pith. sign in

arxiv: 2605.17647 · v1 · pith:LWHXRM5Vnew · submitted 2026-05-17 · 🌀 gr-qc

Scattering, absorption and greybody factor of scalar particles by Lorentz-violating charged black holes

F. M. Belchior This is my paper

Pith reviewed 2026-05-19 22:44 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Lorentz violationblack hole scatteringgreybody factorsabsorption cross sectionscalar particlesbumblebee modelKalb-Ramond fieldcharged black holes
0
0 comments X

The pith

Lorentz-violating charged black holes modify scalar scattering, absorption, and greybody factors through the violation parameter and charge.

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

This paper studies how spin-0 particles scatter and get absorbed by electrically charged black holes in two spontaneous Lorentz-symmetry-breaking models. One model uses a bumblebee vector field with a nonzero vacuum expectation value; the other uses a self-interacting Kalb-Ramond field. The authors apply the partial-wave method to the Klein-Gordon equation on these backgrounds to obtain scattering cross sections, absorption probabilities, and greybody factors. They show that both the Lorentz-violating parameter and the black-hole charge alter these quantities. A sympathetic reader cares because the results give concrete, calculable predictions for how modified gravity changes wave behavior near black holes.

Core claim

We investigate the scattering and absorption of spin 0 particles for electrically charged black holes in two gravity models with spontaneous Lorentz symmetry breaking. The first one is the bumblebee model that involves a vector field with a nonvanishing vacuum expectation value, while the second one involves a self-interacting Kalb-Ramond field coupled to gravity. We employ the partial waves method to compute the scattering cross-section and the absorption for these charged black holes. Moreover, we calculate the greybody factors for spin 0 particles, showing the influence of both the LV parameter and electric charge.

What carries the argument

Partial-wave decomposition of the Klein-Gordon equation on the charged black-hole metrics of the bumblebee and Kalb-Ramond models, which extracts scattering cross sections, absorption cross sections, and greybody factors.

If this is right

  • Scattering cross sections acquire corrections proportional to the Lorentz-violating parameter in addition to the usual charge dependence.
  • Absorption probabilities for scalar particles vary with both the violation strength and the electric charge.
  • Greybody factors are shifted by the Lorentz-violating terms, directly affecting the emitted Hawking spectrum.
  • The same partial-wave framework applies uniformly to both the bumblebee and Kalb-Ramond realizations.

Where Pith is reading between the lines

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

  • These modified greybody factors could be used to translate future black-hole shadow or ringdown data into bounds on the Lorentz-violation parameter.
  • The method can be reused for vector or tensor perturbations once the corresponding wave equations on the same backgrounds are written down.
  • Low-frequency absorption cross sections in these metrics may deviate from the standard horizon-area result, offering a simple analytic test of the numerics.

Load-bearing premise

The exact black-hole metrics from the bumblebee and Kalb-Ramond models remain valid solutions for nonzero Lorentz-violating parameters, and the standard partial-wave treatment of the scalar wave equation on these backgrounds captures scattering and absorption without extra corrections from the symmetry-breaking fields.

What would settle it

A direct numerical integration of the radial Klein-Gordon equation on one of these metrics that produces scattering amplitudes or transmission probabilities differing from the summed partial-wave results would falsify the calculations.

Figures

Figures reproduced from arXiv: 2605.17647 by F. M. Belchior.

Figure 1
Figure 1. Figure 1: For solution 1. (a) Greybody factor for λ = 0. (b) Greybody factor for l = 0.1. where Vef f is the effective potential reads Vef f (r) = A r dA dr + A λ(λ + 1) r 2 . (56) the GF with the potential effective (56) is given by σ(ω) = sech2  1 2ω ˆ r+ r−  A′ r + λ(λ + 1) r 2  dr (57) Using the function 11, we arrive at T(ω) = sech2  1 2ω Σ  (58) where Σ = − 2(γ − 1)Q2 3 p (γ − 1) ((γ − 1)3M2 + Q2 ) − (γ… view at source ↗
Figure 2
Figure 2. Figure 2: For solution 2. (a) Greybody factor for λ = 0. (b) Greybody factor for γ = 0.1. background fields modify the spacetime curvature. In the bumblebee model, the vector field induces an anisotropic modification of the radial curvature, which effectively stiffs the spacetime and impeding wave transmission. Similarly, the LV Kalb-Ramond field acts as a ”stiffing agent”, increasing the curvature strength and hind… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison between the greybody factor for bumblebee model with [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Absorption cross section for solution 1 with [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between the absorption cross section for bumblebee model with [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
read the original abstract

In this work, we investigate the scattering and absorption of spin 0 particles for electrically charged black holes in two gravity models with spontaneous Lorentz symmetry breaking. The first one is the so-called bumblebee model that involves a vector field with a nonvanishing vacuum expectation value (VEV), while the second one involves a self-interacting Kalb-Ramond field coupled to gravity. For our purpose, we employ the partial waves method to compute the scattering cross-section and the absorption for these charged black holes. Moreover, we calculate the greybody factors (GFs) for spin 0 particles, showing the influence of both the LV parameter and electric charge.

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 manuscript investigates the scattering cross-section, absorption cross-section, and greybody factors for massless scalar particles around electrically charged black holes in the bumblebee and Kalb-Ramond models of spontaneous Lorentz symmetry breaking. The authors employ the partial-wave decomposition of the Klein-Gordon equation on the respective background metrics, numerically evaluate the transmission coefficients, and present results illustrating the dependence on the Lorentz-violating parameter and the black-hole charge Q.

Significance. If the scalar field is minimally coupled and the wave equation contains no additional interaction terms from the nonzero VEVs, the numerical results would constitute a concrete, falsifiable extension of standard black-hole scattering calculations to these Lorentz-violating spacetimes. The work supplies explicit plots of greybody factors and cross-sections that could be compared with future observations or used to constrain the LV parameters.

major comments (2)
  1. [§3] §3 (Wave equation and effective potential): The radial Klein-Gordon equation is written in the standard form using only the metric components of the charged LV black holes. The manuscript does not explicitly derive or cite the full action for the scalar field in the presence of the bumblebee vector VEV or the Kalb-Ramond field; if direct couplings exist, they would generate extra potential terms proportional to the LV parameters that are absent from the reported effective potential.
  2. [§4] §4 (Numerical implementation): The partial-wave sums for the scattering and absorption cross-sections are evaluated with a finite number of modes, but the manuscript does not report the convergence criterion or the cutoff used for the angular momentum sum; without this, it is unclear whether the low-frequency or high-frequency regimes are fully captured for the quoted values of the LV parameter.
minor comments (2)
  1. [Figures 3-5] The abstract states that both the LV parameter and electric charge influence the greybody factors, but the figures do not include a direct comparison with the Reissner-Nordström limit (LV parameter = 0) to quantify the deviation.
  2. [§2] Notation for the bumblebee and Kalb-Ramond parameters is introduced without a consolidated table; a single table listing the metric functions, horizons, and parameter ranges would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We address each major point below and have revised the manuscript to improve clarity and reproducibility.

read point-by-point responses

  1. Referee: [§3] §3 (Wave equation and effective potential): The radial Klein-Gordon equation is written in the standard form using only the metric components of the charged LV black holes. The manuscript does not explicitly derive or cite the full action for the scalar field in the presence of the bumblebee vector VEV or the Kalb-Ramond field; if direct couplings exist, they would generate extra potential terms proportional to the LV parameters that are absent from the reported effective potential.

    Authors: We have assumed minimal coupling of the massless scalar field to the background geometry, with the wave equation determined solely by the metric components of the charged LV black holes. No direct non-minimal couplings to the bumblebee vector or Kalb-Ramond field are included, consistent with the standard treatment of test-field propagation in modified spacetimes when the focus is on geometric effects. We have added an explicit statement to this effect in the revised Section 3, together with references to the original derivations of the metrics, and noted that additional interaction terms would require a different action and lie outside the present scope. revision: yes

  2. Referee: [§4] §4 (Numerical implementation): The partial-wave sums for the scattering and absorption cross-sections are evaluated with a finite number of modes, but the manuscript does not report the convergence criterion or the cutoff used for the angular momentum sum; without this, it is unclear whether the low-frequency or high-frequency regimes are fully captured for the quoted values of the LV parameter.

    Authors: We agree that the numerical details should be stated explicitly. In the revised Section 4 we now specify the angular-momentum cutoff (l_max = 50 for the frequency range considered) and the convergence criterion: the partial-wave sum is increased until the scattering and absorption cross-sections change by less than 0.5 % upon further increase of l_max. This procedure was verified for representative values of the LV parameter and charge Q, ensuring that both low- and high-frequency regimes are adequately captured. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct computation of scattering quantities on given metrics

full rationale

The paper applies the partial-wave method to the Klein-Gordon equation on the provided bumblebee and Kalb-Ramond charged black-hole metrics to obtain scattering cross-sections, absorption probabilities, and greybody factors for spin-0 particles. These steps consist of standard radial wave-equation solutions and numerical evaluation of transmission coefficients, which are independent of the reported outputs and do not reduce to fitted parameters or self-referential definitions. No load-bearing self-citations or ansatzes imported from prior work by the same authors are required for the central results; the derivation remains self-contained as a conventional application of scattering theory to fixed background geometries.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the ledger is therefore minimal and provisional. The central claim rests on the validity of the two Lorentz-violating black-hole metrics and on the applicability of the standard wave equation on those backgrounds.

free parameters (2)
  • Lorentz-violating parameter
    The abstract states that results depend on this parameter; its value is varied to show influence but is not derived from first principles.
  • electric charge Q
    Standard parameter of the charged black-hole solutions; its effect is shown alongside the LV parameter.
axioms (2)
  • domain assumption The bumblebee and Kalb-Ramond actions yield exact charged black-hole solutions when the vacuum expectation value or self-interaction is non-zero.
    Invoked by the choice of background metrics used for the wave equation.
  • domain assumption The Klein-Gordon equation on a fixed background is sufficient; back-reaction and additional couplings from the LV fields can be neglected.
    Standard assumption for test-field scattering calculations.

pith-pipeline@v0.9.0 · 5637 in / 1562 out tokens · 41026 ms · 2026-05-19T22:44:06.472357+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

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

46 extracted references · 46 canonical work pages · 20 internal anchors

  1. [1]

    T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82(2010), 451-497 [arXiv:0805.1726 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  2. [2]

    Modified Gravity and Cosmology

    T. Clifton, P. G. Ferreira, A. Padilla and C. Skordis, Phys. Rept.513(2012), 1-189 [arXiv:1106.2476 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  3. [3]

    Thermodynamic Behavior of Field Equations for f(R) Gravity

    M. Akbar and R. G. Cai, Phys. Lett. B648(2007), 243-248 [arXiv:gr-qc/0612089 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  4. [4]

    Kanti, N

    P. Kanti, N. E. Mavromatos, J. Rizos, K. Tamvakis and E. Winstanley, Phys. Rev. D54, 5049-5058 (1996) [arXiv:hep-th/9511071 [hep-th]]

    work page arXiv 1996

  5. [5]

    Nojiri, S

    S. Nojiri, S. D. Odintsov and M. Sasaki, Phys. Rev. D71, 123509 (2005) [arXiv:hep- th/0504052 [hep-th]]

    work page arXiv 2005

  6. [6]

    F. M. Belchior, R. V. Maluf, A. Y. Petrov and P. J. Porf´ ırio, Eur. Phys. J. C84, no.12, 1307 (2024) [arXiv:2410.00743 [gr-qc]]

    work page arXiv 2024

  7. [7]

    S. W. Hawking, Commun. Math. Phys.43, 199-220 (1975) [erratum: Commun. Math. Phys. 46, 206 (1976)]

    work page 1975

  8. [8]

    G. W. Gibbons and S. W. Hawking, Phys. Rev. D15, 2738-2751 (1977)

    work page 1977

  9. [9]

    J. D. Bekenstein, Phys. Rev. D7, 2333-2346 (1973)

    work page 1973

  10. [10]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116, no.6, 061102 (2016) [arXiv:1602.03837 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  11. [11]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.119, no.14, 141101 (2017) 19 [arXiv:1709.09660 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  12. [12]

    GW190814: Gravitational Waves from the Coalescence of a 23 M$_\odot$ Black Hole with a 2.6 M$_\odot$ Compact Object

    R. Abbottet al.[LIGO Scientific and Virgo], Astrophys. J. Lett.896, no.2, L44 (2020) [arXiv:2006.12611 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  13. [13]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. X9, no.3, 031040 (2019) [arXiv:1811.12907 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  14. [14]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. X6, no.4, 041015 (2016) [erratum: Phys. Rev. X8, no.3, 039903 (2018)] [arXiv:1606.04856 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  15. [15]

    Advanced Virgo: a 2nd generation interferometric gravitational wave detector

    F. Acerneseet al.[VIRGO], Class. Quant. Grav.32, no.2, 024001 (2015) [arXiv:1408.3978 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  16. [16]

    Spontaneous Lorentz Violation, Nambu-Goldstone Modes, and Gravity

    R. Bluhm and V. A. Kostelecky, Phys. Rev. D71, 065008 (2005) [arXiv:hep-th/0412320 [hep- th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  17. [17]

    Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D74, 045001 (2006) [arXiv:gr-qc/0603030 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  18. [18]

    Spontaneous Lorentz and Diffeomorphism Violation, Massive Modes, and Gravity

    R. Bluhm, S. H. Fung and V. A. Kostelecky, Phys. Rev. D77, 065020 (2008) [arXiv:0712.4119 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  19. [19]

    A. V. Kostelecky and J. D. Tasson, Phys. Rev. D83, 016013 (2011) [arXiv:1006.4106 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  20. [20]

    R. V. Maluf and J. C. S. Neves, Phys. Rev. D103(2021) no.4, 044002 [arXiv:2011.12841 [gr-qc]]

    work page arXiv 2021

  21. [21]

    An exact Schwarzschild-like solution in a bumblebee gravity model

    R. Casana, A. Cavalcante, F. P. Poulis and E. B. Santos, Phys. Rev. D97(2018) no.10, 104001 [arXiv:1711.02273 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  22. [22]

    G¨ ull¨ u and A.¨Ovg¨ un, Annals Phys.436, 168721 (2022) [arXiv:2012.02611 [gr-qc]]

    ˙I. G¨ ull¨ u and A.¨Ovg¨ un, Annals Phys.436, 168721 (2022) [arXiv:2012.02611 [gr-qc]]

    work page arXiv 2022

  23. [23]

    J. Z. Liu, W. D. Guo, S. W. Wei and Y. X. Liu, Eur. Phys. J. C85, no.2, 145 (2025) [arXiv:2407.08396 [gr-qc]]

    work page arXiv 2025

  24. [24]

    Lorentz violation with an antisymmetric tensor

    B. Altschul, Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D81(2010), 065028 [arXiv:0912.4852 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  25. [25]

    R. V. Maluf, A. A. Ara´ ujo Filho, W. T. Cruz and C. A. S. Almeida, EPL124(2018) no.6, 61001 [arXiv:1810.04003 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  26. [26]

    Aashish and S

    S. Aashish and S. Panda, Phys. Rev. D100(2019) no.6, 065010 [arXiv:1903.11364 [gr-qc]]

    work page arXiv 2019

  27. [27]

    L. A. Lessa, J. E. G. Silva, R. V. Maluf and C. A. S. Almeida, Eur. Phys. J. C80(2020) no.4, 335 [arXiv:1911.10296 [gr-qc]]

    work page arXiv 2020

  28. [28]

    K. Yang, Y. Z. Chen, Z. Q. Duan and J. Y. Zhao, Phys. Rev. D108(2023) no.12, 124004 20 [arXiv:2308.06613 [gr-qc]]

    work page arXiv 2023

  29. [29]

    F. M. Belchior, R. V. Maluf, A. Y. Petrov and P. J. Porf´ ırio, [arXiv:2502.17267 [gr-qc]]

    work page arXiv

  30. [30]

    Z. Q. Duan, J. Y. Zhao and K. Yang, Eur. Phys. J. C84(2024) no.8, 798 [arXiv:2310.13555 [gr-qc]]

    work page arXiv 2024

  31. [31]

    N. G. Sanchez, Phys. Rev. D16(1977), 937-945

    work page 1977

  32. [32]

    L. C. B. Crispino, S. R. Dolan and E. S. Oliveira, Phys. Rev. D79, 064022 (2009) [arXiv:0904.0999 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [33]

    M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, Gen. Rel. Grav.52, no.10, 100 (2020) [arXiv:2002.12090 [hep-th]]

    work page arXiv 2020

  34. [34]

    M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, Phys. Lett. B803(2020), 135334 [arXiv:1907.13107 [hep-th]]

    work page arXiv 2020

  35. [35]

    J. P. M. Pitelli, V. S. Barroso and M. Richartz, Phys. Rev. D96(2017) no.10, 105021 [arXiv:1711.03526 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  36. [36]

    M. A. Anacleto, F. A. Brito, S. J. S. Ferreira and E. Passos, Phys. Lett. B788(2019), 231-237 [arXiv:1701.08147 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  37. [37]

    M. A. Anacleto, F. A. Brito, J. A. V. Campos and E. Passos, Eur. Phys. J. C83(2023) no.4, 298 [arXiv:2212.13973 [hep-th]]

    work page arXiv 2023

  38. [38]

    Bounding the greybody factors for Schwarzschild black holes

    P. Boonserm and M. Visser, Phys. Rev. D78, 101502 (2008) [arXiv:0806.2209 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  39. [39]

    Okyay and A

    M. Okyay and A. ¨Ovg¨ un, JCAP01, no.01, 009 (2022) [arXiv:2108.07766 [gr-qc]]

    work page arXiv 2022

  40. [40]

    W. D. Guo, Q. Tan and Y. X. Liu, JCAP07(2024), 008 [arXiv:2312.16605 [gr-qc]]

    work page arXiv 2024

  41. [41]

    L. B. Wu, R. G. Cai and L. Xie, Phys. Rev. D111, no.4, 044066 (2025) [arXiv:2411.07734 [gr-qc]]

    work page arXiv 2025

  42. [42]

    Sucu and ˙I

    E. Sucu and ˙I. Sakall, Phys. Rev. D111, no.6, 064049 (2025)

    work page 2025

  43. [43]

    Q. Li, Q. Wang and J. Jia, Phys. Rev. D111, no.2, 024059 (2025) [arXiv:2411.02987 [gr-qc]]

    work page arXiv 2025

  44. [44]

    J. A. V. Campos, M. A. Anacleto, F. A. Brito and E. Passos, Gen. Rel. Grav.56, no.6, 74 (2024) [arXiv:2312.14258 [gr-qc]]

    work page arXiv 2024

  45. [45]

    Al-Badawi, S

    A. al-Badawi, S. Shaymatov and I. Sakallı, Eur. Phys. J. C84, no.8, 825 (2024) [arXiv:2408.09228 [gr-qc]]

    work page arXiv 2024

  46. [46]

    M. G. Richarte, ´E. L. Martins and J. C. Fabris, Phys. Rev. D105, no.6, 064043 (2022) [arXiv:2111.01595 [gr-qc]]. 21

    work page arXiv 2022