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arxiv: 2606.13217 · v1 · pith:QDADXUP7new · submitted 2026-06-11 · 🌀 gr-qc · hep-th

Absorption cross section of a Schwarzschild black hole for a massive vector field

Kaustubh Mukund Vispute , Rajesh Karmakar This is my paper

Pith reviewed 2026-06-27 06:23 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords absorption cross sectionmassive vector fieldSchwarzschild black holeProca fieldFKKS basistransmission coefficientparity degeneracylongitudinal mode
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The pith

Massive vector fields on Schwarzschild black holes acquire a longitudinal mode and break even-odd parity degeneracy in absorption.

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

The paper establishes how to define and compute the absorption cross section for a massive vector field incident on a Schwarzschild black hole at any frequency. It uses the FKKS basis to decompose the Proca field into modes and extract reflection and transmission coefficients from the conserved flux. The mass of the field introduces a longitudinal degree of freedom beyond the transverse modes. It also creates a scalar-type branch within the even-parity transverse modes, which eliminates the degeneracy between even and odd parity sectors that exists when the field is massless. These mass-induced features are visible in the resulting transmission and absorption spectra.

Core claim

Working in the FKKS basis, the conserved flux of the normalized Proca field directly yields the usual definitions of absorption cross section in terms of reflection and transmission coefficients. The mass term adds a longitudinal mode and produces a scalar-type branch in the even parity transverse modes, breaking the degeneracy with the odd parity sector that holds for massless fields. These effects are illustrated in the numerically computed transmission and absorption spectra over arbitrary frequencies.

What carries the argument

The FKKS basis decomposition of the Proca field on Schwarzschild spacetime, which separates the field into modes allowing direct computation of reflection and transmission coefficients from conserved flux.

If this is right

  • The absorption cross section receives separate contributions from longitudinal and transverse modes.
  • Even and odd parity sectors no longer share the same transmission coefficients due to the scalar-type branch.
  • Numerical spectra reveal mass-dependent modifications at all frequencies.
  • The analysis applies directly to the transmission properties of ultralight vector bosons around black holes.

Where Pith is reading between the lines

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

  • These parity-breaking effects could influence the growth rates of vector boson clouds around rotating black holes.
  • Similar mode splitting may appear in the quasinormal mode spectrum of massive vector fields.
  • Future calculations could extend this to Kerr black holes to assess observational signatures in gravitational waves.

Load-bearing premise

The FKKS basis remains a valid decomposition for the Proca field on Schwarzschild spacetime that permits clean separation into modes whose conserved flux directly yields standard reflection/transmission coefficients.

What would settle it

A direct numerical solution of the Proca wave equation on Schwarzschild for fixed mass and frequency whose computed flux at infinity and horizon fails to reproduce the reflection coefficient obtained from the FKKS mode analysis.

Figures

Figures reproduced from arXiv: 2606.13217 by Kaustubh Mukund Vispute, Rajesh Karmakar.

Figure 1
Figure 1. Figure 1: FIG. 1. Potential barrier for the monopole mode of the Proca [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Potential barrier for the odd parity mode of the Proca [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The effective potential barrier for the positive even parity mode of the Proca field is plotted as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Effective potential barrier for the negative even parity mode of the Proca field as a function of [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Greybody factor for the monopole mode of the Proca [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Greybody factor for the odd parity mode of the Proca field has been plotted with the frequency, [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Greybody factor for the positive even parity mode of the Proca field has been plotted with the frequency, [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Greybody factor for the negative even parity mode of the Proca field has been plotted with the frequency, [PITH_FULL_IMAGE:figures/full_fig_p008_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Partial ACS frequency spectrum has been plotted [PITH_FULL_IMAGE:figures/full_fig_p011_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Total ACS frequency spectrum has been plotted [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The total ACS frequency spectrum for the even parity modes has been plotted. In the [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Total ACS, combining the contribution of all the [PITH_FULL_IMAGE:figures/full_fig_p013_12.png] view at source ↗
read the original abstract

In the present work, we study the absorption cross section of a Schwarzschild black hole for a massive vector field over arbitrary frequencies. Working in the Frolov-Krtou\v{s}-Kubiz\v{n}\'ak-Santos (FKKS) basis, we show how the conserved flux of the normalized Proca field naturally leads to the usual definitions of the absorption cross section in terms of the reflection and transmission coefficients. We then numerically compute these quantities over arbitrary frequencies. In contrast to the massless (photon) case, massive vector bosons exhibit new features arising from the field mass. In particular, the mass term introduces a longitudinal degree of freedom in addition to the transverse modes. Furthermore, it leads to a scalar-type branch in the even parity transverse modes, thereby breaking the usual degeneracy with the odd parity sector found in the massless case. We illustrate how these characteristics manifest themselves in the transmission and absorption spectrum. Given the recent developments in the study of ultralight bosonic fields in black hole spacetimes, the present analysis of the transmission properties of massive vector bosons bears particular significance.

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 / 1 minor

Summary. The manuscript computes the absorption cross section of a Schwarzschild black hole for a massive vector (Proca) field over arbitrary frequencies. Working in the FKKS basis, the authors show that the conserved flux of the normalized field yields the standard definitions of reflection and transmission coefficients, which are then evaluated numerically. They report new mass-induced features, including an additional longitudinal degree of freedom and a scalar-type branch in the even-parity transverse modes that breaks the odd/even parity degeneracy present in the massless case, and illustrate these in the transmission and absorption spectra.

Significance. If the central numerical results hold, the work is significant for studies of ultralight bosonic fields in black hole spacetimes, as it extends the known massless vector absorption spectrum and identifies qualitative changes due to the mass term. The explicit connection between the FKKS decomposition and standard flux-derived coefficients, if verified, would provide a useful technical foundation for related calculations involving massive fields.

major comments (2)
  1. [Abstract, paragraph on working in the FKKS basis] Abstract, paragraph on working in the FKKS basis: the claim that the FKKS basis yields independent radial equations for the Proca field whose conserved flux directly produces the usual reflection/transmission coefficients requires explicit verification. The mass term can modify asymptotic behavior and potentially couple components, so it is not automatic that the Wronskian or current normalization carries over unchanged from the massless case; this underpins the reported new spectral features.
  2. [Numerical computation] Numerical results (throughout): no details are given on the integration method, radial grid, convergence tests, error bars, or direct comparison to the known massless vector limit. Without these, the claimed breaking of odd/even degeneracy and the longitudinal-mode contributions cannot be assessed for robustness.
minor comments (1)
  1. [Abstract] The abstract refers to 'arbitrary frequencies' but does not specify the frequency range actually computed or any analytic limits recovered at low or high frequency.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will incorporate clarifications and additional material in a revised version.

read point-by-point responses

  1. Referee: [Abstract, paragraph on working in the FKKS basis] Abstract, paragraph on working in the FKKS basis: the claim that the FKKS basis yields independent radial equations for the Proca field whose conserved flux directly produces the usual reflection/transmission coefficients requires explicit verification. The mass term can modify asymptotic behavior and potentially couple components, so it is not automatic that the Wronskian or current normalization carries over unchanged from the massless case; this underpins the reported new spectral features.

    Authors: We agree that an explicit verification strengthens the foundation. Section 2 of the manuscript derives the decoupled radial equations in the FKKS basis for the massive Proca field and shows that the mass term does not induce mixing between the chosen components; the conserved current is constructed from the Proca Lagrangian and reduces to the standard Wronskian form at both horizons and infinity. To address the concern directly, the revised manuscript will add an appendix that recomputes the asymptotic fluxes including the mass-dependent terms, confirms the absence of cross-component contributions, and verifies that the reflection/transmission coefficients retain their usual definitions. This will make the connection fully explicit. revision: yes

  2. Referee: [Numerical computation] Numerical results (throughout): no details are given on the integration method, radial grid, convergence tests, error bars, or direct comparison to the known massless vector limit. Without these, the claimed breaking of odd/even degeneracy and the longitudinal-mode contributions cannot be assessed for robustness.

    Authors: We acknowledge that the numerical implementation details were insufficiently documented. The revised manuscript will include a dedicated subsection (likely Section 4.1) specifying the integration technique (a combination of series solutions near the horizon and asymptotic matching at infinity), the radial grid resolution and spacing, convergence criteria with respect to grid size and truncation, estimated numerical errors, and a direct comparison plot of the massive results against the massless vector absorption cross section in the m o0 limit. These additions will allow independent assessment of the reported parity breaking and longitudinal-mode effects. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained numerical computation

full rationale

The paper works in the externally defined FKKS basis and shows that the conserved flux of the normalized Proca field yields the standard absorption cross-section formulas in terms of reflection and transmission coefficients. It then performs numerical evaluation over frequencies, highlighting new features from the mass term such as the longitudinal mode and even-parity scalar branch. No steps reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central results follow from direct application of standard flux methods to the decomposed modes without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on standard Proca equation in curved spacetime and validity of FKKS decomposition; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The Proca field obeys the massive vector wave equation derived from the Proca Lagrangian in Schwarzschild geometry.
    Invoked when defining the normalized field and its conserved flux.
  • domain assumption The FKKS basis diagonalizes the vector perturbation equations sufficiently to allow independent mode analysis and flux conservation.
    Stated as the working basis for deriving reflection/transmission coefficients.

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discussion (0)

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Reference graph

Works this paper leans on

73 extracted references · 2 canonical work pages

  1. [1]

    De Felice, L

    A. De Felice, L. Heisenberg, R. Kase, S. Mukohyama, S. Tsujikawa, and Y.-l. Zhang, JCAP06, 048 (2016), arXiv:1603.05806 [gr-qc]

    Pith/arXiv arXiv 2016

  2. [2]

    Langacker, Rev

    P. Langacker, Rev. Mod. Phys.81, 1199 (2009), arXiv:0801.1345 [hep-ph]

    Pith/arXiv arXiv 2009

  3. [3]

    Essiget al., inSnowmass 2013: Snowmass on the Mississippi(2013) arXiv:1311.0029 [hep-ph]

    R. Essiget al., inSnowmass 2013: Snowmass on the Mississippi(2013) arXiv:1311.0029 [hep-ph]

    Pith/arXiv arXiv 2013

  4. [4]

    L. A. Anchordoqui, I. Antoniadis, K. Benakli, and D. Lust, Phys. Lett. B810, 135838 (2020), arXiv:2007.11697 [hep-th]

    arXiv 2020

  5. [5]

    Goodsell and A

    M. Goodsell and A. Ringwald, Fortsch. Phys.58, 716 (2010), arXiv:1002.1840 [hep-th]

    Pith/arXiv arXiv 2010

  6. [6]

    A. E. Nelson and J. Scholtz, Phys. Rev. D84, 103501 (2011), arXiv:1105.2812 [hep-ph]

    Pith/arXiv arXiv 2011

  7. [7]

    Arvanitaki and S

    A. Arvanitaki and S. Dubovsky, Phys. Rev. D83, 044026 (2011), arXiv:1004.3558 [hep-th]

    Pith/arXiv arXiv 2011

  8. [8]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837 [gr-qc]

    Pith/arXiv arXiv 2016

  9. [9]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J. Lett.875, L1 (2019), arXiv:1906.11238 [astro-ph.GA]

    Pith/arXiv arXiv 2019

  10. [10]

    Akiyamaet al.(Event Horizon Telescope), Astrophys

    K. Akiyamaet al.(Event Horizon Telescope), Astrophys. J.Lett.930,L12(2022),arXiv:2311.08680[astro-ph.HE]

    Pith/arXiv arXiv 2022

  11. [11]

    Battaner and E

    E. Battaner and E. Florido, Fund. Cosmic Phys.21, 1 (2000), arXiv:astro-ph/0010475

    Pith/arXiv arXiv 2000

  12. [12]

    Lacroix, Astron

    T. Lacroix, Astron. Astrophys.619, A46 (2018), arXiv:1801.01308 [astro-ph.GA]

    Pith/arXiv arXiv 2018

  13. [13]

    Shen, G.-W

    Z.-Q. Shen, G.-W. Yuan, C.-Z. Jiang, Y.-L. S. Tsai, Q. Yuan, and Y.-Z. Fan, Mon. Not. Roy. Astron. Soc. 527, 3196 (2023), arXiv:2303.09284 [astro-ph.GA]

    arXiv 2023

  14. [14]

    Karmakar, D

    R. Karmakar, D. Maity, and K. M. Vispute, (2025), arXiv:2512.06807 [gr-qc]

    arXiv 2025

  15. [15]

    Hancock and H

    F. Hancock and H. Witek, (2025), arXiv:2506.06554 [gr- qc]

    Pith/arXiv arXiv 2025

  16. [16]

    Kitajima and K

    N. Kitajima and K. Nakayama, JCAP07, 014 (2023), arXiv:2303.04287 [hep-ph]

    arXiv 2023

  17. [17]

    Zi and C

    T. Zi and C. Zhang, Phys. Rev. D111, 104062 (2025), arXiv:2406.11724 [gr-qc]

    arXiv 2025

  18. [18]

    Traykova, R

    D. Traykova, R. Vicente, K. Clough, T. Helfer, E. Berti, P. G. Ferreira, and L. Hui, Phys. Rev. D108, L121502 (2023), arXiv:2305.10492 [gr-qc]

    arXiv 2023

  19. [19]

    Oshita, Phys

    N. Oshita, Phys. Rev. D109, 104028 (2024), arXiv:2309.05725 [gr-qc]

    arXiv 2024

  20. [20]

    R. A. Konoplya and A. Zhidenko, JCAP09, 068 (2024), arXiv:2406.11694 [gr-qc]

    Pith/arXiv arXiv 2024

  21. [21]

    R. F. Rosato, K. Destounis, and P. Pani, Phys. Rev. D 110, L121501 (2024), arXiv:2406.01692 [gr-qc]

    arXiv 2024

  22. [22]

    Oshita and V

    N. Oshita and V. Cardoso, Phys. Rev. D111, 104043 (2025), arXiv:2407.02563 [gr-qc]

    arXiv 2025

  23. [23]

    R. K. L. Lo, L. Sabani, and V. Cardoso, Phys. Rev. D 111, 124002 (2025), arXiv:2504.00084 [gr-qc]

    arXiv 2025

  24. [24]

    Chandrasekhar and S

    S. Chandrasekhar and S. L. Detweiler, Proc. Roy. Soc. Lond. A344, 441 (1975)

    1975

  25. [25]

    T.ReggeandJ.A.Wheeler,Phys.Rev.108,1063(1957)

    1957

  26. [26]

    F. J. Zerilli, Phys. Rev. Lett.24, 737 (1970)

    1970

  27. [27]

    S. A. Teukolsky, Phys. Rev. Lett.29, 1114 (1972)

    1972

  28. [28]

    S. A. Teukolsky, Astrophys. J.185, 635 (1973)

    1973

  29. [29]

    W. H. Press and S. A. Teukolsky, Astrophys. J.185, 649 (1973)

    1973

  30. [30]

    Chandrasekhar, Proc

    S. Chandrasekhar, Proc. Roy. Soc. Lond. A343, 289 (1975)

    1975

  31. [31]

    Jung and D

    E. Jung and D. K. Park, Class. Quant. Grav.21, 3717 (2004), arXiv:hep-th/0403251

    Pith/arXiv arXiv 2004

  32. [32]

    Dolan, C

    S. Dolan, C. Doran, and A. Lasenby, Phys. Rev. D74, 064005 (2006), arXiv:gr-qc/0605031

    Pith/arXiv arXiv 2006

  33. [33]

    Doran, A

    C. Doran, A. Lasenby, S. Dolan, and I. Hinder, Phys. Rev. D71, 124020 (2005), arXiv:gr-qc/0503019

    Pith/arXiv arXiv 2005

  34. [34]

    Matsas, Phys

    L.C.B.Crispino, E.S.Oliveira, A.Higuchi, andG.E.A. Matsas, Phys. Rev. D75, 104012 (2007)

    2007

  35. [35]

    L. C. B. Crispino, S. R. Dolan, and E. S. Oliveira, Phys. Rev. Lett.102, 231103 (2009), arXiv:0905.3339 [gr-qc]

    Pith/arXiv arXiv 2009

  36. [36]

    Arbey and J

    A. Arbey and J. Auffinger, Eur. Phys. J. C79, 693 (2019), arXiv:1905.04268 [gr-qc]

    arXiv 2019

  37. [37]

    Arbey and J

    A. Arbey and J. Auffinger, Eur. Phys. J. C81, 910 (2021), arXiv:2108.02737 [gr-qc]

    arXiv 2021

  38. [38]

    M. H. Chan and C. M. Lee, Astrophys. J. Lett.943, L11 (2023), arXiv:2212.05664 [astro-ph.HE]

    arXiv 2023

  39. [39]

    Cheek, L

    A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, and J. Turner, Phys. Rev. D106, 103012 (2022), arXiv:2207.09462 [astro-ph.CO]

    arXiv 2022

  40. [40]

    Herdeiro, M

    C. Herdeiro, M. O. P. Sampaio, and M. Wang, Phys. Rev. D85, 024005 (2012), arXiv:1110.2485 [gr-qc]

    Pith/arXiv arXiv 2012

  41. [41]

    D. V. Gal’tsov, G. V. Pomerantseva, and G. A. Chizhov, Sov. Phys. J.27, 697 (1984)

    1984

  42. [42]

    R. A. Konoplya, Phys. Rev. D73, 024009 (2006), arXiv:gr-qc/0509026

    Pith/arXiv arXiv 2006

  43. [43]

    J. G. Rosa and S. R. Dolan, Phys. Rev. D85, 044043 (2012), arXiv:1110.4494 [hep-th]

    Pith/arXiv arXiv 2012

  44. [45]

    V. P. Frolov, P. Krtouš, D. Kubizňák, and J. E. Santos, Phys. Rev. Lett.120, 231103 (2018), arXiv:1804.00030 [hep-th]

    Pith/arXiv arXiv 2018

  45. [46]

    Karmakar, Phys

    R. Karmakar, Phys. Lett. B870, 139951 (2025), arXiv:2408.05310 [gr-qc]

    arXiv 2025

  46. [47]

    Bunjusuwan and C.-H

    S. Bunjusuwan and C.-H. Chen, (2025), arXiv:2508.19761 [gr-qc]

    Pith/arXiv arXiv 2025

  47. [48]

    Classical electrodynamics,

    J. D. Jackson and R. F. Fox, “Classical electrodynamics,” (1999)

    1999

  48. [49]

    T. V. Fernandes, D. Hilditch, J. P. S. Lemos, and V. Car- doso,Phys.Rev.D105,044017(2022),arXiv:2112.03282 18 [gr-qc]

    arXiv 2022

  49. [50]

    X.-H. Ge, M. Matsumoto, and K. Zhang, Phys. Rev. D 112, 046024 (2025), arXiv:2502.15627 [hep-th]

    arXiv 2025

  50. [51]

    L. Hui, D. Kabat, X. Li, L. Santoni, and S. S. C. Wong, JCAP06, 038 (2019), arXiv:1904.12803 [gr-qc]

    Pith/arXiv arXiv 2019

  51. [52]

    Sanchez, Physical Review D18, 1030 (1978)

    N. Sanchez, Physical Review D18, 1030 (1978)

    1978

  52. [53]

    Percival and S

    J. Percival and S. R. Dolan, Phys. Rev. D102, 104055 (2020), arXiv:2008.10621 [gr-qc]

    arXiv 2020

  53. [54]

    C. L. Benone and L. C. Crispino, Physical Review D99 (2019), 10.1103/physrevd.99.044009

    work page doi:10.1103/physrevd.99.044009 2019

  54. [55]

    L. C. S. Leite, S. R. Dolan, and L. C. B. Crispino, Phys. Lett. B774, 130 (2017), arXiv:1707.01144 [gr-qc]

    Pith/arXiv arXiv 2017

  55. [56]

    D. N. Page, Phys. Rev. D13, 198 (1976)

    1976

  56. [57]

    C. L. Benone, E. S. de Oliveira, S. R. Dolan, and L. C. B. Crispino, Phys. Rev. D89, 104053 (2014), arXiv:1404.0687 [gr-qc]

    Pith/arXiv arXiv 2014

  57. [58]

    Unruh, Physical Review D14, 3251 (1976)

    W. Unruh, Physical Review D14, 3251 (1976)

    1976

  58. [59]

    Greiner and J

    W. Greiner and J. Reinhardt,Field quantization (Springer Science & Business Media, 2013)

    2013

  59. [60]

    Padmanabhan,Gravitation: foundations and frontiers (Cambridge University Press, 2010)

    T. Padmanabhan,Gravitation: foundations and frontiers (Cambridge University Press, 2010)

    2010

  60. [61]

    D. J. Griffiths and D. F. Schroeter,Introduction to quan- tum mechanics(Cambridge university press, 2018)

    2018

  61. [62]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathe- matical Functions with Formulas, Graphs, and Mathe- matical Tables, National Bureau of Standards Applied Mathematics Series, Vol. 55 (U.S. Government Printing Office, Washington, D.C., 1964) reprinted by Dover Pub- lications, 1965

    1964

  62. [63]

    Cardoso and R

    V. Cardoso and R. Vicente, Phys. Rev. D100, 084001 (2019), arXiv:1906.10140 [gr-qc]

    arXiv 2019

  63. [64]

    S. R. Das, G. W. Gibbons, and S. D. Mathur, Phys. Rev. Lett.78, 417 (1997), arXiv:hep-th/9609052

    Pith/arXiv arXiv 1997

  64. [65]

    Liao, J.-H

    H. Liao, J.-H. Chen, and Y.-J. Wang, Chin. Phys. B21, 080402 (2012)

    2012

  65. [66]

    V. P. Frolov, P. Krtouš, D. Kubizňák, and J. E. San- tos, Physical Review Letters120(2018), 10.1103/phys- revlett.120.231103

    work page doi:10.1103/phys- 2018

  66. [67]

    Krtouš, V

    P. Krtouš, V. P. Frolov, and D. Kubizňák, Nucl. Phys. B934, 7 (2018), arXiv:1803.02485 [hep-th]

    Pith/arXiv arXiv 2018

  67. [68]

    Cayuso, O

    R. Cayuso, O. J. C. Dias, F. Gray, D. Kubizňák, A. Mar- galit, J.E.Santos, R.GomesSouza, andL.Thiele,JHEP 04, 159 (2020), arXiv:1912.08224 [hep-th]

    arXiv 2020

  68. [69]

    C. M. Harris and P. Kanti, JHEP10, 014 (2003), arXiv:hep-ph/0309054

    Pith/arXiv arXiv 2003

  69. [70]

    Cheek, L

    A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, and J. Turner, Phys. Rev. D105, 015022 (2022), arXiv:2107.00013 [hep-ph]

    arXiv 2022

  70. [71]

    Cheek, L

    A. Cheek, L. Heurtier, Y. F. Perez-Gonzalez, and J. Turner, Phys. Rev. D108, 015005 (2023), arXiv:2212.03878 [hep-ph]

    arXiv 2023

  71. [72]

    B. E. Taylor, C. M. Chambers, and W. A. Hiscock, Phys. Rev. D58, 044012 (1998), arXiv:gr-qc/9801044

    Pith/arXiv arXiv 1998

  72. [73]

    Chandrasekhar,The mathematical theory of black holes(1985)

    S. Chandrasekhar,The mathematical theory of black holes(1985)

    1985

  73. [74]

    Goldstein, C

    H. Goldstein, C. Poole, and J. Safko,Classical Mechan- ics, Addison-Wesley series in physics (Addison Wesley, 2002)

    2002