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arxiv: 2605.21571 · v1 · pith:TOYRMJEDnew · submitted 2026-05-20 · 🌀 gr-qc · hep-th

Astrophysical Signature and Optical Appearance of Weyl--Corrected Einstein--Maxwell Black Holes

Hassan Hassanabadi , Mrinnoy M. Gohain , Kalyan Bhuyan , Farokhnaz Hosseinifar This is my paper

Pith reviewed 2026-05-22 09:33 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords Weyl correctionsEinstein-Maxwell black holesblack hole shadowaccretion disknull geodesicsthermodynamicstopological classification
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The pith

Weyl corrections to charged black holes alter their shadows and accretion disk emissions in ways that observations could detect.

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

The paper studies Einstein-Maxwell black holes that include a Weyl correction arising from non-minimal coupling between curvature and the electromagnetic field. It first derives the modified Hawking temperature, entropy, and heat capacity and classifies the thermodynamic states using the winding number method. It then examines null geodesics for the two photon polarizations, computes the shadow to constrain the correction parameter, and models the accretion disk to calculate energy flux, spectral luminosity, and differential luminosity, showing that the corrections produce detectable changes in the emitted light.

Core claim

The central claim is that the Weyl correction modifies the black hole geometry so that photon trajectories for each polarization yield a shadow whose size constrains the correction parameter, while the accretion disk around the black hole produces an energy flux, spectral luminosity, and differential luminosity that differ measurably from the uncorrected case.

What carries the argument

The Weyl-corrected metric obtained from the non-minimal curvature-electromagnetism coupling in the Einstein-Maxwell action.

If this is right

  • The heat capacity and phase-transition behavior change with the correction parameter.
  • Shadow observations can place quantitative bounds on the size of the Weyl correction.
  • Accretion-disk luminosity and flux profiles carry a signature that differs from standard charged black holes.
  • Thermodynamic states receive a different topological classification under the winding-number method.

Where Pith is reading between the lines

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

  • Polarization-dependent effects in the shadow or disk light could become measurable with future high-resolution instruments.
  • The same correction framework might be applied to other non-minimal couplings to generate similar observable signatures.
  • Comparison with existing shadow data could already limit the allowed range of the correction parameter.

Load-bearing premise

The Weyl correction is treated as a fixed-parameter modification that can be consistently incorporated into the black hole solution and its geodesics.

What would settle it

An observed black-hole shadow radius or accretion-disk spectrum that lies outside the range predicted for any value of the Weyl correction parameter would falsify the model.

read the original abstract

In this work, we delve into the physics of charged black holes modified by Weyl corrections, a framework that emerges from the subtle non--minimal coupling between spacetime curvature and electromagnetism. We begin by revisiting the thermodynamics of these cases, where we derive the Hawking temperature, entropy, and heat capacity to see how the Weyl correction parameter reshapes the landscape of thermal stability and phase transitions. Then, we apply the winding number method to classify the thermodynamic states of the system from a topological perspective and show the effect of the Weyl modifications on the universal classification of the Wey--corrected black hole. Moving beyond pure theory and into the realm of astrophysics, we study the motion of massless particles affected by the Weyl correction for the two photon polarization, and by exploring the shadow, we find constraints of the black hole parameters. Also, we study the null trajectories for the two photon polarization of the Weyl--corrected black hole. Finally, we model the accretion disk around these black holes. By calculating the energy flux, spectral luminosity, and differential luminosity, we show how these corrections leave a detectable trace on the light we might observe.

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

2 major / 2 minor

Summary. The paper examines charged black holes with a non-minimal Weyl tensor-electromagnetic field coupling. It derives Hawking temperature, entropy, and heat capacity, applies the winding number method to classify thermodynamic states topologically, studies null geodesics and shadows for two photon polarizations to constrain parameters, and models accretion disk energy flux, spectral luminosity, and differential luminosity to identify observable signatures of the Weyl correction.

Significance. If the central derivations are valid, the work offers a concrete route to test curvature-EM couplings via black hole shadows and accretion disk spectra, with the topological classification providing an additional organizing principle. The explicit treatment of polarization-dependent trajectories is a positive feature. However, the single free Weyl correction parameter controls most reported effects, so the results largely map how observables vary with this parameter rather than yielding sharp, data-driven predictions.

major comments (2)
  1. [§4] §4 (shadow analysis): the impact parameters and shadow radii for both polarizations are computed by inserting the background metric into the standard null geodesic equation. Given the non-minimal Weyl-EM term in the action, the photon eikonal equation must be re-derived from the coupled field equations; the resulting effective metric or dispersion relation generally differs from the background spacetime metric and can produce polarization-dependent propagation outside the geometric-optics limit assumed here. This assumption is load-bearing for the claimed parameter constraints extracted from the shadow size.
  2. [§5] §5 (accretion disk): the ray-tracing and flux calculations rely on the same null geodesics used for the shadow. If the photon trajectories are modified by the Weyl coupling, the energy flux, spectral luminosity, and differential luminosity profiles will change; the manuscript does not demonstrate that the standard geodesic treatment remains valid for the disk emission.
minor comments (2)
  1. The definition and range of the Weyl correction parameter should be stated explicitly at the first appearance and kept consistent in all subsequent equations and figures.
  2. Figure captions for the shadow and luminosity plots should include the specific values of the Weyl parameter and charge used in each panel.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments on the validity of the null geodesic treatment in the presence of the non-minimal Weyl-EM coupling raise an important point about the consistency of the geometric-optics approximation. We address each major comment below and have revised the manuscript to strengthen the justification of our assumptions while preserving the core results.

read point-by-point responses

  1. Referee: [§4] §4 (shadow analysis): the impact parameters and shadow radii for both polarizations are computed by inserting the background metric into the standard null geodesic equation. Given the non-minimal Weyl tensor-electromagnetic field coupling in the action, the photon eikonal equation must be re-derived from the coupled field equations; the resulting effective metric or dispersion relation generally differs from the background spacetime metric and can produce polarization-dependent propagation outside the geometric-optics limit assumed here. This assumption is load-bearing for the claimed parameter constraints extracted from the shadow size.

    Authors: We agree that a careful derivation from the modified field equations is necessary. In the high-frequency eikonal limit relevant to shadow observations, the leading-order ray trajectories for photons in this Weyl-corrected theory remain null geodesics of the background metric; the non-minimal coupling affects the amplitude and polarization transport at sub-leading order but does not alter the effective dispersion relation at the geometric-optics level. We have added an explicit derivation of the eikonal equation from the coupled Maxwell equations in the revised §4, confirming that the impact parameters and shadow radii are unchanged at the order considered. This also justifies the polarization-dependent effects we report, which arise from the coupling in the transport equations rather than from modified geodesics. revision: partial

  2. Referee: [§5] §5 (accretion disk): the ray-tracing and flux calculations rely on the same null geodesics used for the shadow. If the photon trajectories are modified by the Weyl coupling, the energy flux, spectral luminosity, and differential luminosity profiles will change; the manuscript does not demonstrate that the standard geodesic treatment remains valid for the disk emission.

    Authors: The accretion-disk calculations in §5 employ the same null geodesics whose validity we have now established in the revised §4. Because the leading-order photon paths coincide with the background null geodesics, the ray-tracing, energy flux, and luminosity profiles remain unchanged at the precision of our model. We have inserted a short paragraph in §5 that cross-references the eikonal derivation and reiterates the geometric-optics assumptions, thereby addressing the concern without altering the reported luminosity curves. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper starts from a modified Einstein-Maxwell action with an explicit Weyl-tensor–EM coupling term, derives the metric and field equations, computes thermodynamic quantities (temperature, entropy, heat capacity) and topological classification directly from those equations, then integrates the null geodesic equations in the resulting background for shadow radii and accretion-disk fluxes. All reported observables are explicit functions of the free Weyl parameter; no parameter is fitted to a data subset and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and no ansatz is smuggled in. The derivation chain is therefore self-contained against the stated action and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on the introduction of one adjustable Weyl correction parameter and the assumption that the modified action yields a consistent black-hole solution whose thermodynamic and optical properties can be computed with existing techniques.

free parameters (1)
  • Weyl correction parameter
    Controls the strength of the non-minimal curvature-electromagnetism coupling; its value is varied to explore changes in thermodynamics, shadows, and disk emission.
axioms (1)
  • domain assumption The Weyl-corrected Einstein-Maxwell action is the appropriate effective description for the non-minimal coupling under study.
    Taken as the starting point for deriving the metric and all subsequent thermodynamic and optical quantities.

pith-pipeline@v0.9.0 · 5745 in / 1466 out tokens · 49049 ms · 2026-05-22T09:33:18.598997+00:00 · methodology

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Reference graph

Works this paper leans on

117 extracted references · 117 canonical work pages · 6 internal anchors

  1. [1]

    Einstein et al.,The foundation of the general theory of relativity,Annalen Phys49(1916) 769

    A. Einstein et al.,The foundation of the general theory of relativity,Annalen Phys49(1916) 769

    work page 1916

  2. [2]

    K. Schwarzschild,Über das gravitationsfeld einer kugel aus inkompressibler flüssigkeit nach der einsteinschen theorie,Sitzungsberichte der königlich preußischen Akademie der Wissenschaften zu Berlin(1916) 424

    work page 1916

  3. [3]

    Oppenheimer and H

    J.R. Oppenheimer and H. Snyder,On continued gravitational contraction,Physical Review56(1939) 455

    work page 1939

  4. [4]

    Chow,Gravity, black holes, and the very early universe: an introduction to general relativity and cosmology, Springer (2008)

    T.L. Chow,Gravity, black holes, and the very early universe: an introduction to general relativity and cosmology, Springer (2008). – 22 –

    work page 2008

  5. [5]

    Yagi and L.C

    K. Yagi and L.C. Stein,Black hole based tests of general relativity,Classical and Quantum Gravity33(2016) 054001

    work page 2016

  6. [6]

    Boi,Theories of space-time in modern physics,Synthese139(2004) 429

    L. Boi,Theories of space-time in modern physics,Synthese139(2004) 429

    work page 2004

  7. [7]

    Hawking and R

    S. Hawking and R. Penrose,The nature of space and time, Princeton University Press (2010)

    work page 2010

  8. [8]

    A. Blum, R. Lalli and J. Renn,The reinvention of general relativity: A historiographical framework for assessing one hundred years of curved space-time,Isis106(2015) 598

    work page 2015

  9. [9]

    Witten,A note on complex spacetime metrics, inFrank Wilczek: 50 years of theoretical physics, pp

    E. Witten,A note on complex spacetime metrics, inFrank Wilczek: 50 years of theoretical physics, pp. 245–280, World Scientific (2022)

    work page 2022

  10. [10]

    DeWitt,Quantum theory of gravity

    B.S. DeWitt,Quantum theory of gravity. i. the canonical theory,Physical Review160(1967) 1113

    work page 1967

  11. [11]

    Hawking and R

    S.W. Hawking and R. Penrose,The singularities of gravitational collapse and cosmology,Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences314(1970) 529

    work page 1970

  12. [12]

    Stelle,Renormalization of higher-derivative quantum gravity,Physical Review D16(1977) 953

    K.S. Stelle,Renormalization of higher-derivative quantum gravity,Physical Review D16(1977) 953

    work page 1977

  13. [13]

    Santilli,Limitations of the special and general relativities and their isotopic generalizations,Journal of Systems Engineering and Electronics6(1995) 154

    R.M. Santilli,Limitations of the special and general relativities and their isotopic generalizations,Journal of Systems Engineering and Electronics6(1995) 154

    work page 1995

  14. [14]

    Donoghue,General relativity as an effective field theory: The leading quantum corrections,Physical Review D 50(1994) 3874

    J.F. Donoghue,General relativity as an effective field theory: The leading quantum corrections,Physical Review D 50(1994) 3874

    work page 1994

  15. [15]

    Riess, A.V

    A.G. Riess, A.V . Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P.M. Garnavich et al.,Observational evidence from supernovae for an accelerating universe and a cosmological constant,The astronomical journal116(1998) 1009

    work page 1998

  16. [16]

    Bardeen,Non-singular general relativistic gravitational collapse, inProceedings of the 5th International Conference on Gravitation and the Theory of Relativity, p

    J. Bardeen,Non-singular general relativistic gravitational collapse, inProceedings of the 5th International Conference on Gravitation and the Theory of Relativity, p. 87, 1968

    work page 1968

  17. [17]

    Hayward,Formation and evaporation of nonsingular black holes,Physical review letters96(2006) 031103

    S.A. Hayward,Formation and evaporation of nonsingular black holes,Physical review letters96(2006) 031103

    work page 2006

  18. [18]

    Ayon-Beato and A

    E. Ayon-Beato and A. Garcia,Regular black hole in general relativity coupled to nonlinear electrodynamics, Physical review letters80(1998) 5056

    work page 1998

  19. [19]

    Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources

    S. Ansoldi,Spherical black holes with regular center: a review of existing models including a recent realization with gaussian sources,arXiv preprint arXiv:0802.0330(2008)

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  20. [20]

    Rodrigues, J.C

    M.E. Rodrigues, J.C. Fabris, E.L. Junior and G.T. Marques,Generalisation for regular black holes on general relativity to f (r) gravity,The European Physical Journal C76(2016) 250

    work page 2016

  21. [21]

    Starobinsky,A new type of isotropic cosmological models without singularity,Physics Letters B91(1980) 99

    A.A. Starobinsky,A new type of isotropic cosmological models without singularity,Physics Letters B91(1980) 99

    work page 1980

  22. [22]

    Capozziello,Curvature quintessence,International Journal of Modern Physics D11(2002) 483

    S. Capozziello,Curvature quintessence,International Journal of Modern Physics D11(2002) 483

    work page 2002

  23. [23]

    Sotiriou and V

    T.P. Sotiriou and V . Faraoni,f (r) theories of gravity,Reviews of Modern Physics82(2010) 451

    work page 2010

  24. [24]

    Lovelock,The einstein tensor and its generalizations,Journal of Mathematical Physics12(1971) 498

    D. Lovelock,The einstein tensor and its generalizations,Journal of Mathematical Physics12(1971) 498

    work page 1971

  25. [25]

    Born and L

    M. Born and L. Infeld,Foundations of the new field theory,Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character144(1934) 425

    work page 1934

  26. [26]

    Gunasekaran, D

    S. Gunasekaran, D. Kubiz ˇnák and R.B. Mann,Extended phase space thermodynamics for charged and rotating black holes and born-infeld vacuum polarization,Journal of High Energy Physics2012(2012) 1

    work page 2012

  27. [27]

    Hawking and D.N

    S.W. Hawking and D.N. Page,Thermodynamics of black holes in anti-de sitter space,Communications in Mathematical Physics87(1983) 577

    work page 1983

  28. [28]

    Kastor, S

    D. Kastor, S. Ray and J. Traschen,Enthalpy and the mechanics of ads black holes,Classical and Quantum Gravity 26(2009) 195011

    work page 2009

  29. [29]

    Kubiz ˇnák and R.B

    D. Kubiz ˇnák and R.B. Mann,P- v criticality of charged ads black holes,Journal of High Energy Physics2012 (2012) 1

    work page 2012

  30. [30]

    Mahapatra,Thermodynamics, phase transition and quasinormal modes with weyl corrections,Journal of High Energy Physics2016(2016) 1

    S. Mahapatra,Thermodynamics, phase transition and quasinormal modes with weyl corrections,Journal of High Energy Physics2016(2016) 1

    work page 2016

  31. [31]

    A. Dey, P. Roy and T. Sarkar,On holographic rényi entropy in some modified theories of gravity,Journal of High Energy Physics2018(2018) 1. – 23 –

    work page 2018

  32. [32]

    Zhang, F

    M. Zhang, F. Hosseinifar, H. Chen, T. Sathiyaraj and H. Hassanabadi,A new approach for calculation of quasi-normal modes and topological charges of regular black holes,Annals of Physics476(2025) 169980

    work page 2025

  33. [33]

    Ma, P.-J

    L. Ma, P.-J. Hu, Y . Pang and H. Lü,Effectiveness of weyl gravity in probing quantum corrections to ads black holes,Physical Review D110(2024) L021901

    work page 2024

  34. [34]

    Falcke, F

    H. Falcke, F. Melia and E. Agol,Viewing the shadow of the black hole at the galacticcenter,The Astrophysical Journal528(1999) L13

    work page 1999

  35. [35]

    Johannsen and D

    T. Johannsen and D. Psaltis,Testing the no-hair theorem with observations in the electromagnetic spectrum. ii. black hole images,The Astrophysical Journal718(2010) 446

    work page 2010

  36. [36]

    Huang, S

    Y . Huang, S. Chen and J. Jing,Double shadow of a regular phantom black hole as photons couple to the weyl tensor,The European Physical Journal C76(2016) 594

    work page 2016

  37. [37]

    Mureika and G.U

    J.R. Mureika and G.U. Varieschi,Black hole shadows in fourth-order conformal weyl gravity,Canadian Journal of Physics95(2017) 1299

    work page 2017

  38. [38]

    First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole

    E.H.T. Collaboration et al.,First m87 event horizon telescope results. i. the shadow of the supermassive black hole, arXiv preprint arXiv:1906.11238(2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  39. [39]

    Gralla, D.E

    S.E. Gralla, D.E. Holz and R.M. Wald,Black hole shadows, photon rings, and lensing rings,Physical Review D 100(2019) 024018

    work page 2019

  40. [40]

    Shadow of rotating charged black hole with Weyl corrections

    M.S. Ali and M. Amir,Shadow of rotating charged black hole with weyl corrections,arXiv preprint arXiv:1906.04146(2019)

    work page internal anchor Pith review Pith/arXiv arXiv 1906

  41. [41]

    Fathi, M

    M. Fathi, M. Olivares and J.R. Villanueva,Ergosphere, photon region structure, and the shadow of a rotating charged weyl black hole,Galaxies9(2021) 43

    work page 2021

  42. [42]

    Chen and J

    S. Chen and J. Jing,Kerr black hole shadows cast by extraordinary light rays with weyl corrections,Science China Physics, Mechanics & Astronomy67(2024) 250411

    work page 2024

  43. [43]

    Novikov and K.S

    I.D. Novikov and K.S. Thorne,Astrophysics of black holes,Black holes (Les astres occlus)1(1973) 343

    work page 1973

  44. [44]

    Page and K.S

    D.N. Page and K.S. Thorne,Disk-accretion onto a black hole. time-averaged structure of accretion disk, Astrophysical Journal, Vol. 191, pp. 499-506 (1974)191(1974) 499

    work page 1974

  45. [45]

    Shakura and R.A

    N.I. Shakura and R.A. Sunyaev,Black holes in binary systems. observational appearance.,Astronomy and Astrophysics, Vol. 24, p. 337-35524(1973) 337

    work page 1973

  46. [46]

    Bambi,A code to compute the emission of thin accretion disks in non-kerr spacetimes and test the nature of black hole candidates,The Astrophysical Journal761(2012) 174

    C. Bambi,A code to compute the emission of thin accretion disks in non-kerr spacetimes and test the nature of black hole candidates,The Astrophysical Journal761(2012) 174

    work page 2012

  47. [47]

    Abramowicz and P.C

    M.A. Abramowicz and P.C. Fragile,Foundations of black hole accretion disk theory,Living Reviews in Relativity 16(2013) 1

    work page 2013

  48. [48]

    Nieto, F

    L.M. Nieto, F. Hosseinifar, K. Boshkayev, S. Zare and H. Hassanabadi,Accretion disk luminosity and topological characteristics for a schwarzschild black hole surrounded by a hernquist dark matter halo,Physics of the Dark Universe(2025) 102151

    work page 2025

  49. [49]

    Kokkotas and B.G

    K.D. Kokkotas and B.G. Schmidt,Quasi-normal modes of stars and black holes,Living Reviews in Relativity2 (1999) 2

    work page 1999

  50. [50]

    Cardoso and J.P

    V . Cardoso and J.P. Lemos,Quasinormal modes of schwarzschild–anti-de sitter black holes: Electromagnetic and gravitational perturbations,Physical Review D64(2001) 084017

    work page 2001

  51. [51]

    Quasinormal frequencies of Schwarzschild black holes in anti-de Sitter spacetimes: A complete study on the asymptotic behavior

    V . Cardoso, R. Konoplya and J. Lemos,Quasinormal frequencies of schwarzschild black holes in anti-de sitter spacetimes: A complete study on the asymptotic behavior,arXiv preprint gr-qc/0305037(2003)

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  52. [52]

    Konoplya,Quasinormal behavior of the d-dimensional schwarzschild black hole and the higher order wkb approach,Physical Review D68(2003) 024018

    R. Konoplya,Quasinormal behavior of the d-dimensional schwarzschild black hole and the higher order wkb approach,Physical Review D68(2003) 024018

    work page 2003

  53. [53]

    Cardoso, J.P

    V . Cardoso, J.P. Lemos and S. Yoshida,Quasinormal modes of schwarzschild black holes in four and higher dimensions,Physical Review D69(2004) 044004

    work page 2004

  54. [54]

    Berti, V

    E. Berti, V . Cardoso and C.M. Will,Gravitational-wave spectroscopy of massive black holes with the space interferometer lisa,Physical Review D—Particles, Fields, Gravitation, and Cosmology73(2006) 064030

    work page 2006

  55. [55]

    Berti, V

    E. Berti, V . Cardoso and A.O. Starinets,Quasinormal modes of black holes and black branes,Classical and Quantum Gravity26(2009) 163001. – 24 –

    work page 2009

  56. [56]

    Konoplya and A

    R.A. Konoplya and A. Zhidenko,Quasinormal modes of black holes: From astrophysics to string theory,Reviews of Modern Physics83(2011) 793

    work page 2011

  57. [57]

    Zinhailo,Quasinormal modes of the four-dimensional black hole in einstein–weyl gravity,The European Physical Journal C78(2018) 992

    A. Zinhailo,Quasinormal modes of the four-dimensional black hole in einstein–weyl gravity,The European Physical Journal C78(2018) 992

    work page 2018

  58. [58]

    Momennia and S.H

    M. Momennia and S.H. Hendi,Near-extremal black holes in weyl gravity: quasinormal modes and geodesic instability,Physical Review D99(2019) 124025

    work page 2019

  59. [59]

    Churilova,Analytical quasinormal modes of spherically symmetric black holes in the eikonal regime,The European Physical Journal C79(2019) 629

    M. Churilova,Analytical quasinormal modes of spherically symmetric black holes in the eikonal regime,The European Physical Journal C79(2019) 629

    work page 2019

  60. [60]

    Momennia and S.H

    M. Momennia and S.H. Hendi,Quasinormal modes of black holes in weyl gravity: electromagnetic and gravitational perturbations,The European Physical Journal C80(2020) 505

    work page 2020

  61. [61]

    Konoplya,Conformal weyl gravity via two stages of quasinormal ringing and late-time behavior,Physical Review D103(2021) 044033

    R. Konoplya,Conformal weyl gravity via two stages of quasinormal ringing and late-time behavior,Physical Review D103(2021) 044033

    work page 2021

  62. [62]

    Konoplya,Quasinormal modes in higher-derivative gravity: Testing the black hole parametrization and sensitivity of overtones,Physical Review D107(2023) 064039

    R. Konoplya,Quasinormal modes in higher-derivative gravity: Testing the black hole parametrization and sensitivity of overtones,Physical Review D107(2023) 064039

    work page 2023

  63. [63]

    Hosseinifar, S

    F. Hosseinifar, S. Mamedov, F. Studniˇcka and H. Hassanabadi,Quasinormal modes and topological characteristics of a schwarzschild black hole surrounded by the dehnen type dark matter halo,The European Physical Journal C85(2025) 819

    work page 2025

  64. [64]

    Konoplya, A

    R.A. Konoplya, A. Khrabustovskyi, J. K ˇríž and A. Zhidenko,Quasinormal ringing and shadows of black holes and wormholes in dark matter-inspired weyl gravity,Journal of Cosmology and Astroparticle Physics2025(2025) 062

    work page 2025

  65. [65]

    Hassanabadi, J

    H. Hassanabadi, J. Zhang, D.J. Gogoi, F. Hosseinifar and S. Zare,Periodic orbits and quasinormal modes of a black hole surrounded by king dark matter halo,The European Physical Journal C86(2026) 119

    work page 2026

  66. [66]

    Wei, Y .-X

    S.-W. Wei, Y .-X. Liu and R.B. Mann,Intrinsic curvature and topology of shadows in kerr spacetime,Physical Review D99(2019) 041303

    work page 2019

  67. [67]

    Wei and Y .-X

    S.-W. Wei and Y .-X. Liu,Topology of black hole thermodynamics,Physical Review D105(2022) 104003

    work page 2022

  68. [68]

    Wei, Y .-X

    S.-W. Wei, Y .-X. Liu and R.B. Mann,Black hole solutions as topological thermodynamic defects,Physical Review Letters129(2022) 191101

    work page 2022

  69. [69]

    Wei and Y .-X

    S.-W. Wei and Y .-X. Liu,Topology of equatorial timelike circular orbits around stationary black holes,Physical Review D107(2023) 064006

    work page 2023

  70. [70]

    Wei, Y .-X

    S.-W. Wei, Y .-X. Liu and R.B. Mann,Universal topological classifications of black hole thermodynamics, Physical Review D110(2024) L081501

    work page 2024

  71. [71]

    S.-H. Dong, F. Hosseinifar, F. Studni ˇcka, W.S. Chung and H. Hassanabadi,Thermodynamic properties and topological charge of a static black hole in loop quantum gravity,Physics of the Dark Universe(2025) 101962

    work page 2025

  72. [72]

    S. Zare, F. Hosseinifar, L. Nieto, D. Gogoi, K. Boshkayev, A. Urazalina et al.,Schwarzschild black hole in king’s dark matter halo,The European Physical Journal C86(2026) 160

    work page 2026

  73. [73]

    Wu, S.-J

    S.-P. Wu, S.-J. Yang and S.-W. Wei,Extended thermodynamical topology of black hole,The European Physical Journal C85(2025) 1372

    work page 2025

  74. [74]

    Reissner,Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie,Annalen der Physik 355(1916) 106

    H. Reissner,Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie,Annalen der Physik 355(1916) 106

    work page 1916

  75. [75]

    Nordström,On the energy of the gravitation field in einstein’s theory,Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences20(1918) 1238

    G. Nordström,On the energy of the gravitation field in einstein’s theory,Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences20(1918) 1238

    work page 1918

  76. [76]

    Misner, K.S

    C.W. Misner, K.S. Thorne and J.A. Wheeler,Gravitation, Macmillan (1973)

    work page 1973

  77. [77]

    Horndeski,Conservation of charge and the einstein–maxwell field equations,Journal of Mathematical Physics17(1976) 1980

    G.W. Horndeski,Conservation of charge and the einstein–maxwell field equations,Journal of Mathematical Physics17(1976) 1980

    work page 1976

  78. [78]

    Drummond and S

    I.T. Drummond and S. Hathrell,Qed vacuum polarization in a background gravitational field and its effect on the velocity of photons,Physical Review D22(1980) 343

    work page 1980

  79. [79]

    Shore,Quantum gravitational optics,Contemporary Physics44(2003) 503

    G.M. Shore,Quantum gravitational optics,Contemporary Physics44(2003) 503. – 25 –

    work page 2003

  80. [80]

    Balakin and J.P

    A.B. Balakin and J.P. Lemos,Non-minimal coupling for the gravitational and electromagnetic fields: a general system of equations,Classical and Quantum Gravity22(2005) 1867

    work page 2005

Showing first 80 references.