants and the numerical comparisons in one pass.

Referee Report

0 major / 2 minor

Summary. The manuscript applies the Kosambi-Cartan-Chern (KCC) theory to perform an extensive examination of Jacobi stability of geodesics around rotating black hole solutions to dynamical Chern-Simons gravity. It presents a comparative study between Jacobi and Lyapunov stability, pointing out the advantages of the more geometrical method over the usual Lyapunov approach.

Significance. If the results hold, the work demonstrates the direct applicability of the KCC formalism to geodesic equations in modified gravity, where only the background metric changes while the Levi-Civita connection (and thus the form of the geodesic ODE) remains unchanged. This yields a parameter-free geometrical analysis via nonlinear connection and curvature invariants, with all theory-specific differences appearing solely in the numerical values of those invariants. The explicit comparison to Lyapunov stability is a strength that could clarify practical advantages of the geometrical approach.

minor comments (2)

[Abstract] Abstract: the phrase 'extensive examination' is used without indicating the range of spin parameters, coupling strengths, or specific black-hole solutions considered; adding this would clarify the scope of the claimed advantages.
[Introduction] The manuscript would benefit from a brief statement early in the text confirming that the geodesic equation retains its standard autonomous form f^i = −Γ^i_jk v^j v^k with the Levi-Civita connection, so that readers see immediately why the KCC construction applies verbatim.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our manuscript applying KCC theory to Jacobi stability of geodesics in dynamical Chern-Simons black holes, along with the comparison to Lyapunov stability. The recommendation for minor revision is noted. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity; external KCC framework applied to standard geodesic equations

full rationale

The paper applies the Kosambi-Cartan-Chern (KCC) theory—an external geometrical formalism for second-order ODE systems—to the geodesic deviation equations on dynamical Chern-Simons black hole backgrounds. The geodesic equation retains its standard autonomous form f^i = −Γ^i_jk (dx^j/dt)(dx^k/dt) fixed solely by the Levi-Civita connection of the metric; the scalar field enters only by determining the numerical values of the metric components. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or derivation outline. The stability comparison between Jacobi (KCC) and Lyapunov methods follows directly from substituting the specific metric into the KCC curvature invariants, which is an independent calculation rather than a reduction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the work relies on standard application of KCC theory to geodesic equations.

pith-pipeline@v0.9.0 · 5601 in / 976 out tokens · 24019 ms · 2026-05-23T06:24:35.418390+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We apply the Kosambi-Cartan-Chern theory to perform an extensive examination of Jacobi stability of geodesics... The second KCC invariant P^i_j ... determines the stability
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

P^i_j = −2∂G^i/∂x^j − ... (deviation curvature tensor)

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

Works this paper leans on

35 extracted references · 35 canonical work pages · 2 internal anchors

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[1] [1]

Physical Review D83(10) (2011) https://doi.org/10.1103/physrevd.83.104002

Yunes, N., Stein, L.C.: Nonspinning black holes in alternative theories of gravity. Physical Review D83(10) (2011) https://doi.org/10.1103/physrevd.83.104002

work page doi:10.1103/physrevd.83.104002 2011

[2] [2]

Physical Review D 68(10) (2003) https://doi.org/10.1103/physrevd.68.104012

Jackiw, R., Pi, S.-Y.: Chern-simons modification of general relativity. Physical Review D 68(10) (2003) https://doi.org/10.1103/physrevd.68.104012

work page doi:10.1103/physrevd.68.104012 2003

[3] [3]

Physical Review D79(8) (2009) https://doi.org/10.1103/physrevd.79.084043

Yunes, N., Pretorius, F.: Dynamical chern-simons modified gravity: Spinning black holes in the slow-rotation approximation. Physical Review D79(8) (2009) https://doi.org/10.1103/physrevd.79.084043

work page doi:10.1103/physrevd.79.084043 2009

[4] [4]

Grumiller, D., Yunes, N.: How do black holes spin in chern-simons modified gravity? Physical Review D77(4) (2008) https://doi.org/10.1103/physrevd.77. 044015

work page doi:10.1103/physrevd.77 2008

[5] [6]

Chern, S.-S.: An invariant theory of three-webs on r-dimensional manifolds inr2r. Abh. Math. Semin. Hamb. Univ.11(1) https://doi.org/10.1007/BF02940731

work page doi:10.1007/bf02940731

[6] [7]

Mathematische Zeitschrift37, 608– 618 (1933)

Kosambi, D.D.: Parallelism and path-spaces. Mathematische Zeitschrift37, 608– 618 (1933)

work page 1933

[7] [8]

Mathematische Zeitschrift 37, 619–622 (1933)

Cartan, E.: Observations sur le mémoire précédent. Mathematische Zeitschrift 37, 619–622 (1933)

work page 1933

[8] [9]

Physical Review D79(6) (2009) https://doi.org/10.1103/physrevd.79.064016

Cardoso, V., Miranda, A.S., Berti, E., Witek, H., Zanchin, V.T.: Geodesic sta- bility, lyapunov exponents, and quasinormal modes. Physical Review D79(6) (2009) https://doi.org/10.1103/physrevd.79.064016

work page doi:10.1103/physrevd.79.064016 2009

[9] [10]

Advances in Theoretical and Mathematical Physics16 (2010) https://doi.org/10.4310/ATMP.2012.v16.n4.a2 19

Boehmer, C., Harko, T., Sabau, S.: Jacobi stability analysis of dynamical sys- tems – applications in gravitation and cosmology. Advances in Theoretical and Mathematical Physics16 (2010) https://doi.org/10.4310/ATMP.2012.v16.n4.a2 19

work page doi:10.4310/atmp.2012.v16.n4.a2 2010

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Aceña, A., López, E., Aldás, F.: Circular geodesics stability in a static black hole in new massive gravity (2019) arXiv:1903.04339 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2019

[11] [12]

Journal of Dynamical Systems and Geometric Theories 10(1), 13–32 (2012) https://doi.org/10.1080/ 1726037X.2012.10698604

Abolghasem, H.: Liapunov stability versus jacobi stability. Journal of Dynamical Systems and Geometric Theories 10(1), 13–32 (2012) https://doi.org/10.1080/ 1726037X.2012.10698604

work page arXiv 2012

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Springer Verlag, New York (2001)

Perko, L.: Differential Equations and Dynamical Systems, 3rd edn. Springer Verlag, New York (2001)

work page 2001

[13] [14]

Springer, Berlin (1990)

Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, 1st edn. Springer, Berlin (1990)

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Kuznetsov, Y.A.: Elements of applied bifurcation theory (2023)

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American Mathematical Society, Providence, Rhode Island (2011)

Kloeden, P.E., Martin, R.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence, Rhode Island (2011)

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Kloeden, P.E., Pötzsche, C.: Nonautonomous dynamical systems in the life sciences (2013)

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Alianza Editorial, España (1983)

Hirsch,M.W.,Smale,S.:EcuacionesDiferenciales,SistemasDinámicosYAlgebra Lineal, 1st edn. Alianza Editorial, España (1983)

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Antonelli, P.L.: Handbook of finsler geometry (2014)

work page 2014

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Bao, D., Chern, S.S., Shen, Z.: An introduction to riemann-finsler geometry (2012)

work page 2012

[20] [21]

International Journal of Geometric Methods in Modern Physics 13(02), 1650014 (2016) https://doi.org/10.1142/ s0219887816500146

Harko, T., Pantaragphong, P., Sabau, S.V.: Kosambi–cartan–chern (kcc) the- ory for higher-order dynamical systems. International Journal of Geometric Methods in Modern Physics 13(02), 1650014 (2016) https://doi.org/10.1142/ s0219887816500146

work page 2016

[21] [22]

Nonlinear Studies8(3) (2001)

Antonelli: transient states analysis of a 2-species dynamical ecological model via kcc-theory. Nonlinear Studies8(3) (2001)

work page 2001

[22] [23]

Antonelli, V.S.S

P.L. Antonelli, V.S.S. S. Rutz: A transient-state analysis of tyson’s model for the cell division cycle by means of kcc-theory. Open Syst. Inf. Dyn. 9 (3), 223–238 (2002) https://doi.org/10.1023/A:1019752327311

work page doi:10.1023/a:1019752327311 2002

[23] [24]

Symetry, 15–2329 (2023) https://doi.org/10.3390/sym15020329

Blaga, C., Blaga, P., Harko, T.: Jacobi and lyapunov stability analysis of circular geodesics around a spherically symmetric dilaton black hole. Symetry, 15–2329 (2023) https://doi.org/10.3390/sym15020329

work page doi:10.3390/sym15020329 2023

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Strogratz, S.H.: Nonlinear dynamical systems and chaos, with applications to physics, biology, chemistry and engineering (2018) 20

work page 2018

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Physics Reports 480(1–2), 1–55 (2009) https://doi.org/10.1016/j.physrep.2009.07.002

Alexander, S., Yunes, N.: Chern–simons modified general relativity. Physics Reports 480(1–2), 1–55 (2009) https://doi.org/10.1016/j.physrep.2009.07.002

work page doi:10.1016/j.physrep.2009.07.002 2009

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Physical Review D78(8) (2008) https://doi.org/ 10.1103/physrevd.78.081502

Grumiller, D., Mann, R., McNees, R.: Dirichlet boundary-value problem for chern-simons modified gravity. Physical Review D78(8) (2008) https://doi.org/ 10.1103/physrevd.78.081502

work page doi:10.1103/physrevd.78.081502 2008

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1103/physrevd.76.024009

Konno, K., Matsuyama, T., Tanda, S.: Does a black hole rotate in chern- simons modified gravity? Physical Review D 76(2) (2007) https://doi.org/10. 1103/physrevd.76.024009

work page 2007

[28] [29]

Progress of Theoretical Physics122(2), 561–568 (2009) https://doi.org/10.1143/ptp.122.561

Konno, K., Matsuyama, T., Tanda, S.: Rotating black hole in extended chern- simons modified gravity. Progress of Theoretical Physics122(2), 561–568 (2009) https://doi.org/10.1143/ptp.122.561

work page doi:10.1143/ptp.122.561 2009

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Visser, M.: The kerr spacetime: A brief introduction (2008) arXiv:0706.0622 [gr- qc]

work page internal anchor Pith review Pith/arXiv arXiv 2008

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Chandrasekhar, S.: The mathematical theory of black holes (1983)

work page 1983

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https://arxiv.org/abs/2403.13062

Rodríguez,B.,Chagoya,J.,Ortiz,C.:ShadowsofblackholesindynamicalChern- Simons modified gravity (2024). https://arxiv.org/abs/2403.13062

work page arXiv 2024

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Collaboration, E.H.T.: First m87 event horizon telescope results. i. the shadow of the supermassive black hole. The Astrophysical Journal Letters875(1), 1 (2019) https://doi.org/10.3847/2041-8213/ab0ec7

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