Pith. sign in

REVIEW 2 major objections 3 minor 60 references

Geodesic dynamics in brane-de Sitter wormholes

T0 review · 2 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper argues that in an asymptotically de Sitter wormhole on a Randall-Sundrum brane, the throat is the unique photon sphere and the unique unstable fixed point of the geodesic dynamics, with radial null geodesics marking a…

desk verdict Solid geodesic mechanics in a speculative wormhole, with a wrong trig identity in the shadow formula and a Bogdanov-Takens mislabel that need fixing. read the letter →

arxiv 2504.17003 v2 pith:LXL5LYPH submitted 2025-04-23 gr-qc hep-th

W. S. Klën , R. C. Barboza , C. Molina This is my paper · ORCID
classification gr-qchep-th
keywords wormholebraneworldgeodesicdynamicsstabilitycriteriaBogdanov-Takensbifurcationphotonsphereshadow
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

The paper sets out to show that in an asymptotically de Sitter wormhole living on a Randall-Sundrum brane, the throat is not merely a geometric bottleneck but the single most important object for geodesic dynamics: it is the unique photon sphere and the unique fixed point of the reduced two-dimensional dynamical system. The authors argue this by rewriting the geodesic equations in a quasilocal radial coordinate $u$ and showing that the effective potential has a single maximum at the throat. They then show the fixed point is unstable under both the Lyapunov eigenvalue criterion and the Jacobi deviation criterion, with explicit hyperbolic solutions for geodesics near the throat. They further claim that the radial-null limit marks a Bogdanov-Takens bifurcation, a qualitative change in the phase portrait as the angular momentum passes through zero. If the paper is right, the result is a unified dynamical picture of wormhole lensing, with null and timelike orbits behaving in the same qualitative way.

What carries the argument

The machinery is a reduction of the geodesic equations to an effective two-dimensional autonomous system in the quasilocal radial coordinate $u$, defined by $du/dr=\sqrt{A/B}$, so that the throat lies at $u=0$. From the conserved energy $E$ and angular momentum $L$, the equation of motion reduces to $\frac{1}{2}(du/d\lambda)^2+V(u)=E^2/2$ with $V(u)=A(u)(L^2/(2r(u)^2)+\kappa)$, and differentiating yields $du/d\lambda=w$, $dw/d\lambda=-dV/du$. The throat is the unique extremum of $V$, expanding as $V(u)=V_0+V_2u^2+O(u^3)$ with $V_2<0$, which determines the Jacobian at the fixed point. The Jacobi-stability test is applied through the Kosambi-Cartan-Chern (KCC) formalism, whose deviation curvature reads $P(0,0)=-V''(0)$.

What would settle it

Perform a two-parameter unfolding of the system (46)-(47) near $(u,w)=(0,0)$ with $L$ and one additional parameter varied, or numerically continue the fixed points as $L$ crosses zero: if the $L=0$ fixed-point set is a line segment, not an isolated equilibrium, and no second parameter is varied, strict Bogdanov-Takens behavior cannot occur. A second, independent check is to compute the first Lyapunov coefficient or normal form on the center manifold and see whether it matches the standard Bogdanov-Takens normal form.

Watch

Extended reading notes

Core claim

The central claim is that the throat at $u=0$ of this asymptotically de Sitter brane-world wormhole is simultaneously the unique photon sphere and the unique fixed point of the geodesic dynamical system for every nonradial null geodesic and every timelike geodesic. The fixed point requires $E^2=2V_0$ with $V_0=A_0(L^2/(2r_{\rm thr}^2)+\kappa)$, and the critical impact parameter for photons is $D_{\rm crit}=r_{\rm thr}/\sqrt{A_0}$. Linearizing around the throat gives eigenvalues $\nu_\pm=\pm\sqrt{-2V_2}$ with $V_2<0$, so the fixed point is a Lyapunov-unstable saddle; the KCC deviation curvature $P(0,0)=-V''(0)=-2V_2>0$ gives the same verdict under the Jacobi criterion. The paper also derives hyperbolic near-throat geodesic solutions, a near-throat shadow formula that tends to $\sin^2\alpha=1$ at the throat, and identifies the $L=0$ radial-null limit as a Bogdanov-Takens bifurcation. It concludes that null and timelike dynamics are qualitatively similar, with timelike trajectories approaching null ones at high energy.

Load-bearing premise

The load-bearing premise is that a Jacobian with a double-zero eigenvalue and one-dimensional eigenspace at $L=0$ is sufficient evidence for a codimension-two Bogdanov-Takens bifurcation; the paper gives no two-parameter unfolding or normal form, and in the radial-null case the fixed points form a continuum, so if double-zero degeneracy alone is not sufficient the bifurcation claim is unsupported.

Editorial extensions

If this is right

  • If the throat is the unique photon sphere and it is unstable, then light with impact parameter exactly $D_{\rm crit}=r_{\rm thr}/\sqrt{A_0}$ asymptotically circles the throat, while rays with $|D|<D_{\rm crit}$ pass through to the other side and those with $|D|>D_{\rm crit}$ bounce back.
  • The agreement between Lyapunov and Jacobi stability criteria at the fixed point gives a consistency check that the throat is a saddle point in the phase space of both null and timelike geodesics, so particles and photons spiral in or out.
  • The near-throat shadow formula, $\sin^2\alpha = r_{\rm thr}^2(A_0+A_2u_\odot^2)/(2r_{\rm thr}u_\odot^2(A_2r_{\rm thr}-A_0K)-A_0K^2u_\odot^4+A_0r_{\rm thr}^2)$, says an observer at the throat sees exactly half the sky illuminated by each wormhole mouth.
  • Timelike geodesics at high energy approach the null geodesic behavior, so the photon-sphere description extends to massive particles in the ultrarelativistic limit.

Reading between the lines

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

  • Editorial extension: I read the Bogdanov-Takens claim as heuristic rather than strict: the codimension-two bifurcation would require varying two parameters and an isolated equilibrium, whereas the paper exhibits a continuum of fixed points at $L=0$; the qualitative message that radial null geodesics organize the phase-portrait change survives even if the strict label does not.
  • Editorial extension: The near-throat shadow analysis suggests a distant-observer shadow program in which $D_{\rm crit}$ sets the leading shadow radius, but the paper does not compute that distant-observer shadow.
  • Editorial extension: A natural next step would connect the Lyapunov exponent $\sqrt{-2V_2}$ of the photon sphere to quasinormal modes in the eikonal limit, following the established black-hole correspondence; the paper does not perform this calculation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The manuscript studies null and timelike geodesics in the asymptotically de Sitter brane wormhole of Ref. [14], using the quasilocal radial coordinate u in which the throat is u=0. It reduces the geodesic equations to an effective one-dimensional potential problem and then to a two-dimensional autonomous system. The paper argues that the wormhole throat is the unique maximum of the effective potential, hence the unique photon sphere and the unique fixed point of the geodesic dynamics; that this fixed point is unstable under both Lyapunov and Jacobi stability criteria; that near-throat geodesics are hyperbolic and can be written explicitly; that radial null geodesics display a Bogdanov-Takens bifurcation at zero angular momentum; and that a near-throat observer sees a shadow whose boundary is given by an analytic formula. It concludes that null and timelike geodesic dynamics are qualitatively similar.

Significance. The core dynamical-systems content is a clean and mostly correct treatment: the effective potential V(u) has a single maximum at the throat, the fixed-point condition E^2=2V0 selects the unique photon sphere, the Jacobian eigenvalues are ±sqrt(-2V2) with V2<0, and the KCC/Jacobi criterion gives P(0,0)=-V''(0)>0, consistently with the Lyapunov result. These steps are analytic and internally consistent. The near-throat hyperbolic solutions and the null/timelike comparison are useful. If the shadow formula is corrected and the Bogdanov-Takens claim is either properly established or reclassified, the remaining paper would be a solid contribution to geodesic dynamics in brane wormholes. However, as printed, two load-bearing advertised results are not supported: the Bogdanov-Takens bifurcation and the shadow angle formula, which violates sin^2(alpha)<=1.

major comments (2)
  1. [Sec. IV D, Eq. (73); also Abstract and Sec. V] The identification of a Bogdanov-Takens bifurcation from the nilpotent Jacobian J=[[0,1],[0,0]] alone is not justified. A genuine BT bifurcation is a codimension-two phenomenon requiring a two-parameter unfolding, a center-manifold reduction, and verification of nondegeneracy conditions in the normal form. Here the family is effectively one-parameter (L, with E fixed by E^2=2V0), and at L=0 the system is dot u = w, dot w = 0, so the equilibrium set is the entire u-axis rather than an isolated codim-2 equilibrium. The authors themselves note this continuum in Sec. IV D, and Eq. (73) does not impose the fixed-point constraint E=0 that follows from Eq. (52) at L=0. The observation that the Jacobian degenerates is correct, but the advertised conclusion that a Bogdanov-Takens bifurcation is observed is unsupported. Since the abstract and Sec. V present the BT bifurcation as a main result, this is load-bearing and must be either established with a proper unfolding/normal-form computation or removed and replaced by a precise statement about a degenerate non-isolated equilibrium.
  2. [Sec. IV E, Eq. (81)] The trigonometric identity used to pass from Eq. (80) to Eq. (81) is wrong. The text states sin^2(alpha) = tan^2(alpha)[tan^2(alpha)-1]^{-1}, but the correct identity is sin^2(alpha) = tan^2(alpha)[1+tan^2(alpha)]^{-1}. With Eq. (80) and D^2 = r_thr^2/A0, the correct expression is sin^2(alpha) = D^2(A0+A2 u_sun^2)/(K u_sun^2+r_thr)^2, which is manifestly bounded by 1. The printed Eq. (81) is not bounded by 1; for small u_sun it behaves as 1 + (|A2|/A0 + 2K/r_thr)u_sun^2 + O(u_sun^4), which exceeds 1 for u_sun != 0. Thus Eq. (81) cannot be the sine squared of a real angle. This is a load-bearing error because the shadow boundary is one of the paper's advertised results. The authors should correct the identity and the resulting shadow formula and re-derive the subsequent discussion.
minor comments (3)
  1. [Secs. III and IV D] The same glyph L is used for the angular momentum constant and for the affine-parametrization constant 2L appearing in Eqs. (25), (29), and (32). In Sec. IV D, the phrase 'L=0 and L=0' is consequently very hard to parse. Please distinguish these symbols, e.g. by writing the Lagrangian constant as \mathcal{L} throughout.
  2. [Sec. V] The closing statement that the absence of homoclinic and heteroclinic trajectories 'suggests that these dynamical systems are structurally stable' is not established. Structural stability is a stronger property and, in particular, the radial-null system at L=0 has a continuum of fixed points and is not structurally stable. If this remark is kept, it needs a precise definition and supporting argument, or it should be softened.
  3. [Figs. 2 and 5] The numerical integration used to produce the dashed curves is not described. A sentence giving the integration method, tolerances, and parameter values would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geodesic results are derived from the explicitly given wormhole metric and effective potential, not fitted or renamed inputs.

full rationale

The paper's central claims are derived from the effective potential V(u) in Eq. (32), obtained directly from the wormhole line element in Eq. (18), and from the near-throat Taylor expansions Eqs. (38)-(42). The fixed-point condition Eq. (52), the critical impact parameter Eq. (67), the Lyapunov eigenvalues Eq. (56), and the Jacobi curvature Eq. (63) all follow from those equations rather than being assumed. The wormhole metric is taken from [14], which has a coauthor overlap, but that is the spacetime being analyzed, not a target result being predicted; citing the source of an input is not circular. The Bogdanov-Takens identification in Sec. IV D uses the nilpotent Jacobian of Eq. (73) and cites [26,41,44]; even if the sufficiency of a double-zero eigenvalue alone for a genuine BT bifurcation is questionable without a normal-form or unfolding calculation, that would be an unsupported correctness claim rather than a circular reduction. There is no parameter fitting to data, no fitted quantity renamed as a prediction, and no uniqueness theorem imported from the authors' prior work to forbid alternatives. The only potentially load-bearing self-citation is the geometric property in Sec. II B that the largest extremum of A(r) lies below rthr [14], which helps establish the sign of dA/du; that property concerns the input metric and is not equivalent to the derived geodesic claims. The paper also explicitly limits its shadow analysis to near-throat observers, avoiding an overclaim that would disguise an input as a result. Overall, the derivation chain is self-contained once the wormhole spacetime is accepted as input.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central argument uses the [14] wormhole as an input and standard stability theory. No parameters are fitted; the plot parameters are illustrative. The main unproven ad hoc assumption is the inference of a Bogdanov-Takens bifurcation from the degeneracy of the Jacobian.

assumptions (6)
  • standard math Standard geodesic and Hamiltonian formalism: Lagrangian (24), constants E and L (28), and effective potential (31)-(32).
    Invoked throughout Sec. III and IV A; standard differential geometry.
  • standard math Lyapunov stability is determined by eigenvalues of the Jacobian of the 2D system (46)-(47).
    Sec. IV B, Eqs. (54)-(56).
  • standard math Jacobi (KCC) stability criteria as formulated in Refs. [52,53], with deviation curvature scalar P of Eq. (60).
    Sec. IV B, Eqs. (57)-(63).
  • domain assumption The metric functions A(r), B(r) from [14] with 0<C<1 describe a wormhole with A,B>0 analytic for rthr<r<rc, B(rthr)=0, and a smooth extension across the throat.
    Sec. II B, Eqs. (6)-(7), (15)-(22); accepted from prior literature without independent check in this paper.
  • domain assumption Near-throat Taylor expansions r(u)=rthr+K u^2+O(u^3), A(u)=A0+A2 u^2+O(u^3) with 2K>0 and A2<0.
    Sec. III, Eqs. (36)-(44); derived from properties of the [14] solution.
  • ad hoc to paper A Jacobian with a double-zero eigenvalue and one-dimensional eigenspace is sufficient to identify a codimension-two Bogdanov-Takens bifurcation.
    Sec. IV D, Eq. (73); the paper provides no unfolding or normal-form computation, so this premise is assumed and is not justified.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geodesic dynamics in brane-de Sitter wormholes." pith.science (2026). https://pith.science/paper/LXL5LYPH

@misc{pith2026250417003,
  author       = {Pith},
  title        = {Pith review of: Geodesic dynamics in brane-de Sitter wormholes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LXL5LYPH}},
  note         = {Machine review of arXiv:2504.17003}
}
read the original abstract

We present a dynamical analysis of the null and timelike geodesics around an asymptotically de Sitter wormhole in a Randall-Sundrum brane. In this framework, the wormhole throat is interpreted both as a photon sphere and as a fixed point of the associated dynamical system. The stability of this structure is evaluated using Lyapunov and Jacobi criteria with consistent results. A Bogdanov-Takens bifurcation is observed in the null-geodesic dynamics, highlighting critical changes in the behavior of light around the wormhole. Explicit solutions are derived for geodesics near the throat, providing insight into the optical appearance of the wormhole shadow. These results show qualitatively similar behavior for null and timelike orbits, suggesting universal features of geodesic dynamics in brane-de Sitter wormholes.

Figures

Figures reproduced from arXiv: 2504.17003 by the authors.

Figure 1
Figure 1. FIG. 1. Typical orbits in the two-dimensional phase portrai [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Comparison between the approximate expression (sol [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Light ray received by an observer and relevant quanti [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Two-dimensional phase portrait [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Comparison between the approximate expression (sol [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 16 canonical work pages

  1. [14]

    Molina and J

    C. Molina and J. C. S. Neves, Wormholes in de Sitter brane s, Phys. Rev. D 86, 024015 (2012). arXiv:1204.1291

    arXiv 2012

  2. [1]

    B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), O bservation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116, 061102 (2016). arXiv:1602.03837

    arXiv 2016

  3. [2]

    Akiyama et al

    K. Akiyama et al. (The Event Horizon Telescope Collaboration), First M87 eve nt horizon telescope results. IV. Imaging the central supermassive black hole, Astrophys. J. 875, L4 (2019). arXiv:1906.11241

    arXiv 2019

  4. [3]

    Randall and R

    L. Randall and R. Sundrum, Large mass hierarchy from a sma ll extra dimension, Phys. Rev. Lett. 83, 3370 (1999). arXiv:hep-ph/9905221

    arXiv 1999

  5. [4]

    Randall and R

    L. Randall and R. Sundrum, An alternative to compactifica tion, Phys. Rev. Lett. 83, 4690 (1999). arXiv:hep-th/9906064

    arXiv 1999

  6. [5]

    Germani and R

    C. Germani and R. Maartens, Stars in the brane world, Phys . Rev. D 64 124010 (2001). arXiv:hep-th/0107011

    arXiv 2001

  7. [6]

    Casadio, A

    R. Casadio, A. Fabbri, and L. Mazzacurati, New black hole s in the brane world?, Phys. Rev. D 65, 084040 (2002). arXiv:gr-qc/0111072

    arXiv 2002

  8. [7]

    J. P. S. Lemos, F. S. N. Lobo, and S. Q. de Oliveira, Morris- Thorne wormholes with a cosmological constant, Phys. Rev. D 68, 064004 (2003). arXiv:gr-qc/0302049

    arXiv 2003

Show all 60 references
  1. [8]

    K. A. Bronnikov and S.-W. Kim, Possible wormholes in a bra ne world Phys. Rev. D 67 064027 (2003). arXiv:gr-qc/0212112

    2003 arXiv

  2. [9]

    K. A. Bronnikov, V. N. Melnikov andd H. Dehnen, On a genera l class of braneworld black holes, Phys. Rev. D 68, 024025 (2003). arXiv:gr-qc/0304068

    2003 arXiv

  3. [10]

    Abdalla, B

    E. Abdalla, B. Cuadros-Melgar, A. B. Pavan, and C. Molin a, Stability and thermodynamics of brane black holes, Nucl. Phys. B 752 40 (2006). arXiv:gr-qc/0604033

    2006 arXiv

  4. [11]

    F. S. N. Lobo, General class of braneworld wormholes, Ph ys. Rev. D 75, 064027 (2007). arXiv:gr-qc/0701133

    2007 arXiv

  5. [12]

    Molina and J

    C. Molina and J. C. S. Neves, Black holes and wormholes in AdS branes, Phys. Rev. D 82, 044029 (2010). arXiv:1005.1319

    2010 arXiv

  6. [13]

    Molina, P

    C. Molina, P. Martín-Moruno, and P. F. González-Díaz, I sotropic extensions of the vacuum solutions in general rela tivity, Phys. Rev. D 84, 104013 (2011). arXiv:1107.4627

    2011 arXiv

  7. [15]

    J. C. S. Neves and C. Molina, Rotating black holes in a Ran dall-Sundrum brane with a cosmological constant, Phys. Rev . D 86, 124047 (2012). arXiv:1211.2848

    2012 arXiv

  8. [16]

    Molina, Deformations of the vacuum solutions of gene ral relativity subjected to linear constraints, Phys

    C. Molina, Deformations of the vacuum solutions of gene ral relativity subjected to linear constraints, Phys. Rev. D 88, 127501 (2013). arXiv:1311.7137

    2013 arXiv

  9. [17]

    Parsaei and N

    F. Parsaei and N. Riazi, New wormhole solutions on the br ane, Phys. Rev. D 91, 024015 (2015). DOI:10.1103/PhysRevD.91.024015

    2015 doi

  10. [18]

    Molina, A

    C. Molina, A. B. Pavan, and T. E. Medina Torrejón, Electr omagnetic perturbations in new brane world scenarios, Phys . Rev. D 93, 124068 (2016). arXiv:1604.02461

    2016 arXiv

  11. [19]

    Ghosh and S

    B. Ghosh and S. Mitra, A new shape function and some speci fic wormhole solutions in braneworld scenario, Mod. Phys. Lett. A 36, 2150167 (2021). arXiv:2109.03885

    2021 arXiv

  12. [20]

    Visser Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, Melville, 1995)

    M. Visser Lorentzian Wormholes: From Einstein to Hawking (American Institute of Physics, Melville, 1995)

    1995

  13. [21]

    M. S. Morris and K. S. Thorne, Wormholes in spacetime and their use for interstellar travel: A tool for teaching gener al relativity, Am. J. Phys. 56, 395 (1988). DOI:10.1119/1.15620

    1988 doi

  14. [22]

    M. S. Morris, K. S. Thorne, and U. Yurtsever, Wormholes, time machines, and the weak energy condition, Phys. Rev. Lett. 61, 1446 (1988). DOI:10.1103/PhysRevLett.61.1446

    1988 doi

  15. [23]

    Övgün and M

    A. Övgün and M. Halilsoy, Existence of traversable worm holes in the spherical stellar systems, Astrophys. Space Sc i. 361, 214 (2016). arXiv:1509.01237

    2016 arXiv

  16. [24]

    R. A. Konoplya and A. Zhidenko, Traversable wormholes i n general relativity, Phys. Rev. Lett. 128, 091104 (2022). arXiv:2106.05034

    2022 arXiv

  17. [25]

    R. A. Konoplya and C. Molina, The ringing wormholes, Phy s. Rev. D 71, 124009 (2005). arXiv:gr-qc/0504139

    2005 arXiv

  18. [26]

    W. S. Klën and C. Molina, Dynamical analysis of null geod esics in braneworld spacetimes, Phys. Rev. D 102, 104051 (2020). arXiv:2011.03054

    2020 arXiv

  19. [27]

    J. C. S. Neves, Five-dimensional regular black holes in a brane world, Phys. Rev. D 104, 084019 (2021). arXiv:2107.04072

    2021 arXiv

  20. [28]

    D. A. Frizo, C. A. M. de Melo, L. G. Medeiros, and J. C. S. Ne ves, Viable wormhole solution in Bopp–Podolsky electro- dynamics, Ann. Phys. 457 169411 (2023). arXiv:2210.09938

    2023 arXiv

  21. [29]

    J. C. S. Neves, Wormholes from beyond, Eur. Phys. J. C 85, 333 (2025). arXiv:2412.11947 16

    2025 arXiv

  22. [30]

    Cardoso, A

    V. Cardoso, A. S. Miranda, E. Berti, H. Witek, and V. T. Za nchin, Geodesic stability, Lyapunov exponents, and quasinormal modes, Phys. Rev. D 79, 064016 (2009). arXiv:0812.1806

    2009 arXiv

  23. [31]

    C. Y. Chen, H. W. Chiang, and J. S. Tsao, Eikonal quasinor mal modes and photon orbits of deformed Schwarzschild black holes, Phys. Rev. D 106, 044068 (2022). arXiv:2205.02433

    2022 arXiv

  24. [32]

    M. A. Anacleto, J. A. V. Campos, F. A. Brito, and E. Passos , Quasinormal modes and shadow of a Schwarzschild black hole with GUP, Ann. Phys. 434, 168662 (2021). arXiv:2108.04998

    2021 arXiv

  25. [33]

    R. A. Konoplya and Z. Stuchlík, Are eikonal quasinormal modes linked to the unstable circular null geodesics?, Phys . Lett. B 771, 597 (2017). arXiv:1705.05928

    2017 arXiv

  26. [34]

    Morgan, V

    J. Morgan, V. Cardoso, A. S. Miranda, C. Molina, and V. T. Zanchin, Quasinormal modes of black holes in anti-de Sitter space: a numerical study of the eikonal limit, Phys. Rev. D 80, 024024 (2009). arXiv:0906.0064

    2009 arXiv

  27. [35]

    Chiba and M

    T. Chiba and M. Kimura, A note on geodesics in the Hayward metric, Prog. Theor. Exp. Phys. 2017, 043E01 (2017). arXiv:1701.04910

    2017 arXiv

  28. [36]

    Heydari-Fard and M

    M. Heydari-Fard and M. Heydari-Fard, Null geodesics an d shadow of 4D Einstein-Gauss-Bonnet black holes surrounde d by quintessence, Int. J. Mod. Phys. D 31, 2250066 (2022). arXiv:2109.02059

    2022 arXiv

  29. [37]

    Heydari-Fard, M

    M. Heydari-Fard, M. Heydari-Fard, and H. R. Sepangi, Nu ll geodesics and shadow of hairy black holes in Einstein-Max well- dilaton gravity, Phys. Rev. D 105, 124009 (2022). arXiv:2110.02713

    2022 arXiv

  30. [38]

    Conroy, A

    A. Conroy, A. S. Koshelev, and A. Mazumdar, Geodesic com pleteness and homogeneity condition for cosmic inflation, Phys. Rev. D 90, 123525 (2014). arXiv:1408.6205

    2014 arXiv

  31. [39]

    W. J. Cunningham, D. Rideout, J. Halverson, and D. Kriou kov, Exact geodesic distances in FLR W spacetimes, Phys. Rev . D 96, 103538 (2017). arXiv:1705.00730

    2017 arXiv

  32. [40]

    Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (Springer, New York, 2023). DOI:10.1007/978-3-031-22007 -4

    2023 doi

  33. [41]

    Y. A. Kuznetsov, Practical computation of normal forms on center manifolds at degenerate Bogdanov-Takens bifurca tions, Int. J. Bif. Cha. 15, 3535 (2005). DOI:10.1142/S0218127405014209

    2005 doi

  34. [42]

    I. S. Kohli and M. C. Haslam, Einstein’s field equations a s a fold bifurcation, J. Geom. Phys. 123, 434 (2018). arXiv:1607.05300

    2018 arXiv

  35. [43]

    S. S. Kokarev, Structural instability of Friedmann–Ro bertson–Walker cosmological models, Gen. Relativ. Gravit . 41, 1777 (2009). arXiv:0810.5080

    2009 arXiv

  36. [44]

    A. A. Azim, A. Awad, and E. I. Lashin, Degenerate Bogdano v–Takens bifurcations in a bulk viscous cosmology, Eur. Phy s. J. C 80, 868 (2020). arXiv:1607.02401

    2020 arXiv

  37. [45]

    Antoniou, A

    G. Antoniou, A. Bakopoulos, and P. Kanti, Black-hole so lutions with scalar hair in Einstein-scalar-Gauss-Bonnet theories, Phys. Rev. D 97, 084037 (2018). arXiv:1711.07431

    2018 arXiv

  38. [46]

    D. D. Doneva and S. S. Yazadjiev, New Gauss-Bonnet black holes with curvature-induced scalarization in extended sc alar- tensor theories, Phys. Rev. Lett. 120, 131103 (2018). arXiv:1711.01187

    2018 arXiv

  39. [47]

    Aydiner, Chaotic interaction between dark matter an d dark energy, Int

    E. Aydiner, Chaotic interaction between dark matter an d dark energy, Int. J. Theor. Phys. 64, 1 (2025). arXiv:2304.06614

    2025 arXiv

  40. [48]

    C. A. R. Herdeiro and E. Radu, Kerr black holes with scala r hair, Phys. Rev. Lett. 112, 221101 (2014). arXiv:1403.2757

    2014 arXiv

  41. [49]

    C. A. R. Herdeiro, E. Radu, N. Sanchis-Gual, and J. A. Fon t, Spontaneous scalarization of charged black holes, Phys. Rev. Lett. 121, 101102 (2018). arXiv:1806.05190

    2018 arXiv

  42. [50]

    Bombelli and E

    L. Bombelli and E. Calzetta, Chaos around a black hole, C lassical Quantum Gravity 9, 2573 (1992). DOI:10.1088/0264- 9381/9/12/004

    1992 doi

  43. [51]

    Shiromizu, K

    T. Shiromizu, K. I. Maeda, and M. Sasaki, The Einstein eq uations on the 3-brane world, Phys. Rev. D 62, 024012 (2000). arXiv:gr-qc/9910076

    2000 arXiv

  44. [52]

    C. G. Boehmer, T. Harko, and S. V. Sabau, Jacobi stabilit y analysis of dynamical systems: Applications in gravitati on and cosmology, Adv. Theor. Math. Phys. 16, 1145 (2012). arXiv:1010.5464

    2012 arXiv

  45. [53]

    S. V. Sabau, Some remarks on Jacobi stability, Nonlinea r Anal. 63, e143 (2005). DOI:10.1016/j.na.2005.02.061

    2005 doi

  46. [54]

    Hossein, Jacobi stability of circular orbits in a cen tral force, J

    A. Hossein, Jacobi stability of circular orbits in a cen tral force, J. Dyn. Syst. Geom. Theor. 10, 197 (2012). DOI:10.1080/1726037X.2012.10698621

    2012 arXiv

  47. [55]

    F. S. Khoo and Y. C. Ong, Lux in obscuro: Photon orbits of e xtremal black holes revisited, Classical Quantum Gravity 33, 235002 (2016). arXiv:1605.05774

    2016 arXiv

  48. [56]

    C. M. Claudel, K. S. Virbhadra, and G. F. Ellis, The geome try of photon surfaces, J. Math. Phys. 42, 818 (2001). arXiv:gr-qc/0005050

    2001 arXiv

  49. [57]

    Perlick, O

    V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Bl ack hole shadow in an expanding universe with a cosmological constant, Phys. Rev. D 97, 104062 (2018). arXiv:1804.04898

    2018 arXiv

  50. [58]

    Roy and S

    R. Roy and S. Chakrabarti, Study on black hole shadows in asymptotically de Sitter spacetimes, Phys. Rev. D 102, 024059 (2020). arXiv:2003.14107

    2020 arXiv

  51. [59]

    Ohgami and N

    T. Ohgami and N. Sakai, Wormhole shadows, Phys. Rev. D 91, 124020 (2015). arXiv:1704.07065

    2015 arXiv

  52. [60]

    Bouhmadi-López, C.-Y

    M. Bouhmadi-López, C.-Y. Chen, X. Y. Chew, Y. C. Ong, and D.-h. Yeom, Traversable wormhole in Einstein 3-form theory with self-interacting potential, J. Cosmol. Astrop art. Phys. 10, 059 (2021). arXiv:2108.07302

    2021 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.