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arxiv: 2603.09169 · v2 · submitted 2026-03-10 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Photon spheres and bulk probes in AdS₃/CFT₂: the quantum BTZ black hole

Oscar Lasso Andino , Axel Le\'on-Arteaga , Guillermo Ram\'irez-Ulloa
Authors on Pith no claims yet

Pith reviewed 2026-05-15 14:20 UTC · model grok-4.3

classification ✦ hep-th
keywords quantum BTZ black holeAdS3/CFT2entanglement entropygeodesicsphoton spheresbulk probes
0
0 comments X

The pith

The quantum BTZ black hole allows boundary-anchored geodesics in all branches, with photon spheres ensuring timelike points connect via spacelike or null paths.

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

The paper analyzes the conditions for geodesics with two ends on the boundary in the quantum BTZ black hole and its charged version. It determines the separation type between boundary points for these geodesics across all solution branches. The work uses light ring criteria to support a conjecture that spherically symmetric spacetimes with photon spheres always permit spacelike or null connections between timelike-separated boundary points. This matters for computing entanglement entropy in the dual CFT via minimal surfaces in the bulk.

Core claim

The quantum BTZ black hole admits geodesics anchored in the boundary in all branches, and for spherically symmetric spacetimes with a photon sphere there are always timelike-separated points connectable by spacelike or null geodesics.

What carries the argument

Photon spheres in the quantum BTZ metric, used as a criterion to classify the distance type between boundary points linked by geodesics.

If this is right

  • If a photon ring is present, the timelike entanglement entropy in AdS3/CFT2 has no imaginary part.
  • Space-like geodesics can connect timelike-separated points by winding around the horizon.
  • The existence conditions hold for both the neutral and charged quantum BTZ black holes.
  • The results extend the known behavior from Schwarzschild-AdS to the quantum BTZ case.

Where Pith is reading between the lines

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

  • The quantum backreaction in the BTZ solution does not obstruct the standard application of holographic entanglement entropy calculations.
  • Similar geodesic analyses could be performed in other three-dimensional black hole solutions to test the generality of the photon sphere conjecture.
  • These findings imply that bulk probes like geodesics remain reliable even in quantum-corrected geometries for AdS3/CFT2.

Load-bearing premise

The standard Ryu-Takayanagi prescription applies without modifications from quantum backreaction in the quantum BTZ geometry.

What would settle it

Discovery of a quBTZ branch lacking any boundary-anchored geodesics for timelike-separated points despite having a photon sphere would disprove the conditions and the supported conjecture.

read the original abstract

The entanglement entropy in $d+1$ dimensional conformal field theories can be calculated using the area of $d$ dimensional minimal surfaces in $AdS_{d+2}$. Therefore, the existence of surfaces anchored in the boundary of an asymptotically anti-de Sitter (AdS) spacetime is crucial for the calculation of entanglement entropy. In particular, in $d=3$ the extremal surfaces are geodesics with two ends in the boundary. In the Schwarzschild-AdS black hole the space-like geodesics can connect timelike-separated points by winding around the horizon multiple times. This result can be extended to other asymptotically AdS spacetimes. For geodesics joining time-like separated points, if there is a photon ring then the timelike entanglement entropy in the $\text{AdS}_3/\text{CFT}_2$ will not have an imaginary part. We present an exhaustive analysis about the existence of geodesics anchored in the boundary of the three dimensional quantum BTZ (quBTZ) black hole and its charged counterpart. We found conditions for the existence of geodesics with two ends in the boundary in all branches of the quBTZ and determine the type of distance between the points in the boundary. We use a criteria for the existence of light rings to shed some light over the conjecture for spacetimes that are spherically symmetric and have a photon sphere: there are always points with time-like separation that can be connected by space-like or null geodesics.

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.

Referee Report

0 major / 2 minor

Summary. The paper investigates the existence of boundary-anchored geodesics in the quantum BTZ black hole and its charged counterpart within the AdS3/CFT2 framework. It derives conditions for such geodesics in all branches of the quBTZ metric, classifies the separation between boundary points, and applies light-ring criteria to a conjecture on photon spheres in spherically symmetric spacetimes, concluding that timelike-separated points can be connected by spacelike or null geodesics. This analysis is motivated by holographic entanglement entropy calculations using minimal surfaces.

Significance. Should the central claims be verified through the detailed calculations, this manuscript would contribute meaningfully to the study of holographic entanglement in quantum black hole geometries. It offers specific conditions for geodesic existence and supports a general conjecture linking photon spheres to geodesic connectivity, which may aid in understanding timelike entanglement entropy without imaginary parts. The use of the provided quBTZ metric for direct geodesic analysis is a positive aspect.

minor comments (2)
  1. The notation used for the different branches of the quBTZ black hole should be introduced more clearly with explicit metric forms at the outset.
  2. A formal statement of the conjecture being addressed would help readers follow the argument in the light-ring section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of our work on boundary-anchored geodesics in the quantum BTZ black hole and its charged version. We appreciate the recommendation for minor revision and the recognition that the analysis supports the conjecture linking photon spheres to geodesic connectivity in spherically symmetric spacetimes. We will incorporate clarifications to strengthen the presentation of the conditions for geodesic existence across all branches.

Circularity Check

0 steps flagged

Standard geodesic analysis on supplied quBTZ metric; no reduction to fitted inputs or self-definitions

full rationale

The derivation applies the classical geodesic equation and Ryu-Takayanagi prescription directly to the given quantum BTZ line element (including charged branch). Existence conditions for boundary-anchored geodesics and the photon-sphere implication for timelike separation follow from solving the geodesic ODE on this fixed background. No parameters are fitted inside the paper, no central claim is renamed from prior results, and no uniqueness theorem or ansatz is smuggled via self-citation. The quantum label is already encoded in the supplied metric, so the analysis remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Ryu-Takayanagi prescription for entanglement entropy and the geodesic equation in the quBTZ metric, which are taken from prior literature.

axioms (2)
  • domain assumption The Ryu-Takayanagi formula equates entanglement entropy to the area of minimal surfaces in the bulk
    Standard assumption in AdS/CFT literature
  • standard math Geodesics in the quBTZ metric can be found by solving the geodesic equation with boundary conditions
    Mathematical method for extremal surfaces

pith-pipeline@v0.9.0 · 5591 in / 1324 out tokens · 95290 ms · 2026-05-15T14:20:03.719668+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

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

44 extracted references · 44 canonical work pages · 12 internal anchors

  1. [1]

    The Large N Limit of Superconformal Field Theories and Supergravity

    J. M. Maldacena, “The LargeNlimit of superconformal field theories and supergravity,” Adv. Theor. Math. Phys.2(1998), 231-252DOI:10.4310/ATMP.1998.v2.n2.a1 arXiv:hep-th/9711200[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/atmp.1998.v2.n2.a1 1998

  2. [2]

    Anti De Sitter Space And Holography

    E. Witten, “Anti de Sitter space and holography,” Adv. Theor. Math. Phys.2(1998), 253-291 DOI:10.4310/ATMP.1998.v2.n2.a2arXiv:hep-th/9802150[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/atmp.1998.v2.n2.a2arxiv:hep-th/9802150 1998

  3. [3]

    On geodesic propagators and black hole holography

    J. Louko, D. Marolf and S. F. Ross, “On geodesic propagators and black hole holography,” Phys. Rev. D62(2000), 044041,DOI:10.1103/PhysRevD.62.044041arXiv:hep-th/0002111[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.62.044041arxiv:hep-th/0002111 2000

  4. [4]

    Late-time correlators and complex geodesics in de Sitter space,

    L. Aalsma, M. M. Faruk, J. P. van der Schaar, M. R. Visser and J. de Witte, “Late-time correlators and complex geodesics in de Sitter space,” SciPost Phys.15(2023) no.1, 031, DOI:10.21468/SciPostPhys.15.1.031, arXiv:2212.01394[hep-th]. – 31 –

    work page doi:10.21468/scipostphys.15.1.031 2023

  5. [5]

    Holographic Derivation of Entanglement Entropy from AdS/CFT

    S. Ryu and T. Takayanagi, “Holographic derivation of entanglement entropy from AdS/CFT,” Phys. Rev. Lett.96(2006), 181602,DOI:10.1103/PhysRevLett.96.181602, arXiv:hep-th/0603001[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.96.181602 2006

  6. [6]

    Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki, “Pseudoentropy in dS/CFT and Timelike Entanglement Entropy,” Phys. Rev. Lett.130(2023) no.3, 031601, DOI:10.1103/PhysRevLett.130.031601, arXiv:2210.09457[hep-th]

    work page doi:10.1103/physrevlett.130.031601 2023

  7. [7]

    Timelike entanglement entropy,

    K. Doi, J. Harper, A. Mollabashi, T. Takayanagi and Y. Taki, “Timelike entanglement entropy,” JHEP05(2023), 052,DOI:10.1007/JHEP05(2023)052, arXiv:2302.11695[hep-th]

    work page doi:10.1007/jhep05(2023)052 2023

  8. [8]

    New holographic generalization of entanglement entropy,

    Y. Nakata, T. Takayanagi, Y. Taki, K. Tamaoka and Z. Wei, “New holographic generalization of entanglement entropy,” Phys. Rev. D103(2021) no.2, 026005, DOI:10.1103/PhysRevD.103.026005, arXiv:2005.13801[hep-th]]

    work page doi:10.1103/physrevd.103.026005 2021

  9. [9]

    On holographic time-like entanglement entropy,

    Z. Li, Z. Q. Xiao and R. Q. Yang, “On holographic time-like entanglement entropy,” JHEP04 (2023), 004,DOI:10.1007/JHEP04(2023)004arXiv:2211.14883[hep-th]

    work page doi:10.1007/jhep04(2023)004arxiv:2211.14883 2023

  10. [10]

    Black hole singularity and timelike entanglement,

    T. Anegawa and K. Tamaoka, “Black hole singularity and timelike entanglement,” JHEP10 (2024), 182,DOI:10.1007/JHEP10(2024)182, arXiv:2406.10968[hep-th]]

    work page doi:10.1007/jhep10(2024)182 2024

  11. [11]

    A duality of Ryu-Takayanagi surfaces inside and outside the horizon,

    W. z. Guo and J. Xu, “A duality of Ryu-Takayanagi surfaces inside and outside the horizon,” arXiv:2502.16774[hep-th]

    work page arXiv

  12. [12]

    The Black Hole in Three Dimensional Space Time

    M. Banados, C. Teitelboim and J. Zanelli, “The Black hole in three-dimensional space-time,” Phys. Rev. Lett.69(1992), 1849-1851DOI:10.1103/PhysRevLett.69.1849 arXiv:hep-th/9204099[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.69.1849 1992

  13. [13]

    Quantum BTZ black hole,

    R. Emparan, A. M. Frassino and B. Way, “Quantum BTZ black hole,” JHEP11(2020), 137 DOI:10.1007/JHEP11(2020)137arXiv:2007.15999[hep-th]

    work page doi:10.1007/jhep11(2020)137arxiv:2007.15999 2020

  14. [14]

    Quantum black holes as holograms in AdS brane worlds,

    R. Emparan, A. Fabbri and N. Kaloper, “Quantum black holes as holograms in AdS brane worlds,” JHEP08, 043 (2002),DOI:10.1088/1126-6708/2002/08/043

    work page doi:10.1088/1126-6708/2002/08/043 2002

  15. [15]

    Exact Description of Black Holes on Branes II: Comparison with BTZ Black Holes and Black Strings

    R. Emparan, G. T. Horowitz and R. C. Myers, “Exact description of black holes on branes. 2. Comparison with BTZ black holes and black strings,” JHEP01, 021 (2000), DOI:10.1088/1126-6708/2000/01/021, arXiv:hep-th/9912135[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2000/01/021 2000

  16. [16]

    Critical phenomenon of quantum BTZ black holes,

    H. M. Cui and Z. Y. Fan, “Critical phenomenon of quantum BTZ black holes,” Eur. Phys. J. C 85(2025) no.9, 1044DOI:10.1140/epjc/s10052-025-14768-9arXiv:2505.23188[hep-th]

    work page doi:10.1140/epjc/s10052-025-14768-9arxiv:2505.23188 2025

  17. [17]

    Quantum charged black holes,

    Y. Feng, H. Ma, R. B. Mann, Y. Xue and M. Zhang, “Quantum charged black holes,” JHEP08 (2024), 184,DOI:10.1007/JHEP08(2024)184arXiv:2404.07192[hep-th]

    work page doi:10.1007/jhep08(2024)184arxiv:2404.07192 2024

  18. [18]

    Uniformly accelerating charged mass in general relativity,

    W. Kinnersley and M. Walker, “Uniformly accelerating charged mass in general relativity,” Phys. Rev. D2(1970), 1359-1370 doi:10.1103/PhysRevD.2.1359

    work page doi:10.1103/physrevd.2.1359 1970

  19. [19]

    Thermodynamic bounce effect in quantum BTZ black hole,

    Z. M. Xu, P. P. Zhang, B. Wu and X. Zhang, “Thermodynamic bounce effect in quantum BTZ black hole,” JHEP12(2024), 181,DOI:10.1007/JHEP12(2024)181, arXiv:2407.08241[gr-qc]

    work page doi:10.1007/jhep12(2024)181 2024

  20. [20]

    Chemical potential and charge in quantum black holes,

    A. Climent, R. Emparan and R. A. Hennigar, “Chemical potential and charge in quantum black holes,” JHEP08(2024), 150,DOI:10.1007/JHEP08(2024)150, arXiv:2404.15148[hep-th]

    work page doi:10.1007/jhep08(2024)150 2024

  21. [21]

    Thermodynamical topology of quantum BTZ black hole,

    S. P. Wu and S. W. Wei, “Thermodynamical topology of quantum BTZ black hole,” Phys. Rev. D110(2024) no.2, 024054,DOI:10.1103/PhysRevD.110.024054, arXiv:2403.14167[gr-qc]. – 32 –

    work page doi:10.1103/physrevd.110.024054 2024

  22. [22]

    Criticality and thermodynamic geometry of quantum BTZ black holes,

    S. A. Hosseini Mansoori, J. F. Pedraza and M. Rafiee, “Criticality and thermodynamic geometry of quantum BTZ black holes,” Phys. Rev. D111(2025) no.2, 024012, DOI:10.1103/PhysRevD.111.024012, arXiv:2403.13063[hep-th]

    work page doi:10.1103/physrevd.111.024012 2025

  23. [23]

    Bulk-cone singularities & signatures of horizon formation in AdS/CFT

    V. E. Hubeny, H. Liu and M. Rangamani, “Bulk-cone singularities & signatures of horizon formation in AdS/CFT,” JHEP01(2007), 009,DOI:10.1088/1126-6708/2007/01/009, arXiv:hep-th/0610041[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/1126-6708/2007/01/009 2007

  24. [24]

    Geodesics connecting endpoints of timelike interval in an asymptotically AdS spacetime,

    J. H. He and R. Q. Yang, “Geodesics connecting endpoints of timelike interval in an asymptotically AdS spacetime,” Phys. Rev. D111(2025) no.2, 026024, DOI:10.1103/PhysRevD.111.026024, arXiv:2408.04783[hep-th]

    work page doi:10.1103/physrevd.111.026024 2025

  25. [25]

    Specific heats for quantum BTZ black holes in extended thermodynamics,

    C. V. Johnson and R. Nazario, “Specific heats for quantum BTZ black holes in extended thermodynamics,” Phys. Rev. D110(2024) no.10, 106004, DOI:10.1103/PhysRevD.110.106004, arXiv:2310.12212[hep-th]

    work page doi:10.1103/physrevd.110.106004 2024

  26. [26]

    Extremal surfaces as bulk probes in AdS/CFT

    V. E. Hubeny, “Extremal surfaces as bulk probes in AdS/CFT,” JHEP07, 093 (2012), DOI:10.1007/JHEP07(2012)093, arXiv:1203.1044[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep07(2012)093 2012

  27. [27]

    Theorems on gravitational time delay and related issues

    S. Gao and R. M. Wald, “Theorems on gravitational time delay and related issues,” Class. Quant. Grav.17, 4999-5008 (2000) doi:10.1088/0264-9381/17/24/305 [arXiv:gr-qc/0007021 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0264-9381/17/24/305 2000

  28. [28]

    Geometric approach to circular photon orbits and black hole shadows,

    C. K. Qiao and M. Li, “Geometric approach to circular photon orbits and black hole shadows,” Phys. Rev. D106(2022) no.2, L021501,DOI:10.1103/PhysRevD.106.L021501 arXiv:2204.07297[gr-qc]

    work page doi:10.1103/physrevd.106.l021501 2022

  29. [29]

    Curvatures, photon spheres, and black hole shadows,

    C. K. Qiao, “Curvatures, photon spheres, and black hole shadows,” Phys. Rev. D106(2022) no.8, 084060,DOI:10.1103/PhysRevD.106.084060, arXiv:2208.01771[gr-qc]

    work page doi:10.1103/physrevd.106.084060 2022

  30. [30]

    Massive particle surfaces, partial umbilicity, and circular orbits,

    B. Bermúdez-Cárdenas and O. L. Andino, “Massive particle surfaces, partial umbilicity, and circular orbits,” Phys. Rev. D111(2025) no.6, 064001,DOI:10.1103/PhysRevD.111.064001 arXiv:2409.10789[gr-qc]

    work page doi:10.1103/physrevd.111.064001 2025

  31. [31]

    Massive particle surfaces and black hole shadows from intrinsic curvature

    B. Bermúdez-Cárdenas and O. L. Andino, “Massive particle surfaces and black hole shadows from intrinsic curvature,” Eur. Phys. J. C85(2025) no.11, 1266, DOI:doi:10.1140/epjc/s10052-025-15009-9, arXiv:2503.21203[gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epjc/s10052-025-15009-9 2025

  32. [32]

    A Riemannian geometric approach for timelike and null spacetime geodesics,

    M. A. Argañaraz and O. L. Andino, “A Riemannian geometric approach for timelike and null spacetime geodesics,” Gen. Rel. Grav.56(2024) no.10, 121 DOI:10.1007/s10714-024-03314-9, arXiv:2112.10910[gr-qc]

    work page doi:10.1007/s10714-024-03314-9 2024

  33. [33]

    Light rings, timelike circular orbits and curvature of traversable wormholes,

    B. Bermúdez-Cárdenas and O. Lasso Andino, “Light rings, timelike circular orbits and curvature of traversable wormholes,” arXiv:2504.10732[gr-qc]

    work page arXiv

  34. [34]

    Timelike entanglement and central charge for quantum BTZ black holes,

    D. Roychowdhury, “Timelike entanglement and central charge for quantum BTZ black holes,” Phys. Lett. B869, 139846 (2025)DOI:10.1016/j.physletb.2025.139846, arXiv:2507.19813 [hep-th]

    work page doi:10.1016/j.physletb.2025.139846 2025

  35. [35]

    Holographic complexity of rotating quantum black holes,

    B. Chen, Y. Liu and B. Yu, “Holographic complexity of rotating quantum black holes,” JHEP 01, 055 (2024)DOI:10.1007/JHEP01(2024)055, arXiv:2307.15968[hep-th]

    work page doi:10.1007/jhep01(2024)055 2024

  36. [36]

    Quantum Inequalities for Quantum Black Holes,

    A. M. Frassino, R. A. Hennigar, J. F. Pedraza and A. Svesko, “Quantum Inequalities for Quantum Black Holes,” Phys. Rev. Lett.133, no.18, 181501 (2024) DOI:10.1103/PhysRevLett.133.181501, arXiv:2406.17860[hep-th]. – 33 –

    work page doi:10.1103/physrevlett.133.181501 2024

  37. [37]

    Geometric Interpretation of Timelike Entanglement Entropy,

    M. P. Heller, F. Ori and A. Serantes, “Geometric Interpretation of Timelike Entanglement Entropy,” Phys. Rev. Lett.134, no.13, 131601 (2025)DOI:10.1103/PhysRevLett.134.131601, arXiv:2408.15752[hep-th]

    work page doi:10.1103/physrevlett.134.131601 2025

  38. [38]

    Einstein Rings in Holography

    K. Hashimoto, S. Kinoshita and K. Murata, “Einstein Rings in Holography,” Phys. Rev. Lett. 123, no.3, 031602 (2019)DOI:10.1103/PhysRevLett.123.031602, arXiv:1906.09113[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.123.031602 2019

  39. [39]

    Imaging black holes through the AdS/CFT correspondence,

    K. Hashimoto, S. Kinoshita and K. Murata, “Imaging black holes through the AdS/CFT correspondence,” Phys. Rev. D101, no.6, 066018 (2020)DOI:10.1103/PhysRevD.101.066018, arXiv:1811.12617[hep-th]

    work page doi:10.1103/physrevd.101.066018 2020

  40. [40]

    Holographic Einstein ring of a charged AdS black hole,

    Y. Liu, Q. Chen, X. X. Zeng, H. Zhang, W. L. Zhang and W. Zhang, “Holographic Einstein ring of a charged AdS black hole,” JHEP10, 189 (2022)DOI:10.1007/JHEP10(2022)189, arXiv:2201.03161[hep-th]

    work page doi:10.1007/jhep10(2022)189 2022

  41. [41]

    Extremal Surface Barriers

    N. Engelhardt and A. C. Wall, “Extremal Surface Barriers,” JHEP03, 068 (2014) DOI:10.1007/JHEP03(2014)068, arXiv:1312.3699[hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2014)068 2014

  42. [42]

    Polynomial Transformations of Tschirnhaus, Bring and Jerrard,

    V. S. Adamchik and D. J. Jeffrey, “Polynomial Transformations of Tschirnhaus, Bring and Jerrard,” ACM SIGSAM Bulletin37(2003) no. 3, 90–94

    work page 2003

  43. [43]

    I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky,Discriminants, Resultants, and Multidimensional Determinants, Birkhäuser, Boston (1994).DOI:10.1007/978-0-8176-4771-1

    work page doi:10.1007/978-0-8176-4771-1 1994

  44. [44]

    Gravitational Capture Cross-Section of Particles by Schwarzschild–Tangherlini Black Holes,

    B. Ahmedov, O. Rahimov and B. Toshmatov, “Gravitational Capture Cross-Section of Particles by Schwarzschild–Tangherlini Black Holes,” Universe7(2021) no. 8, 307 doi:10.3390/universe7080307 – 34 –

    work page doi:10.3390/universe7080307 2021