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Periodic orbits of neutral test particles in Reissner-Nordstr\"{o}m naked singularities

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Pith's one-line read This paper charts periodic orbits of neutral test particles around a Reissner-Nordström naked singularity and establishes that outer and inner orbits can share identical taxonomy, energy, and angular momentum while having different sizes.

desk verdict Two real results: the shared-(L,E) orbit pair is solid, but the full (L,E) atlas rests on an unvalidated λ-continuation heuristic. read the letter →

arxiv 2502.03082 v1 pith:76A4DCOC submitted 2025-02-05 gr-qc

Zoe C S Chan , Yen-Kheng Lim This is my paper · ORCID
classification gr-qc MSC 83C1083C5783C75
keywords periodicorbitsnakedsingularityReissner-Nordströmspacetimetimelikegeodesicszoom-whirlJacobiellipticfunctionsquarticradialpolynomialstablecircular
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

Neutral test particles around a Reissner-Nordström naked singularity — a charged, horizonless point mass with $Q>M$ — can move on two families of bound time-like orbits, one outer and one inner. This paper extends the Levin taxonomy of zoom-whirl periodic orbits to that setting and charts, in the conserved energy–angular-momentum plane, which rational orbits exist for every charge ratio. Its central finding is that an outer and an inner periodic orbit can carry identical labels $(z,w,v)$ and identical values of $L$ and $E$ while tracing out different-size loops around the singularity. The chart is organized by the roots of the quartic radial polynomial $P(u)$, which partition $(L,E)$-space into eight domains $\mathcal{D}_1$ through $\mathcal{D}_8$; each domain calls for a different Jacobi elliptic-function solution when drawing the orbit. A reader should care because the $(L,E)$ map reveals where periodic orbits live, which domains require different analytic solutions, and how the inner region's circular orbits constrain the orbits.

What carries the argument

The load-bearing object is the quartic radial polynomial $P(u)=-Q^2u^4+2Mu^3-(1+Q^2/L^2)u^2+(2M/L^2)u-(1-E^2)/L^2$, whose roots are the turning points of radial motion for a particle with energy $E$ and angular momentum $L$ in the inverse radial coordinate $u=1/r$. Its factored form $Q^2(a-u)(b-u)(c-u)(u-d)$ controls whether one or two bound-orbit domains exist: the $c,d$ pair parameterizes the outer region and the $a,b$ pair the inner region. The central computational relation is Eq. (24), which links the rational $q=w+v/z$ from the orbit taxonomy to the complete elliptic integral of the first kind through the geometric parameters eccentricity $e$ and semi-latus rectum $\lambda$. Solving it for $\lambda$ and inserting the result in Eq. (14a) gives the $(L,E)$ coordinates of each periodic orbit, and the domain classification $\mathcal{D}_1$–$\mathcal{D}_8$ records which root pattern is present at each point.

What would settle it

Pick an inner $q$-branch in Case 2 near its claimed tip (for example $Q=1.08M$ with a rational label having $w\ge 3$), track all solutions of the elliptic-integral equation for $\lambda$ continuously instead of using the monotonicity heuristic, integrate Eq. (9a) with the resulting $L,E$, and check whether the trajectory closes after $\Delta\phi=2\pi(q+1)$; if it does not close, or if another $\lambda$ root yields a closed orbit at the same $(L,E)$, the branch-length claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that for an RN naked singularity it is possible for a time-like neutral test particle to have a pair of periodic orbits with the same $(z,w,v)$ and $(L,E)$ values but different radial size, one in the outer bound region and one in the inner bound region. This happens when the four roots of $P(u)$ are all real, because the $c,d$ root pair and the $a,b$ root pair each define a closed radial oscillation domain; the $e,\lambda$ parametrization then assigns the same $L$ and $E$ to both. The paper generalizes this into a full atlas: $q$-branches emanate from the stable circular-orbit segments of each region, and the $(L,E)$ plane is divided into domains $\mathcal{D}_1$–$\mathcal{D}_8$ according to whether four real roots, two real plus two complex roots, or degenerate roots are present. Analytical solutions for $r(\phi)$ are supplied for each configuration, from Jacobi elliptic sine and cosine forms to the degenerate-root formula used at the ISCO/OSCO critical points. For the three charge intervals $M<Q<\sqrt{9/8}M$, $\sqrt{9/8}M\le Q<\sqrt{5/2}M$, and $Q\ge\sqrt{5/2}M$, the distribution of these branches is mapped, including newly reported branch-termination trends for large-whirl orbits in Cases 2 and 3.

Load-bearing premise

The load-bearing premise is a branch-selection heuristic: for each orbit label and eccentricity, the paper chooses the solution for the orbit-size parameter $\lambda$ that increases monotonically from the circular-orbit limit and avoids large jumps, and if the true physical branch does not follow that rule the domain assignments and branch charts could misrepresent actual periodic orbits.

Editorial extensions

If this is right

  • Every rational periodic orbit around an RN naked singularity can be placed on the same $(L,E)$ chart, so locating such an orbit reduces to reading off a domain and a branch point rather than integrating individual geodesics.
  • Because an outer and an inner orbit can share $(L,E)$, energy and angular momentum measurements alone cannot tell which region a particle occupies; a size or orbital-period measurement is also needed.
  • The termination of high-whirl branches in Cases 2 and 3 implies that large-$w$ periodic orbits are absent or suppressed near the ISCO and OSCO of a naked singularity, which would narrow the expected zoom-whirl signals in gravitational-wave searches.
  • The domain decomposition forces a piecewise description: one orbit family can require different Jacobi elliptic solutions as it crosses from $\mathcal{D}_1$ into $\mathcal{D}_3$ or $\mathcal{D}_4$.
  • The degenerate-root solutions cover homoclinic orbits and the chaotic orbits produced by perturbing them, so the atlas includes the full bound-orbit zoo rather than only periodic orbits.

Reading between the lines

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

  • If the $\lambda$ branch rule is replaced by a rigorous global root-continuation algorithm, some claimed $q$-branch tips in Cases 2 and 3 may shift or disappear, so the reported 'shrinking branch' trend should be re-examined before being treated as physical.
  • The same $\mathcal{D}_k$ construction should apply to any spherically symmetric spacetime whose radial polynomial is quartic; testing a black-bounce or quantum-corrected metric for shared-$(L,E)$ orbit pairs would show whether this degeneracy is generic.
  • If naked singularities exist, an observed feature tied to a single $(L,E)$ value could actually be a superposition of two different-size orbits, giving a distinctive two-radius signature in timing or imaging data.
  • Extending the analysis to charged test particles or to a slowly rotating background would reveal whether the shared-$(L,E)$ degeneracy survives when the quartic root symmetry is broken.
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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 / 3 minor

Summary. This paper extends Levin's periodic-orbit taxonomy to neutral timelike geodesics around Reissner-Nordström naked singularities. The authors parametrize bound orbits by eccentricity e and latus rectum λ, derive L(e,λ,Q) and E(e,λ,Q) via Vieta's formulas, and solve the periodicity condition (24) numerically to chart q-branches in (L,E)-space. They classify (L,E)-space into domains D1–D8 according to the real/complex root structure of the quartic P(u), provide Jacobi-elliptic orbit solutions for each domain, and report that for a four-real-root configuration the same (L,E) can support two periodic orbits of different radial size. They discuss three charge ranges (Cases 1–3) and report a branch-shrinking phenomenon for large w.

Significance. If correct, this is a useful atlas of periodic orbits in RN naked singularities, extending the Schwarzschild methodology of [37] to a quartic radial polynomial and giving explicit analytical solutions per domain. The Vieta derivation, the elliptic-integral construction, and the concrete numerical example of a shared-(L,E) pair are clearly presented and independently checkable. The paper's broader claims, however, rest on two unproven steps: the λ-root selection rule used to generate all q-branch charts, and the equality of the two radial periods that underlies the 'same (z,w,v)' statement. Neither issue is fatal in principle, but both need support before the atlas can be accepted.

major comments (3)
  1. [Sec. 2.3 and Sec. 3.2] The procedure for selecting the physical root of Eq. (24) is not a well-defined algorithm. For fixed (Q,e,z,w,v), Eq. (24) can admit several real λ solutions; the rule 'take the closest monotonically increasing (e,λ) values starting from e=0 and avoid nonsensical large jumps' is a plotting heuristic rather than a mathematical criterion, and no uniqueness or continuity argument is provided. Because every q-branch in Figs. 4, 6, 8, 14, 15 and the domain assignments D1–D8 are generated with this rule, the reported branch terminations in Sec. 4.2 (the 'shrinking branch' phenomenon, whose origin the paper says it has not identified, including in the four-real-root domains) could be numerical artifacts of the root-tracking rule. Please validate the selected λ sheet with an independent bracketing or homotopy continuation for representative (Q,z,w,v), and state whether the reported endpoints correspond to loss of real λ solutions for all branches or only for the chosen sheet.
  2. [Sec. 3.1, Eqs. (30)-(31)] The inner-orbit formula r(φ)_II uses the same g^{-1} and k as the outer-orbit formula r(φ)_I. This is equivalent to asserting that the two complete elliptic integrals over the two real cycles of P(u), ∫_b^a du/√P and ∫_d^c du/√P, are equal. That equality is not automatic for a quartic with four real roots, so the central conclusion that the same (z,w,v) and (L,E) yield two periodic orbits of different size is not established. The authors should either prove the equality from Eqs. (12) and (24) or verify directly (e.g., by numerical integration of Eq. (9a)) that the inner orbit in the example (26b)-(27b) closes after Δφ=2π(q+1) with q=w+v/z. If only the equality of (L,E) is guaranteed, the Conclusion's 'same (z,w,v)' statement must be weakened.
  3. [Sec. 4.2] The branch-shrinking trend is presented as a main result, but the paper's own text states that it is not known whether it is a mathematical consequence of the root structure, and the proposed complex-root explanation is acknowledged not to cover the case where all four roots are real. As it stands, the claim that the q-branches 'terminate' at the reported e values is an unresolved observation rather than a demonstrated property. Please either identify the mechanism, including for the four-real-root domains, or explicitly recast the statement as an open numerical observation separate from the verified part of the atlas.
minor comments (3)
  1. [Throughout] There are numerous typographical errors that should be corrected before publication, including 'unrealisitc', 'exemplify', 'similiar', 'censorshhip', 'chemcial', 'inifinite', 'strucutre', 'arbitary', and 'coalesece'.
  2. [Eq. (39a)] The displayed formula for r(ϕ)_VI has unclear parentheses and typesetting; please rewrite it so that the numerator and denominator are unambiguous.
  3. [Fig. 8 caption] The caption states a root-configuration transition at e≈0.864 without a derivation or formula; a brief explanation of how this threshold is obtained would improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the periodic-orbit atlas is generated from first integrals and root configurations, not from fitted targets or self-citation chains.

full rationale

The paper's derivation is self-contained in the sense required by the circularity pass. It starts from the RN metric, defines the conserved E and L, constructs the effective potential and quartic P(u), then parametrizes the roots via latus rectum and eccentricity using Vieta's formulas in Appendix A. Equation (24) is a transcendental condition obtained by equating the elliptic-integral periastron advance to the rational q=(w+v)/z; solving it for λ for fixed (z,w,v;e) is an inverse construction, not a fit to the target chart. The (L,E) values of each branch are then computed from the exact formulas (14), so the q-branches are not fitted inputs relabeled as predictions. The central pair claim follows from the root configuration of one quartic: when four real roots exist, the same P(u) admits two bound radial domains, giving two orbits with identical (L,E), which is a mathematical consequence rather than a circular definition. Reliance on the authors' prior Schwarzschild paper [37] is methodological (e,λ parametrization and branch-selection practice) and concerns a different spacetime; it is not an unverified self-citation used to forbid alternatives, and no uniqueness theorem is imported. The admitted uncertainty about the shrinking-branch trend ('We had yet to pinpoint whether it is purely a mathematical consequence...') is a correctness and robustness caveat about the numerical continuation heuristic, not evidence of circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical entities. Its central construction rests on standard GR geodesic equations, Vieta algebra, and tabulated elliptic integrals, plus two imported domain assumptions: the Levin q taxonomy and the Pugliese circular-orbit classification. The only hand-chosen element is the λ-branch selection rule, which affects the numerical charts.

free parameters (1)
  • λ root-branch selection rule
    For inner-region branches and some outer branches, Eq. (24) has multiple real λ solutions. The paper selects the branch by the heuristic of closest monotonic increase from e=0 and avoiding large jumps (Sec. 2.3), which is a hand-chosen convention that shapes the reported q-branch distribution.
assumptions (5)
  • standard math The geodesic equations for a neutral test particle follow from the RN metric with conserved E and L (Sec. 2.1).
    Standard Lagrangian derivation of geodesics in a static spherically symmetric spacetime.
  • domain assumption The Levin periodic-orbit taxonomy relation q = w + v/z and Δφ/2π = q + 1 holds for these orbits (Sec. 2.3).
    Imported from prior literature [10,37]; the paper does not re-derive it.
  • standard math The root parametrization (12) obtained from Vieta's formulas is algebraically valid for the quartic P(u) (Appendix A).
    Pure algebraic manipulation; no external empirical input.
  • standard math Elliptic integral reduction formulas from Byrd and Friedman [43] and Gradshteyn and Ryzhik [44] are correct and applicable in the stated root orderings (Secs. 3.1-3.3).
    Standard handbooks; the paper does not prove them but cites page numbers.
  • domain assumption The classification of circular-orbit stability, r*, rγ±, ISCO/OSCO for RN naked singularities is taken from Pugliese et al. [38] (Sec. 2.2, Appendix B).
    Imported from cited prior work; the paper uses these to define emanation ranges and domains.

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Cite this review

Pith. "Pith review of Periodic orbits of neutral test particles in Reissner-Nordstr\"{o}m naked singularities." pith.science (2026). https://pith.science/paper/76A4DCOC

@misc{pith2026250203082,
  author       = {Pith},
  title        = {Pith review of: Periodic orbits of neutral test particles in Reissner-Nordstr\"om naked singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76A4DCOC}},
  note         = {Machine review of arXiv:2502.03082}
}
abstract

We conduct studies on Levin's taxonomy of periodic orbits for neutral test particles around a Reissner-Nordstr\"{o}m naked singularity. It was known that naked singularities could harbor two distinct regions of time-like bound orbits and thus we expect periodic orbits to appear in both regions. It is possible for a pair of periodic orbits from both regions to possess the exact same angular momentum $L$ and energy $E$ values. We chart the sets of periodic orbits in $(L,E)$-parameter space and highlight the general distribution pattern of these sets for three possible scenarios. Regions within $(L,E)$-space can be partitioned into multiple domains $\mathcal{D}_k$ based on the roots configuration of the quartic polynomial $P(u)$ where $u$ is the inverse radial coordinate. Consequently, each domain and interestingly enough, portions of certain periodic orbits sets that lie in different $\mathcal{D}_k$ require different analytical solutions to plot the resulting orbit. Furthermore, we uncover physical properties of some hypothetical circular orbits residing in the inner region from analysing the $(L,E)$-space.

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Reference graph

Works this paper leans on

51 extracted references · 21 canonical work pages · cited by 3 Pith papers

  1. [37]

    Lim, Y.-K., Yeo, Z.-C.: Energies and angular momenta of periodic Schwarzschild geodesics. Phys. Rev. D 109, 024037 (2024) https://doi.org/10.1103/PhysRevD. 109.024037 arXiv:2401.13894 [gr-qc]

    arXiv 2024

  2. [1]

    Annalen der Physik 355, 106–120 (2006) https://doi.org/10.1002/ andp.19163550905

    Reissner, H.: ¨Uber die Eigengravitation des elektrischen Feldes nach der Einstein- schen Theorie. Annalen der Physik 355, 106–120 (2006) https://doi.org/10.1002/ andp.19163550905

    work page 2006

  3. [2]

    Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 20, 1238–1245 (1918)

    Nordstr¨ om, G.: On the Energy of the Gravitation field in Einstein’s Theory. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences 20, 1238–1245 (1918)

    work page 1918

  4. [3]

    Zajaˇ cek, A.E

    M. Zajaˇ cek, A.E. A. Tursunov, Britzen, S.: On the charge of the Galactic centre black hole. Monthly Notices of the Royal Astronomical Society 480, 4408–4423 (2018) https://doi.org/10.1093/mnras/sty2182 arXiv:1808.07327 [astro-ph.GA]

    arXiv 2018

  5. [4]

    Abbott, B.P.e.a.: Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 116, 061102 (2016) https://doi.org/10.1103/ PhysRevLett.116.061102 arXiv:1602.03837 [gr-qc]

    arXiv 2016

  6. [5]

    Symmetry 15(2), 537 (2023) https: //doi.org/10.3390/sym15020537 arXiv:2209.00874 [gr-qc]

    Benavides-Gallego, C.A., Han, W.-B.: Gravitational Waves and Electromagnetic Radiation from Charged Black Hole Binaries. Symmetry 15(2), 537 (2023) https: //doi.org/10.3390/sym15020537 arXiv:2209.00874 [gr-qc]

    arXiv 2023

  7. [6]

    Event Horizon Telescope Collaboration, K.A.e.a., First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. The Astrophys- ical Journal Letters 875(L1) (2019) https://doi.org/10.3847/2041-8213/ab0ec7 arXiv:1906.11238 [astro-ph.GA]

    arXiv 2019

  8. [7]

    Event Horizon Telescope Collaboration, K.A.e.a.: First Sagittarius A* Event Hori- zon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way. The Astrophysical Journal Letters 930(L12) (2022) https://doi.org/10.3847/2041-8213/ac6674 arXiv:2311.08680 [astro-ph.HE]

    arXiv 2022

Show all 51 references
  1. [8]

    Hsieh, T., Lee, D.-S., Lin, C.-Y.: Strong gravitational lensing by Kerr and Kerr- Newman black holes. Phys. Rev. D 103, 104063 (2021) https://doi.org/10.1103/ PhysRevD.103.104063 arXiv:2101.09008 [gr-qc]

    2021 arXiv

  2. [9]

    Glampedakis, K., Kennefick, D.: Zoom and whirl: Eccentric equatorial orbits around spinning black holes and their evolution under gravitational radiation reaction. Phys. Rev. D 66, 044002 (2002) https://doi.org/10.1103/PhysRevD.66. 044002 arXiv:gr-qc/0203086

    2002 arXiv

  3. [10]

    Levin, J., Perez-Giz, G.: A Periodic Table for Black Hole Orbits. Phys. Rev. D 77, 103005 (2008) https://doi.org/10.1103/PhysRevD.77.103005 arXiv:0802.0459 [gr-qc] 32

    2008 arXiv

  4. [11]

    Levin, J., Grossman, B.: Dynamics of Black Hole Pairs I: Periodic Tables. Phys. Rev. D 79, 043016 (2009) https://doi.org/10.1103/PhysRevD.79.043016 arXiv:0809.3838 [gr-qc]

    2009 arXiv

  5. [12]

    Grossman, R., Levin, J.: A Periodic Table for Black Hole Orbits. Phys. Rev. D 79, 043017 (2009) https://doi.org/10.1103/PhysRevD.79.043017 arXiv:0811.3798 [gr-qc]

    2009 arXiv

  6. [13]

    Levin, J., Perez-Giz, G.: Homoclinic Orbits around Spinning Black Holes. I. Exact Solution for the Kerr Separatrix. Phys. Rev. D 79, 124013 (2009) https://doi. org/10.1103/PhysRevD.79.124013 arXiv:0811.3814 [gr-qc]

    2009 arXiv

  7. [14]

    Perez-Giz, G., Levin, J.: Homoclinic Orbits around Spinning Black Holes II: The Phase Space Portrait. Phys. Rev. D 79, 124014 (2009) https://doi.org/10.1103/ PhysRevD.79.124014 arXiv:0811.3815 [gr-qc]

    2009 arXiv

  8. [15]

    Healy, J.L., Shoemaker, D.: Zoom-Whirl Orbits in Black Hole Binaries

    J. Healy, J.L., Shoemaker, D.: Zoom-Whirl Orbits in Black Hole Binaries. Phys. Rev. Lett. 103, 131101 (2009) https://doi.org/10.1103/PhysRevLett.103.131101 arXiv:0907.0671 [gr-qc]

    2009 arXiv

  9. [16]

    Levin, J.: Energy Level Diagrams for Black Hole Orbits. Class. Quant. Grav. 26, 35010 (2009) https://doi.org/10.1088/0264-9381/26/23/235010 arXiv:0907.5195 [gr-qc]

    2009 arXiv

  10. [17]

    V.Misra, Levin, J.: Rational Orbits around Charged Black Holes. Phys. Rev. D 82, 083001 (2010) https://doi.org/10.1103/PhysRevD.82.083001 arXiv:1007.2699 [gr-qc]

    2010 arXiv

  11. [18]

    Grossman, J.L., Perez-Giz, G.: The harmonic structure of generic Kerr orbits

    R. Grossman, J.L., Perez-Giz, G.: The harmonic structure of generic Kerr orbits. Phys. Rev. D 85, 023012 (2012) https://doi.org/10.1103/PhysRevD.85.023012 arXiv:1105.5811 [gr-qc]

    2012 arXiv

  12. [19]

    G. Z. Babar, A.Z.B., Lim, Y.-K.: Periodic orbits around a spherically symmet- ric naked singularity. Phys. Rev. D 96, 084052 (2017) https://doi.org/10.1103/ PhysRevD.96.084052 arXiv:1710.09581 [gr-qc]

    2017 arXiv

  13. [20]

    Liu, C.D., Jing, J.: Periodic orbits around Kerr Sen black holes

    C. Liu, C.D., Jing, J.: Periodic orbits around Kerr Sen black holes. Commun. Theor. Phys. 71(12 1461) (2019) https://doi.org/10.1088/0253-6102/71/12/1461 arXiv:1804.05883 [gr-qc]

    2019 arXiv

  14. [21]

    Azreg-A ¨ ınou, B.D.M.J.T.Z.Q.W

    M. Azreg-A ¨ ınou, B.D.M.J.T.Z.Q.W. Z. Chen, Lim, Y.-K.: Orbital mechanics and quasiperiodic oscillation resonances of black holes in Einstein-Æther theory. Phys. Rev. D 102, 044028 (2020) https://doi.org/10.1103/PhysRevD.102.044028 arXiv:2004.02602 [gr-qc]

    2020 arXiv

  15. [22]

    Zhang, J., Xie, Y.: Probing a black-bounce-Reissner-Nordstr¨ om spacetime with precessing and periodic motion. Eur. Phys. J. C 82(10 854) (2022) https://doi. 33 org/10.1140/epjc/s10052-022-10846-4

    2022 doi

  16. [23]

    Wang, F.G., Chen, H.: Periodic orbits around a static spherically symmetric black hole surrounded by quintessence

    R. Wang, F.G., Chen, H.: Periodic orbits around a static spherically symmetric black hole surrounded by quintessence. Annals Phys. 447(1 169167) (2022) https: //doi.org/10.1016/j.aop.2022.169167

    2022

  17. [24]

    Tu, T.Z., Wang, A.: Periodic orbits and their gravitational wave radiations in a polymer black hole in loop quantum gravity

    Z.-Y. Tu, T.Z., Wang, A.: Periodic orbits and their gravitational wave radiations in a polymer black hole in loop quantum gravity. Phys. Rev. D 108, 024035 (2023) https://doi.org/10.1103/PhysRevD.108.024035 arXiv:2304.14160 [gr-qc]

    2023 arXiv

  18. [25]

    Deng, X.-M.: Periodic orbits around brane-world black holes. Eur. Phys. J. C 80(6 489) (2020) https://doi.org/10.1140/epjc/s10052-020-8067-7

    2020 doi

  19. [26]

    Deng, X.-M.: Geodesics and periodic orbits around quantum-corrected black holes. Phys. Dark Univ. 30(100629) (2020) https://doi.org/10.1016/j.dark.2020. 100629

    2020 doi

  20. [27]

    Universe 8(5 278) (2022) https://doi.org/10.3390/universe8050278

    Lin, H.-Y., Deng, X.-M.: Bound Orbits and Epicyclic Motions around Renor- malization Group Improved Schwarzschild Black Holes. Universe 8(5 278) (2022) https://doi.org/10.3390/universe8050278

    2022 doi

  21. [28]

    Li, X.-M.K.: Precessing and periodic timelike orbits and their potential applications in Einsteinian cubic gravity

    Y.-Z. Li, X.-M.K.: Precessing and periodic timelike orbits and their potential applications in Einsteinian cubic gravity. The European Physical Journal C 84(5

  22. [29]

    Penrose, R.: Gravitational Collapse: The Role of General Relativity. Riv. Nuovo Cim 1, 252–276 (1969)

    1969

  23. [30]

    Patil, M.K

    M. Patil, M.K. P. S. Joshi, Nakao, K.: Acceleration of particles and shells by Reissner-Nordstr¨ om naked singularities. Phys. Rev. D 86, 084023 (2012) https: //doi.org/10.1103/PhysRevD.86.084023 arXiv:1108.0288 [gr-qc]

    2012 arXiv

  24. [31]

    R. S. S. Vieira, W.K.Z.S. J. Schee, Abramowicz, M.: Circular geodesics of naked singularities in the Kehagias-Sfetsos metric of Hoˇ rava’s gravity. Phys. Rev. D 90, 024035 (2014) https://doi.org/10.1103/PhysRevD.90.024035 arXiv:1311.5820 [gr-qc]

    2014 arXiv

  25. [32]

    Monthly Notices of the Royal Astronomical Society 523, 4615–4623 (2023) https://doi.org/10.1093/mnras/stad1718 arXiv:2304.05932 [astro-ph.HE]

    Vieira, R.S.S., Klu´ zniak, W.: Astrophysical cloaking of a naked singularity. Monthly Notices of the Royal Astronomical Society 523, 4615–4623 (2023) https://doi.org/10.1093/mnras/stad1718 arXiv:2304.05932 [astro-ph.HE]

    2023 arXiv

  26. [33]

    Proceedings of RAGtime (23-25), 151–166 (2023) arXiv:2312.01827 [gr-qc]

    Mishra, R., Klu´ zniak, W.: Equilibrium tori orbiting Reissner-Nordstr¨ om naked singularities. Proceedings of RAGtime (23-25), 151–166 (2023) arXiv:2312.01827 [gr-qc]

    2023 arXiv

  27. [34]

    Shaikh, R.N.P.S.J

    R. Shaikh, R.N.P.S.J. P. Kocherlakota: Shadows of spherically symmetric black 34 holes and naked singularities. Monthly Notices of the Royal Astronomical Soci- ety 482, 52–64 (2019) https://doi.org/10.1093/mnras/sty2624 arXiv:1802.08060 [astro-ph.HE]

    2019 arXiv

  28. [35]

    Chakraborty, S.B.: Near- and sub-solar-mass naked singularities and black holes from transmutation of white dwarfss arXiv:2401.08462 [astro-ph.HE]

    C. Chakraborty, S.B.: Near- and sub-solar-mass naked singularities and black holes from transmutation of white dwarfss arXiv:2401.08462 [astro-ph.HE]

  29. [36]

    P. S. Joshi, S.B.: Primordial naked singularities arXiv:2401.14431 [gr-qc]

  30. [38]

    Pugliese, H.Q., Ruffini, R.: Circular motion of neutral test particles in Reissner- Nordstr¨ om spacetime

    D. Pugliese, H.Q., Ruffini, R.: Circular motion of neutral test particles in Reissner- Nordstr¨ om spacetime. Phys. Rev. D83, 024021 (2011) https://doi.org/10.1103/ PhysRevD.83.024021 arXiv:1012.5411 [astro-ph.HE]

    2011 arXiv

  31. [39]

    The international series of monographs on physics

    Chandrasekhar, S.: The Mathematical Theory of Black Holes. The international series of monographs on physics. Oxford University Press, Chicago (1983)

    1983

  32. [40]

    Schroven, K., Grunau, S.: Innermost stable circular orbit of charged particles in Reissner-Nordstr¨ om, Kerr-Newman, and Kerr-Sen spacetimes. Phys. Rev. D103, 024016 (2021) https://doi.org/10.1103/PhysRevD.103.024016 arXiv:2007.08823 [gr-qc]

    2021 arXiv

  33. [41]

    Zhang, M., Jiang, J.: Internal structures and circular orbits for test particles. Phy. Let. B. 834, 137476 https://doi.org/10.1016/j.physletb.2022.137476 (2022)

    2022

  34. [42]

    Cohen, J.M., Gautreau, R.: Naked singularities, event horizons, and charged par- ticles. Phys. Rev. D 19(8), 2273–2279 (1979) https://doi.org/10.1103/PhysRevD. 19.2273

    1979 doi

  35. [43]

    Grundlehren der mathematischen Wissenschaften

    Bryd, P.F., Friedman, M.D.: Handbook of Elliptic Integrals for Engineers and Sci- entists. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (1971)

    1971

  36. [44]

    Elsevier Science, Amsterdam (2002)

    Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier Science, Amsterdam (2002)

    2002

  37. [45]

    PhD thesis, University of Waikato (2006)

    Slez´ akov´ a, G.: Geodesic Geometry of Black Holes. PhD thesis, University of Waikato (2006)

    2006

  38. [46]

    Grunau, S., Kagramanova, V.: Geodesics of electrically and magnetically charged test particles in the Reissner-Nordstr¨ om space-time: Analytical solutions. Phys. Rev. D 83, 044009 (2011) https://doi.org/10.1103/PhysRevD.83.044009 arXiv:1011.5399 [gr-qc] 35

    2011 arXiv

  39. [47]

    Soltani, F.: Global Kruskal-Szekeres coordinates for Reissner-Nordstr¨ om space- time. Phys. Rev. D 108, 124002 (2023) https://doi.org/10.1103/PhysRevD.108. 124002 arXiv:2307.11026 [gr-qc]

    2023 arXiv

  40. [48]

    Y.-T Li, D.-S.L. C.-Y. Wang, Lin, C.-Y.: Homoclinic orbits in Kerr-Newman black holes. Phys. Rev. D 108, 044010 (2023) https://doi.org/10.1103/PhysRevD.108. 044010 arXiv:2302.09471 [gr-qc]

    2023 arXiv

  41. [49]

    Bombelli, L., Calzetta, E.: Chaos around a black hole. Class. Quant. Grav. 9, 2573–2599 (1992) https://doi.org/10.1088/0264-9381/9/12/004

    1992 doi

  42. [50]

    From MathWorld -A Wolfram Web Resource

    Weisstein, E.W.: Vieta’s Formulas. From MathWorld -A Wolfram Web Resource. https://mathworld.wolfram.com/VietasFormulas.html. Accessed: 2024-05-13 36

    2024

  43. [529]

    (2024) https://doi.org/10.1140/epjc/s10052-024-12895-3 arXiv:2401.16071 [gr-qc]

    2024 arXiv

Pith tools

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