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arxiv: 2605.16461 · v1 · pith:VSO4QPLZnew · submitted 2026-05-15 · 🌀 gr-qc · astro-ph.HE· hep-th

Particle Dynamics, Shadow and Hawking Sparsity of a Kalb-Ramond Black Hole Coupled to Nonlinear Electrodynamics

Faizuddin Ahmed , Ahmad Al-Badawi , \.Izzet Sakall{\i} This is my paper

Pith reviewed 2026-05-20 17:40 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords Kalb-Ramond fieldnonlinear electrodynamicsblack hole shadowHawking sparsitygeodesicsphoton spherequasinormal modes
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The pith

A Kalb-Ramond black hole with nonlinear electrodynamics emits Hawking radiation that is over three times sparser than a Schwarzschild black hole.

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

This paper studies the paths of massive particles and light rays near a black hole that includes a Kalb-Ramond field coupled with nonlinear electrodynamics. It calculates the effective potentials for orbits and determines the size of the black hole's shadow. The shadow is checked against observations from the Event Horizon Telescope for M87* and Sgr A*. The main result is that the sparsity of the Hawking radiation spectrum increases substantially because of the additional fields and charges in the model.

Core claim

The combined influence of the Kalb-Ramond field and the magnetic monopole charge raises the Gray-Visser sparsity parameter of Hawking radiation from the Schwarzschild value of 16π³ ≈ 496 to nearly 1.7×10³. This means the radiation is emitted in a more discrete, less continuous manner. At the same time, the radius of the photon ring and the shadow stay within the one-sigma observational limits set by the Event Horizon Telescope for a wide range of the allowed parameters.

What carries the argument

The Gray-Visser sparsity parameter, computed from the Hawking temperature and the horizon area, which indicates how sparse the particle emission is in the radiation cascade.

If this is right

  • The black hole shadow radius fits within EHT 1σ bounds for most of the physically allowed range of q and γ.
  • The Lyapunov exponent of the unstable photon orbits determines the frequencies of eikonal quasinormal modes.
  • Closed-form expressions exist for the specific energy and angular momentum of massive particles in circular orbits.
  • The effective potential governs both timelike and null geodesic motion in this spacetime.

Where Pith is reading between the lines

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

  • If the sparsity increase holds, it could mean that black holes in this model evaporate more slowly or in distinct bursts rather than a steady stream.
  • Future high-precision shadow imaging might further restrict the Lorentz-violating parameters.
  • The model provides a concrete example of how exotic matter fields alter standard black hole thermodynamics predictions.

Load-bearing premise

The geometry is a static and spherically symmetric solution to the field equations with the Kalb-Ramond field coupled to nonlinear electrodynamics and specific parameters for charge and Lorentz violation.

What would settle it

An observation that the Hawking radiation from a black hole is not sparser than the Schwarzschild prediction, or EHT data that places the shadow radius outside the viable parameter space for this model.

Figures

Figures reproduced from arXiv: 2605.16461 by Ahmad Al-Badawi, Faizuddin Ahmed, \.Izzet Sakall{\i}.

Figure 1
Figure 1. Figure 1: FIG. 1. The dependence of e [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Behavior of the ISCO radius for numerous values of BH [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The dependence of the specific energy [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The dependence of the orbital velocity [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the radial behavior of the effective potential Veff for different values of λ and q. Increasing these parameters enhances the effective potential, indicating stronger spacetime curvature effects on [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: confirms that rs and Rsh both shrink as q grows, with γ providing a milder downward shift. The Schwarzschild values rs = 3M and Rsh = 3 √ 3 M ≃ 5.196 M are recovered at q = γ = 0. B. EHT compatibility for M87∗ and Sgr A∗ The 1σ EHT bands on the shadow radius for M87∗ and Sgr A∗ read [6, 72, 74] 4.31 ≤ R M87∗ sh /M ≤ 6.08, 4.55 ≤ R SgrA∗ sh /M ≤ 5.22. (40) The Sgr A∗ band is the more restrictive of the two.… view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The dependence of the e [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The dependence of the squared Lyapunov exponent [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Parametric plots of the photon trajectories, described by [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
read the original abstract

We study the timelike and null geodesic structure of a static, spherically symmetric black hole sourced by a Kalb--Ramond (KR) field coupled to nonlinear electrodynamics (NED). The geometry is characterized by the mass $M$, the magnetic monopole charge $q$, and the Lorentz-violating parameters $(\gamma,\lambda)$. Closed-form expressions are derived for the effective potential, as well as the specific energy and angular momentum of massive particles on circular orbits. We further analyze the photon sphere, black hole shadow, and the Lyapunov exponent associated with unstable null circular geodesics. The latter determines the eikonal quasinormal-mode frequencies through $\omega_{\rm eik}=(\ell+1/2)\,\Omega_c-i(n+1/2)\,|\lambda_L|$. The shadow radius is compared with the Event Horizon Telescope (EHT) observations of M87$^\ast$ and Sgr~A$^\ast$, allowing us to identify the viable region in the $(q,\gamma)$ parameter space. Finally, we compute the Hawking temperature, horizon area, and the Gray--Visser sparsity parameter. We demonstrate that the combined effects of the KR field and magnetic monopole charge increase the sparsity parameter from the Schwarzschild value $16\pi^3 \simeq 496$ to nearly $1.7\times10^3$. This indicates a significantly sparser Hawking cascade compared to the Schwarzschild case, while the photon ring remains consistent with the EHT $1\sigma$ observational bounds across most of the physically allowed parameter range.

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

1 major / 2 minor

Summary. The manuscript studies the geodesic motion of timelike and null particles around a black hole in Kalb-Ramond gravity coupled to nonlinear electrodynamics. It provides closed-form expressions for the effective potential, specific energy and angular momentum for circular orbits, analyzes the photon sphere, black hole shadow compared to EHT data for M87* and Sgr A*, derives eikonal quasinormal modes via Lyapunov exponent, and computes the Hawking temperature, horizon area, and Gray-Visser sparsity parameter, reporting an increase in sparsity from ~496 to ~1700 while the shadow remains consistent with observations.

Significance. Should the derivations prove robust, the work offers concrete predictions for observable quantities like the shadow radius in a modified gravity setting with Lorentz-violating parameters, and highlights how additional fields can affect the sparsity of Hawking radiation. The constraint of parameters using EHT 1σ bounds is a positive aspect, allowing for falsifiable claims.

major comments (1)
  1. [Hawking sparsity computation] In the section on Hawking temperature, horizon area, and the Gray-Visser sparsity parameter: the generalized expression for the sparsity (using the modified Hawking temperature from surface gravity and the horizon area) is not shown to reduce exactly to the Schwarzschild value 16π³ when q = γ = λ = 0. This limit check is load-bearing for the central quantitative claim of an increase to nearly 1.7×10³.
minor comments (2)
  1. [Abstract] The approximation 16π³ ≃ 496 in the abstract is numerically correct but could be stated with more precision or as 'approximately 496'.
  2. [Metric and parameter definitions] The physical interpretation and allowed ranges for the Lorentz-violating parameters γ and λ could be clarified earlier in the setup section to improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment, which helps strengthen the presentation of our results. We address the point below and describe the revisions we will implement.

read point-by-point responses

  1. Referee: In the section on Hawking temperature, horizon area, and the Gray-Visser sparsity parameter: the generalized expression for the sparsity (using the modified Hawking temperature from surface gravity and the horizon area) is not shown to reduce exactly to the Schwarzschild value 16π³ when q = γ = λ = 0. This limit check is load-bearing for the central quantitative claim of an increase to nearly 1.7×10³.

    Authors: We appreciate the referee pointing out the need for an explicit limit check. The sparsity parameter is obtained from the surface-gravity temperature and the horizon area of the KR-NED metric; the functional form was chosen so that the Schwarzschild case is recovered when the parameters vanish. Nevertheless, we agree that a direct substitution was omitted and that this verification is important for supporting the reported increase from 16π³ ≈ 496 to ~1.7×10³. In the revised manuscript we will add a short paragraph (or appendix subsection) that substitutes q = γ = λ = 0 into the generalized expressions for temperature, area, and sparsity, showing the reduction to the exact Schwarzschild value with all intermediate steps displayed. This addition will be placed immediately before the discussion of the numerical results for nonzero parameters. revision: yes

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 2 invented entities

The central claim relies on the existence of this particular black hole solution with three additional parameters beyond mass, which are treated as free and constrained by data.

free parameters (3)
  • q
    Magnetic monopole charge parameter that affects the geometry and is constrained by shadow observations.
  • gamma
    Lorentz-violating parameter characterizing the KR field.
  • lambda
    Lorentz-violating parameter characterizing the KR field.
axioms (2)
  • domain assumption The black hole solution is static and spherically symmetric.
    Stated in the abstract as the geometry studied.
  • standard math Geodesic motion follows the standard GR equations for timelike and null paths.
    Used to derive effective potentials and circular orbits.
invented entities (2)
  • Kalb-Ramond field no independent evidence
    purpose: To source the black hole metric along with NED.
    Part of the model construction, no new evidence provided.
  • Nonlinear electrodynamics coupling no independent evidence
    purpose: To modify the electromagnetic contribution to the stress-energy.
    Specific to this setup.

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Works this paper leans on

77 extracted references · 77 canonical work pages · 2 internal anchors

  1. [1]

    Particle Dynamics, Shadow and Hawking Sparsity of a Kalb-Ramond Black Hole Coupled to Nonlinear Electrodynamics

    explored the propagation of light in strongly curved spacetimes, while later studies by Bardeen and others established the properties of photon trajectories and shadow formation in Schwarzschild and Kerr geometries [8, 9]. A possible violation of local Lorentz symmetry at the particle level arises from the presence of a rank-two antisymmetric tensor field...

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  2. [2]

    Penrose, Phys

    R. Penrose, Phys. Rev. Lett.14, 57 (1965)

    work page 1965

  3. [3]

    S. W. Hawking and R. Penrose, Proc. R. Soc. London, Series A: Math. Phys. Sci.314, 529 (1970)

    work page 1970

  4. [4]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L1 (2019)

    work page 2019

  5. [5]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L6 (2019)

    work page 2019

  6. [6]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.930, L12 (2022)

    work page 2022

  7. [7]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.930, L17 (2022)

    work page 2022

  8. [8]

    J. L. Synge, MNRAS131, 463 (1966)

    work page 1966

  9. [9]

    J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J 178, 347 (1972)

    work page 1972

  10. [10]

    Luminet, Astron

    J.-P. Luminet, Astron. Astrophys.75, 228 (1979)

    work page 1979

  11. [11]

    Kalb and P

    M. Kalb and P. Ramond, Phys. Rev. D9, 2273 (1974)

    work page 1974

  12. [12]

    M. B. Green, J. H. Schwarz, and E. Witten,Superstring Theory, V olume 2(Cambridge University Press, Cambridge, 1987)

    work page 1987

  13. [13]

    Altschul, Q

    B. Altschul, Q. G. Bailey, and V . A. Kostelecký, Phys. Rev. D 81, 065028 (2010)

    work page 2010

  14. [14]

    F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Rev. Mod. Phys.48, 393 (1976)

    work page 1976

  15. [15]

    F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester, Phys. Rep.258, 1 (1995)

    work page 1995

  16. [16]

    L. A. Lessa, J. E. G. Silva, R. V . Maluf, and C. A. S. Almeida, Eur. Phys. J. C80, 335 (2020)

    work page 2020

  17. [17]

    L. A. Lessa, R. Oliveira, J. E. G. Silva, and C. A. S. Almeida, Annals of Physics433, 168604 (2021)

    work page 2021

  18. [19]

    Jumaniyozov, S

    S. Jumaniyozov, S. Murodov, J. Rayimbaev,et al., Eur. Phys. J C85, 797 (2025)

    work page 2025

  19. [20]

    Yang, Y .-Z

    K. Yang, Y .-Z. Chen, Z.-Q. Duan, and J.-Y . Zhao, Phys. Rev. D108, 124004 (2023)

    work page 2023

  20. [21]

    Z. Q. Duan, J. Y . Zhao, and K. Yang, Eur. Phys. J C84, 798 (2024)

    work page 2024

  21. [22]

    Güllü and A

    ˙I. Güllü and A. Övgün, Ann. Phys. (NY)436, 168721 (2022)

    work page 2022

  22. [23]

    F. M. Belchior, R. V . Maluf, A. Y . Petrov,et al., Eur. Phys. J C 85, 658 (2025)

    work page 2025

  23. [24]

    Fathi and A

    M. Fathi and A. Övgün, Eur. Phys. J Plus140, 280 (2025)

    work page 2025

  24. [25]

    Baruah, Y

    A. Baruah, Y . Sekhmani, S. K. Maurya, A. Deshamukhya, and M. K. Jasim, JCAP2025, 023 (2025)

    work page 2025

  25. [26]

    Ahmed and E

    F. Ahmed and E. O. Silva, (2025), arXiv:2511.21374 [hep-th]

    work page arXiv 2025

  26. [27]

    Charged Black Holes in KR-gravity Surrounded by Perfect Fluid Dark Matter

    F. Ahmed, M. Fathi, and E. O. Silva, (2026), arXiv:2604.11357 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  27. [28]

    Ahmed, A

    F. Ahmed, A. Al-Badawi, and ˙I. Sakallı, (2026), arXiv:2601.10303 [gr-qc]

    work page arXiv 2026

  28. [29]

    Ahmed, A

    F. Ahmed, A. Al-Badawi, and E. O. Silva, (2026), arXiv:2602.15570 [gr-qc]

    work page arXiv 2026

  29. [30]

    Ahmed and E

    F. Ahmed and E. O. Silva, (2026), arXiv:2603.11312 [gr-qc]

    work page arXiv 2026

  30. [31]

    Born and L

    M. Born and L. Infeld, Nature132, 1004 (1933)

    work page 1933

  31. [32]

    Born and L

    M. Born and L. Infeld, Proc. Roy. Soc. Lond. A144, 425 (1934)

    work page 1934

  32. [33]

    Heisenberg and H

    W. Heisenberg and H. Euler, Z. Phys.98, 714 (1936)

    work page 1936

  33. [34]

    Schwinger, Phys

    J. Schwinger, Phys. Rev.82, 664 (1951)

    work page 1951

  34. [35]

    H. H. Soleng, Phys. Rev. D52, 6178 (1995)

    work page 1995

  35. [36]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Lett, B493, 149 (2000)

    work page 2000

  36. [37]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Gen. Relativ. Gravit.31, 629 (1999)

    work page 1999

  37. [38]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia, Phys. Rev. Lett.80, 5056 (1998)

    work page 1998

  38. [39]

    A. He, J. Tao, P. Wang, Y . Xue, and L. Zhang, Eur. Phys. J C 82, 683 (2022)

    work page 2022

  39. [40]

    K. A. Bronnikov, Phys. Rev. D63, 044005 (2001)

    work page 2001

  40. [41]

    Allahyari, M

    A. Allahyari, M. Khodadi, S. Vagnozzi, and D. F. Mota, JCAP 2020, 003 (2020)

    work page 2020

  41. [42]

    Güllü and S

    I. Güllü and S. H. Mazharimousavi, Phys. Scr.96, 095213 (2021)

    work page 2021

  42. [43]

    S. I. Kruglov, Int. J Mod. Phys. A33, 1850023 (2018)

    work page 2018

  43. [44]

    S. I. Kruglov, Gravit. Cosmol.25, 190 (2019)

    work page 2019

  44. [45]

    S. I. Kruglov, Eur. Phys. J C80, 250 (2020)

    work page 2020

  45. [46]

    S. I. Kruglov, Int. J. Mod. Phys. D31, 2250025 (2022)

    work page 2022

  46. [47]

    S. I. Kruglov, Ann. Phys. (NY)353, 299 (2015)

    work page 2015

  47. [48]

    S. I. Kruglov, Int. J Mod. Phys. A32, 1750147 (2017)

    work page 2017

  48. [49]

    S. I. Kruglov, Commun. Theor. Phys.66, 59 (2016)

    work page 2016

  49. [50]

    S. I. Kruglov, Ann. Physik (Berlin)527, 397 (2015)

    work page 2015

  50. [51]

    M.-S. Ma, Ann. Phys. (NY)362, 529 (2015)

    work page 2015

  51. [52]

    Dymnikova and E

    I. Dymnikova and E. Galaktionov, J. Phys.: Conf. Series2103, 012078 (2021)

    work page 2021

  52. [53]

    S. H. Mazharimousavi, Phys. Dark Univ.43, 101413 (2024)

    work page 2024

  53. [54]

    Al-Badawi and F

    A. Al-Badawi and F. Ahmed, Chin. J Phys.94, 185 (2025)

    work page 2025

  54. [55]

    F. F. d. Nascimento, V . B. Bezerra, and J. d. M. Toledo, Uni- verse10, 430 (2024)

    work page 2024

  55. [56]

    Dymnikova, Class

    I. Dymnikova, Class. Quantum Grav.19, 725 (2002)

    work page 2002

  56. [57]

    Dymnikova, Class

    I. Dymnikova, Class. Quantum Grav.21, 4417 (2004)

    work page 2004

  57. [58]

    Z. Y . Fan and X. Wang, Phys. Rev. D94, 124027 (2016)

    work page 2016

  58. [59]

    M. E. Rodrigues, E. L. B. Junior, G. T. Marques, and V . T. Zanchin, Phys. Rev. D94, 024062 (2016)

    work page 2016

  59. [60]

    Balart and E

    L. Balart and E. C. Vagenas, Phys. Rev. D90, 124045 (2014)

    work page 2014

  60. [61]

    D. V . Singh, S. Upadhyay, P. Paul, and K. Myrzakulov, Eur. Phys. J Plus141, 364 (2026)

    work page 2026

  61. [62]

    Atamurotov, D

    F. Atamurotov, D. Ortiqboev, A. Abdujabbarov,et al., Eur. Phys. J C82, 659 (2022)

    work page 2022

  62. [63]

    S. A. Hayward, Phys. Rev. Lett.96, 031103 (2006)

    work page 2006

  63. [64]

    Reissner, Annalen der Physik50, 106 (1916)

    H. Reissner, Annalen der Physik50, 106 (1916)

    work page 1916

  64. [65]

    Nordström, Proceedings of the Royal Netherlands Academy of Arts and Sciences20, 1238 (1918)

    G. Nordström, Proceedings of the Royal Netherlands Academy of Arts and Sciences20, 1238 (1918)

    work page 1918

  65. [66]

    K. K. Nandi, R. N. Izmailov, R. K. Karimov, and A. A. Potapov, Eur. Phys. J C83, 984 (2023)

    work page 2023

  66. [67]

    Al-Badawi, F

    A. Al-Badawi, F. Ahmed, and I. Sakallı, Phys. Dark Univ.50, 102076 (2025)

    work page 2025

  67. [68]

    Ahmed, A

    F. Ahmed, A. Al-Badawi, and I. Sakallı, Mod. Phys. Lett. A 41, 2650061 (2026)

    work page 2026

  68. [69]

    Ahmed, A

    F. Ahmed, A. Al-Badawi, and I. Sakalli, Phys. Dark Univ.52, 102315 (2026)

    work page 2026

  69. [70]

    Chandrasekhar,The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983)

    S. Chandrasekhar,The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983)

    work page 1983

  70. [71]

    R. M. Wald,General Relativity(University of Chicago Press, Chicago, IL, 1984)

    work page 1984

  71. [72]

    Cardoso, A

    V . Cardoso, A. S. Miranda, E. Berti, H. Witek, and V . T. Zanchin, Phys. Rev. D79, 064016 (2009). 12

    work page 2009

  72. [73]

    Vagnozziet al., Class

    S. Vagnozziet al., Class. Quant. Grav.40, 165007 (2023)

    work page 2023

  73. [74]

    Perlick and O

    V . Perlick and O. Y . Tsupko, Phys. Rep.947, 1 (2022)

    work page 2022

  74. [75]

    Kocherlakota and L

    P. Kocherlakota and L. Rezzolla, Phys. Rev. D102, 064058 (2020)

    work page 2020

  75. [76]

    R. A. Konoplya and Z. Stuchlik, Phys. Lett. B771, 597 (2017)

    work page 2017

  76. [77]

    I. Z. Stefanov, S. S. Yazadjiev, and G. G. Gyulchev, Phys. Rev. Lett.104, 251103 (2010)

    work page 2010

  77. [78]

    F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, Class. Quan- tum Grav.33, 115003 (2016)

    work page 2016